MATERIAL FOR A MASTERCLASS ON HYPERBOLIC GEOMETRY Timeline 10 minutes Introduction and History 10 minutes Exercise session: Introducing curved spaces 5 minutes Talk: spherical lines and polygons 15 minutes Exercise session: Areas of Spherical triangles 10 minutes Talk: Mapping the sphere 10 minutes Exercise session: Sphere projections 5 minutes Introducing Hyperbolic Geometry 10 minutes Exercise session: Constructing Hyperbolic planes 15 minutes BREAK (resume by 1hr 30) 10 minutes Talk: The Poincaré disc, with demonstration 15 minutes Exercises on the Poincaré disc 10 minutes Talk: Hyperbolic Geometry in art and philosophy 15 minutes Exercise: Tiling the Poincaré disk 10 minutes Finish: Applications and generalisations. 1
2 MASTERCLASS MATERIAL Introductory problems: A first look at curved space In this exercise you will make (approximations of) two different kinds of curved space, as well as the flat space you are more familiar with. (1) Take 6 of the triangles you have been given and arrange them around a common corner, there is no need to glue or tape these - they should lie flat. (2) Now take 5 triangles and tape them together around a common corner to form a cap shape. (3) Finally take 7 triangles and tape these together around a common corner. This should make a saddle shape. Set this aside - we will add to it later. Figure 1. An Icosahedron - one of the Platonic Solids (1) Why do six triangles lie flat and other numbers do not? (2) Working in small groups can you combine your cap shapes (5 triangles meeting at a corner) into an icosahedron? (You will need to either take some more triangles to fill in some gaps or break up one of the caps into its triangles). (3) An icosahedron has 5 triangle faces arranged around each corner. If I tried constructing a shape with 3 triangles arranged each corner what would I make? What about 4 triangles meeting at each corner? (4) The icosahedron is an approximation of a sphere by flat pieces (triangles). How could you make a better approximation? (Hint: Can you divide an equilateral triangle into smaller ones?)
MASTERCLASS MATERIAL 3 Lines and triangles on a sphere In this exercise we will work towards a formula for calculating the area of a triangle on a sphere, also called a spherical triangle. Figure 2. A triangle formed of lines on a sphere Hopefully you have been told that a sphere of radius R has area 4πR 2. I will assume for the whole session that my sphere has radius 1, so the area of my whole sphere is simply 4π. (1) What is the longest length a line can be on a sphere without it meeting itself? Can this happen in normal geometry? (Hint: Remember that lines on a sphere are really parts of circles with the same radius as the sphere.) (2) Draw out the following triangle on a sphere: Start at the north pole and draw a line to the equator, from there go a quarter of the way around the equator and close up the triangle. What are the angles of this triangle? (3) If I start at a point on a sphere and take two lines in different directions, do these lines meet again? Do you know a name for this point? Figure 3. A lune on a sphere
4 MASTERCLASS MATERIAL A lune is the shape enclosed by pair of lines leaving the same point in different directions, see Figure 3. (1) Is this shape possible in normal geometry (a shape with two straight edges and two corners)? If not, why not? (2) Do any two lines on the sphere enclose a lune? If so, how many? (3) What does this mean about parallel lines in spherical geometry? The area of a lune is proportional to the angle between the lines (for example if I double the angle I double the area of the lune). Therefore the formula for the area of a lune between lines on a sphere making angle a is k a for some number k. Can you work out this number out, and so write down a formula for the area of a lune? (Hint: What is the area of the lune when a = 360, remember the sphere has radius 1.) Figure 4. A triangle formed of lines on a sphere (1) The triangle shown above sits inside three lunes, using your formula add up the areas of these three lunes. (2) The area covered by these lunes is 2π (half the area of the sphere), can you prove this? (3) Thinking about how the three lunes overlap can you work out the area of the triangle shown?
MASTERCLASS MATERIAL 5 Mapping the sphere We ll think about one kind of map in this exercise, the gnomonic projection. This is a way of representing geometry of a sphere on flat space. Since (as we ve seen) these geometries are different: the distances, lines and angles on a sphere aren t all the same as on the plane, but using a map we can make precise statements about the geometry of the sphere. Figure 5. Gnomonic projection (1) What do the concentric circles on the map represent? (2) Can you find a country in the world at a lower latitude than the UK but where the shortest flight path involves going to a higher latitude? (3) Does the equator ever appear in this map? (4) Where do distances/lengths on the sphere get most stretched by the projection? Where does the projection affect lengths least? (5) Are angles between lines on the map the same as angles between the corresponding lines on the sphere? Are they sometimes the same? Do you know, or can you think of, any other map projections of the sphere? What do spherical lines look like on your map?
6 MASTERCLASS MATERIAL The Hyperbolic plane Starting from your paper models with seven triangles meeting at a corner build a larger piece of the hyperbolic plane by taping on more triangles so that there are exactly seven triangles meeting at any corner. Continue until you have used at least 20 triangles. On your paper models of the hyperbolic plane construct the following shapes. (1) Two lines which start parallel, but which diverge (move away from each other). (2) As large a hyperbolic triangle (triangle with edges hyperbolic lines) as you can. (3) Two lines which are not parallel but increasingly close to parallel. Is the sum of the interior angles of your hyperbolic triangle equal to, less than or greater than 180? Are any of these constructions possible in normal (flat) geometry? You might also like to attempt an annular model: taking two identical annuli tape the outside edge of one to the inside of the next and repeat. You find the surface you form is forced to bend and twist. Figure 6. A different paper model of the hyperbolic plane: hyperbolic football Just as with the sphere we can make precise statements in hyperbolic geometry by using maps rather than models. One of the most famous and useful maps of the hyperbolic plane is the Poincaré Disc. In this exercise you will explore the geometry fo the hyperbolic plane using this map. This map has the following basic properties (1) Hyperbolic lines are (arcs of) circles which meet the boundary circle at 90. This includes diameters of the disc. (2) Angles between hyperbolic lines are the (usual) angles between tangents to the circles where they meet.
MASTERCLASS MATERIAL 7 Figure 7. The triangles of your models, see how area is stretched near the edge Using the Poincaré disc model we turn statements about hyperbolic lines into statements about circles in a disc. That means we can prove things about hyperbolic geometry using things we know about circles, the subject of the rest of the exercises in this section. First we need to learn how to draw straight (hyperbolic) lines in the disc map of Hyperbolic plane. (1) Draw a circle with center P representing the hyperbolic disc (large, but not reaching the edges of the page). (2) Choose a point A on the circle and draw the line P A. (3) Draw the line through A at right angles to the line P A (this should be tangent to your large circle). (4) Choose a point B on this perpendicular line. (5) Draw the arc of the circle with center B passing through A (P A should then be a tangent line). (6) Draw a few different hyperbolic lines using this method. They should like the circles on page 8.
MASTERCLASS MATERIAL 9 (1) Given two points A, A on the boundary circle can you find the unique hyperbolic line between them? (Recall this is an arc which meets the circle at right angles at both A and A.) (2) Given a point A on the boundary circle and a point A inside the circle, can you construct the unique hyperbolic line between them? (3) Given two points A, A inside the circle can you construct the unique hyperbolic line joining them? Figure 8. Some lines in the Poincaré disc Now we know how to draw lines in the plane we can compare properties of these lines with lines in usual geometries. (1) Draw a pair of hyperbolic lines which meet at the boundary of the disc. What would these look like on the paper model? (2) Given a hyperbolic line and a point not on that line how many lines go through that point but do not meet the line? What happens in normal geometry? What happened on the sphere? (3) How small can the interior angle sum of a triangle in hyperbolic space become? Can you find a triangle with minimal angle sum? Symmetries A key feature of spherical, flat and hyperbolic geometries is their large groups of symmetries. A symmetry of a space is a transformation which preserves all angles and distances. In normal geometry there are three important kinds of symmetry: (1) Translation: in which every point is moved by a fixed amount. (2) Rotation: in which every point moves by a fixed angle around a central point. (3) Reflection: in which every point is sent it its mirror in a given line of reflection. These can be used to generate tilings of the plane, starting from certain shapes and using certain symmetries we can fill out the plane. For example, the tiling by hexagons shown.
10 MASTERCLASS MATERIAL Figure 9. Hexagon tiling (1) Construct two more tilings, using one or two shapes. Can you find all the symmetries of your tiling? (2) Given one tiling you can make another one by putting points in the middle of faces and edges which cross the edges of the old tiling once. This is the dual tiling. What do you get in your examples? Can you find a tiling whose dual tiling is itself? (3) How do the symmetries of a tiling and its dual tiling compare? (4) If your tiling is regular (all angles the same), its Schläfli symbol is the pair {number of edges of each tile, number of tiles meeting at a corner}. An important value we can work out from this is, 1 number of edges of each tile + 1 number of tiles meeting at a corner For example, the hexagonal tiling has symbol {6, 3} and 1/6 + 1/3 = 1/2. Work these out for your regular tilings. There are analogs of symmetries and tilings on the sphere and in hyperbolic space.
MASTERCLASS MATERIAL 11 Figure 10. Ball and stick representation of a tiling of a sphere (1) Does a sphere have any translation symmetry? (2) How many symmetries of the sphere can you find? (Hint: Think about rotations and reflections in 3D space) (3) Can you find tilings of a sphere? What are the symmetries of these tilings? (Hint: Start from a Platonic solid, e.g. a cube or tetrahedron.) (4) What are the Schläfli symbols {n, k} of these tilings, and what is 1/n+1/k? (5) We know a tiling of the hyperbolic plane by triangles - we used it to make a model. What is its Schläfli symbol?