Spherical Triangles Exploration

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.

Larger triangles have larger defect. In fact, the defect of a triangle is proportional to its area:

(1) For a spherical triangle, ${\text{defect}}=X*({\text{fraction of sphere's area covered}})$

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 , , and symmetries, which give spheres.

1. Explain the (flat) vs. (curved) styles.
2. Switch to the flat style, and use the symmetry group. Move the control point until you get an octahedron. Switch to curved style, and move the control point slightly to show the polygonal shapes.
   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 triangles cover?
   3. What are the corner angles of these triangles?
   4. What is the defect of one of these triangles?
   5. What would $X$ need to be to make formula (1) correct?
3. Repeat question 2, but use the group and the tetrahedron.
4. Repeat question 2, but use the group and the icosahedron.
5. Return to the symmetry group and the tetrahedron. On the View menu, turn on the "Cut Along Mirror Lines" option. You should have a tessellation by small triangles with dark edges. Answer questions a-e for these triangles.
6. Re-write formula (1) to give the area fraction in terms of the defect.
7. 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?

Handin: A sheet with answers to all questions.