Explorations For Spherical Geometry Exploration: Spherical Geometry Materials: Smooth inflatable ball, magic marker, wet sponge, string. Geodesics 1. Pick two points on a sphere. What is the shortest way to get from one to the other? 2. You live on a sphere. If you walk "straight" (or fly), what will your path look like? 3. A piece of string pulled tight between two points is "straight". Try this with a ball. Describe the "straight" lines on a sphere. These are called geodesics. Draw some geodesics. Do they work well with all three of the above ideas? 4. How many geodesics are there connecting the North and the South Pole? Between In the plane, if three points are on a line then one is always between the other two. Is this true for on a sphere? 5. Can you give a meaningful definition of "between"? 6. Consider the geodesic that is the Earth’s equator. Pick any two points on the equator. With your definition, what points on the Earth lie between these two points? 7. Two points directly opposite each other are called antipodal points, for example the North and South poles of the Earth are antipodal. With your definition of between, which points on the Earth are between the North and South poles? Circles A circle is the curve of all points which are the same distance from a given point, called the centre. We can use the same definition in spherical geometry. 8. Draw some circles on the sphere, by marking a center point and using a piece of string to find all points at a fixed distance from it. 9. As the radius gets larger, what happens to the circle? Then what happens? Then what happens? 10. In the plane, a circle has an inside which is finite and an outside which is infinite. Do circles have insides and outsides on the sphere? Explain. Postulates The five postulates (assumptions) for Euclidean geometry are: I. There is exactly one straight line joining any two points. II. Any straight line can be extended forever. III. There is a circle with any given center and radius. IV. The plane looks the same at every point. V. Given a line and a point not on the line, there is exactly one line through the given point which is parallel to the given line. 11. If you replace “straight line” with “geodesic”, most of these are wrong for spherical geometry. How would you adapt them to spherical geometry? 12. In plane geometry, two triangles which have equal side lengths are congruent. This is called Side-Side-Side. Is SSS still true in spherical geometry? What about SideAngle-Side, is SAS still true? ASA? In plane geometry, there's no AAA.. why not? Is there AAA in spherical geometry? Exploration: Drawing polygons on the Sphere We have seen before that a polygon in the plane is defined as follows: A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others. On the sphere we will use a similar definition: Definition: A polygon on the sphere is a closed figure made by joining geodesic segments, where each geodesic segment intersects at most two others. 1. In Euclidean (Planar) Geometry there are 3-, 4-, 5-gons etc., but there are no 1- and 2gons. Are there 1-,2-, 3-, 4-, 5-gons etc. on the sphere? Draw some examples of those that exist. 2. Focus on the 3-gons for a moment. Are there regular triangles? If so, what are their angle measures? 3. Draw 4 different triangles (different sizes) and measure the sum of their angles. 4. Based on your experiment in question 3, what can we say about the sum of the angles in a triangle on the sphere? Give a convincing argument. 5. Are there quadrilaterals on the sphere? If there are, how would you construct one? 6. Are there any squares or rectangles on the sphere? Why or why not? 7. What kind of tessellations can you draw on the sphere? Which of the planar tessellations may be adapted to the sphere? Which ones can definitely not be adapted (if any)? Exploration: Isometries on the Sphere 1. On the sphere below draw an example of a translation. 2. On the sphere below draw an example of a rotation. Center of rotation; 45 degrees counter clockwise 3. On the sphere below draw an example of a reflection. 4. On the sphere below draw an example of a glide-reflection. Spherical Tessellations 5. Below, draw examples of tessellations of the sphere. Experiment with 2-gons (also called lunes), triangles, and other polygonal shapes. 6. For the spherical tessellation to the right, answer the following: a. What is the highest order rotation? b. Are there reflections? c. Are there glide-reflections? d. What is the underlying geometric tessellation? 7. For the spherical tessellation to the right, answer the following: a. What is the highest order rotation? b. Are there reflections? c. Are there glide-reflections? d. What is the underlying geometric tessellation? Images by Andrew Crompton http://www.cromp.com/tess/home.html Exploration: Introduction to Spherical Tessellations Regular Tessellations by Triangles Let's build a regular tessellation of the sphere by demanding that 4 equilateral triangles meet at each vertex. 1. What corner angles will each triangle have? 2. What defect will each triangle have? 3. What fraction of the sphere will each triangle cover? 4. How many such triangles will we need to cover the sphere? Draw on a ball this regular tessellation of the sphere. 5. Answer questions 1-4 assuming three equilateral triangles meet at a vertex. 6. What are other possibilities for number of triangles meeting at a vertex? Do these give spherical tessellations? Kaliedotile Kaleidotile is a free computer program, available at http://geometrygames.org Play with Kaleidotile. It's pretty cool. Try changing the basepoint, spinning the picture, and playing with the view options and symmetry groups. Today, we're mostly interested in the (2,3,3), (2,3,4), and (2,3,5) symmetry groups, which give spheres. 7. Explain the flat vs. curved option. 8. Use Kaleidotile to find your regular tessellations from part I. What are the names of the flat versions? 9. Which of the other named tessellations are regular tessellations? 10. On the View menu, you can choose More Symmetry Groups. Try /2, /2, /2. Try /2, /2, /3. What happens for other choices? Exploration: Tessellations of the Sphere with Kaleidotile Assume the base point triangle in Kaleidotile is labeled as indicated above. 1. Complete the following table Symmetry BaseRegular or semigroup point regular tessellation? (2,3,3) A Regular B Semi-regular C Regular D E F G Semi-regular (2,3,4) A B C D E F G (2,3,5) A B C D E F G There are 3 regular tessellations of the plane: Triangles: (3,3,3,3,3,3) Squares: (4,4,4,4) Vertex type (3,3,3) - all triangles (3,6,6) - triangles and hexagons (3,3,3,3) - all triangles (4,6,6) - quadrilateral and hexagons Hexagons: (6,6,6) There are 8 semi-regular tessellations (Page Error! Bookmark not defined.) of the plane: (3,3,3,4,4) (3,6,3,6) (3,3,4,3,4) (3,3,3,3,6) (4,8,8) (3,4,6,4) (3,12,12) (4,6,12) 2. How many regular tessellations of the sphere did you find? List their vertex types. 3. How many semi-regular tessellations of the sphere did you find? List their vertex types. Exploration: Area on the Sphere All spherical triangles have angles adding up to more than 180°. We called the amount over 180° the defect of the triangle. This project should help convince you that a triangle covers a fraction of the sphere equal to defect 720° . Run Kaliedotile. 1. Use the (2,3,3) symmetry group and move the basepoint until you get a tetrahedron. Set Kaliedotile to show the curved tessellation. Turn on the mirror lines. The sphere is now covered with congruent (identical) triangles. a. How many of these triangles are there (it may help to look at the flat view). b. What fraction of the sphere does one of these cover? c. What are the corner angles of these triangles? d. What is the defect of one of these triangles? e. Does the formula for area fraction in terms of defect check out? 2. Repeat question 1, but use the (2,3,4) group and the cube with mirror lines. 3. Repeat question 1, but use the (2,3,5) group and the icosahedron, with mirror lines turned off. The area of a sphere is 4π R2, where R is the radius of the sphere. 4. If we have a sphere of radius 1 inch, what is the area of the sphere? What are the units? 5. If we have a sphere of radius 5 inches, what is the area of the sphere? What are the units? 6. Suppose we have a 90°-90°-90° triangle. Sketch one. What fraction of the sphere does this triangle cover? Suppose it lies on the sphere of radius 5, what it the surface area of the triangle? What's the area on a sphere of radius R? 7. Suppose we have a 60° lune. Sketch one. What fraction of the sphere does this lune cover? Suppose it lies on the sphere of radius 1, what is the surface area of the lune? What's the area on a sphere of radius R? Exploration: Platonic Solids 1. Look at Escher’s Reptiles (Schatt. pg 113, Magic pg 175). What platonic solid is the focal point of this picture? 2. Look at Escher’s Crystal (Magic pg. 102). It is built out of two intersecting platonic solids. Which two are they? 3. Look at Salvador Dali’s The Sacrament of the Last Supper (1955). What platonic solid forms the window? 4. Fill in the following table: Shape # of vertices Tetrahedron Cube Octahedron Dodecahedron Icosahedron Can you find any patterns in this table? # of edges #of faces Explorations For Hyperbolic Geometry Exploration: Hyperbolic Paper Equipment needed: Triangle paper, scissors, tape. You can make Euclidean space by gluing equilateral triangles together so that six touch at each vertex (this is just the usual tessellation by triangles). You made an icosahedron by gluing equilateral triangles together so that five touch at each vertex. This corresponds to a tessellation of the sphere, and makes a pretty good model for an actual curved sphere. Gluing four, three, or two triangles also makes a sphere, of sorts. Using triangle paper, cut and tape triangles together so that seven triangles meet at every vertex. Your group might want to make more than one of these models. Some tips: • The less taping you do, the better. Try to tape on chunks of connected triangles at a time, folding along the edges for flexibility. • To start, it's helpful to cut out an entire hexagon of six triangles, then make a slot to the center. Keep building until you've got at least one vertex which is surrounded by two rings of triangles. Each triangle in your model is flat, so geodesics are the usual straight lines. When a line crosses a fold in the model, flatten out the fold and continue the line in the usual straight way. Try drawing a long straight line or two on your model. Use light pencil, or things will get cluttered later. 1. Start two "parallel" lines as shown: Continue the two lines in both directions. What happens? Do they stay the same distance from each other? 2. Draw a triangle on the hyperbolic paper. How does its angle sum compare to 180°? 3. What happens to the angles of triangles when the triangles get larger? Exploration: Escher’s Circle Limit Series Recall that in Spherical Geometry we determined that shortest distance was measured by great circles. We called the distance minimizing curves geodesics, and these geodesics play the same role as lines do in Euclidean Geometry. "Circle Limit III" by M.C.Escher In Circle Limit III, the white lines are meant to represent straight lines in this new geometry. These curves are also called geodesics. Segments of these geodesics will form the sides of polygons. 1. What type of polygons do you see in this figure? NOTE: In this picture we should interpret all fish as having the same size. A smaller figure does not mean that it is physically smaller. Think of it as being farther away. 2. Looking at the angles in a triangle, what will the angle sum be? Equal to, greater than or less than 180o? Why? "Circle Limit I" by M.C.Escher 3. What is the highest degree of rotation? 4. Use a colored marker, and draw in some geodesics (the hyperbolic version of a straight line). The Easiest way of doing this is to follow the spines of the fish. What is the underlying geometric tessellation? "Circle Limit II" by M.C.Escher 5. What is the highest degree of rotation? 6. Draw geodesics in this figure. What is the underlying geometric tessellation? "Circle Limit IV" by M.C.Escher 7. What is the highest degree of rotation? What other degrees of rotation are present? 8. Draw geodesics in this figure. What is the underlying geometric tessellation? 9. Draw a geodesic NOT passing through the center point. 10. How many geodesics can you draw through the center point, so that the new geodesic does not meet the geodesic you picked? Another way to ask the same question: How many geodesics pass through the point so that the new geodesic is parallel to the first geodesic? Exploration: Isometries of the Hyperbolic Plane We have seen that a disk may model the hyperbolic plane. Technically, the disk will not contain its boundary. In this geometry, the geodesics (curves which give you the shortest distance between two points) are line segments passing through the center, and semicircles meeting the boundary at a right angle. We have seen that the hyperbolic plane contains 3-, 4-, 5-, … , n -gons. There are no 1 and 2- gons in this geometry. There are however ideal n- gons: these are n-gons whose vertices lie on the boundary, which we call the circle at infinity. Next we want to investigate the behavior of isometries in hyperbolic space. We have seen that the isometries on the sphere behave as expected. Translations move figures around, and rotations rotate shapes around. Reflections over geodesics exist, as well as glidereflections. There were no major surprises there. Now we need to think about translations, rotations, reflections and glide-reflections in the hyperbolic plane. You should think carefully about the following questions. Your answers should be at least a paragraph for each, and include drawings to illustrate the ideas.. Some answers may require more than a paragraph. Be thorough in answering the questions. 1. For each of the four hyperbolic tessellations Escher created, determine and sketch the underlying geometric tessellation. 2. For each tessellation determine what the translations look like. Describe carefully what hyperbolic translations look like. 3. For each tessellation determine what the rotations look like. Describe carefully what hyperbolic rotations look like. 4. For each tessellation determine what the reflections look like. Describe carefully what hyperbolic reflections look like. Exploration: Computer Assisted Hyperbolic Geometry NonEuclid is a Java program, written by Joel Castellanos, for doing geometry in hyperbolic space. You can run it from his web page, at: http://cs.unm.edu/~joel/NonEuclid When you have the web page showing, click the big button to run NonEuclid Applet. It should display a large black circle - that's the Poincaré disk. Like Geometer's Sketchpad, NonEuclid lets you draw points, lines, and circles. The menus "File" (#2), "Edit" (#2), "View", "Constructions", "Measurements", and "Help" are for the applet.. these are to the right of the normal menus for the browser. When you start, you should be in "Draw Line Segment" mode, as shown in the upper left corner. Draw some line segments. 1. What do lines that go through the center of the disk look like? To clear your drawing, use New on the File menu. Now draw a triangle. On the Edit menu, choose "Move Point". Now you can drag the corners of your triangle and see how it changes. On the Measurements menu, select "Measure Triangle", and click the three corners of your triangle. You should see the side and angle measurements of the triangle in the box on the left. Go back to "Move Point" mode. Move your triangle around to get a feel for the lengths of its sides and the sum of its angles. 2. Draw a triangle which appears large but really has sides of length under 4. Draw a triangle which appears small but really has sides of length over 10. (The "micro-move" feature may help: hold down Shift while you move the vertices). 3. Try to make the angle sum 180°. What do you have to do? 4. Try to make all three sides of the triangle large. What happens to the angle sum? Draw an "Infinite Line" (on the Constructions menu). Now use "Reflect" to make the reflection of your triangle across your infinite line. Move the line around, and notice the position and size of the reflection. The reflection is the same size and shape as the original, it just appears different. If you have time, try the “Hyperbolic Applet” at http://www.math.umn.edu/~garrett/a02/H2.html. Watch the hyperbolic animations at: http://www.josleys.com/animationsindex3.htm (#2,4, and 5 are the best) Exploration: Ideal Hyperbolic Tessellations 1. Draw an example of a 4-gon (vertices in the interior of hyperbolic space) and an ideal 4-gon in hyperbolic space. Label your drawings. 2. Draw an example of a 5-gon (vertices in the interior of hyperbolic space) and an ideal 5-gon in hyperbolic space. Label your drawings. 3. Draw an example of a 6-gon (vertices in the interior of hyperbolic space) and an ideal 6-gon in hyperbolic space. Label your drawings. Creating an ideal tessellation: 4. Create an ideal tessllation with the following steps: Step 1: draw a regular, ideal 4-gon. Step 2: Draw in both diagonals so that the polygon consists of 4 triangles. Color the triangles alternately black and white. Step 3: Between two adjacent vertices put 2 equally spaced vertices. Construct another 4-gon with the 4 vertices you thus created. Step 4: Draw in both diagonals so that the polygon consists of 4 triangles. Color the triangles alternately black and white. Now repeat Step 3 and 4 5. Create an ideal tessllation with the following steps: Step 1: draw a regular, ideal 4-gon. Step 2: Connect opposite midpoints so that the polygon consists of 4 quadrilateral. Color the quadrilaterals alternately black and white. Step 3: Between two adjacent vertices put 2 equally spaced vertices. Construct another 4-gon with the 4 vertices you thus created. Step 4: Divide the quadrilateral into 4 smaller ones, as in step 2. Color the quadrilaterals alternately black and white. Now repeat Step 3 and 4 6. Construct an ideal tessellation based on an ideal 6-gon. Use some consistent coloring scheme to “decorate” the tessellation. Exploration: Tessellations of the Hyperbolic Plane 1. Below, sketch an ideal tessellation for a triangle, a quadrilateral, a pentagon, and a hexagon. Triangle Quadrilateral Pentagon Hexagon 2. Generalizing from the work so far, what ideal polygons tessellate the hyperbolic plane? How many regular tessellations of the hyperbolic plane do we get? *** These are not regular tessellations - they’re ideal. 3. Below, sketch a tessellation for a triangle with quadrilaterals surrounding it, a quadrilateral surrounded by triangles, a pentagon with triangles, and a hexagon with quadrilaterals. Triangle surrounded by quadrilaterals Quadrilateral surrounded by triangles Pentagon surrounded by triangles Hexagon surrounded by quadrilaterals 4. Generalizing from the work so far, how many semi-regular tessellations of the hyperbolic plane do we get? *** These are not semi-regular, they’re ideal 5. Now create a tessellation with polygons that are not ideal: The following website has hyperbolic tessellations: http://www.josleys.com/creatures38.htm 6. What polygons appear in these tessellations? List the degrees of rotational symmetry you see (and record which tessellation has that degree of rotational symmetry). Example: hyp.01 consists of triangles, and has 6-fold symmetry at the center. It possibly also has 4-fold symmetry. Hyperbolic 01 Triangles 6-fold symmetry at the center; possibly also has 4fold symmetry. Hyperbolic 02 Hyperbolic 03 Hyperbolic 07 Hyperbolic 10 Hyperbolic 11 Hyperbolic 12 Hyperbolic 16 Hyperbolic 21 Hyperbolic 28 Hyperbolic 29 Hyperbolic 35 Exploration: Hyperbolic Tessellations Go to David Joyce’s hyperbolic tessellations web page: http://aleph0.clarku.edu/~djoyce/poincare and explore the regular tessellations of the Poincare disk. 1. In the Schläfli symbol {n,k}, what do n and k represent? 2. For these hyperbolic tessellations by triangles, what are the corner angles of each triangle? What is the defect of each triangle? a. {3,7} b. {3,8} c. {3,10} d. {3,12} Does there appear to be a relationship between area and defect? 3. Experiment with the (2,3,7) symmetry group in Kaliedotile. 4. Look at Salvador Dali’s Santiago el Grande (1957). Is the background hyperbolic? The table below has one entry for each Schläfli symbol {n,k}. In each box, put an S, E, or H depending on whether the corresponding regular tessellation is Spherical, Euclidean (flat) or Hyperbolic. k 2 3 4 5 6 7 8 n 2 3 4 5 6 7 8 Exploration: New Hyperbolic Tessellations From Old There are a variety of different methods for constructing new tessellations from old. We can find the midpoint of each side and connect adjacent points. We can find the dual, just as in Euclidean geometry: Place a vertex inside each polygon, and connect two points if and only if there is an edge between them. 1. In the two tessellations below, find the midpoint of all of the sides. Connect adjacent points. What polygons is the new tessellation made out off? 2. Find the dual of the two tessellations below. That is to say: Place a vertex inside each polygon, and connect two points if and only if there is an edge between them. What polygons is the new tessellation made out off? Exploration: Comparison Between the 3 Geometries Compare and contrast the three geometries based on the following topics: Euclidean Spherical Hyperbolic Geometry Geometry Geometry Geodesics (What do they look like?) Parallel Lines (How many, if any?) Polygons (Do we get new ones? Which ones don’t exist?) Sum of the angles in a triangle. Regular tessellations (How many different ones are there?) Semi-regular tessellations. (How many different ones are there?) Isometries (Which ones have we encountered?) Area of a triangle. (What is the formula?) Escher’s work based on each geometry. (How extensively did he work on each of the geometries?) Other Artists? (Did other artists work in any of these geometries?)