Spherical Geometry Exercises

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  1. Does every point on a sphere have an antipodal point? How many antipodal points does any given point on the sphere have?
  2. Why was Escher unhappy with tessellations of the plane as a means to display infinity? Look at Escher’s spherical tessellations on pg. 91,92 of Magic and on pg. 244-245 of Schatt. Do you think these give a good sense of infinity? Why or why not?
  3. What might “between” mean for points on a sphere? Write a definition you are happy with. With your definition, is St. Louis between the North Pole and the South Pole? Is the North Pole between the South Pole and St. Louis?
  4. Draw a picture of sphere. Draw a triangle on it with three 90° angles.
  5. Same as (1), but draw a triangle with one angle larger than 180°.
  6. Draw three straight lines on a sphere so each one is perpendicular to the others.
  7. Fill in the empty (lettered) places in this table:
    Angles Defect Area Fraction
    90° 90° 90°
    a: b:
    120° 80° 70°
    c: d:
    72° 72° 72°
    e: f:
    90° 45° g:
    45° h:
    135° 135° i:
    j: 1/4


  8. What is the largest defect a triangle can achieve on the sphere? Hint 1: a really "big" triangle looks like the outside of a small triangle. Hint 2: What fraction of the sphere could it cover?
  9. What is the largest defect a spherical polygon with n sides can achieve?
  10. What is the upper limit for the corner angle of an equilateral spherical triangle?
  11. What is the upper limit for the corner angle of a spherical regular polygon with n sides?
  12. Use your knowledge of spherical triangles to explain why the sum of the angles of a quadrilateral on a sphere is always larger than 360°.
  13. The state of Colorado has four 90° corners. However, we know that no spherical rectangle can have four right angles. What is going on with Colorado?
  14. A "biangle" is a polygon with two sides and two angles. They don’t exist in Euclidean geometry, but they do on the sphere. Draw some biangles. (Hint: put one of the corners at the north pole).
  15. Find a formula relating the angles of a biangle to the fraction of the sphere covered by the biangle.
  16. Hiram of Tyre helped build King Solomon’s temple. From 1 Kings 7.23: Then he made the molten sea; it was round, ten cubits from brim to brim, and five cubits high. A line of thirty cubits would encircle it completely. This presumably describes a sort of circular above-ground pool, with a diameter of ten cubits and circumference of thirty cubits.
    1. Discuss the plausibility of these measurements.
    2. Explain how the pool could be built exactly as specified on the surface of a sphere.
    3. On what diameter sphere (i.e. on what size of planet) could this pool be built precisely?
    (Mathematics Magazine, Vol 76 #3, June 2003)
  17. Does every tessellation of the sphere by polygons give a corresponding polyhedron?
  18. Build yourself a set of five platonic solids using “nets”. Nets are flat cutouts that can be folded and glued into polyhedra, and are readily available for the Platonic solids.
  19. In Double Planetoid (Magic pg. 94) what shape are the two parts of the planet? (And by the way, check out Escher’s comment about this picture on page 97).
  20. In Stars (Magic pg 100), the main structure is built out of three intersecting platonic solids. What are they?
  21. In Gravitation (Magic pg 101), how many lizards are there, counting ones which would be hidden on the back? Hint: Pay close attention to the colors. Hint: This shape is called a stellated dodecahedron.
  22. How many seashells are on Escher’s Verblifa candy tin (Schatt. pg 295)?
  23. Draw the tessellation of the plane by hexagons (three hexagons meet at each vertex). Its dual is the tessellation by equilateral triangles (six at a vertex). Draw pictures to illustrate this.
  24. What is the dual of the regular tessellation of the plane by squares?
    1. What is the dual of the tetrahedron?
    2. What is the dual of the cube?
    3. What is the dual of the octahedron?
    4. What is the dual of the dodecahedron?
    5. What is the dual of the icosahedron?
  25. Johannes Kepler (1571-1630) writes in The Harmonies Of The World (Book V): However, there are, as it were, two noteworthy weddings of these figures, made from different classes: the males, the cubes and the dodecahedron, among the primary; the females, the octahedron and the icosahedron, among the secondary, to which is added one, as it were, bachelor or hermaphrodite, the tetrahedron, because it is inscribed in itself, just as those female solids are inscribed in the males and are, as it were, subject to them, and have the signs of the feminine sex, opposite the masculine, namely, angles opposite planes. What is Kepler saying? What do you think of it?
  26. A tetrahedron has six edges. Sketch a picture of a tetrahedron, and color the edges with three colors so no two edges with the same color touch each other.
  27. Use the defect and area fraction to check that an icosahedron has 20 faces.
  28. Check out the rhombic dodecahedron and the spherical tessellation shown on the bottom of page 246 (Schatt), as well as on page 307 (Schatt). This is not a platonic solid, or a regular tessellation of the sphere. Why not?
  29. By looking at the sphere on page 246 (Schatt), determine the corner angles of the rhombus on the sphere. Remember, you can do this by deciding how many fit together to make 360° at a vertex.
  30. What is the defect of the rhombus from the previous problem? (Remember, defect for a 4 sided shape is the amount over 360°) What fraction of the sphere does it cover?
  31. Count the number of vertices, number of faces, and number of edges in the rhombic dodecahedron. The Euler characteristic should still be 2.. is that what you got?
  32. Look at Escher’s Concentric Rinds on pg. 165 of Magic, and focus on the outer sphere. It’s tessellated by many copies of one triangle.
    1. What are the corner angles of this triangle?
    2. What is it’s defect?
    3. How many copies of the triangle cover the sphere?
    4. How many vertices, edges, and faces does this tessellation have?
  33. The “pentakis dodecahedron” is a spherical tessellation by 60°-60°-72° triangles.
    1. How many vertices, edges, and faces does this tessellation have?
    2. The dual of the pentakis dodecahedron is a well known spherical tessellation. Where have you seen it before?
  34. Look at Escher’s Möbius Strip II (Red Ants) on Magic pg. 186. Assume the strip is a flat surface with a tessellation by squares. How many vertices, edges, and faces (squares) does it have? What is the Euler characteristic of the strip?
  35. How many vertices, edges, and faces does the donut shown in the Euler Characteristic section have? What is its Euler characteristic?