The Sphericon

This page describes what happens when sphericons roll around each other. It deals with the original sphericon, i.e. that which is described in the introduction. How other shapes in the series can roll together has yet to be investigated. If you would like to play with this yourself, a template for a sphericon can be found here.

If two sphericons are placed squarely together so that they are touching at the same point, they can roll around each other (Figure 1), covering all of the area of both sphericons.

Two sphericons roll around each other
Figure 1: Two sphericons roll around each other*

In fact, eight sphericons can simultaneously roll around a central sphericon, as shown in Figure 2. This displays some fractal properties; if we call 8 sphericons rolling around a central one a 'group', then 8 groups will roll around a central group, with 64 sphericons constantly rolling.

Eight sphericons roll around a central one   Eight sphericons roll around a central one
Figure 2: Eight sphericons roll around a central one

Four sphericons can be placed together as in Figure 3(a), they can all roll around each other and return to where they started. Interestingly, at Figure 3(c), there is an arbitrary decision to make. They can either roll horizontally, as in the figure, or vertically. Up until this point, four lines of symmetry have been maintained. At this point, two are lost. Click here to see them move (this is a large file, so you may need to wait for it to load).

Four sphericons roll around each other
Figure 3: Four sphericons roll around each other. Click here for them to move.

If eight groups of four sphericons are placed together with their centres on a plane, all sphericons can roll around each other until their centres form a truncated octahedron (see Figure 4).

24 sphericons in a sheet roll into a truncated octahedron
Figure 4: Twenty-four sphericons in a sheet roll into a truncated octahedron.

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* Image created by Tom Longtin. Used with permission.