Ideal Hyperbolic Tessellations Exploration

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Objective: Create Ideal Hyperbolic Tessellations


Remember that geodesics in hyperbolic space are semicircles perpendicular to the boundary of the disc or lines passing through the center of the circle. Below I have drawn some examples. Geodesics can pass through the center, in which case they resemble Euclidean straight lines. Geodesics that do not pass through the center appear curved.

Geodesics-hyperbolic.jpg

Below are two triangles. The one on the left has its vertices on the interior of hyperbolic space. The one on the right has its vertices on the boundary of hyperbolic space; hence the vertices lie at infinity (they are called ideal points, not real points). The triangle on the right is called an ideal triangle.

Triangles-hyperbolic.jpg

1. Draw an example of a 4-gon (vertices in the interior of hyperbolic space) and an ideal 4-gon (vertices are ideal points) in hyperbolic space. Label your drawings.

Circles.jpg

2. Draw an example of a 5-gon (vertices in the interior of hyperbolic space) and an ideal 5-gon in hyperbolic space. Label your drawings.

Circles.jpg

3. Draw an example of a 6-gon (vertices in the interior of hyperbolic space) and an ideal 6-gon in hyperbolic space. Label your drawings.

Circles.jpg


Creating an ideal tessellation:

4. Create an ideal tessllation with the following steps:

Step 1: Draw a regular, ideal 4-gon.

Step 2: Draw in both diagonals so that the polygon consists of 4 triangles. Color the triangles alternately black and white.

Step 3: Between two adjacent vertices (the ideal points on the boundary) put 2 equally spaced vertices. Construct another ideal 4-gon with the 4 vertices you thus created.

Step 4: Draw in both diagonals (as hyperbolic geodesics) to that new 4-gon so that tit also consists of 4 triangles. Color the triangles alternately black and white.

Step 5: Repeat Step 3 and 4 in the analogous positions for the other sides of the first 4-gon (from Step 1).

Big-circle.jpg


5. Create an ideal tessllation with the following steps:

Step 1: Draw a regular, ideal 4-gon.

Step 2: Connect opposite midpoints so that the polygon consists of 4 quadrilaterals (partly ideal). Color the quadrilaterals alternately black and white.

Step 3: Between two adjacent ideal-point vertices put 2 equally spaced vertices (also ideal points). Construct another 4-gon with the 4 vertices you thus created.

Step 4: Divide the quadrilateral into 4 smaller ones, as in Step 2 (using hyperbolic geodesics). Color the quadrilaterals alternately black and white.

Step 5: Now repeat Steps 3 and 4 in the analogous positions for the other sides of the first 4-gon (from Step 1).

Big-circle.jpg

6. Construct an ideal tessellation based on an ideal 6-gon. Use some consistent coloring scheme to “decorate” the tessellation.

Big-circle.jpg


Handin: All answers plus all drawings.