Tait a mathematical physicist, offered a solution to the problem. Independently, Tait had established that maps where an even number of boundary lines meet at every point, could be coloured with two colours, although this result had appeared earlier in Kempe's papers. During Tait became well-known for his study and classification of knots. At that time there were a number of different theories about the structure of atoms. William Thompson Later, Lord Kelvin inspired by the experiments of the German Physicist Hermann von Helmholtz proposed a theory that atoms were knotted tubes of ether.

Kelvin's theory of 'vortex atoms' was taken seriously for about twenty years, and it inspired Tait to undertake a classification of knots. Tait, Thomson and James Clark Maxwell invented many topological ideas during their studies. However, Kelvin's theory was fundamentally mistaken and physicists lost interest in Tait's work. One of the outcomes of Tait's study was his Hamiltonian graph conjecture. A map is regarded as a polyhedron drawn on a sphere, and it can then be projected onto a plane. Tait proposed that any cubic polyhedral map has a Hamiltonian cycle [ see note 3 below ].

Tait's method focused on the edges of the graph and he showed that a Hamiltonian cycle could produce a four-coloring of a map. It was not until that William Tutte found the first counterexample to Tait's conjecture. Tait initiated the study of snarks in , when he proved that the four colour theorem was equivalent to the statement that no snark is planar. A planar graph is one that can be drawn in the plane with no edges crossing.

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It looks as if Tait's idea of non-planar graphs might have come from his study of knots and Hamiltonian paths. The first known snark was the Petersen graph discovered in , and mathematicians began to hunt for more of these kinds of graphs but it was not until that another snark was found. The Hunting of the Snark is a poem written by Lewis Carroll, and Martin Gardner named these graphs Snarks, because they were so elusive.

Although Heawood found the major flaw in Kempe's proof method in , he was unable to go on to prove the four colour theorem, but he made a significant breakthrough and proved conclusively that all maps could be coloured with five colours. Heawood made many important contributions to the problem, shifting the focus of attention from the areas of a map, to the borders between them. By he had proved that if the number of edges around each region is divisible by 3 then the regions could be coloured with four colours.

Cauchy's proof of Euler's formula also included the idea that any net of a polyhedron can be triangulated by adding edges to make non-triangular faces into triangles. He then developed a procedure whereby he deleted the edges one by one, showing that Euler's formula could be maintained at each step. Cauchy's proof of Euler's Formula began with the idea of a projection of a polyhedron to obtain a plane net.

He further demonstrated a that any net could be triangulated, and his proof b of Euler's Formula was accepted at the time. By , mathematicians knew that a planar graph can be constructed from any map using the powerful concept of duality [ see note 5 below ]. In the dual, the regions are represented by vertices and two vertices are joined by an edge if the regions are adjacent.

- Circle theorems.
- The Index Number Problem: Construction Theorems.
- The Four-Color Theorem of Map-Making Proved by Construction!

In these graphs, 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. During the first half of the twentieth century, mathematicians focused on modifying these kinds of techniques to reduce complicated maps to special cases which could be identified and classified, to investigate their particular properties and developed the idea of a minimal set of map configurations that could be tested. In the first instance, the set was thought to contain nearly 9, members which was an enormous task, and so the mathematicians turned to computer techniques to write algorithms that could do the testing for them.

The algorithms used modified versions of Kempe's original idea of chains together with other techniques to reduce the number of members of the minimal set. After collaborating with John Koch on the problem of reducibility, in at the University of Illinois, Kenneth Appel and Wolfgang Haken eventually reduced the testing problem to an unavoidable set with 1, configurations, and a complete solution to the Four Colour Conjecture was achieved.

This problem of checking the reducibility of the maps one by one was double checked with different programs and different computers. Their proof showed that at least one map with the smallest possible number of regions requiring five colours cannot exist. Since the first proof, more efficient algorithms have been found for 4-coloring maps and by , the unavoidable set of configurations had been reduced to Because the proof was done with the aid of a computer, there was an immediate outcry. Many mathematicians and philosophers claimed that the proof was not legitimate.

Some said that proofs should only be 'proved' by people, not machines, while others, of a more practical mind questioned the reliability of both the algorithms and the ability of the machines to carry them out without error. However, many of the proofs written by mathematicians have been found to be faulty, so the argument about reliability seems empty.

## The Index Number Problem, Construction Theorems by Sydney Afriat | | Booktopia

Whatever the opinions expressed, the situation produced a serious discussion about the nature of proof which still continues today. Use the notes tab at the top of this article or click here. Penguin Books. For a more detailed and technical history, the standard reference book is: Biggs, N. Oxford University Press. This one brings us up to date, with more recent foundations and philosophy. Hardly any general history book has much on the subject, but the last chapter in Katz called 'Computers and Applications' has a section on Graph Theory, and the Four Colour Theorem is mentioned twice. Polya G.

This is the classic book about Problem Solving. There have been many editions of this book since it first appeared in the s and it is still easily available.

Curiously, recent editions have been given the subtitle 'A new aspect of Mathematical Method'. Lakatos, I. This is another important book which led to the research into Problem Solving and Investigations in the s. It begins as a classroom discussion between a teacher and a group of students about the proof of Euler's formula, and ranges through the ideas, objections and possibilities that were actually discussed by mathematicians and scientists in the nineteenth century.

It raises some of the most important issues about teaching and learning problem solving and about mathematical methods and proof. I have had a little book on String Games for some time. When I was at school it was called Cat's Cradle, and we played it in our break time. Recently, a French journal has published a paper on the 'algebra' of string figures!

There are some 80 figures described with coloured diagrams. It's spiral bound, so it will stay open while you follow the instructions. It also comes with a couple of string loops! For knot experts, The Ashley Book of Knots is a classic for anyone interested in the hundreds of different kinds of knots and their uses. Amazon has various editions available at different prices. And of course MacTutor biographies of those involved in developing all the different mathematical aspects can be found at the MacTutor Biographies Index. For the mathematically persistent the following website has an intriguing new approach to attacking the problem of constructing a new algorithm for solving the problem, and tying to reduce the reliance on a computer.

For Graph Theory, Wikipedia gives a good overview, and you can skip the really technical stuff. It shows the kinds of modern applications of this area of mathematics. If you go to Graph Colouring and click on the Four Colour Theorem, then you will find a lot more information. Go to their website and browse the alphabetical list of resources. Find out all about Knots on the Knot Atlas!

More artistic and colourful - but no less mathematical is the Knot Plot Site. Main menu Search. The Four Colour Theorem. In the Beginning The conjecture that any map could be coloured using only four colours first appeared in a letter from Augustus De Morgan , first professor of mathematics at the new University College London, to his friend William Rowan Hamilton the famous Irish mathematician in Augustus De Morgan and William Rowan Hamilton The problem, so simply described, but so tantalizingly difficult to prove, caught the imagination of many mathematicians at the time.

A copy of De Morgan's original sketch in his letter to Hamilton and a simple four colour map. Arthur Cayley Arthur Cayley showed that if four colours had already been used to colour a map, and a new region was added, it was not always possible to keep the original colouring. Above, all four colours have been used on the original map, and a new region is drawn to surround it.

## The Index Number Problem: Construction Theorems

In this case, a red region is changed to blue, so that red can be used on the new surrounding region. The Patch demonstration. Imagine that at some place in a map a number of countries meet at a point. Now put a patch over the meeting point, and all the new meeting points will have three borders emanating from them. These are cubic maps, and a fourth colour can be used for the central region.

On removing the patch, we can return to the original colouring. Imagine squashing the red cube down onto a plane so that its base is opened out to form the outside edge of the green net. To show that a Minimal Criminal cannot contain a region with two edges a digon. Suppose there is a minimal criminal that contains a digon. Proof of Theorem 13 - If two triangles are similar then Construction 17 - Incentre and incircle of a given triangle Construction 19 - Tangent to a given circle Construction 21 - Centroid of a triangle.

Construction 22 - Orthocentre of a triangle. Reviewing constructions. Reflection in a point. Reflection in axes. Circle with centre 0,0. Circle with centre h,k. Circle with centre -g,-f. GeoGebra File. Intersection of a line and a circle. Tangents to a circle. Comparing the functions sin x and cos x. Investigating sin x and cos x along the unit circle.

Is the function a sine or a cosine function? Area of a triangle quiz. Circle quizes. Arithmetic sequences and series. Geometric sequences and series. The sum to infinity of a geometric sequence. PowerPoint File. Arithmetic sequences. Arithmetic series. Geometric sequences. Geometric series. The sum of the first n natural numbers. The sum to a geometric sequence. The Argand Diagram and modulus of a complex number. Addition of complex numbers. Subtraction of complex numbers. The conjugate of a complex number.

Multiplying a number by i. Multiplication of complex numbers. Division of complex numbers. The area of a trapezium. Estimating the area of a semi-circle using the trapezoidal rule. Finding the area of a circular pond and its surrounding path. Finding the area of a rectangular picture frame.

Race-track problem. Investigating the rate of change of height as volume changes. Relations without formulae. Changing the base of a logarithm. Understanding an ESB bill. Understanding exchange rates. Understanding value added tax VAT. Amortisation of a loan. PowerPoint Show. Understanding the rules of indices. Net of a cube. Net of a cylinder. Net of a cone. Net of a pyramid. Net of an octahedron. Solving a linear and quadratic equation simultaneously. Understanding absolute-value functions. Absolute value - less than. Absolute value - greater than.

Writing complex numbers in polar form.

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Proof of De Moivre's Theorem. De Moivre's Theorem. Proof of Sin 3x and Cos 3x formulae. Solving quadratic inequalities.