One-Sided Toroidal Polyhedra

In 1968, Professor Gonzalo Vélez–Jahn (Universidad Central de Venezuela) discovered three-dimensional bodies with Möbius strip–like characteristics. Specifically, a trajectory along the cube faces behaves as if it were on a one-sided Möbius strip, in the sense that the end of the path will touch all of the polyhedron’s faces and eventually meet its beginning.

Martin Gardner later described these sorts of figures as prismatic rings and toroidal polyhedrons.

As a non-example, here’s a torus made of cubes.

cube torus

The figure has four faces (with two, plus a bit of a third, highlighted with arrows).

cube torus with arrow

However, introducing various twists results in a figure with a single face, in the Möbius strip sense. Here’s a single-sided figure made of 10 cubes:

single-sided figure made of 10 cubes

Or the same thing with a color gradient showing the path instead of an arrow:

same figure as previous, with colors same as previous, different viewpoint

When you slice through the cubes, you still have a connected figure:

first slice

The sliced figure is also one-sided, as it turns out:

first slice with arrow

To run it into the ground, here’s a second slice:

second slice

And a third:

third slice

And a fourth:

fourth slice

Funny enough, a slice through the figure parallel to one of the axes looks like Cantor dust.

slice looking like Cantor dust

While we’re at it, a 22-cube single-sided figure:

single-sided figure made of 22 cubes

Here it is, sliced:

22-cube figure, sliced

By introducing more twists, we can create a two-sided figure:

two-sided figure made of 18 cubes

Sliced. I’d hoped this would result in a disconnected figure, but nope:

18-cube figure, sliced

Without whom not: Professor Gonzalo Vélez–Jahn, Luis Vélez, and Roger Bagula.

See Martin Gardner, Fractal Music, Hypercards and More…, pp. 81–86, 1992.

See also Moebius 3D Wiki.

Figures created with Mathematica 10.

Robert Dickau, 2016.

[ home ] || [ 2016-04-24 ]

robertdickau.com/toroidalpolyhedra.html