Hyperbolic Geometry. Thomas Prince. Imperial College London. 21 January 2017
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1 Hyperbolic Geometry Thomas Prince Imperial College London 21 January 2017 Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
2 Introducing Geometry What does the word geometry mean to you? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
3 Introducing Geometry What does the word geometry mean to you? Can you think of any results in geometry you ve covered in class? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
4 Introducing Geometry What does the word geometry mean to you? Can you think of any results in geometry you ve covered in class? Pythagorus theorem, Circle theorems, Trigonometry. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
5 A little history The geometry you will have seen in classes is some of the most ancient material taught in schools (including in history classes). Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
6 A little history The geometry you will have seen in classes is some of the most ancient material taught in schools (including in history classes). There is a clay tablet from around 1800BC listing a collection of triples a, b, c such that a 2 = b 2 + c 2. Euclid s Elements written around 300BC contains many results on circles, cones and cylinders that are well known to you. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
7 A little history The geometry you will have seen in classes is some of the most ancient material taught in schools (including in history classes). There is a clay tablet from around 1800BC listing a collection of triples a, b, c such that a 2 = b 2 + c 2. Euclid s Elements written around 300BC contains many results on circles, cones and cylinders that are well known to you. Here are some faces associated to results you will know... Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
8 What is geometry? What is geometry? According to Euclid it is like algebra: starting from some very basic notions we combine them and apply logical rules to obtain more sophisticated statements. But are the basic rules obvious? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
9 What is geometry? What is geometry? According to Euclid it is like algebra: starting from some very basic notions we combine them and apply logical rules to obtain more sophisticated statements. But are the basic rules obvious? According to Euclid these lines must meet at some point. Is this controversial? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
10 Where now? Can we think of shapes which obey almost all of Euclid s rules, but not all of them? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
11 Where now? Can we think of shapes which obey almost all of Euclid s rules, but not all of them? We re going to focus on two examples of curved geometry: 1 Spherical geometry (looking at shapes on the surface of a sphere). 2 Hyperbolic geometry (shapes in the hyperbolic plane - the sphere s evil twin). We ll study these geometries by making models, and by using maps to turn statements about the geometry in curved space into statements in flat space. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
12 Where now? Can we think of shapes which obey almost all of Euclid s rules, but not all of them? We re going to focus on two examples of curved geometry: 1 Spherical geometry (looking at shapes on the surface of a sphere). 2 Hyperbolic geometry (shapes in the hyperbolic plane - the sphere s evil twin). We ll study these geometries by making models, and by using maps to turn statements about the geometry in curved space into statements in flat space. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
13 Exercises The first set of exercises is to give you an idea about curved spaces. In particular you will: Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
14 Exercises The first set of exercises is to give you an idea about curved spaces. In particular you will: Models of curved space 1 Make a model of a sphere using triangles. 2 Make models of patches of spaces with different curvature. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
15 Introducing spheres Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
16 Introducing spheres Sphere A sphere of radius R is the set of points at distance R from a given point, or origin in 3 dimensions. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
17 Introducing spheres Sphere A sphere of radius R is the set of points at distance R from a given point, or origin in 3 dimensions. The first thing I need to know is how far away two points are from each other, this is the length of the line connecting them, but... Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
18 Distances on a sphere What is a line? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
19 Distances on a sphere What is a line? A line is the shortest path between two points. So we need to know distances to know lines. How can we resolve this paradox? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
20 Distances on a sphere What is a line? A line is the shortest path between two points. So we need to know distances to know lines. How can we resolve this paradox? Well, we can measure distances in 3D space, so we can imagine taking a piece of string fixed to sphere to find the shortest path. Distances are then the length of these bits of string. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
21 Great circles There is a nice way of describing these shortest paths, or spherical lines. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
22 Great circles There is a nice way of describing these shortest paths, or spherical lines. Great circles A great circle is a circle in 3D space obtained by intersecting a sphere with a plane passing through the centre of the sphere. Alternatively it is a circle contained in the sphere of the same radius as the sphere. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
23 Flight paths These are useful in navigation. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
24 Exercises In these exercises you ll explore the geometry of a sphere in a bit more detail. These exercises work through the notion of spherical area, in particular the areas of triangles formed using great circles on a sphere - not an insubstantial result! Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
25 Maps of the sphere Great circles are useful, but it can be awkward to work out distances on a globe. A very useful tool in geometry is to make maps of curved regions. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
26 Maps of the sphere Great circles are useful, but it can be awkward to work out distances on a globe. A very useful tool in geometry is to make maps of curved regions. Some maps Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
27 Maps of the sphere Great circles are useful, but it can be awkward to work out distances on a globe. A very useful tool in geometry is to make maps of curved regions. Some maps We can make maps by projecting the sphere to a plane. We ll study two projections in more detail. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
28 Gnomonic projection How the projection works Place the map at the bottom of the sphere and choose a point P on the sphere. There s a line from P to the center of the sphere. Extending this to the plane tells you how to project this point. Notice that lines (great circles) on the sphere are projected to lines - this is a very special property. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
29 Stereographic projection How the projection works Place the map at the bottom of the sphere and choose a point P on the sphere and a north pole. There s a (straight) line between the north pole and P. Extending this line to the map tells you how to project the point. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
30 Exercises: Can you invent a map projection? In this exercise session you ll study the gnomonic projection. Can you find any other projections? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
31 The hero: the Hyperbolic plane Up to now we ve looked at positively curved space. We ve seen angles of triangles are bigger than expected, areas of circles are smaller than expected. Can we arrange for the opposite: is there a space with triangles whose angles sum to less than 180? Or where circles have bigger areas? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
32 The hero: the Hyperbolic plane Up to now we ve looked at positively curved space. We ve seen angles of triangles are bigger than expected, areas of circles are smaller than expected. Can we arrange for the opposite: is there a space with triangles whose angles sum to less than 180? Or where circles have bigger areas? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
33 Hyperbolic geometry The problem is that in a very real sense the answer is no. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
34 Hyperbolic geometry The problem is that in a very real sense the answer is no. Hilbert s theorem (1901) There is no smooth surface of constant negative curvature in three dimensional space. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
35 Hyperbolic geometry The problem is that in a very real sense the answer is no. Hilbert s theorem (1901) There is no smooth surface of constant negative curvature in three dimensional space. There are two major caveats to this: first it doesn t mean that we can t make models in 3D space, like the icosahedron, they might not be perfect, but we can still see a lot of the geometry. Second, it doesn t mean there is no such space, just that we can t find it in 3D space like we did for a sphere. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
36 Exercises: a model of hyperbolic space Now it is time to make your own model of hyperbolic space. Your sheets explain constructions of models of the hyperbolic plane and a few starter exercises to explore what geometry is like on this shape. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
37 Break Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
38 Maps of hyperbolic space In fact there are hyperbolic planes. I won t tell you exactly what they are, since we d need to develop the theory of Riemannian geometry a little. However just like with the sphere we can make maps, and I can tell you what hyperbolic lines look like on a map. That way we can still make precise statements. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
39 Poincaré disc The model on the left is called the Poincaré disc model. It maps the entire hyperbolic plane into a disc. Although, as you can see greatly distorts areas and distances near the boundary. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
40 Poincaré disc The model on the left is called the Poincaré disc model. It maps the entire hyperbolic plane into a disc. Although, as you can see greatly distorts areas and distances near the boundary. Properties 1 Like stereographic projection, this map is conformal: angles are all the same as in the disc. 2 Hyperbolic lines are mapped onto (arcs of) circles which meet the bounary at 90. Including diameters. 3 Distances are nearly correct near the center, but distorted greatly farther out. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
41 Poincaré disc On the online demonstration ( joel/noneuclid/noneuclid.html) construct the following. Poincaré disc constructions 1 Lines which move increasingly close to being parallel. 2 Two different lines neither of which never meet a third. 3 Triangle with zero angle sum? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
42 Exercises In this set of exercises you ll compare lines on your model with hyperbolic lines in the Poincaré disc. You will also study how to use ruler and compasses to construct hyperbolic lines in your disc map of hyperbolic space. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
43 Why Hyperbolic geometry? As we mentioned at the beginning curved geometries may or may not satisfy each of Euclid s Postulates for geometry. The most controversial (even for Euclid) was the fifth, or parallel postulate, which implies that given a line and point not on the line there is a unique parallel line which it never meets. Does this postulate depend on the other four? If we can only find a geometry which obeys all four others, but not the fifth we will resolve this question. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
44 Escher and the infinite The artist M. C. Escher used hyperbolic geometry to capture the infinite inside a two dimensional plane. His idea was to use a hyperbolic tiling to produce an endless repetition of the same pattern contained in a finite space. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
45 Symmetries These images made heavy use of the symmetries of the hyperbolic plane. This is a broad and important subject in its own right. For any tessellation of the plane there is an associated group of symmetries. For example the tessellation shown has rotational symmetry of order four with center in the middle of a square. Can you think of any others? Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
46 Exercises Constructing your own tilings. In this exercise you will investigate symmetries and tilings of flat and spherical geometry Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
47 Applications: Maths Hyperbolic geometry is unavoidable in any study of modern mathematics and underlies many major theorems. Maybe most dramatically the uniformization theorem, which says there are three kinds of closed up smooth surfaces. 1 Spheres: positively curved. 2 Tori: flat (glued from a square) 3 Many-holed tori: (glued from patches of the hyperbolic plane) So apart from the sphere and torus any surface is hyperbolic. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
48 Applications: Maths Hyperbolic geometry is unavoidable in any study of modern mathematics and underlies many major theorems. Maybe most dramatically the uniformization theorem, which says there are three kinds of closed up smooth surfaces. 1 Spheres: positively curved. 2 Tori: flat (glued from a square) 3 Many-holed tori: (glued from patches of the hyperbolic plane) So apart from the sphere and torus any surface is hyperbolic. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
49 Applications: Thought and Philosophy We ve touched on how hyperbolic geometry gave new ways to represent the infinite, and settled the problem of Euclid s fifth postulate. However it s implications were much broader, clarifying the relationship between mathematics and observation of the world. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
50 Applications: Thought and Philosophy We ve touched on how hyperbolic geometry gave new ways to represent the infinite, and settled the problem of Euclid s fifth postulate. However it s implications were much broader, clarifying the relationship between mathematics and observation of the world. Non euclidean geometries impose a divison between what is obvious and what is true. Dostoyevsky even makes an analogy with the problem of evil: Non Euclidean Geometry in Dostoyevsky Let the parallel lines even meet before my own eyes: I shall look and say, yes, they meet, and still I will not accept it. - the Brothers Karamazov Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
51 Generalisations: Toward Higher dimensions In the same way we can have negative curvature in two dimensions, we can have it in three, or more. This plays an essential role in modern physics. Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
52 The End Thomas Prince (Imperial College London) Hyperbolic Planes 21 January / 31
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