Spherical Geometry Exercises

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The numbering of these exercises has changed as of November 20, 2013. The old version is still available at [1]


Geometry on the Sphere

  1. Does every point on a sphere have an antipodal point? How many antipodal points does any given point on the sphere have?
  2. 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?
  3. Draw a picture of sphere. Draw a triangle on it with three 90° angles.
  4. Draw a picture of sphere. Draw a triangle on it with one angle larger than 180°.
  5. Draw three geodesics on a sphere so each one is perpendicular to the others.
  6. What is the upper limit for the corner angle of an equilateral spherical triangle?
  7. What is the upper limit for the corner angle of a spherical regular polygon with n sides?
  8. 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°.
  9. The state of Colorado has four 90° corners. However, we know that no spherical quadrilateral can have four right angles. What is going on with Colorado?
  10. 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).
  11. Consider the following picture of a soccer ball. What kind of polygons are used to tessellate the sphere?
  12. Soccer-ball.svg
  13. Consider the following picture of juggling balls. What kind of polygons are used to tessellate these spheres? What are the angles?
  14. Juggling-balls.jpg
  15. Escher's Ivory Ball Study shows a cardboard model of a rhombic dodecahedron and a spherical tessellation of the same pattern on a plastic ball. The ball is tessellated by identical rhombuses, which are four sided polygons with all sides equal but with two different angle measures in each rhombus.
    Determine the corner angles of each rhombus on the sphere. Hint: How many fit together to make 360° at each vertex?
  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? [1]

Defect and Area

  1. Give a definition for the defect of a triangle on the sphere. How do we compute the area of a triangle if we know what the defect is?
  2. Give a definition for the defect of a square on the sphere. What is the defect of a general n-gon? How do we compute the area of an n-gon?
  3. For spherical triangles, 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
  4. What is the upper limit for the defect of a triangle 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?
  5. What is the upper limit for the defect of a spherical polygon with n sides?
  6. Find a formula relating the angles of a biangle to the fraction of the sphere covered by the biangle.
  7. Consider the sphere in Escher's Ivory Ball Study.
    1. Determine the defect of the rhombus.
    2. What fraction of the sphere does one rhombus cover?
  8. Examine Concentric Rinds, 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?

Polyhedra

  1. Does every tessellation of the sphere by polygons give a corresponding polyhedron?
  2. Build yourself a set of paper models of the five platonic solids, using the nets from Spherical_Geometry#Platonic_Solids
  3. In Double Planetoid what shape are the two parts of the planet? (And by the way, check out Escher’s comment about this picture on page 97 of Magic of M.C. Escher).
  4. In Stars, the main structure is built out of three intersecting platonic solids. What are they?
  5. In Gravitation, 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.
  6. How many seashells are on Escher’s Verblifa Candy Tin?
  7. 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.
  8. Use the defect and area fraction to check that an icosahedron has 20 faces.
  9. Consider the rhombic dodecahedron and the spherical tessellation in Escher's Ivory Ball Study. They do not represent a platonic solid or a regular tessellation of the sphere. Why not?

Duality

  1. 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.
  2. 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?
  3. Besides mathematical duality, Escher includes dual themes in many of his works. For each work, describe the main thematic dualities.
    1. Day and Night
    2. Sun and Moon
    3. Order and Chaos
    4. Sky and Water I
    5. Double Planetoid
    6. Bond of Union
  4. 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?

Euler Characteristic

  1. Count the number of vertices, number of faces, and number of edges in the rhombic dodecahedron in Escher's Ivory Ball Study. The Euler characteristic should still be 2. Is that what you got?
  2. Count the number of vertices, number of faces, and number of edges in the outer rind of Escher's Concentric Rinds. What is the Euler characteristic?
  3. Pentakis Dodecahedron
  4. The pentakis dodecahedron is a polyhedra whose corresponding spherical tessellation is made of 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?
  5. Look at Escher’s Möbius Strip II (Red Ants). 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?
  6. How many vertices, edges, and faces are there on the torus from Spherical Geometry#Torus? What is the Euler characteristic of the torus?

Escher and Spherical Geometry

  1. Why was Escher unhappy with tessellations of the plane as a means to display infinity? Look at Escher’s Sphere with Fish, Sphere with Angels and Devils, Sphere with Eight Grotesques, and Sphere with Reptiles (pages 244 and 245 Visions of Symmetry ). Do you think these balls would give a good sense of infinity? Why or why not?
  2. Consider Escher's carved spheres Sphere with Fish, Sphere with Angels and Devils, Sphere with Eight Grotesques, and Sphere with Reptiles, all shown on pages 244 and 245 of Visions of Symmetry. Describe the rotational symmetry present in each of the patterns. Record the degree(s) of rotation and what the center of rotation looks like. For instance Sphere with Angels and Devils has a 2-fold rotation about the bottom of the feet of the bats and angels.
  3. Go to the Escher Gallery on the website maintained by Leys
    • Compare Escher's Sketch #21 (Imp) to Leys' spherical tessellation Clowns 04. What rotational symmetries are present in the flat tessellation? What rotational symmetries are present in the spherical tessellation?
    • Compare Escher's Sketch #15 (Lizard) with the flat tessellation created by Leys Dragons 01 and the spherical tessellation Dragons 03. What did Leys do to adapt the tessellation to the sphere?

Notes

  1. From Mathematics Magazine, Vol 76 #3, June 2003

Instructor:Spherical Geometry Exercises Solutions