Spherical Triangles Exploration

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Time-30.svg

Objective: Check the relationship between defect and area fraction for some nice spherical triangles.

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° .

Kaleidotile is a free computer program. Run Kaleidotile and play with it a bit. Try changing the control point, spinning the picture, and playing with the controls and menu options. Today, we're interested in the Kt-sym3.svg, Kt-sym4.svg, and Kt-sym5.svg symmetries, which give spheres.

  1. Explain the Kt-flat-style.svg (flat) vs. Kt-curved-style.svg (curved) styles.
  2. Switch to the flat style, and use the Kt-sym3.svg symmetry group. Move the control point until you get a tetrahedron (you'll want to turn on the "Say Names" menu option). Switch to curved style and move the cursor slightly to show the underlying polyhedra.
    1. How many of these triangles are there (it may help to look at the picture in flat style).
    2. What fraction of the sphere does one of these cover?
    3. What are the corner angles of these triangles?
    4. What is the defect of one of these triangles?
    5. Does the formula for area fraction in terms of defect check out?
  3. Repeat question 2, but use the Kt-sym4.svg group.
  4. Repeat question 2, but use the Kt-sym5.svg group and the icosahedron.
  5. Look at Concentric Rinds. What are the corner angles of the triangles in the outer rind? What is the defect of one of these triangles? Use the formula to calculate the fraction of the sphere covered by each triangle. How many triangles are there in the outer rind?
  6. How many regular tessellations of the sphere can you find? List their vertex configuration. For example there is a regular tessellation by triangles labeled (3, 3, 3, 3, 3, 3)
  7. How many regular tessellations of the sphere can you find? List their vertex configuration.

Handin: A sheet with answers to all questions.