The four colour theorem
The Four Colour Conjecture first seems to have been made by Francis Guthrie. He was a student at University College London where he studied under De Morgan. After graduating from London he studied law but by this time his brother Frederick Guthrie had become a student of De Morgan. Francis Guthrie showed his brother some results he had been trying to prove about the colouring of maps and asked Frederick to ask De Morgan about them.
De Morgan was unable to give an answer but, on 23 October 1852, the same day he was asked the question, he wrote to Hamilton in Dublin. De Morgan wrote:-
A student of mine asked me today to give him a reason for a fact which I did not know was a fact – and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured – four colours may be wanted, but not more – the following is the case in which four colours are wanted. Query cannot a necessity for five or more be invented. …… If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did….
Hamilton replied on 26 October 1852 (showing the efficiency of both himself and the postal service):-
I am not likely to attempt your quaternion of colour very soon.
Before continuing with the history of the Four Colour Conjecture we will complete details of Francis Guthrie. After practising as a barrister he went to South Africa in 1861 as a Professor of Mathematics. He published a few mathematical papers and became interested in botany. A heather (Erica Guthriei) is named after him.
De Morgan kept asking if anyone could find a solution to Guthrie’s problem and several mathematicians worked on it. Charles Peirce in the USA attempted to prove the Conjecture in the 1860’s and he was to retain a lifelong interest in the problem. Cayley also learnt of the problem from De Morgan and on 13 June 1878 he posed a question to the London Mathematical Society asking if the Four Colour Conjecture had been solved. Shortly afterwards Cayley sent a paper On the colouring of maps to the Royal Geographical Society and in was published in 1879. The paper explains where the difficulties lie in attempting to prove the Conjecture.
On 17 July 1879 Alfred Bray Kempe announced in Nature that he had a proof of the Four Colour Conjecture. Kempe was a London barrister who had studied mathematics under Cayley at Cambridge and devoted some of his time to mathematics throughout his life. At Cayley‘s suggestion Kempe submitted the Theorem to the American Journal of Mathematics where it was published in 1879. Story read the paper before publication and made some simplifications. Story reported the proof to the Scientific Association of Johns Hopkins University in November 1879 and Charles Peirce, who was at the November meeting, spoke at the December meeting of the Association of his own work on the Four Colour Conjecture.
Kempe used an argument known as the method of Kempe chains. If we have a map in which every region is coloured red, green, blue or yellow except one, say X. If this final region X is not surrounded by regions of all four colours there is a colour left for X. Hence suppose that regions of all four colours surround X. If X is surrounded by regions A, B, C, D in order, coloured red, yellow, green and blue then there are two cases to consider.
(ii) There is a chain of adjacent regions from A to C alternately coloured red and green.
If (i) holds there is no problem. Change A to green, and then interchange the colour of the red/green regions in the chain joining A. Since C is not in the chain it remains green and there is now no red region adjacent to X. Colour X red.
If (ii) holds then there can be no chain of yellow/blue adjacent regions from B to D. [It could not cross the chain of red/green regions.] Hence property (i) holds for B and D and we change colours as above.
Kempe received great acclaim for his proof. He was elected a Fellow of the Royal Society and served as its treasurer for many years. He was knighted in 1912. He published two improved versions of his proof, the second in 1880 aroused the interest of P G Tait, the Professor of Natural Philosophy at Edinburgh. Tait addressed the Royal Society of Edinburgh on the subject and published two papers on the (what we should now call) Four Colour Theorem. They contain some clever ideas and a number of basic errors.
The Four Colour Theorem returned to being the Four Colour Conjecture in 1890. Percy John Heawood, a lecturer at Durham England, published a paper called Map colouring theorem. In it he states that his aim is
rather destructive than constructive, for it will be shown that there is a defect in the now apparently recognised proof.
Although Heawood showed that Kempe‘s proof was wrong he did prove that every map can be 5-coloured in this paper. Kempe reported the error to the London Mathematical Society himself and said he could not correct the mistake in his proof. In 1896 de la Vallée Poussin also pointed out the error in Kempe‘s paper, apparently unaware of Heawood‘s work.
Heawood was to work throughout his life on map colouring, work which spanned nearly 60 years. He successfully investigated the number of colours needed for maps on other surfaces and gave what is known as the Heawood estimate for the necessary number in terms of the Euler characteristic of the surface.
Heawood‘s other claim to fame is raising money to restore Durham Castle as Secretary of the Durham Castle Restoration Fund. For his perseverance in raising the money to save the Castle from sliding down the hill on which it stands Heawood received the O.B.E.
Heawood was to make further contributions to the Four Colour Conjecture. In 1898 he proved that if the number of edges around each region is divisible by 3 then the regions are 4-colourable. He then wrote many papers generalising this result.
To understand the later work we need to define some concepts.
Clearly a graph can be constructed from any map the regions being represented by the vertices and two vertices being joined by an edge if the regions corresponding to the vertices are adjacent. The resulting graph is planar, that is can be drawn in the plane without any edges crossing. The Four Colour Conjecture now asks if the vertices of the graph can be coloured with 4 colours so that no two adjacent vertices are the same colour.
From the graph a triangulation can be obtained by adding edges to divide any non-triangular face into triangles. A configuration is part of a triangulation contained within a circuit. An unavoidable set is a set of configurations with the property that any triangulation must contain one of the configurations in the set. A configuration is reducible if it cannot be contained in a triangulation of the smallest graph which cannot be 4-coloured.
The search for avoidable sets began in 1904 with work of Weinicke. Renewed interest in the USA was due to Veblen who published a paper in 1912 on the Four Colour Conjecture generalising Heawood‘s work. Further work by G D Birkhoff introduced the concept of reducibility (defined above) on which most later work rested.
Franklin in 1922 published further examples of unavoidable sets and used Birkhoff‘s idea of reducibility to prove, among other results, that any map with ≤ 25 regions can be 4-coloured. The number of regions which resulted in a 4-colourable map was slowly increased. Reynolds increased it to 27 in 1926, Winn to 35 in 1940, Ore and Stemple to 39 in 1970 and Mayer to 95 in 1976.
However the final ideas necessary for the solution of the Four Colour Conjecture had been introduced before these last two results. Heesch in 1969 introduced the method of discharging. This consists of assigning to a vertex of degree i the charge 6 – i. Now from Euler‘s formula we can deduce that the sum of the charges over all the vertices must be 12. A given set S of configurations can be proved unavoidable if for a triangulation T which does not contain a configuration in S we can redistribute the charges (without changing the total charge) so that no vertex ends up with a positive charge.
Heesch thought that the Four Colour Conjecture could be solved by considering a set of around 8900 configurations. There were difficulties with his approach since some of his configurations had a boundary of up to 18 edges and could not be tested for reducibility. The tests for reducibility used Kempe chain arguments but some configurations had obstacles to prevent reduction.
The year 1976 saw a complete solution to the Four Colour Conjecture when it was to become the Four Colour Theorem for the second, and last, time. The proof was achieved by Appel and Haken, basing their methods on reducibility using Kempe chains. They carried through the ideas of Heesch and eventually they constructed an unavoidable set with around 1500 configurations. They managed to keep the boundary ring size down to ≤ 14 making computations easier that for the Heesch case. There was a long period where they essentially used trial and error together with unbelievable intuition to modify their unavoidable set and their discharging procedure. Appel and Haken used 1200 hours of computer time to work through the details of the final proof. Koch assisted Appel and Haken with the computer calculations.
The Four Colour Theorem was the first major theorem to be proved using a computer, having a proof that could not be verified directly by other mathematicians. Despite some worries about this initially, independent verification soon convinced everyone that the Four Colour Theorem had finally been proved. Details of the proof appeared in two articles in 1977. Recent work has led to improvements in the algorithm.
A history of Topology
Topological ideas are present in almost all areas of today’s mathematics. The subject of topology itself consists of several different branches, such as point set topology, algebraic topology and differential topology, which have relatively little in common. We shall trace the rise of topological concepts in a number of different situations.
Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler. In 1736 Euler published a paper on the solution of the Königsberg bridge problem entitled Solutio problematis ad geometriam situs pertinentis which translates into English as The solution of a problem relating to the geometry of position. The title itself indicates that Euler was aware that he was dealing with a different type of geometry where distance was not relevant.
Here is a diagram of the Königsberg bridges
The paper not only shows that the problem of crossing the seven bridges in a single journey is impossible, but generalises the problem to show that, in today’s notation,
A graph has a path traversing each edge exactly once if exactly two vertices have odd degree.
The next step in freeing mathematics from being a subject about measurement was also due to Euler. In 1750 he wrote a letter to Christian Goldbach which, as well as commenting on a dispute Goldbach was having with a bookseller, gives Euler‘s famous formula for a polyhedron
where v is the number of vertices of the polyhedron, e is the number of edges and f is the number of faces. It is interesting to realise that this, really rather simple, formula seems to have been missed by Archimedes and Descartes although both wrote extensively on polyhedra. Again the reason must be that to everyone before Euler, it had been impossible to think of geometrical properties without measurement being involved.
Euler published details of his formula in 1752 in two papers, the first admits that Euler cannot prove the result but the second gives a proof based dissecting solids into tetrahedral slices. Euleroverlooks some problems with his remarkably clever proof. In particular he assumed that the solids were convex, that is a straight line joining any two points always lies entirely within the solid.
The route started by Euler with his polyhedral formula was followed by a little known mathematician Antoine-Jean Lhuilier (1750 -1840) who worked for most of his life on problems relating to Euler‘s formula. In 1813 Lhuilier published an important work. He noticed that Euler‘s formula was wrong for solids with holes in them. If a solid has g holes the Lhuilier showed that
This was the first known result on a topological invariant.
Möbius published a description of a Möbius band in 1865. He tried to describe the ‘one-sided’ property of the Möbius band in terms of non-orientability. He thought of the surface being covered by oriented triangles. He found that the Möbius band could not be filled with compatibly oriented triangles.
Johann Benedict Listing (1802-1882) was the first to use the word topology. Listing‘s topological ideas were due mainly to Gauss, although Gauss himself chose not to publish any work on topology. Listing wrote a paper in 1847 called Vorstudien zur Topologie although he had already used the word for ten years in correspondence. The 1847 paper is not very important, although he also introduces the idea of a complex, since it is extremely elementary. In 1861 Listing published a much more important paper in which he described the Möbius band (4 years before Möbius) and studied components of surfaces and connectivity.
Listing was not the first to examine connectivity of surfaces. Riemann had studied the concept in 1851 and again in 1857 when he introduced the Riemann surfaces. The problem arose from studying a polynomial equation f (w, z) = 0 and considering how the roots vary as w and z vary. Riemann introduced Riemann surfaces, determined by the function f (w, z), so that the function w(z) defined by the equation f (w, z) = 0 is single valued on the surfaces.
Jordan introduced another method for examining the connectivity of a surface. He called a simple closed curve on a surface which does not intersect itself an irreducible circuit if it cannot be continuously transformed into a point. If a general circuit c can be transformed into a system of irreducible circuits a1, a2, …., an so that c describes ai mi times then he wrote
The circuit c is reducible if
A system of irreducible circuits a1, a2, …., an is called independent if they satisfy no relation of the form (*) and complete if any circuit can be expressed in terms of them. Jordan proved that the number of circuits in a complete independent set is a topological invariant of the surface.
Listing had examined connectivity in three dimensional Euclidean space but Betti extended his ideas to n dimensions. This is not as straightforward as it might appear since even in three dimensions it is possible to have a surface that cannot be reduced to a point yet closed curves on the surface can be reduced to a point. Betti‘s definition of connectivity left something to be desired and criticisms were made by Heegaard.
The idea of connectivity was eventually put on a completely rigorous basis by Poincaré in a series of papers Analysis situs in 1895. Poincaré introduced the concept of homology and gave a more precise definition of the Betti numbers associated with a space than had Betti himself. Euler‘s convex polyhedra formula had been generalised to not necessarily convex polyhedra by Jonquières in 1890 and now Poincaré put it into a completely general setting of a p-dimensional variety V.
Also while dealing with connectivity Poincaré introduced the fundamental group of a variety and the concept of homotopy was introduced in the same 1895 papers.
A second way in which topology developed was through the generalisation of the ideas of convergence. This process really began in 1817 when Bolzano removed the association of convergence with a sequence of numbers and associated convergence with any bounded infinite subset of the real numbers.
Cantor in 1872 introduced the concept of the first derived set, or set of limit points, of a set. He also defined closed subsets of the real line as subsets containing their first derived set. Cantor also introduced the idea of an open set another fundamental concept in point set topology.
Weierstrass in 1877 in a course of unpublished lectures gave a rigorous proof of the Bolzano–Weierstrass theorem which states
A bounded infinite subset S of the real numbers possesses at least one point of accumulation p, i.e. p satisfies the property that given any ε > 0 there is an infinite sequence (pn) of points of S with | p – pn | < ε.
Hence the concept of neighbourhood of a point was introduced.
Hilbert used the concept of a neighbourhood in 1902 when he answered in the affirmative one of his own questions, namely
Is a continuous transformation group differentiable?
In 1906 Fréchet called a space compact if any infinite bounded subset contains a point of accumulation. However Fréchet was able to extend the concept of convergence from Euclidean space by defining metric spaces. He also showed that Cantor‘s ideas of open and closed subsets extended naturally to metric spaces.
Riesz, in a paper to the International Congress of Mathematics in Rome (1909), disposed of the metric completely and proposed a new axiomatic approach to topology. The definition was based on an set definition of limit points, with no concept of distance. A few years later in 1914 Hausdorff defined neighbourhoods by four axioms so again there were no metric considerations. This work of Riesz and Hausdorff really allows the definition of abstract topological spaces.
There is a third way in which topological concepts entered mathematics, namely via functional analysis. This was a topic which arose from mathematical physics and astronomy, brought about because the methods of classical analysis were somewhat inadequate in tackling certain types of problems. Jacob Bernoulli and Johann Bernoulli invented the calculus of variations where the value of an integral is thought of as a function of the functions being integrated.
Hadamard introduced the word ‘functional’ in 1903 when he studied linear functionals F of the form
where the limit is taken as n → ∞ and the integral is from a to b. Fréchet continued the development of functional by defining the derivative of a functional in 1904.
Schmidt in 1907 examined the notion of convergence in sequence spaces, extending methods which Hilbert had used in his work on integral equations to generalise the idea of a Fourier series. Distance was defined via an inner product. Schmidt‘s work on sequence spaces has analogues in the theory of square summable functions, this work being done also in 1907 by Schmidt himself and independently by Fréchet.
A further step in abstraction was taken by Banach in 1932 when he moved from inner product spaces to normed spaces. Banach took Fréchet‘s linear functionals and showed that they had a natural setting in normed spaces.
Poincaré developed many of his topological methods while studying ordinary differential equations which arose from a study of certain astronomy problems. His study of autonomous systems
involved looking at the totality of all solutions rather than at particular trajectories as had been the case earlier. The collection of methods developed by Poincaré was built into a complete topological theory by Brouwer in 1912.
Non-Euclidean geometry
In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. Euclid stated five postulates on which he based all his theorems:
- To draw a straight line from any point to any other.
- To produce a finite straight line continuously in a straight line.
- To describe a circle with any centre and distance.
- That all right angles are equal to each other.
- That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
It is clear that the fifth postulate is different from the other four. It did not satisfy Euclid and he tried to avoid its use as long as possible – in fact the first 28 propositions of The Elements are proved without using it. Another comment worth making at this point is that Euclid, and many that were to follow him, assumed that straight lines were infinite.
Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false ‘proof’. Proclus then goes on to give a false proof of his own. However he did give the following postulate which is equivalent to the fifth postulate.
Playfair‘s Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
Although known from the time of Proclus, this became known as Playfair’s Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid‘s fifth postulate by this axiom.
Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods of time until the mistake was found. Invariably the mistake was assuming some ‘obvious’ property which turned out to be equivalent to the fifth postulate. One such ‘proof’ was given by Wallis in 1663 when he thought he had deduced the fifth postulate, but he had actually shown it to be equivalent to:-
To each triangle, there exists a similar triangle of arbitrary magnitude.
One of the attempted proofs turned out to be more important than most others. It was produced in 1697 by Girolamo Saccheri. The importance of Saccheri‘s work was that he assumed the fifth postulate false and attempted to derive a contradiction.
Here is the Saccheri quadrilateral
In this figure Saccheri proved that the summit angles at D and C were equal.The proof uses properties of congruent triangles which Euclid proved in Propositions 4 and 8 which are proved before the fifth postulate is used. Saccheri has shown:
a) The summit angles are > 90° (hypothesis of the obtuse angle).
b) The summit angles are < 90° (hypothesis of the acute angle).
c) The summit angles are = 90° (hypothesis of the right angle).
Euclid‘s fifth postulate is c). Saccheri proved that the hypothesis of the obtuse angle implied the fifth postulate, so obtaining a contradiction. Saccheri then studied the hypothesis of the acute angle and derived many theorems of non-Euclidean geometry without realising what he was doing. However he eventually ‘proved’ that the hypothesis of the acute angle led to a contradiction by assuming that there is a ‘point at infinity’ which lies on a plane.
In 1766 Lambert followed a similar line to Saccheri. However he did not fall into the trap that Saccheri fell into and investigated the hypothesis of the acute angle without obtaining a contradiction. Lambert noticed that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased.
Legendre spent 40 years of his life working on the parallel postulate and the work appears in appendices to various editions of his highly successful geometry book Eléments de Géométrie. Legendre proved that Euclid‘s fifth postulate is equivalent to:-
The sum of the angles of a triangle is equal to two right angles.
Legendre showed, as Saccheri had over 100 years earlier, that the sum of the angles of a triangle cannot be greater than two right angles. This, again like Saccheri, rested on the fact that straight lines were infinite. In trying to show that the angle sum cannot be less than 180° Legendre assumed that through any point in the interior of an angle it is always possible to draw a line which meets both sides of the angle. This turns out to be another equivalent form of the fifth postulate, but Legendre never realised his error himself.
Elementary geometry was by this time engulfed in the problems of the parallel postulate. D’Alembert, in 1767, called it the scandal of elementary geometry.
The first person to really come to understand the problem of the parallels was Gauss. He began work on the fifth postulate in 1792 while only 15 years old, at first attempting to prove the parallels postulate from the other four. By 1813 he had made little progress and wrote:
In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics…
However by 1817 Gauss had become convinced that the fifth postulate was independent of the other four postulates. He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line. Perhaps most surprisingly of all Gauss never published this work but kept it a secret. At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought and Gauss disliked controversy.
Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made several false proofs of the parallel postulate. Farkas Bolyai taught his son, János Bolyai, mathematics but, despite advising his son not to waste one hour’s time on that problem of the problem of the fifth postulate, János Bolyai did work on the problem.
In 1823 János Bolyai wrote to his father saying I have discovered things so wonderful that I was astounded … out of nothing I have created a strange new world. However it took Bolyai a further two years before it was all written down and he published his strange new world as a 24 page appendix to his father’s book, although just to confuse future generations the appendix was published before the book itself.
Gauss, after reading the 24 pages, described János Bolyai in these words while writing to a friend: I regard this young geometer Bolyai as a genius of the first order . However in some sense Bolyai only assumed that the new geometry was possible. He then followed the consequences in a not too dissimilar fashion from those who had chosen to assume the fifth postulate was false and seek a contradiction. However the real breakthrough was the belief that the new geometry was possible. Gauss, however impressed he sounded in the above quote with Bolyai, rather devastated Bolyai by telling him that he (Gauss) had discovered all this earlier but had not published. Although this must undoubtedly have been true, it detracts in no way from Bolyai‘s incredible breakthrough.
Nor is Bolyai‘s work diminished because Lobachevsky published a work on non-Euclidean geometry in 1829. Neither Bolyai nor Gauss knew of Lobachevsky‘s work, mainly because it was only published in Russian in the Kazan Messenger a local university publication. Lobachevsky‘s attempt to reach a wider audience had failed when his paper was rejected by Ostrogradski.
In fact Lobachevsky fared no better than Bolyai in gaining public recognition for his momentous work. He published Geometrical investigations on the theory of parallels in 1840 which, in its 61 pages, gives the clearest account of Lobachevsky‘s work. The publication of an account in French in Crelle‘s Journal in 1837 brought his work on non-Euclidean geometry to a wide audience but the mathematical community was not ready to accept ideas so revolutionary.
In Lobachevsky‘s 1840 booklet he explains clearly how his non-Euclidean geometry works.
All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two classes – into cutting and non-cutting. The boundary lines of the one and the other class of those lines will be called parallel to the given line.
Here is the Lobachevsky’s diagram
Hence Lobachevsky has replaced the fifth postulate of Euclid by:-
Lobachevsky’s Parallel Postulate. There exist two lines parallel to a given line through a given point not on the line.
Lobachevsky went on to develop many trigonometric identities for triangles which held in this geometry, showing that as the triangle became small the identities tended to the usual trigonometric identities.
Riemann, who wrote his doctoral dissertation under Gauss‘s supervision, gave an inaugural lecture on 10 June 1854 in which he reformulated the whole concept of geometry which he saw as a space with enough extra structure to be able to measure things like length. This lecture was not published until 1868, two years after Riemann‘s death but was to have a profound influence on the development of a wealth of different geometries. Riemann briefly discussed a ‘spherical’ geometry in which every line through a point P not on a line AB meets the line AB. In this geometry no parallels are possible.
It is important to realise that neither Bolyai‘s nor Lobachevsky‘s description of their new geometry had been proved to be consistent. In fact it was no different from Euclidean geometry in this respect although the many centuries of work with Euclidean geometry was sufficient to convince mathematicians that no contradiction would ever appear within it.
The first person to put the Bolyai – Lobachevsky non-Euclidean geometry on the same footing as Euclidean geometry was Eugenio Beltrami (1835-1900). In 1868 he wrote a paper Essay on the interpretation of non-Euclidean geometry which produced a model for 2-dimensional non-Euclidean geometry within 3-dimensional Euclidean geometry. The model was obtained on the surface of revolution of a tractrix about its asymptote. This is sometimes called a pseudo-sphere.
You can see the graph of a tractrix and what the top half of a Pseudo-sphere looks like.
In fact Beltrami‘s model was incomplete but it certainly gave a final decision on the fifth postulate of Euclid since the model provided a setting in which Euclid‘s first four postulates held but the fifth did not hold. It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry.
Beltrami‘s work on a model of Bolyai – Lobachevsky‘s non-Euclidean geometry was completed by Klein in 1871. Klein went further than this and gave models of other non-Euclidean geometries such as Riemann‘s spherical geometry. Klein‘s work was based on a notion of distance defined by Cayley in 1859 when he proposed a generalised definition for distance.
Klein showed that there are three basically different types of geometry. In the Bolyai – Lobachevsky type of geometry, straight lines have two infinitely distant points. In the Riemann type of spherical geometry, lines have no (or more precisely two imaginary) infinitely distant points. Euclidean geometry is a limiting case between the two where for each line there are two coincident infinitely distant points.
Topology and Scottish mathematical physics
The Scottish mathematical physicists referred to in the title are Thomson, Maxwell and Tait. These three became involved in topological concepts, in particular knot theory, because it entered their physical considerations in a natural way. In 1847 Listing published Vorstudien zur Topologie and then Riemann published important papers on complex analysis in 1851 and 1857 which investigated connectivity and Riemann surfaces. The Scottish mathematical physicists did not know of these papers until much later but Helmholtz, whose paper of 1858 directly influenced them, built much on Riemann‘s ideas.
In 1858 Helmholtz published his important paper in Crelle’s Journal on the motion of a perfect fluid. Helmholtz‘s paper Uber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen began by decomposing the motion of a perfect fluid into translation, rotation and deformation. It was this aspect which first interested Tait who saw that by using Hamilton‘s quaternions he could express the fluid velocity as a “vector function”. However the ideas in the paper which eventually led the Scottish mathematical physicists to topological considerations concerned vortex lines and vortex tubes. Helmholtz defined vortex lines as lines coinciding with the local direction of the axis of rotation of the fluid, and vortex tubes as bundles of vortex lines through an infinitesimal element of area. Helmholtz showed that the vortex tubes had to close up and also that the particles in a vortex tube at any given instant would remain in the tube indefinitely so no matter how much the tube was distorted it would retain its shape.
Helmholtz was aware of the topological ideas in his paper, particularly the fact that the region outside a vortex tube was multiply connected which led him to consider many-valued potential functions. He described his theoretical conclusions regarding two circular vortex rings with a common axis of symmetry in the following way:-
If they both have the same direction of rotation they will proceed in the same sense, and the ring in front will enlarge itself and move slower, while the second one will shrink and move faster, if the velocities of translation are not too different, the second will finally reach the first and pass through it. Then the same game will be repeated with the other ring, so the ring will pass alternately one through the other.
As we have mentioned Tait‘s first interest in Helmholtz‘ paper was because he saw applications of quaternions there. It was not until 1867 that Tait verified Helmholtz‘ theoretical claims regarding two circular vortex rings with experiments with smoke rings. He used two boxes each with a rubber diaphragm which shot out white smoke rings when the diaphragm was struck. Thomson wrote to Helmholtz on 22 January 1867:-
… a few days ago Tait showed me in Edinburgh a magnificent way of producing [vortex rings]. We sometimes can make one ring shoot through another, illustrating perfectly your description; when one ring passes near another, each is much disturbed, and is seen to be in a state of violent vibration for a few seconds, till it settles again into its circular form. … The vibrations make a beautiful subject for mathematical work.
These experiments were to have a major influence on Thomson who saw the permanence of form as a possible explanation for atoms and therefore explain the way that the different elements could be built. It is remarkable that Thomson was able to develop his ideas quickly enough that he could publish On vortex atoms in the Proceedings of the Royal Society of Edinburgh still in 1867. In this paper he wrote:-
Leucretius’s atom does not explain any of the properties of matter without attributing them to the atom itself … The possibility of founding a theory of elastic solids and liquids on the dynamics of closely packed vortex atoms may be reasonably anticipated.
Although was now know that Thomson was completely wrong, there is much in the way of correct reasoning in the quote we have just given. It was an idea which also led to interesting mathematical questions. As he wrote in the same paper:-
A full mathematical investigation of the mutual action between two vortex rings of any given velocities passing one another in any two lines, so directed that they never come nearer to one one another than a large multiple of the diameter of either, is a perfect mathematical problem; and the novelty of the circumstances contemplated presents difficulties of an exciting character. Its solution will be the proposed new kinetic theory of gases.
Thomson, as most scientists of his time, viewed space as being filled with a perfect fluid, the aether. The vortex atoms were then knotted tubes of aether which, by Helmholtz‘s theory, retained their form despite being distorted. This stability of vortices explained the stability of atoms. The vortex atoms, being built from the aether, required no special material. The different elements were accounted for by atoms composed of different knots or links and oscillations of the knots would, Thomson believed, explain the spectral lines which were characteristic of the different elements.
Tait began to think about knots and Thomson‘s second paper on vortex atoms, which appeared in 1869, included diagram of knots and links drawn by Tait. Long before this, however, Maxwell had entered the discussions which went on in letters exchanged by the three Scottish mathematical physicists. He was interested in knots because of electromagnetic considerations and in a letter to Tait written on the 4 December 1867 he rediscovered an integral formula counting the linking number of two closed curves which Gauss had discovered, but had not published, in 1833. Maxwell also gave equations in three dimensions which represented knotted curves.
However just when their work was proceeding rapidly, a major problem arose. Bertrand claimed that Helmholtz‘s 1858 paper, on which the idea of vortex atoms was based, was wrong. Maxwell was already intrigued by the problems and was not convinced that Bertrand‘s objections dealt a serious blow to Helmholtz‘s paper. In September 1868 Maxwell wrote several manuscripts which study knots and links. He set out the basic problem of the classification of knots and links as follows:-
Let any system of closed curves in space be given and let them be supposed capable of having their forms changed in any continuous manner, provided that no two curves or branches of a curve ever pass through the same point of space, we propose to investigate the necessary relations between the positions of the curves and the degree of complication of the different curves of the system.
Maxwell considered two-dimensional projections of links and devised a way of coding the diagrams to indicate which curve was above and which below at crossings on the projections.
He then looked at ways of modifying the diagrams without changing the link or knot. For a region bounded by one arc Maxwell noted that the region could be eliminated by uncoiling the curve. For regions bounded by two arcs, he noted that there were two cases, one where the arcs could be separated and the region eliminated, the other where this could not be done without making changes in other parts of the diagram. For regions bounded by three arcs Maxwell noted that again there were two cases:-
In the first case any one curve can be moved past the intersection of the other two without disturbing them. In the second case this cannot be done and the intersection of two curves is a bar to the motion of the third in that direction.
Although his approach contained no mathematical rigour, still it is interesting to note that at this early stage Maxwell had defined the “Reidemeister moves” which would be shown to be the fundamental moves in modifying knots in the 1920s.
In a a second manuscript Maxwell considered a region of space bounded by one external surface of genus n and m internal surfaces of genus n1, n, …, nm and showed that the region possessed N = n+n1+n+ …+ nm cycles and was (N+1)-ly connected. Now in modern terminology Maxwell was claiming that the first Betti number of the region was N. Again we should note that Maxwell did not give precise mathematical definitions of the concepts he was dealing with so no rigorous proof was possible. It is reasonable to ask how he then found the correct answer. The reason was that Maxwell, and for that matter Thomson too, reached their correct results using correct physical understanding, rather than mathematical intuition.
These manuscripts by Maxwell were not published at the time they were written despite Tait asking him to submit his ideas on knot theory to the Royal Society of Edinburgh for publication. However, more than 100 years after they were written these manuscripts were published in [2]. There are three manuscripts on knots and some time between the second, which Maxwell wrote in October 1868, and the third, which he wrote on 29 December 1868, he had read Listing‘s 1847 paper Vorstudien zur Topologie for in the third manuscript he lists Listing‘s main results. In February 1869 Maxwell presented an account of Listing‘s topological ideas to the London Mathematical Society.
In 1869 Thomson tried to clarify the topological ideas that he was using. The problem really came down to the fact that, perhaps not surprisingly, he had confused what we know today are two different concepts. He wrote:-
I shall call a finite portion of space n-ply continuous when its bounding surface is such that there are n irreconcilable paths between any two points on it.
He explains that two irreconcilable paths between points P and Q are paths which cannot be smoothly transformed into each other by paths which remain within the portion of space considered. This definition does not really work although one can see Thomson struggling to reach a definition of homotopy. He does not compose his paths, however, a vital ingredient in the definition of homotopy. In the proofs he gave in the paper Thomson does not even try to use his definition but rather resorts to arguments involving virtual barriers which would stop fluid flowing. Again his topology is driven by physical ideas of fluid flow but this notion, similar to the idea of a “cutting surface” which Riemann had introduced and Helmholtz had used, relates to our present day idea of homology, not homotopy.
By 1876 Thomson had made little progress with his ideas of vortex atoms. There were many problems in his way and indeed by this stage he had not succeeded in mathematically describing how two vortex rings would interact if they did not have a common axis of symmetry much more the way that knotted vortices would interact. Also there was no insight into vortex atoms through lists of knots which, in Thomson‘s theory, would explain the chemical elements. Tait decided to embark on a classification of plane closed curves in 1876, writing in a report to the British Association for the Advancement of Science:-
The development of this subject promises absolutely endless work – but work of a very interesting and useful kind – because it is intimately connected with the theory of knots, which (especially as applied in Sir W Thomson‘s Theory of Vortex Atoms) is likely soon to become an important branch of mathematics.
Now by looking at plane closed curves Tait was considering alternating knots, namely those which when traversing the projection in 2-dimensional space the crossings go alternately over and under. Choosing a starting point and a direction to traverse the path, he labelled the first, third, fifth etc. points by A, B, C etc. A knot with n crossings A, B, C, … would then be described by the sequence of crossings of length 2n where each of A, B, C, … occurred exactly twice when the knot was traversed. Tait called the sequence the “scheme of the knot”.
A knot with scheme ACBDCADB.
There were then two basic problems to solve. Firstly which sequences of the above type correspond to a knot, and secondly how could it be determined when two knots described by such sequences were the same. However there were some other problems, for example although a sequence of length 10, say, might represent a knot it might be one with less than 5 crossings. It might be a knot which could be reduced to one with fewer crossings. For example if the projection contained a crossing which divided the curve into two parts which did not intersect, then this was a nugatory crossing which could be removed by a twist.
Example of a nugatory crossing.
Tait conjectured that an alternating diagram without nugatory crossings would contain the minimum number of crossings. This became known as Tait‘s first conjecture. He gave a “proof” which showed that only nugatory crossings allowed the number of crossings to be reduced. However this is not good enough for there might be a sequence of moves which first increase the number of crossings, then further moves reduce to a fewer number of crossings than were there originally. If we interpret Tait in a form that he seems to have used the conjecture, namely that two alternating diagrams without nugatory crossings representing the same prime knot are related by a sequence of twists, then we get what has been called Tait‘s second conjecture. This was not finally proved until 1993.
Without any rigorous theory, which would have been well beyond nineteenth century mathematics, Tait began to classify knots using his mathematical and geometrical intuition. He knew that what was really required was a knot invariant, that is something which would be independent of the way that the knot was represented in two dimensions. First he looked for numerical invariants and considered the minimal number of crossings that a given knot might have in a two dimensional representation. This would lead him to Tait‘s first conjecture for alternating knots.
Another idea which seemed promising to Tait was the “beknottedness” which he defined as follows. Travel round the knot diagram and immediately after each crossing throw a copper coin to the left and a silver coin to the right if the crossing was above, or throw a silver coin to the left and a copper coin to the right if the crossing was below.
This led to:
Tait then defined “beknottedness” (now known as the twist number) as the excess of silver crossings over copper ones. If only diagrams without nugatory crossings were considered then Tait believed that this was a knot invariant. In fact it is not, but for alternating knots, it is an invariant and this fact is a consequence of Tait‘s second conjecture (a theorem since 1993). He tried other more obviously physical ideas such as considering the knot as a circuit and looking at the work done by a magnetic particle carried by a current in the knot. He tried another idea which at first looked very promising to him, namely the minimal number of crossings which required to be changed for under to over (or visa-versa) to unknot the knot. His first thought was that this would be half the beknottedness. He soon saw that this was not so. Seeing that the two concepts were distinct Tait changed his definitions and called the minimal number of crossings which required to be changed to unknot the diagram the beknottedness and he called the minimal number of crossings the knottiness.
Listing had introduced polynomials in to variables associated with a knot. These were produced by marking the four corners of a crossing l or r (for left of right) according to a rule which again related to over of under crossings. These polynomials were not invariants, however. Tait tried a similar idea where he marked the regions of the graph l or r and then connected r regions with a multiple bond of the order of the number of crossings on the boundary between the two regions. He was inspired to do this by analogy with chemical representations invented by his colleague (and brother-in-law) Alexander Crum Brown.
By 1877 Tait had classified all knots with seven crossings but he stopped there. He returned to the topic of knots in his address to the Edinburgh Mathematical Society in 1883:-
We find that it becomes a mere question of skilled labour to draw all the possible knots having any assigned number of crossings. The requisite labour increases with extreme rapidity as the number of crossings is increased. … I have not been able to find time to carry out this process further than the knots with seven crossings. … It is greatly desired that someone, with the requisite leisure, should try to extend this list, if possible up to 11 …
Kirkman read the text of Tait‘s address and began to work on classifying knots with more than seven crossings. He sent Tait his results on knot projections with up to nine crossings in May 1884 but he had not looked at the problem of deciding which of the projections led to equivalent knots. Tait worked on this side of the problem and, considering only alternating knots, solved the equivalence problems within a few weeks. Tait seemed to know how to tell whether two knots were equivalent without rigorous methods. He states this quite clearly in the paper he wrote tabulating the knots where he says that his methods have:-
… the disadvantage of being to a greater or less extent tentative. Not that the rules laid down … leave any room for mere guessing, but they are too complex to be always completely kept in view. Thus we cannot be absolutely certain that by means of such processes we have obtained all the essentially different forms which the definition we employ comprehends.
Despite the problems Tait knew exactly what he was doing for, remarkably, his tables are correct. When Kirkman sent him all knot projections with 10 crossings in January 1885 again Tait found all in equivalent knots. The tables were printed in September 1885 and again they are completely correct. By then he had received from Kirkman 1581 knot projections with 11 crossings and this time Tait felt that he did not have the time to solve the equivalence problem for these. However by this time an American mathematician and engineer Charles N Little had sent Tait knot tables which he had calculated and Little began to extend the tables to knots other than alternating ones, and to knots with eleven crossings.
Cubic surfaces
An algebraic surface is one of the form f(x,y,z) = 0 where f(x,y,z) is a polynomial in x, y and z. The order of the surface is the degree of the polynomial. A surface of order one is a plane. A surface of order two is called a quadric surface and consists of surfaces such as ellipsiods and hyperboloids. These include cones, cylinders and paraboloids. The surface whose history we are interested in for this short article is a surface of order three which is called a cubic surface.
In 1849 Salmon and Cayley published the results of their correspondence on the number of straight lines on a cubic surface. It was Cayley who, in a letter to Salmon, first showed that there could be only a finite number of straight lines on a cubic surface while it was Salmon who then proved that there were exactly 27 such straight lines in general. At the end of his 1865 treatise The Geometry of Three Dimensions Salmon described how the two had collaborated over finding the Cayley–Salmon theorem.
Steiner already knew of Cayley–Salmon theorem about 27 straight lines when he started his own work on cubic surfaces. He wrote an important article which gave results that allowed a purely geometrical treatment of cubic surfaces. He proved in 1856 that:-
The nine straight lines in which the surfaces of two arbitrarily given trihedra intersect each other determine, together with one given point, a cubic surface.
He introduced the notion of a “nuclear surface” and investigated its properties. Many results on cubic surfaces were stated by Steiner without proof and we shall comment later how Cremona and Rudolf Sturm proved many of these ten years after Steiner‘s paper.
Clebsch described the plane representations of various rational surfaces, he was especially interested in that of the general cubic surface. Using the Hessian surface, he gave the first proof that any given cubic surface could be written in the pentahedral form which had been proposed by Sylvester. Other results on cubic surfaces were proved by Clebsch which included: there exists a covariant of order nine which intersects the cubic surface in exactly 27 lines; and every smooth cubic surface can be represented in the plane using four plane cubic surfaces through six points and vice-versa.
It was Steiner who communicated to Schläfli the Cayley–Salmon theorem on 27 lines on a cubic surface. In 1858 Schläfli became the first to classify the cubic surfaces with respect to the number of real straight lines and tritangent planes on them, finding that there were exactly five types in his classification. Schläfli then found 36 “double sixes” on this surface. He divided cubic surfaces into 23 species according to the nature of their singularities in 1863 and he published the classification in his paper On the distribution of surfaces of the third order into species, in reference to the presence or absence of singular points and the reality of their lines. In his lengthy Memoir on Cubic Surfaces Cayley presented Schläfli’s complete classification of cubic surfaces into 23 distinct species and he also added further investigations of his own.
In March 1866 Cremona published Memoire de géometrie pure sur les surfaces du troisième ordre. In this memoir he established many of the properties that had only been stated by Steiner. He also established connections between the Cayley–Salmon theorem on 27 lines on a cubic surface and Pascal‘s Mystic Hexagram:-
If a hexagon is inscribed in any conic section, then the points where opposite sides meet are collinear.
For his memoir Cremona was awarded a share of the Steiner Prize. He shared the Prize with Rudolf Sturm who studied third degree surfaces in their projective representations and also proved theorems stated, but not proved, by Steiner.
In 1869, at Clebsch‘s suggestion, Christian Wiener constructed plaster of Paris models of cubic surfaces which, together with other models of surfaces he had constructed, were exhibited in London in 1876, Munich in 1893, and Chicago also in 1893. Klein investigated cubic surfaces in 1870 and his work shows a special concern for geometric intuition regarding spatial constructions.
Karl Geiser‘s great uncle was Steiner so he set out on his mathematical career already having links to one of the important figures in the development of the theory of cubic surfaces. Perhaps, therefore, it is not surprising that he should make his most important research contribution on cubic surfaces. One of his results explains how the 28 double tangents of the plane quadric are related to the 27 straight lines of the cubic surface.
Le Paige spent his whole career at the University of Liège where he worked on the theory of algebraic forms, a topic whose study had been initiated by Boole in 1841 and then developed by Cayley, Sylvester, Hermite, Clebsch and Aronhold. In particular Le Paige studied the geometry of algebraic curves and surfaces, building on this earlier work. He is best known for his construction of a cubic surface given by nineteen points. Starting from the construction of a cubic surface given by a straight line, three groups of three points on a line, and six other points, Le Paige was led to the construction of a cubic surface given by a line, three points on a line and twelve other points. By means of this construction he then constructed a cubic surface given by three points on a line and sixteen other points, finally arriving at a cubic surface given by nineteen points.
The last person we will mention in this short history of the study of cubic surfaces is Gino Fano. Fano studied with Klein in 1893 and did an Italian translation of Klein‘s Erlanger Program (1872), which gave his synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations. It was later in his career that Fano became interested in cubic surfaces and algebraic surfaces in general. Already by this time, however, interest in topics of this type had somewhat declined.
Mathematics and art – perspective
This article looks at some of the interactions between mathematics and art in western culture. There are other topics which will look at the interaction between mathematics and art in other cultures. Before beginning the discussion of perspective in western art, we should mention the contribution by al-Haytham. It was al-Haytham around 1000 A.D. who gave the first correct explanation of vision, showing that light is reflected from an object into the eye. He studied the complete science of vision, called perspectiva in medieval times, and although he did not apply his ideas to painting, the Renaissance artists later made important use of al-Haytham‘s optics.
There is little doubt that a study of the development of ideas relating to perspective would be expected to begin with classical times, and in particular with the ancient Greeks who used some notion of perspective in their architecture and design of stage sets. However, although Hellenistic painters could create an illusion of depth in their works, there is no evidence that they understood the precise mathematical laws which govern correct representation. We chose to begin this article, therefore, with the developments in the understanding of perspective which took place during the Renaissance. First let us state the problem: how does one represent the three-dimensional world on a two-dimensional canvass? There are two aspects to the problem, namely how does one use mathematics to make realistic paintings and secondly what is the impact of the ideas for the study of geometry.
By the 13th Century Giotto was painting scenes in which he was able to create the impression of depth by using certain rules which he followed. He inclined lines above eye-level downwards as they moved away from the observer, lines below eye-level were inclined upwards as they moved away from the observer, and similarly lines to the left or right would be inclined towards the centre. Although not a precise mathematical formulation, Giotto clearly worked hard on how to represent depth in space and examining his pictures chronologically shows how his ideas developed. Some of his last works suggest that he may have come close to the correct understanding of linear perspective near the end of his life.
The person who is credited with the first correct formulation of linear perspective is Brunelleschi. He appears to have made the discovery in about 1413. He understood that there should be a single vanishing point to which all parallel lines in a plane, other than the plane of the canvas, converge. Also important was his understanding of scale, and he correctly computed the relation between the actual length of an object and its length in the picture depending on its distance behind the plane of the canvas. Using these mathematical principles, he drew two demonstration pictures of Florence on wooden panels with correct perspective. One was of the octagonal baptistery of St John, the other of the Palazzo de Signori. To give a more vivid demonstration of the accuracy of his painting, he bored a small hole in the panel with the baptistery painting at the vanishing point. A spectator was asked to look through the hole from behind the panel at a mirror which reflected the panel. In this way Brunelleschi controlled precisely the position of the spectator so that the geometry was guaranteed to be correct. These perspective paintings by Brunelleschi have since been lost but a “Trinity” fresco by Masaccio from this same period still exists which uses Brunelleschi‘s mathematical principles.
Here is a picture of Masaccio’s Holy Trinity
It is reasonable to think about how Brunelleschi came to understand the geometry which underlies perspective. Certainly he was trained in the principles of geometry and surveying methods and, since he had a fascination with instruments, it is reasonable to suppose that he may have used instruments to help him survey buildings. He had made drawing of the ancient buildings of Rome before he came to understand perspective and this must have played an important role.
Now although it is clear that Brunelleschi understood the mathematical rules involving the vanishing point that we have described above, he did not write down an explanation of how the rules of perspective work. The first person to do that was Alberti in his treatise On painting. Now in fact Alberti wrote two treatises, the first was written in Latin in 1435 and entitled De pictura while the second, dedicated to Brunelleschi, was an Italian work written in the following year entitled Della pittura. Certainly these books are not simply the same work translated into two different languages. Rather Alberti addresses the books to different audiences, the Latin book is much more technical and addressed to scholars while his Italian version is aimed at a general audience.
De pictura is in three parts, the first of which gives the mathematical description of perspective which Alberti considers necessary to a proper understanding of painting. It is, Alberti writes:-
… completely mathematical, concerning the roots in nature from which arise this graceful and noble art.
In fact he gives a definition of a painting which shows just how fundamental he considers the notion of perspective to be:-
A painting is the intersection of a visual pyramid at a given distance, with a fixed centre and a defined position of light, represented by art with lines and colours on a given surface.
Alberti gives background on the principles of geometry, and on the science of optics. He then sets up a system of triangles between the eye and the object viewed which define the visual pyramid referred to above. He gives a precise concept of proportionality which determines the apparent size of an object in the picture relative to its actual size and distance from the observer.
One of the most famous examples used by Alberti in his text was that of a floor covered with square tiles. For simplicity we take the centric point, as Alberti calls it (today it is called the vanishing point), in the centre of the square picture.
Here is a Alberti’s construction of perspective for a tiled floor
In our diagram the centric point is C. The square tiles are assumed to have one edge parallel to the bottom of the picture. The other edges which in reality are perpendicular to these edges, will appear in the picture to converge to the centric point C. The diagonals of the squares will all converge to a point D on a line through the centric point parallel to the bottom of the picture. The length of CD determines the correct viewing distance, that is the distance the observer has to be from the picture to obtain the correct perspective effect. Alberti chooses not to give mathematical proofs, however, writing:-
We have talked as much as seems necessary about the pyramid, the triangle, the intersection. I usually explain these things to my friends with certain tedious geometrical proofs, which in this commentary it seems to me better to omit for the sake of brevity.
Pictures from this period which include a square tiled floor are called pavimento (Italian for floor) pictures. There are many examples of such pictures in the years following Alberti‘s book which had a huge influence on painting.
Of course the pavimento provides a type of Cartesian coordinate system. Alberti shows how to use the grid to obtain the correct shape for a circle. Place a circle on a square grid and mark where the squares cut the circle. Construct the perspective view of the square grid as above and reconstruct the circle by seeing the positions of the points of intersection in the projected view. The circle will project into an ellipse, but it would be a long time before the importance of projecting conic sections was realised.
Next we should mention Lorenzo Ghiberti who was born in Pelago, Italy around 1378. He is famed as a sculptor and his most famous work is the bronze doors on the east side of the baptistery in Florence. He created two sets of doors and before he designed the second set he had become familiar with the new ideas on perspective as set out by Alberti. The doors contain ten panels which, Ghiberti wrote, exhibit:-
… architectural settings in the relation with which the eye measures them, and real to such a degree that … one sees the figures which are near appear larger, and those that are far off smaller, as reality shows it.
Ghiberti is also important for his treatise I Commentarii, written around 1447, in three volumes. The work contains a history of art in ancient times, a history of thirteenth century artists, an autobiography, and a compilation of medieval texts on the theory of vision such as that by al-Haytham. This was important since, as we mentioned at the beginning of this article, al-Haytham and others had studied optics and vision without relating the ideas to painting, while now Ghiberti showed the relevance of the earlier ideas on optics to art.
The most mathematical of all the works on perspective written by the Italian Renaissance artists in the middle of the 15th century was by Piero della Francesca. In some sense this is not surprising since as well as being one of the leading artists of the period, he was also the leading mathematician writing some fine mathematical texts. In Trattato d’abaco which he probably wrote around 1450, Piero includes material on arithmetic and algebra and a long section on geometry which was very unusual for such texts at the time. It also contains original mathematical results which again is very unusual in a book written in the style of a teaching text (although in the introduction Piero does say that he wrote the book at the request of his patron and friends and not as a school book). Is there a connection with perspective? Yes there is, for Piero illustrates the text with diagrams of solid figures drawn in perspective.
Here is Piero’s illustration of a dodecahedron
Continuing the theme of the regular solids, we note that a later text by Piero is Short book on the five regular solids. However, it is his three volume treatise On perspective for painting (some believe written in the mid 1470s, others believe written in the 1460s) which is of most interest to us in this article. His book begins with a description of painting:-
Painting has three principal parts, which we say are drawing, proportion and colouring. Drawing we understand as meaning outlines and contours contained in thing. Proportion we say is these outlines and contours positioned in proportion in their places. Colouring we mean as giving the colours as shown in the things, light and dark according as the light makes them vary. Of the three parts I intend to deal only with proportion, which we call perspective, mixing in with it some part of drawing, because without this perspective cannot be shown in action; colouring we shall leave out, and we shall deal with that part which can be shown by means of lines, angles and proportion, speaking of points, lines, surfaces and bodies.
We see from this introduction that Piero intends to concentrate on the mathematical principles. Perhaps it is most accurate to say that he is studying the geometry of vision which he later makes clearer:-
First is sight, that is to say the eye; second is the form of the thing seen; third is the distance from the eye to the thing seen; fourth are the lines which leave the boundaries of the object and come to the eye; fifth is the intersection, which comes between the eye and the thing seen, and on which it is intended to record the object.
Piero begins by establishing geometric theorems in the style of Euclid but, unlike Euclid, he also gives numerical examples to illustrate them. He then goes on to give theorems which relate to the perspective of plane figures. In the second of the three volumes Piero examines how to draw prisms in perspective. Although less interesting mathematically than the first volume, the examples he chooses to examine in the volume are clearly important to him since they appear frequently in his own paintings. The third volume deals with more complicated objects such as the human head, the decoration on the top of columns, and other “more difficult shapes”. For this Piero uses a method which involves a very large amount of tedious calculation. He uses two rulers, one to determine width, the other to determine height. In fact he is using a coordinate system and computing the correct perspective position of many points of the “difficult shape” from which the correct perspective of the whole can be filled in.
Piero della Francesca‘s works were heavily relied on by Luca Pacioli for his own publications. In fact the third book of Pacioli‘s Divina proportione is an Italian translation of Piero‘s Short book on the five regular solids. The illustrations in Pacioli‘s work were by Leonardo da Vinci and include some fine perspective drawings of regular solids.
Here is a Leonardo’s illustration
Now in Leonardo‘s early writings we find him echoing the precise theory of perspective as set out by Alberti and Piero. He writes:-
… Perspective is a rational demonstration by which experience confirms that the images of all things are transmitted to the eye by pyramidal lines. Those bodies of equal size will make greater or lesser angles in their pyramids according to the different distances between the one and the other. by a pyramid of lines I mean those which depart from the superficial edges of bodies and converge over a distance to be drawn together in a single point.
He developed mathematical formulas to compute the relationship between the distance from the eye to the object and its size on the intersecting plane, that is the canvas on which the picture will be painted:-
If you place the intersection one metre from the eye, the first object, being four metres from the eye, will diminish by three-quarters of its height on the intersection; and if it is eight metres from the eye it will diminish by seven-eighths and if it is sixteen metres away it will diminish by fifteen-sixteenths, and so on. As the distance doubles so the diminution will double.
Not only did Leonardo study the geometry of perspective but he also studied the optical principles of the eye in his attempts to create reality as seen by the eye. By around 1490 Leonardo had moved forward in his thinking about perspective. He was one of the first people to study the converse problem of perspective: given a picture drawn in correct linear perspective compute where the eye must be placed to see this correct perspective. Now he was led to realise that a picture painted in correct linear perspective only looked right if viewed from one exact location. Brunelleschi had been well aware of this when he arranged his demonstration of perspective through a hole. However for a painting on a wall, say, many people would not view it from the correct position, indeed for many paintings it would be impossible for someone viewing them to have their eye in this correct point, as it may have been well above their heads.
Leonardo distinguished two different types of perspective: artificial perspective which was the way that the painter projects onto a plane which itself may be seen foreshortened by an observer viewing at an angle; and natural perspective which reproduces faithfully the relative size of objects depending on their distance. In natural perspective, Leonardo correctly claims, objects will be the same size if they lie on a circle centred on the observer. Then Leonardo looked at compound perspective where the natural perspective is combined with a perspective produced by viewing at an angle. Perhaps in Leonardo, more than any other person we mention in this article, mathematics and art were fused in a single concept.
The story we have told up to this point has been very much an understanding of perspective in Italy by artists and mathematicians learning personally from each other. By 1500, however, Dürer took the development of the topic into Germany. He did so only after learning much from trips to Italy where he learned at first hand from mathematicians such as Pacioli. He published Unterweisung der Messung mit dem Zirkel und Richtscheit in 1525, the fourth book of which contains his theory of shadows and perspective. Geometrically his theory is similar to that of Piero but he made an important addition stressing the importance of light and shade in depicting correct perspective. An excellent example of this is in the geometrical shape he sketched in 1524.
Here is a Dürer’s shaded geometrical design
Another contribution to perspective made by Dürer in his 1525 treatise was the description of a variety of mechanical aids which could be used to draw images in correct perspective.
Let us consider a number of other contributions to the study of perspective over the following 200 years. We mention first Federico Commandino who published Commentarius in planisphaerium Ptolemaei in 1558. In this work he gave an account of Ptolemy‘s stereographic projection of the celestial sphere, but its importance for perspective is that he broadened the study of that topic which had up until then been concerned almost exclusively with painting. Commandino was more interested in the use of perspective in the making of stage scenery principally because his main interest was in classic texts and, unlike many earlier treatises he was writing for mathematicians rather than artists.
Wentzel Jamnitzer wrote a beautiful book on the Platonic solids in 1568 called Perspectiva corporum regularium. This is not a book designed to teach perspective drawing but, nevertheless, contains many illustrations superbly drawn in perspective. He is clear in his intention:-
All superfluity will be avoided and, in contrast to the old fashioned way of teaching, no line or point will be drawn needlessly.
Daniele Barbaro’s La Practica della perspectiva published in 1569, the year after Jamnitzer’s treatise, complained that painters had stopped using perspective. Taken at face value this is not true, but what he undoubtedly meant was that painters were not painting architectural scenes. Barbaro was interested in perspective in stage sets mainly because he had published an Italian translation of Vitruvius’s On architecture in 1556 and his interest had been aroused by this work. His 1569 treatise shows that he had studied the work of Piero and Dürer carefully and the methods he gave for perspective constructions were variations on their methods.
Egnatio Danti, like so many of the others we have mentioned in this article, was both an excellent mathematician and artist. His preface to Le due regole della prospettiva pratica di M Iacomo Barozzi da Vignola was published in 1583, three years before his death. In his introduction to this work Danti wrote a brief history of perspective:-
… we know of no book or written document which has come down to us from ancient practitioners, although they were mot excellent, as is convincingly shown by the descriptions of the stage scenery they made, which was much prized both in Athens among the Greeks and in Rome among the Latins. But in our own time, among those who have left something of note in this art, the earliest, and one who wrote with best method and form, was Messer Pietro della Francesca dal Borgo Sansepolcro, from whom we have today three books in manuscript, most excellently illustrated; and whoever wants to know how excellent they are should look to Daniele Barbaro, who has transcribed a great part of them in his book on Perspective.
Not only did Danti write an introduction to his edition of Vignola’s treatise, but he also added considerably to its content by giving mathematical justification where Vignola simply states a rule to be applied.
The next contributor we mention is Giovanni Battista Benedetti who was a pupil of Tartaglia. He produced a work entitled A book containing various studies of mathematics and physics in 1585 which contains a treatise on arithmetic, some other short works and letters on various scientific topics, as well as a short treatise on perspective De rationibus operationum perspectivae. In his perspective treatise Benedetti was concerned not just with rules for artists working in two dimensions but with the underlying three-dimensional reasons for the rules.
We mentioned Commandino above and the next person who we want to note for his contribution to perspective, Guidobaldo del Monte, was a pupil of Commandino. Del Monte‘s six books on perspective Perspectivae libri sex (1600) contain theorems which he deduces with frequent references to Euclid‘s Elements. The most important result in del Monte‘s treatise is that any set of parallel lines, not parallel to the plane of the picture, will converge to a vanishing point. This treatise represents a major step forward in understanding the geometry of perspective and it was a major contribution towards the development of projective geometry.
In 1636 Desargues published the short treatise La perspective which only contains 12 pages. In this treatise, which consists of a single worked example, Desargues sets out a method for constructing a perspective image without using any point lying outside the picture field. He considers the representation in the picture plane of lines which meet at a point and also of lines which are parallel to each another. In the last paragraph of the work he considered the problem of finding the perspective image of a conic section.
Three years later, in 1639, Desargues wrote his treatise on projective geometry Brouillon project d’une atteinte aux evenemens des rencontres du cone avec un plan. One can see the influence of the work from three years earlier, but Desargues himself gives no motivation for the ideas he introduced. The first part of this treatise deals with the properties of sets of straight lines meeting at a point and ranges of points lying on a straight line. In the second part, the properties of conics are investigated in terms of properties of ranges of points on straight lines. The modern term “point at infinity” appears for the first time in this treatise and pencils of lines are introduced, although that name is not used. In this treatise Desargues shows that he had completely understood the connection between conics and perspective; in fact he treats the fact that any conic can be projected into any other conic as obvious. Although a “cone of vision” had been considered by earlier authors, the significance of this and the way that a study of conics could thus be unified had not been appreciated before.
Following Desargues‘ innovative work it may be surprising that the subject was not developed rapidly in the following years. That it was not may in part have been due to mathematicians failing to recognise the power in what had been put forward. On the other hand the algebraic approach to geometry put forward by Descartes at almost exactly the same time (1637) may have diverted attention from Desargues‘ projective methods. The first person to really carry forward Desargues‘ ideas was Philippe de la Hire. He had written a work on conics in 1673 before he discovered Desargues‘ Brouillon project. In 1679 he made a copy of Desargues‘ book writing:-
In the month of July of the year 1679, I first read this little book by M. Desargues, and copied it out so as to get to know it better. This was more than six years after I had published my first work on conic sections. And I do not doubt that, if I had known anything of this treatise, I should not have discovered the method that I used, for I should never have believed it possible to find any simpler procedure which was also general in application.
In fact la Hire had treated conics from a projective point of view in his 1673 treatise New method of geometry for sections of conics and cylindrical surfaces and there he had introduced the cross ratio of four points before meeting Desargues‘ approach. In 1685 la Hire published Conic sections which is a projective approach to conics which combines the best of the ideas from his earlier work and also those of Desargues.
Before discussing the work of Brook Taylor, with which we will end our article, let us mention that of Humphry Ditton who wrote A treatise on perspective, demonstrative and practical in 1712. This is relevant to Taylor‘s work since it influenced him. Ditton’s book is not particularly original but he did present a geometrical approach to perspective which is carefully constructed and well written. In many ways Brook Taylor‘s Linear perspective: or a new method of representing justly all manners of objects which appeared three years later in 1715, is similar to Ditton’s work in its quality. One notable aspect of Taylor‘s work was that he stated the incidence properties as axioms, making him the first to do so.
In 1719 Taylor published a much modified second edition New principles of linear perspective. The work gives the first general treatment of vanishing points. Taylor had a highly mathematical approach to the subject and, despite being an accomplished amateur artist himself, made no concessions to artists who should have found the ideas of fundamental importance to them. At times this highly condensed work is very difficult for even a mathematician to understand, and Taylor makes it clear that he is interested in the underlying principles rather than their application. The phrase “linear perspective” was invented by Taylor in this work and he defined the vanishing point of a line, not parallel to the plane of the picture, as the point where a line through the eye parallel to the given line intersects the plane of the picture. He also defined the vanishing line to a given plane, not parallel to the plane of the picture, as the intersection of the plane through the eye parallel to the given plane. As we have shown above the term vanishing point was invented long before Taylor‘s time, but he was one of the first to stress the mathematical importance of the vanishing point and vanishing line. The main theorem in Taylor‘s theory of linear perspective is that the projection of a straight line not parallel to the plane of the picture passes through its intersection and its vanishing point.
There is also the interesting inverse problem which is to find the position of the eye in order to see the picture from the viewpoint that the artist intended. Taylor was not the first to discuss this inverse problem as we saw above, one of the first to examining it had been Leonardo nearly 250 years earlier, but Taylor did make innovative contributions to the theory of such perspective problems. One could certainly consider this work as being an important step towards the theory of descriptive and projective geometry as developed by Monge, Chasles and Poncelet.
Let us end by giving examples of artists having fun with the deliberate misuse of perspective. The first is by the famous English artist William Hogarth (1697-1764) whose Perspective absurdities formed the frontispiece to J J Kirby’s book Dr Brook Taylor’s method of perspective made easy in both theory and practice (1754).
Here is Hogarth’s Perspective absurdities
The second examples are by Maurits Escher who is famous for producing impossible pictures using perspective tricks.
Here are Waterfall and Up and down
Euclid’s definitions
Book 1 of The Elements begins with numerous definitions followed by the famous five postulates. Then, before Euclid starts to prove theorems, he gives a list of common notions. The first few definitions are:
Def. 1.1. A point is that which has no part.
Def. 1.2. A line is a breadthless length.
Def. 1.3. The extremities of lines are points.
Def. 1.4. A straight line lies equally with respect to the points on itself.
The postulates are ones of construction such as:
One can draw a straight line from any point to any point.
The common notions are axioms such as:
Things equal to the same thing are also equal to one another.
We should note certain things.
- Euclid seems to define a point twice (definitions 1 and 3) and a line twice (definitions 2 and 4). This is rather strange.
- Euclid never makes use of the definitions and never refers to them in the rest of the text.
- Some concepts are never defined. For example there is no notion of ordering the points on a line, so the idea that one point is between two others is never defined, but of course it is used.
- As we noted in The real numbers: Pythagoras to Stevin, Book V of The Elements considers magnitudes and the theory of proportion of magnitudes. However Euclid leaves the concept of magnitude undefined and this appears to modern readers as though Euclid has failed to set up magnitudes with the rigour for which he is famed.
- When Euclid introduces magnitudes and numbers he gives some definitions but no postulates or common notions. For example one might expect Euclid to postulate a + b = b + a, (a + b) + c = a + (b + c), etc., but he does not.
- When Euclid introduces numbers in Book VII he does make a definition rather similar to the basic ones at the beginning of Book I:A unit is that by virtue of which each of the things that exist are called one.
Some historians have suggested that the difference between the way that basic definitions occur at the beginning of Book I and of Book V is not because Euclid was less rigorous in Book V, rather they suggest that Euclid always left his basic concepts undefined and the definitions at the beginning of Book I are later additions. What is the evidence for this?
The first comment would be that this would explain why Euclid never refers to the basic definitions. If they were not in the text that Euclid wrote then of course he couldn’t refer to them. The next point to note is that they are very similar to the work which is ascribed to Heron called Definitions of terms in geometry. This contains 133 definitions of geometrical terms beginning with points, lines etc. which are very close to those given by Euclid. In [2] Knorr argues convincingly that this work is in fact due to Diophantus. The point here is the following. Is Definitions of terms in geometry based on Euclid‘s Elements or have the basic definitions from this work been inserted into later versions of The Elements?
We have to consider what Sextus Empiricus says about definitions. First note that Sextus wrote about 200 AD and it was believed until comparatively recently that Heron lived later than this. Were this the case, then of course Sextus could not have referred to anything written by Heron. However more recently Heron has been dated to the first century AD and this tells us that Sextus wrote after Heron. The other part of the puzzle we have to consider here is the earliest version of Euclid‘s Elements to be found. When Vesuvius erupted in 79 AD, Herculaneum together with Pompeii and Stabiae, was destroyed. Herculaneum was buried by a compact mass of material about 16 m deep which preserved the city until excavations began in the 18th century. Special conditions of humidity of the ground conserved wood, cloth, food, and in particular papyri which give us important information. One papyrus found there contains fragments of The Elements and was clearly written before 79 AD. Since Philodemus, a student of Zeno of Sidon, took his library of papyri there some time soon after 75 BC the version of The Elements is likely to be of around that date.
Let us go back to Sextus who writes about “mathematicians describing geometrical entities” and it is interesting that the word “describing” is not used in The Elements but is used by Heron in Definitions of terms in geometry. Again the descriptions he gives are closer to the exact words appearing in Heron than those of Euclid. When Sextus give “the definition of a circle” he uses the word “definition” which is that of Euclid. Sextus quotes the precise definition of a circle which appears in the Herculaneum fragment. This does not include a definition of “circumference” although Euclid does use the notion of circumference of a circle. The later versions of The Elements which have come down to us include a definition of “circumference” within the definition of a circle.
None of the above proves whether the basic definitions of geometric objects have been added to The Elements later. They do show fairly convincingly that the definition of a circle has been extended to include the definition of circumference in later editions of the book. The hypothesis is that Sextus has The Elements and Definitions of terms in geometry in front of him when he is writing and he uses the word “describe” when he refers to Heron and “define” when he refers to Euclid. Even if this is correct it still doesn’t prove that the version of The Elements sitting in front of Sextus does not contain basic definitions of geometric objects but it does make such a possibility at least worth debating. What do you think?
One last point to think about. We quoted above:
Def. 1.4. A straight line lies equally with respect to the points on itself.
What does this mean? It does seem a strange description for Euclid to give, for it appears to be meaningless. Compare it with the definition of a straight line in Definitions of terms in geometry:
A straight line is a line that equally with respect to all points on itself lies straight and maximally taut between its extremities.
Again we ask the reader: do you think that the definition appearing in The Elements is a corruption of Heron‘s definition and so was added later, or do you think that Euclid gave a rather poor definition which was improved by Heron? Why do neither use the definition of a straight line as the shortest distance between two points?
A History of Fractal Geometry
Any mathematical concept now well-known to school children has gone through decades, if not centuries of refinement. A typical student will, at various points in her mathematical career — however long or brief that may be — encounter the concepts of dimension, complex numbers, and “geometry”. If the field of mathematics does not particularly interest her, this student might see these concepts as distinct and unrelated and, in particular, she might make the mistake of thinking that the Euclidean geometry taught to her in school encompasses the whole of the field of geometry. However, if she were to pursue mathematics at the university level, she might discover an exciting and relatively new field of study that links the aforementioned ideas in addition to many others: fractal geometry.
While the lion’s share of the credit for the development of fractal geometry goes to Benoît Mandelbrot, many other mathematicians in the century preceding him had laid the foundations for his work. Moreover, Mandelbrot owes a great deal of his advancements to his ability to use computer technology — an advantage that his predecessors distinctly lacked; however, this in no way detracts from his visionary achievements. Nevertheless, while acknowledging and understanding the accomplishments of Mandelbrot, it undoubtedly helps to have some familiarity with the relevant works of Karl Weierstrass, Georg Cantor, Felix Hausdorff, Gaston Julia, Pierre Fatou and Paul Lévy — not only to make Mandelbrot‘s work clearer — but to see its connections to other branches of mathematics. Equally, while most authors will not fail to include at least brief discussion of Mandelbrot‘s rather interesting and slightly unconventional (for a modern mathematician) life in their texts on fractals, it seems only fair to give some, if not equal, consideration to his predecessors.
Until the 19th century, mathematics had concerned itself only with functions that produced differentiable curves. Indeed, the conventional wisdom of the day said that any function with an analytic formula (i.e. sum of a convergent power series) would certainly produce such a curve. [3] However, on July 18, 1872, Karl Weierstrass presented a paper at the Royal Prussian Academy of Sciences showing that for a a positive integer and 0 < b < 1
bncos(anx π)is not differentiable. Using the limit definition of a derivative, he showed that the difference quotient of the function
gets arbitrarily large as the index of summation increases.
As Weierstrass himself pointed out, Riemann had introduced
sin(n2x)/n2as an example of a non-differentiable analytic function, but never published a proof, nor could anyone replicate it. [14] Thus, Weierstrass‘s proof stands as the first rigorously proven example of a function that is analytic, but not differentiable. While Weierstrass, and indeed, much of the mathematical establishment of the time eschewed the use of graphs in favour of symbolic manipulation in order to prove results, future mathematicians such as Helge von Koch and Mandelbrot himself found it useful to represent their results graphically. [5] [7] Indeed, when one has only worked with curves that are differentiable almost everywhere, an obvious question when one encounters a formula for a curve that is not is, “what does it look like?”
While these are both approximations, one can see that these functions lack the smoothness of parabolas or of the sine and cosine functions. These functions resisted traditional analysis and were — though not due to their appearance, which was beyond the ability of mathematicians of the day to represent — labelled “monsters” by Charles Hermite and were largely ignored by the contemporary mathematical community. [2]
In 1883 Georg Cantor, who attended lectures by Weierstrass during his time as a student at the University of Berlin [9] and who is to set theory what Mandelbrot is to fractal geometry, [3] introduced a new function, ψ , for which ψ‘ = 0 except on the set of points, {z}. This set, {z}, is what became known as the Cantor set.
The function ψ is singular, monotone, non-constant and ψ‘ = 0 almost everywhere. It also has the property that
ψ‘ (x) dx = 0 [3]The Cantor set has a Lebesgue measure of zero; however, it is also uncountably infinite. [3] What is more, it has the property of being self-similar, meaning that if one magnifies a section of the set, one obtains the whole set again. Looking at Figure 4, one can easily see that each horizontal line is one third the size of the horizontal line directly above it. In fact, self-similarity is a feature of fractals, and the Cantor set is an early example of a fractal, though self-similarity was not defined until 1905 (by Cesàro, who was analysing the paper by Helge von Koch discussed below) and fractals were not defined until Mandelbrot in 1975, [2] thus Cantor would not have thought of it in those terms.
In a paper published in 1904, Swedish mathematician Helge von Koch constructed using geometrical means the now-famous von Koch curve and hence the Koch snowflake, which is three von Koch curves joined together. In the introduction to his paper he stated the following about Weierstrass‘s 1872 essay [6]:
… it seems to me that his [Weierstrass‘s] example is not satisfactory from the geometrical point of view since the function is defined by an analytic expression that hides the geometrical nature of the corresponding curve and so from this point of view one does not see why the curve has no tangent. Rather it seems that the appearance is actually in contradiction with the factual reality established by Weierstrass in a purely analytic way.
Von Koch‘s curve, like the Cantor set, has the property of self-similarity. It, too, is a fractal, though, like Cantor, von Koch was not thinking in such terms. He merely aimed to provide an alternative way of proving that functions that were non-differentiable (i.e. functions that “have no tangents” in geometric parlance) could exist — a way that involved using “elementary geometry” (reference [6]’s title translates to On a Continuous Curve without Tangent Constructible from Elementary Geometry). In doing so, von Koch expressed a link between these non-differentiable “monsters” of analysis and geometry.
Von Koch himself was a fairly unremarkable mathematician. Many of his other results were derived from those of Henri Poincaré, from whom he knew it was possible to obtain “pathological” results — i.e. these so-called “monsters” — but never really explored them, outside of the aforementioned essay. [5] Poincaré, it should be noted, studied non-linear dynamics in the later 19th century, which eventually led to chaos theory, [2] a field closely related to fractal geometry, though beyond the scope of this paper. It is therefore fitting that a mathematician whose work followed that of Poincaré so closely would turn out to be one of the forefathers of a field that is closely related to the area of study for which Poincaré himself helped lay the foundations.
An absolutely key concept in the study of fractals, aside from the aforementioned self-similarity and non-differentiability, is that of Hausdorff dimension, a concept introduced by Felix Hausdorff in March of 1918. Hausdorff‘s results from the same paper were important to the field of topology, as well; [3] however that his definition of dimension extended the previous definition to allow for sets to have a dimension that is an arbitrary, non-zero value [4] (unlike topological dimension) ended up being integral to the definition of a fractal, as Mandelbrot defined fractals “a set having Hausdorff dimension strictly greater than its topological dimension.” [2]
As soon as Hausdorff introduced this new, expanded definition of dimension, it was the subject of investigation — in particular by Abraham Samilovitch Besicovitch, who, from 1934 to early 1937 wrote no less than three papers referencing Hausdorff‘s work. [3] Sadly, by this time, Hausdorff was experiencing difficulties living as a Jew in Nazi Germany. He was forced to give up his post as a professor at the University of Bonn in 1935, and even though he continued to work on set theory and topology, his work could only be published outside of Germany. Despite temporarily managing to avoid being sent to a concentration camp, the situation in Germany quickly became unbearable and, with nowhere else to go, he, along with his wife and sister-in-law, opted to commit suicide in January 1942. [4]
The Hausdorff dimension, d, of a self-similar set — its connection to fractal geometry, though, as previously stated, there are many other applications of Hausdorff dimension — which is scaled down by ratios r1 , r2 , … , rN (i.e. the first iteration of the set is the whole set, scaled down by a factor of r1) satisfies the following two equations [2]:
These equations, however, do not appear in Hausdorff‘s paper, as they relate directly to fractals (and calculating the dimension of a fractal), which were ideas that would have been unknown to Hausdorff. Still, from these two equations, it is easy to see how one can obtain a dimension that is not a whole number, as [2]
At nearly the same time that Hausdorff did his research, two French mathematicians, Gaston Julia and Pierre Fatou, developed results (though not together) that ended up being important to fractal geometry. They studied mappings of the complex plane and iterative functions. Their work with iterative functions led to the ideas of attractors, points in space which attract other points to them; and repellors, points in space that repel other points, usually to another attractor. These concepts are also important to chaos theory. The boundaries of the various basins of attraction turned out to be very complicated and are known today as Julia sets, [7] an example of which can be seen in Figure 6. A more analytic definition of a Julia set for a function, f (z), is [2]
Namely, “the Julia set of f is the boundary of the set of points z ∈ C that escape to infinity under repeated iteration by f (z).” [2]
Because Fatou and Julia (and, by extension, their work) predated computers, they were unable to generate pictures such as the one on the right, which is the graph of millions of iterations of a function. They were limited to what they could do by hand, which would only be about three or four iterations. [7] Julia published a 199-page paper in 1918 called Mémoire sur l’iteration des fonctions rationelles, which discussed much of his work on iterative functions and describing the Julia set. With this paper, Julia won the Grand Prix of the Académie des Sciences and became extremely famous in mathematical circles throughout the 1920s. However, despite this prominence, his work on iteration fell into obscurity for about fifty years. [11]
Fatou, on the other hand, did not achieve the same level of fame as Julia, even contemporarily, despite discovering very similar results — though in a different manner — and also submitting them to be published. He submitted an announcement of his results to Comptes Rendus, while Julia had chosen to send his opus to the Journal de Mathématiques Pures et Appliquées. Julia, protective of his work, sent letters to Comptes Rendus asking them to investigate whose results had priority. The publication duly launched an investigation and included a note on Julia‘s findings in the same issue as the Fatou‘s announcement. This apparently discouraged Fatou enough to keep him from entering for the Grand Prix. Still, the Académie des Sciences gave him some recognition and awarded him a prize for his paper on the topic. [10]
Julia sets can be completely disconnected, in which case they are “dust” (Figure 7) — similar to the Cantor set (Figure 4) — or they are completely connected (Figure 6). On rare occasions, they can be “dendrites” (Figure 8), where they are “made up completely of continuously sub-branching lines, which are only just connected since the removal of any point from them would split them in two,” [7] at which point, they would be considered “dust”. [7]
The method for deciding whether or not a set is connected is to calculate out the orbit of the starting point. The orbit for a starting point, x0 , is the sequence [2]
If this sequence goes off to infinity, then the set is disconnected. Otherwise, it is connected. [7]
In 1938, the year after Besicovitch‘s last paper on Hausdorff dimension, Paul Lévy produced a comprehensive treatment on the property of self-similarity. He showed that the von Koch curve was just one of many examples of a self-similar curve, though von Koch himself had stated that his curve could be generalized. The curves generated by Lévy (see Figure 9 for an example — the green and blue sets are two smaller copies of the larger set) were iterative and connected and, with enough iterations, covers (or tiles) the plane. Lévy‘s curves, however, are not fractals, as they have both a Hausdorff and a topological dimension of two. [3]
Little did anyone at this time suspect that there was someone, albeit still a very young person, who would unite the works of Lévy and Hausdorff. Benoit Mandelbrot was born in 1924 in Warsaw, Poland and, like Hausdorff, he was also Jewish, though his family managed to escape life under the Third Reich in 1936 by leaving Poland for France, where family and friends helped them set up their new lives. One of Mandelbrot‘s uncles, Szolem Mandelbrojt, was a pure mathematician, who took an interest in the young Mandelbrot and tried to steer him towards mathematics. In fact, in 1945, Mandelbrojt showed his nephew the works of Fatou and Julia, though the young Mandelbrot initially did not take much of an interest. [13]
Mandelbrot‘s education was very uneven, and completely interrupted in 1940, when Mandelbrot and his family were forced to flee the Nazis again. This time they went to central France. Mandelbrot, like Helge von Koch before him, preferred visual representations of mathematical problems, as opposed to the symbolic, [7] though this may also stem from his lack of formal education, due to World War II. [13] Unfortunately, this would bring him into direct conflict with the teaching style of “Bourbaki“, a group of mathematicians whose belief in solving problems analytically (as opposed to visually) dominated the teaching of mathematics in France at the time. [7]
After the war had ended, Mandelbrot took the entrance exams for the École Polytechnique in Paris, despite having no preparation. He did very well in the mathematics section, where he could employ his ability to solve problems through visualisation to answer questions. While this method was not always possible on other sections, he managed to pass [7] and after a one-day career at the École Normale, Mandelbrot started at the École Polytechnique, where he met another of his mentors, Paul Lévy, [13] who was a professor at there from 1920 until his retirement in 1959 [12].
After completing his studies, Mandelbrot moved to New York, where he started work for IBM’s Thomas J. Watson Research Centre. The company gave him a free hand in choosing a topic of study, which allowed him to explore and develop concepts using his own methods, without having to worry about the reaction of the academic community. In 1967, while still there, Mandelbrot wrote his landmark essay, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension [8], in which he linked the idea of previous mathematicians to the real world — namely coastlines, which he claimed were “statistically self-similar”. He argued that [8]
Self-similarity methods are a potent tool in the study of chance phenomena, including geostatics, as well as economics and physics. In fact, many noises have dimensions D contained between 0 and 1 …
After this essay and with the aid of computers, Mandelbrot returned to the work of Julia and Fatou. With the ability to see, for the first time, what these sets looked like in their limits, Mandelbrot came up with the idea of mapping the values of c ∈ C for which the Julia set for the function fc (z) = z2 + c is connected. This creates the Mandelbrot set, M (Figure 10), which is more formally denoted as
The Mandelbrot set is, for many, the quintessential fractal. When one zooms in on some part of the edge, one notices that the Mandelbrot set is, indeed, self-similar. Furthermore, if one zooms in even further on various sections of the edge, one obtains different Julia sets. In fact, it is “asymptotically similar to Julia sets near any point on its boundary,” as proved in a theorem by the Chinese mathematician Tan Lei. [7]
Mandelbrot has managed not only to invent the discipline of fractal geometry, but has also popularized it through its applications to other areas of science. He clearly believed this was important, as he once stated [3]
The rare scholars who are nomads-by-choice are essential to the intellectual welfare of the settled disciplines.
As he hinted in How Long Is the Coast of Britain? fractal geometry comes in useful in representing natural phenomena; things such as coastlines, the silhouette of a tree, or the shape of snowflakes — things are not easily represented using traditional Euclidean geometry. After all, no organic entity comes to mind when one contemplates a square or a circle. Equally, no simple shape from Euclidean geometry comes to mind when contemplating things such as the path of a river. Even the earth is not a perfect sphere, however convenient it may be for one’s calculations to treat it as such. Furthermore, fractal geometry and chaos theory have important connections to physics, medicine, and the study of population dynamics. [7] However, even if the field lacked these links, it would be hard for those so inclined to resist the aesthetic appeal of most fractals.
Mandelbrot‘s non-traditional approach led him to invent an amazing and useful new form of mathematics. However, no mathematician can claim to have developed his results in complete isolation from anyone else’s. Mandelbrot‘s discovery owes a great deal to the mathematicians who preceded him, such as Weierstrass and von Koch, but especially to Julia, Fatou, and Hausdorff. He also benefitted from access to computers, which allowed him not only to build upon the works of others in a new way — one which had definitely not been done before — but to use his preferred method of solving problems — namely visualisation. Furthermore, his invention also makes a case for the importance of the study of pure mathematics: until Mandelbrot came along and united the eclectic ideas of Hausdorff, Julia, et al, they represented very abstract mathematical ideas from varying branches of (pure) mathematics. There is very little that would interest an ordinary biologist about set theory. However, through fractal geometry, many of these seemingly abstract ideas (from mathematicians who are relatively unknown outside of their own spheres of research) develop applications that other scientists and even non-scientists can appreciate. Thus, the work that eventually led to fractals and their applications are an excellent counterexample to the arguments of anyone who would dare to denigrate the study of pure mathematics.
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