Ideal Hyperbolic Tessellations Exploration

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Time-40.svg

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 I have drawn 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. 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 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 put 2 equally spaced vertices. Construct another 4-gon with the 4 vertices you thus created.

Step 4: Draw in both diagonals so that the polygon consists of 4 triangles. Color the triangles alternately black and white. Now repeat Step 3 and 4

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 quadrilateral. Color the quadrilaterals alternately black and white.

Step 3: Between two adjacent vertices put 2 equally spaced vertices. Construct another 4-gon with the 4 vertices you thus created.

Step 4: Divide the quadrilateral into 4 smaller ones, as in step 2. Color the quadrilaterals alternately black and white. Now repeat Step 3 and 4

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:

Plus all drawings.