Strange Geometries

On this page:

Introduction
Flexibility
Geometry on the surfaces
Virtual surfaces
The sphere and curvature

Introduction

We wish to study the geometry of surfaces. You are used to doing geometry on a plane surface, like a sheet of paper. You can draw a line as the shortest path between two points; construct circles and triangles. You also know many properties of these geometrical objects. For example, two straight lines either meet in a single point or they are parallel; the angles of a triangle add up to 180°.
What happens if we try to do geometry on a curved surface, like the surface of a globe, or on a sheet of paper that has been curved? This is important since a lot of modern mathematics uses the geometry of curved surfaces. In physics, Einstein showed how we should think of gravitational forces as things that curved space-time. In many parts of mathematics, we study the symmetries of curved
spaces.

We will look at the simplest cases. So we will study surfaces that are made up by joining together lots of triangles. Each triangle is flat, so we know what the geometry is like on it. However, we need to think about how the geometry changes as we pass from one triangle to its neighbours. We want the surfaces we get to be as symmetric as possible so we will start with the most symmetric
triangles possible --- equilateral triangle all of the same size --- and try to join the triangles together as uniformly as we can. This means that we will join two triangles together by sticking one edge (side) of the first to an edge of the second. We will try to do this so that there are the same number of triangles meeting at each vertex (corner). This means that the surface we produce looks
the same near each vertex.

Let us try this sticking three triangles together at each vertex. We need to use four triangles altogether and we get an object called a (regular) tetrahedron. The ancient Greeks knew about these objects so they gave them the names that we still use. Each is called a polyhedron (poly = many, hedron = side). This one has four sides so it is called a tetrahedron because tetra
is the Greek for four. This is quite a pretty object. It looks the same no matter which vertex we look at so it is highly symmetric. Many of you have probably made models of this before. Let us continue and ask what happens if we fix a different number of triangles together at each vertex.

For two triangles at a vertex, we just get two equilateral triangles back to back. For three triangles at a vertex we get the tetrahedron. For four triangles at a vertex we get a surface made up of eight triangles: a regular octahedron (octa = eight). For five triangles at a vertex we get a surface made up of twenty triangles: a regular icosahedron (icosa = twenty).

This is the point at which teachers generally stop. It is becoming quite complicated to make these surfaces. Moreover, the next example looks rather dull. If we join six triangles at each vertex we just get a flat plane divided up into equilateral triangles. There are infinitely many triangles and they tessellate the plane.

Note that this picture only shows a small part of the surface which continues indefinitely in all directions. The polyhedra all use only a finite number of triangles but for this we need infinitely many. We will call this surface the triangulated plane.

Let us persevere and try joining seven triangles at each vertex. We really can do this. If we take seven triangles we can stick their edges together so that they all meet at one vertex. The triangles have to bend up but they still join as they should. We can continue to add extra triangles ensuring that seven meet at each vertex. This continues for ever, adding more and more triangles to
produce a rather strange surface that twists, bends and folds back on itself more and more as we add further triangles. The surface is still regular, each of the vertices looks exactly the same. In fact this surface is very important indeed. It is a model for the hyperbolic plane which is used in many parts of mathematics. The picture below shows a small part of it.

We will call this surface the (triangulated) hyperbolic plane.

I intend to look at some properties of all these surfaces: the tetrahedron, octahedron, icosahedron, triangulated plane, and hyperbolic plane. Then I will ask you to explore them further in a variety of ways. It is very useful to actually make models of these surfaces yourself. Then you get a feeling for how they behave and how they differ. So part of your task will be to make models like
those I have shown you.

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Flexibility

For two or three triangles joined at each vertex, there is no choice about the surface we produce. However, as we increase the number of triangles at each vertex, so we get more flexibility. For four triangles, we can form a cup shape by pushing one vertex of the octahedron in. We can do a similar thing to the icosahedron to get a variety of twenty-faced polyhedra. Note that these do not look
regular to us because of the way they are seen in space. However, they are regular surfaces since we still have the same number of faces meeting at each vertex, so the geometry of the surface is the same around each vertex.

When we join six triangles at each vertex, we get a great deal more flexibility. We can fold the plane along a line of edges. Sticking edges together we can get a cylinder and a torus.

When we join seven triangles at each vertex, there is even more flexibility. Indeed the surface we get flexes and twists as we try to move it in space. We can not flatten it out and get a ``best'' way to see it. I will ask some of you to investigate this further.

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Geometry on the surfaces

We want to consider lines, triangles, circles on the surfaces we have constructed. First we must think about what we should mean by a straight line or a circle on one of our surfaces.

We already know what we mean by a straight line across one of the triangular faces. For each triangle is just part of the flat plane. So we can construct a straight line on the triangle in the usual way by using a ruler or pulling a string taut between two points. The problem arises when we want to extend our line across an edge of a triangle. How should we do this? Suppose that we want to
draw a straight line across the edge joining two triangles. We can cut these two triangles out of the surface and then flatten them out. This gives a flat rhombus (diamond shape). We know how to draw a straight line on this, so we draw it and then stick the triangles back onto the surface. In this way we can extend our straight lines from one triangle to the next, and the next, for ever by
flattening out the surface just a bit at a time.

There remains a problem. If we are very unlucky, the line we are drawing will go through a vertex of the surface. It is now really difficult to extend the line since we can not flatten out the surface at a vertex without tearing it. There is actually no good way of continuing the line through a vertex. For consider what happens if we move the line a little so it passes to one side or the other
of the vertex. If we move to one side we get the line continuing in one direction but if we move to the other we get a line continuing in a very different direction. If the line went through the vertex, we would want it to continue in both directions at once.

So we will never allow our lines to go through a vertex. Fortunately if you choose a line at random it is very unlikely to go through any vertex.

Having explained what lines are on our surfaces, it is easy to explain triangles and circles. Triangles are formed by drawing three lines on the surface. Each pair of these lines should cross at a point and we can measure the angle between the lines where they cross by flattening out the surface near that point and then using a protractor. If A is one point on the surface, the
distance from any other point P to A is the length of the shortest line on the surface joining P to A. A circle of radius r about A is the set of all points which are at a distance r from A.

We are now in a position to do geometry on our surfaces. Any result you know for the flat plane you can try on our surfaces. You can ask if it is still true or how it changes.

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Virtual Surfaces

When I made the models that I have shown you, I made a net out of card and then stuck the edges together. For example, to make the tetrahedron I made the net shown below and stuck the sides labelled A, B and C together.

If we wish to draw straight lines on this surface, we can do it on the net before we fold it up. So, for example, a straight line on the tetrahedron comes from a straight line on the net. When this line meets the edge of the net, say at the point P, it should continue from the point P' to which P should be joined when we make up the tetrahedron. The last triangular
face the line goes through before P is moved to the dashed position and then we can continue the line as shown by the dashed line below.

This shows that we can do geometry on the surface just by using the net without ever making up the surface itself.

Now consider the net shown below.

It is not possible to make up this surface because we can not join the edges in the way required. The difficulty here is that the surface we are trying to make does not "live" in our usual 3-dimensional space. However, we can still study the geometry of these surfaces even though we can not make them. We might call them virtual surfaces. The one illustrated above is called the
real projective plane.

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The sphere and curvature

The tetrahedron, octahedron and icosahedron all look similar to the sphere. Indeed, all of these polyhedra can be fitted inside a sphere so that all their vertices lie on the surface of the sphere. If we imagine having an intense light at the centre of the sphere, the light will cast a shadow of the edges of our polyhedra onto the surface of the sphere. Below is a picture of this for the
octahedron.

This gives us a pattern on the sphere. The edges now correspond to arcs of ``great circles''. (These are circles on the sphere centred at the centre of the sphere.) Each face is bounded by three of these circular edges. Note that the angles for these triangular faces have been changed. For the octahedron in our picture, the angles are all 90°. So, on the sphere with have an equilateral
triangle with all the sides being of equal length and all the angles being 90°. The great advantage of using the sphere rather than our polyhedra is that it is even more symmetric. It is curved in the same way at every point whereas our polyhedra have all their curvature concentrated at the vertices. So we can think of the tetrahedron, octahedron and icosahedron as being ways of dividing the
surface of the sphere up into triangular pieces.

We can do a similar thing for the triangulated hyperbolic plane although it is a bit more complicated. If we look at our model of the triangulated hyperbolic plane we see that it grow as we move out. There are more and more triangles and, because they are all the same size, they have to fold back on themselves. If we made the triangles get smaller as we moved out, then we could stop this
happening and get a prettier model. A French mathematician, Poincaré, saw how to do this. He fitted all the infinitely many triangles into a disc as shown below.

The artist M.C. Escher used the different geometries of the sphere the flat plane and the hyperbolic plane to give wonderful pictures. These are from The Magic of M.C. Escher, Thames and Hudson, 2000.

All M.C. Escher works (c) 2001 Cordon Art BV - Baarn - the Netherlands. All rights reserved. www.mcescher.com