Ideal Hyperbolic Tessellations Exploration

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Time-40.svg

Objective: Create Ideal Hyperbolic Tessellations

Geodesics in hyperbolic space.

Geodesics play the role of straight lines in hyperbolic geometry. Geodesics which pass through the center of the disc actually look straight, while all others appear curved. A geodesic in hyperbolic space is either:

  • an arc of a circle which is perpendicular to the boundary of the disc

or

  • a line passing through the center of the disc.

Ideal Polygons

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. Points on the boundary are not actually in hyperbolic space, and are called ideal points. Because these points are infintely far away, the geodesics do not actually meet but get closer and closer together as they head towards the edges of space. A polygon with all its vertices on the boundary is called an ideal polygon.

Triangle Ideal Triangle
Hyp-triangle.svg Hyp-ideal-tri.svg
  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.
    Circle.svg Circle.svg
  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.
    Circle.svg Circle.svg
  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.
    Circle.svg Circle.svg

Creating an ideal tessellation

  1. 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).

    Circle.svg

  2. 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).

    Circle.svg

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

Handin: All answers plus all drawings.