Regular Polygons in Hyperbolic Geometry
Theorem: For any integer p>0 and angle measure x in , there is a regular polygon with p vertices and interior angle equal to x.
Theorem: The interior angle of a regular polygon depends on the radius of its circumcircle and vice versa.
No proofs, however explore the following 3 examples which illustrate these theorems.
Example 1: Adjust A, B, E and G to observe different congruent equilateral triangles. Observe that that:
- The triangles remain congruent
- The interior angle measure depends on the radius of the circumcircle
Example 2: Adjust A, B, F and G to observe different congruent equilateral quadrilaterals. Observe that that (like above):
- The quadrilaterals remain congruent
- The interior angle measure depends on the radius of the circumcircle
Example 3: Adjust A, B, G and H to observe different congruent equilateral pentagons. Observe that that (like above):
- The pentagons remain congruent
- The interior angle measure depends on the radius of the circumcircle