From One to Many Geometries

Tuesday, 11 December 2012 - 1:00pm
Museum of London


For 100 years up to the end of the 19th century the study of geometry was completely changed with the development of non-Euclidean geometries and the use of techniques to think of geometries in higher dimensions - a development essential to Einstein in his development of the theory of General Relativity.

This is a part of the lecture series, Shaping Modern Mathematics. The other lectures in the series are as follows:
    Ghosts of Departed Quantities: Calculus and its Limits
    Polynomials and their Roots
    The Queen of Mathematics
    Are Averages Typical?
    Modelling the World

Transcript of the lecture

11 December 2012

From One to Many Geometries

Professor Raymond Flood

Thank you for coming to the third lecture this academic year in my series on shaping modern mathematics. Last time I considered algebra. This month it is going to be geometry, and I want to discuss one of the most exciting and important developments in mathematics – the development of non–Euclidean geometries.

I will start with Greek mathematics and consider its most important aspect - the idea of rigorously proving things. This idea took its most dramatic form in Euclid’s Elements, the most influential mathematics book of all time, which developed and codified what came to be known as Euclidean geometry. We will see that Euclid set out some underlying premises or postulates or axioms which were used to prove the results, the propositions or theorems, in the Elements. Most of these underlying premises or postulates or axioms were thought obvious except one – the parallel postulate – which has been called a blot on Euclid.

This is because many thought that the parallel postulate should be able to be derived from the other premises, axioms or postulates, in other words, that it was a theorem. We will follow some of these attempts over the centuries until in the nineteenth century two mathematicians János Bolyai and Nikolai Lobachevsky showed, independently, that the Parallel Postulate did not follow from the other assumptions and that there was a geometry in which all of Euclid’s assumptions, apart from the Parallel Postulate, were true. In this geometry, called not surprisingly, non-Eulidean geometry, the Parallel postulate is false. 

I will introduce a model of this geometry, called the Poincare Disc, which help us to visualise non-Euclidean geometry.

This changed the study of geometry allowing many different, indeed infinitely many different geometries in one, two three and higher dimensions.

Around 300BC, with the rise to power of Ptolemy I, mathematical activity moved to the Egyptian part of the Greek empire. In Alexandria, Ptolemy founded a university that became the intellectual centre for Greek scholarship for over 800 years. He also started its famous library, which eventually held over half-a-million manuscripts before being destroyed by fire. Alexandria’s Pharos lighthouse was one of the seven wonders of the ancient world.

The first important mathematician associated with Alexandria was Euclid (c.300BC), who is credited with writing on geometry, optics and astronomy. But he is mainly remembered for one work — the Elements, the most influential and widely read mathematical book of all time. It was in use for more than 2000 years and, apart from the Bible, may even be the most printed book ever.

Euclid’s Elements, a model of deductive reasoning, was a compilation of known results organized in a logical order. Starting from initial axioms and postulates, it used rules of deduction to derive each new proposition in a systematic way. It was not the earliest such work, but was the most important. Let me give a brief idea of the contents of Euclid’s Elements.

It consists of thirteen sections, usually called ‘Books’ although they were originally written on rolls of papyrus. They are traditionally divided into three main parts — plane geometry, arithmetic and solid geometry.

The geometrical part (Books I to VI) opens with definitions of such basic terms as point, line and circle, followed by some axioms (or postulates) that permit us to carry out certain geometrical constructions with an  unmarked ruler and a pair of compasses.

These allow us to:
• draw a straight line from any given point to any other,
• draw a circle with any given centre and radius.

Euclid then presented his first result, which gives a construction for an equilateral triangle (a triangle with all three sides equal):

Given a straight line AB, construct an equilateral triangle with AB as its base.

To do this, he used the construction, which allows us to draw a circle with any given centre and radius, to draw two circles, one with centre A and radius AB, and the other with centre B and radius AB. These circles meet at two points C and D, and the triangle ABC (or ABD) is then the required equilateral triangle. Euclid proved why this construction always gives an equilateral triangle. At each stage of his proof he made reference to an appropriate definition or postulate.

Book I continues with results about congruent triangles (those of the same size and shape) and parallel lines. Euclid also proved the ‘angle-sum theorem’, which is that the angles of any triangle add up to 180°, and Book 1 finishes with a proof of the Pythagorean Theorem, named after a mathematician, Pythagoras, who had lived about 200 years earlier, although the result was known to the Mesopotamians 1000 years earlier.

Book II includes various results on rectangles, such as the construction of a rectangle equal in area to a given triangle, while Book III introduces properties of circles, such as Thales’ theorem that the angle in a semicircle is a right angle, and a proof that if a quadrilateral is drawn in a circle then the pairs of opposite angles add up to 180°.

Book IV contains circle constructions, including those of regular 5-sided, 6-sided and 15-sided polygons inside a given circle. Book V may be due to Eudoxus, a student of Plato at the Academy, and concerns ratios of lengths.

These are then applied in Book VI to similar geometrical figures (those of the same shape but not necessarily the same size).

In Books VII to X, we enter the world of arithmetic, but the descriptions are still given in geometrical terms, using lengths of lines to represent numbers.

There are discussions of odd and even numbers, and of what it means for one number to be a factor of another. Included here is the so-called Euclidean algorithm, a systematic method for finding the highest common factor of two numbers. This section of the Elements also includes a discussion of prime numbers. A prime number is a number, greater than 1, whose only factors are itself and 1: the first few are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. They are central to arithmetic because they are the building blocks for numbers: every whole number can be obtained by multiplying prime numbers.

Book IX contains Euclid’s proof of the fact that the list of primes continues for ever: There are infinitely many prime numbers. It is one of the most famous proofs in the whole of mathematics. Book X deals with incommensurables or irrational quantities.

I will have a lot more to say about prime numbers in my lecture in January.

The final three books of Euclid’s Elements deal with aspects of three-dimensional geometry. Of these, Book XIII is the most remarkable. Here, Euclid investigated the five regular solids (tetrahedron, cube, octahedron, dodecahedron and icosahedron) and showed how they can be constructed.

A regular solid is one whose faces all have the same shape and size, and whose vertices all look the same.

Tetrahedron, whose faces are four equilateral triangles,

Cube,or hexahedron, whose six faces are squares,

Octahedron,whose eight faces are equilateral triangles,

Dodecahedron,whose twelve faces are regular pentagons,

Icosahedron,whose twenty faces are equilateral triangles.

They are also called the Platonic solids because Plato In his book Timaeus linked the universe with the dodecahedron and assigned the other four polyhedra to the Greek elements of earth, air, fire and water. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron.

Euclid concluded the Elements by proving that these are the only possible regular solids — there can be no others. This, the first ever ‘classification theorem’ in mathematics, forms a fitting climax to this great work.

Now that I have given a brief idea of the contents of Euclid’s Elements let me turn to the kind of enterprise that it was. We get a clue from the title Elements. By Euclid’s time its mathematical meaning was of a starting point for many other theorems. A later mathematician, Proclus, compares an element in this sense to a letter of the alphabet from which all words are constructed, and, in geometry, the elements are the foundational propositions from which geometrical theorems can be deduced.

Proclus’s comments also show us why Euclid’s Elements were so valued:

“It is a difficult task in any science to select and arrange properly the elements out of which all other matters are produced and into which they can be resolved …”

Proclus continues

“in general many ways of constructing elementary expositions have been individually invented. Such a treatise ought to be free of everything superfluous, for that is a hindrance to learning: the selections chosen must all be coherent and conducive to the end proposed, in order to be of the greatest usefulness for knowledge: it must devote great attention both to clarity and to preciseness, for what lacks these qualities confuses our understanding; It ought to aim at the comprehension of its theorems in a general form, for dividing one’s attention too minutely and teaching it by bits make knowledge of it difficult to attain.”

And he concludes

“Judged by these criteria, you will find Euclid’s introduction superior to others.”

So Euclid’s success lay in the selection and arrangement of the “elements out of which all other matters are produced”. Euclid’s Elements is the first example of an axiomatic approach in mathematics where every result is proved by reference back to the foundational starting points.

Perhaps you can still remember the excitement and pleasure you felt when you first met Euclid’s Elements. Maybe your experience was similar to that of Thomas Hobbes, who was afriend of Francis Bacon, as well as of Galileo, Descartes, and Mersenne, and who was one of the outstanding intellectual figures of his age. This quote is from the life of Thomas Hobbes in John Aubrey's Brief Lives.

He was 40 years old before he looked on Geometry; which happened accidentally.  Being in a Gentleman’s Library, Euclid’s Elements lay open, and ’twas the 47 El. libri I [the ‘Pythagorean Theorem’]. 

And continues

He read the Proposition.  By G—, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is impossible!  So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read.  That referred him back to another, which he also read.  Et sic deinceps [going thus one after another] that at last he was demonstratively convinced of that trueth.  This made him in love with Geometry.

Or, maybe your reaction was more like that of the philosopher Bertrand Russell, who wrote in his autobiography:

‘At the age of eleven, I began Euclid with my brother as my tutor. This was one of the great events of my life, as dazzling as first love.

I had not imagined there was anything so delicious in the world. After I had learned the fifth proposition, my brother told me it was generally considered difficult, but I had found no difficulty whatever. 

This was the first time that it dawned on me that I might have some intelligence.

From that moment until Whitehead and I finished Principia Mathematica when I was thirty-eight, mathematics was my chief interest, and my chief source of happiness.’

So, let us try to see how the Elements caused such rapture!

Let us look at the Propositions of Elements, Book I which enter into the proof of Proposition 47, the Pythagorean Theorem.

An upwards directed line indicates that the proof of the higher numbered proposition directly invokes the lower one. It is a dense and complicated interaction of chains of reasoning.

For example the Pythagorean Theorem invokes, amongst others, prop 46 which amongst others invokes Prop 29 which invokes Prop 16 which amongst others invokes Prop 3 which invokes Prop 1 and this one we have seen – it is how to construct an equilateral triangle. This downward progression has to stop and it does so at the postulates and common notions at the start of the book.

It is now time to look at them, for they are at the heart of the axiomatic structure.

The common notions are general truths or assumptions which could be applied to many areas of investigation.

1.  Things which are equal to the same thing are also equal to one another.
2.  If equals be added to equals, the wholes are equal.
3.  If equals be subtracted from equals, the remainders are equal.
4.  Things which coincide with one another are equal to one another.
5.  The whole is greater than the part.

Whereas we now see that the postulates have geometrical subject matter.

1.  To draw a straight line from any point to any point.
2.  To produce a finite straight line continuously in a straight line.
3.  To describe a circle with any center and distance.
4.  That all right angles are equal to one another.
5.  That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Don’t worry about Postulate 5 yet – it is harder to work out what it is saying than for the other postulates. We will come back to it. In fact, this lecture is all about Postulate 5.

We can think of the Common Notions and Postulates as logical and geometrical axioms respectively. Using these axioms and the 23 definitions of Book I – further definitions were introduced in later books – Euclid proved all his results.

Let us briefly look at some of the definitions – I am picking out the ones to do with points and straight lines.

A point is that which has no part.
A line is breadthless length.
The extremities of a line are points.
A straight line is a line which lies evenly with the points on itself.
A surface is that which has length and breadth only.
The extremities of a surface are lines.

It is important to note that Euclid does not give any definition of point or line – it is in the common notions and the postulates that he tells us how to work with them and gives their properties and interrelationships. Essentially all that he is saying is there are things called points and lines and they have connections.

Most of the other Definitions do define something – they have some content. For example:

10.  When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

Note the diagram If the angles are equal then they are both called right angles.

23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

This is the definition of parallel lines; they are simply lines that do not meet.

Euclid showed remarkable insight and ability in selecting his common notions and postulates so that everything was deducible from them. But one of them stands out as perhaps seemingly different in nature to the others and that is Postulate 5. It is longer and different.

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Here we see from the diagram what the Postulate is saying: If α+ βis less than two right angles then the two lines l and m will meet.

From the first, it appears that mathematicians felt that the fifth Postulate should be deducible from all the others. In other words, it was felt that it was a theorem and not a postulate. However centuries of attempts to prove it as a theorem were in vain and eventually in the nineteenth century it was shown that Euclid had been right all along. It could not be deduced from the other postulates and common notions.

The fifth postulate is not easy to understand but there is an equivalent formulation of it, called the Uniqueness of Parallels Postulate, which is more immediate. It states: 

The parallel to any line through a given point (not on the given line) is unique.

It is also really the uniqueness of Parallels that is used in the Elements rather than Postulate 5. It seems a very reasonable assumption when expressed in this form. Euclid did not use it until half way through Book 1, until Proposition 29, when he used it to prove results such as:

The angles in any triangle add up two right angles i.e. 180 degrees, I, 32
Parallel lines are always equidistant, I, 33
Pythagoras’ Theorem I, 47
Over the centuries many mathematicians tried to deduce the uniqueness of parallels  from the definitions, common notions and other postulates but what they achieved was only to find other equivalent assumptions.

The equidistant curve to a straight line is itself straight - Ibn Al-Haytham, 965 – c.1040

The angle sum of a triangle is 180 degrees – Nasir Eddin, 1201 – 74

We have seen a little earlier that the sum of the angles in a triangle being 180 degrees follows for the Parallel postulate. This result now says it is equivalent to it so the Parallel Postulate follows from the assumption that the sum of the angles in a triangle is 180 degrees.

There exist triangles with the same angles which have different sizes - John Wallis, 1616 – 1703.

We have been looking so far at Euclidean geometry, the geometry of the plane and space and about which we seem to have an intuitive understanding which leads us to the impression that the parallel postulate should be deducible from all the other axioms. Before going on to consider this further I want to introduce another geometry in which many of Euclid’s postulates and not just the parallel postulate fail to hold. This is spherical geometry and I think you will agree that it is a perfectly good geometry, as we live on the earth which is essentially a sphere. It has long been studied because of its importance for navigation.

In spherical geometry, the points are just the points on the surface of the sphere. For our lines we use the analogy that in Euclidean geometry the line between two points connects the points by the shortest distance between them. On a sphere the shortest distance between two points is along the great circle connecting them. A great circle on a sphere is a circle on the sphere whose centre is at the centre of the sphere. The equator is an example of a great circle and the meridians of longitude form exactly half a great circle.

There is a unique great circle connecting two points on the sphere except when the two points are antipodal, like the North and South pole. Two points on the sphere are antipodal if the line joining them passes through the centre of the sphere.

There are an infinite number of great circles connecting two antipodal points, for example, an infinite number of meridians connecting the North and South Poles.

This means that spherical geometry does not satisfy the first Postulate of Euclid which says that there is one and only one line connecting any two points. In spherical geometry there are many lines connecting any two antipodal points.

Also Spherical geometry does not satisfy the Uniqueness of Parallels Postulate of Euclid. This is because if you take a great circle and a point not on it then every great circle through the point meets the first great circle twice. There are no parallel lines! Any two distinct great circles meet twice.

Because of this we might expect spherical geometry to give different results from Euclidean geometry and we can illustrate this by looking at triangles.

A spherical triangle has its sides made up of segments of great circles. It is the case that the sum of the angles of a spherical triangle is always greater than 180 degrees,

α+ β+ γ> π

The amount that it is greater depends on the area of the triangle and the radius of the sphere.

α+ β+ γ= π +  Area (Triangle) where angles are measured in radians.

We can show this formula is correct in the case of a triangle with three right angles.

For example, Take the triangle starting at the North Pole and going down the Greenwich meridian to the equator and then West along the equator to longitude 90 degrees and then up this meridian back to the North Pole. Then all the angles in this triangle are right angles so their sum is 270 degrees or in radians 3π/2.

So here the left hand side is α+ β+ γ= 3π/2
But the area of the triangle is one eight of the surface of the earth, so is one eight of 4πR2, which is  π R2.
Hence the right hand side is π +  Area (Triangle) = π +   π R2 = π +  π = 3π/2

Thee is also a difference in how distance is measured in Euclidean and Spherical geometry. In Euclidean Geometry distance is measured using the Pythagorean Theorem – here it is in the oldest extant copy of Euclid’ Elements.

The Pythagorean Theorem states that:
In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

This copy of the Elements is in the Bodleian Library, Oxford University. It was copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD.

This Pythagorean Theorem allows us to calculate the distance between two points in Euclidean geometry – it is the length of the straight line connecting them. If we have two points, A and B, on the Earth the Euclidean distance between them is length of the line through the Earth connecting them, but in spherical Geometry it is the length of the segment of the great circle connecting them on the surface of the sphere, which is of course always greater.

Having looked at another geometry, spherical geometry, let’s return to the Parallel Postulate and the important work of Gerolama Saccheri, a Jesuit logician, who was Professor of Mathematics at Pavia in Northern Italy from 1697 until his death in 1733.

In the year he died he wrote a book  entitled Euclid freed of all Blemish which contained an ingenious investigation of the Parallel Postulate. He considered the system of the Elements but with the Parallel Postulate removed and replaced with one of the following assumptions put in its place:

1. Hypothesis of the Obtuse Angle
The angle sum of a quadrilateral is greater than 360°
2. Hypothesis of the Right Angle
The angle sum of a quadrilateral is exactly 360°
This is equivalent to the Parallel Postulate
3. Hypothesis of the Acute Angle
The angle sum of a quadrilateral is less than 360°

These are the only possibilities. So Sachieri made three hypothesis from each of which he could make deductions, when used with the other assumptions of Euclid.

He first showed that if the  Hypothesis of the Obtuse Angle was true for one quadrilateral it was true for all of them.

Similarly if the Hypothesis of the Right Angle was true for one quadrilateral it was true for all of them.

Finally if the Hypothesis of the Acute Angle was true for one quadrilateral it was true for all of them.

He then took each case in turn. His aim was to show that in case 1 and in case 3 he would obtain a contradiction leaving case 2 as the only possibility and case 2 was Euclid’s geometry. Note how clever his method proof was: he was going to prove that the Parallel Postulate was true, not by deducing it from the other axioms, but by proving that the alternatives to it led to a contradiction.

Case 1, the  Hypothesis of the Obtuse Angle, did indeed lead to a contradiction – he could show that lines were of finite length – which contradicts another of Euclid’s postulates. As Saccheri said of the Hypothesis of the Obtuse Angle:

‘it is absolutely false because it destroys itself’

Now he started on investigating the consequences of assuming the Hypothesis of the Acute Angle. He thought he’d obtained a contradiction but he was wrong. He had made a mistake.

Indeed he could not have found a contradiction because there is a geometry, called non-Euclidean Geometry, which satisfies the axioms of Euclid but with the Parallel Postulate replaced by the Hypothesis of the Acute Angle.

It is to its discovery that we now turn.

There are ‘non-Euclidean’ geometries that satisfy the first four postulates, but not the fifth. Their existence was first published around 1830 by the Transylvanian János Bolyai (1802–1860) and the Russian Nikolai Lobachevsky (1792–1856).

Bolyai and Lobachevsky independently both constructed a new type of geometry in which the angles of any quadrilateral add up to less than 360°.

Both Bolyai and Lobachevsky were unsuccessful in getting their work widely known, and they enjoyed little of the credit due to them for their remarkable discovery. It was not until after their deaths that their geometries were fully understood.

The non-Euclidean geometries of Bolyai and Lobachevsky were considered very controversial, as they were seen not to correspond to the world we live in. Some years earlier, Gauss had been thinking along similar lines:

I am becoming more and more convinced that the necessity of our [Euclidean] geometry cannot be proved … Perhaps in another life we will be able to obtain insight into the nature of space, which is now unattainable.

However, Gauss was unwilling to publish his startling predictions.

János’s father, Farkas Bolyai, had also worked on the parallel postulate and  earnestly tried to dissuade his son from doing so:

You must not attempt this approach to parallels. I know its way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life… I have travelled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail.

But the son persisted, and when Farkas Bolyai informed his old friend Gauss of his son’s success, Gauss accepted the results but claimed them as his own:

If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years.

János Bolyai never forgave Gauss for this.

One difficulty caused by the non-Euclidean geometry of Bolyai and Lobachevsky was that it was hard to visualize. A number of pictorial representations were suggested, the most successful being the ‘disc model’, discovered by Poincaré in 1880.

Henri Poincaré (1854–1912) is viewed as one of the great geniuses of all time, being probably the last person to cover the entire range of mathematics. He virtually founded the theories of several complex variables and algebraic topology, and one of his conjectures in topology, known as the Poincaré conjecture, was solved only in this century. He made outstanding contributions to differential equations and non-Euclidean geometry, and also worked on electricity, magnetism, quantum theory, hydrodynamics, elasticity, the special theory of relativity and the philosophy  of science. As an active popularizer of his subject, he wrote popular works for  nonmathematicians, stressing the importance of mathematics and science and discussing the psychology of mathematical discovery.

Consider the following picture of a disc (the inside of a circle). We consider a  geometry in which the points are those lying inside the bounding circle, and the lines are either diameters that pass through the centre of the disc or circular arcs that meet the bounding circle at right angles. As we can see,

• some pairs of lines (diameters or circular arcs) do not meet
• some pairs of lines meet at internal points
• some pairs of lines meet on the bounding circle:

With these definitions of point and line, Euclid’s first four axioms are satisfied, but not the fifth, so we do indeed have a non-Euclidean geometry.

This Bolyai–Lobachevsky geometry has some very strange features.

Given any line L and any point P that does not lie on this line, there are infinitely many lines, parallel to L, that pass through P

Moreover, if two triangles are similar (they have the same angles), then they must  also be congruent (they have the same size) — which is also not true in Euclidean geometry.

In this geometry many Euclidean concepts (such as size and shape) are no longer appropriate; for example, all the grey and white triangles in the picture turn out to be congruent to each other! The Dutch artist Maurits Escher based some of his woodcuts (such as Circle Limit IV) on this picture of a non-Euclidean geometry.


and area of triangle α+ β+ γ= π -  Area (Triangle)

There are two features of this geometry that I would like to show you. First is for the sum of the angles in a triangle in non –Euclidean geometry. It is  always less that 180 degrees

α+ β+ γ  < π

and depends on the area of the triangle. The second gives the distance between two close points in this geometry.

Another mathematician, Bernhard Riemann (1826–1866), had obtained a remarkable generalization of the idea of ‘geometry’, both Euclidean and non-Euclidean. He studied the way that surfaces can ‘curve’ inwards or outwards (like a globe or a cooling tower) and proposed generalized ideas of distance for such surfaces, not only in three dimensions, but also for their higher-dimensional analogues (called manifolds). By disregarding the surrounding higher-dimensional space, he could study manifolds in their own right and measure distances on them. Arising from this work, he was able to describe infinitely many different geometries, each one equally valid, and each one a candidate for the physical space we live in.

Years later, one of his geometries proved to be the natural setting for Einstein’s theory of relativity, when Hermann Minkowski developed a new view of space and time and laid the mathematical foundations of the theory of relativity. Minkowski described his approach as follows:

Henceforth space by itself, and time by itself, are doomed to fade away into  mere shadows, and only a kind of union of the two will preserve an independent reality.

The kind of union that Minkowski mentions is now known as space-time and is a four dimensional non-Euclidean geometry that incorporates the three dimensions of space with the one of time. It comes with a way of measuring the distance between two different points of space-time. Space and time are now no longer separate, as Newton had thought, but are intermixed.

In Euclidean geometry the distance of each point (x, t) to the origin is 


but the requirements of relativity replace this in spacetime with the distance 


Where c is the speed of light.

The minus sign implies that events in space-time, such as the one labelled ‘here and now’, are associated with two cones. With just one space dimension, these cones are now triangles, with one representing the future of the ‘here and now’ and the other its past.

Einstein initially thought little of Minkowski’s approach to space-time, but later found it invaluable, indeed essential, when he was trying to extend his theory to include gravity.

Einstein’s general theory of gravity, building also on Riemann’s geometrical ideas, produced space-time that was curved as a result of the presence of mass and  energy. The curvature increased near to massive bodies, and it was the curvature of space-time that controlled the motion of bodies. The theory predicted that light rays would be bent by the curvature of space-time produced by the sun, an effect that was observed during the 1919 eclipse of the sun.

To finish:
There was a flowering of Greek mathematics with the founding, around 387 BC, of Plato’s Academy in a suburb of Athens called ‘Academy’ – that is where the word Academy came from. This slide shows Raphael’s fresco, The School of Athens, with representations of Plato and Aristotle at the top of the steps. It is said that over the entrance to Plato’s Academy were inscribed the words ‘Let no one ignorant of Geometry enter these doors’ and that seems a fitting motto to end this lecture.


© Professor Raymond Flood2012