Spirallohedra Rule!

The k-armed rhombic spirallohedra are approximations to ruled surfaces, and certain plane sections give regular k-gons, where k=3, 4, 5, ... . When k=3 or k=4, spirallohedra close-pack to fill space. We create arrays of such spirallohedra, and take the proper sections needed to give regular k-gons. Thus we obtain an array of k-gons in the sectioning plane. These are exactly those "uniform tilings which are not edge-to-edge" described in Grunbaum & Shepherd's "Tilings and Patterns," on page 74, Figure 2.4.2.

Category:

Film & Animation

Tags:

• geometry
• tilings
• uniform
• spirallohedra
• plane
• section
• ruled
• surface

[LokiClock]

Since you posted this, have you looked into the 4D analogues you mentioned and found any specific examples?

[rufus16180339887]

No, I have not. I feel sure they exist. There are various ways the 2D and 3D forms can be constructed. I use a looping algortihm to make the 3D spirallohedra, but I use the Generalized Dual Method (GDM) to make the 2D "bitten zonogons." If I could use the GDM to make spirallohedra, I would then be able to extend to 4D, 5D, etc., quite freely. I believe a 3D spirallohedron is very similar to a 3D section of its 4D analogue.

[OpiatedBliss]

Spirallohedra Rules!

[doseryder]

The nomenclature system for these polyhedrons/polytopes always seemed so complicated

[mnykamp]

Gheh lovely, especially with the completing jingle at the end. What did you use to animate the figures with? The lighting effects really look great!