Developable roller

A developable roller is a kind of convex curved 3D solid that can roll indefinitely on a flat plane without stopping, such that eventually every point on its surface will come in contact with the plane. The surface of a developable roller consists of a single curved "face" that always touches the plane at a line segment, which may vary in length.

Classes

There are several known classes of developable rollers.

Prime polysphericons

A view of a cul-de-sac on the (5,1)-polysphericon.

The (p,k)-polysphericon is constructed by taking the solid of rotation of a regular p-gon, bisecting it along an p-gon, rotating one of the halves by 2kπ/n, and recombining the two halves along their p-gonal faces. A (2p,k)-polysphericon is a developable roller iff $\gcd(p,k)=1$. A (p,l)-polysphericon where p is odd, is never a developable roller. The result can still have a single developable face, however it will not roll indefinitely because the solid of rotation has a circular face where the axis of rotation intersects an edge which results in culs-de-sac preventing the polysphericon from rolling.

The polysphericons which are developable rollers are called prime polysphericons. The prime polysphericons correspond 1-to-1 with the regular polygons (including the degenerate digon) with the n/k-gon corresponding to the (2n,k)-polysphericon.

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  Sphericon

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  Hexasphericon

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  Octasphericon

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  (10,1)-polysphericon

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  (10,2)-polysphericon

Polycons

Not to be confused with Polygon.

The polycons are an infinite class of developable rollers that generalize the construction of the sphericon. The n-polycon can be constructed as follows:

• Start with a cone with apex angle equal to the internal angle of a regular 2n-gon.
• Arrange n copies of the cone such that their apices lie on the vertices of a regular n-gon and their axes pass through the origin.
• Take the intersection of the cones.
• Divide the result in half along the plane which intersects the apices of the cones.
• Rotate one of the halves by π/n.
• Recombine the two halves to form the polycon.

This construction always results in a developable roller of n > 1. The smallest polycon is the sphericon with n=2. The edges of polycons are conic sections, with the sphericon's edges being circular, the hexacon's edges being parabolic and all other polycons having hyperbolic edges.

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  Sphericon

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  Hexacon

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  Octacon

Platonicons

The platonicons are a class of developable rollers based on the platonic solids.

External links

• Wikipedia Contributors. "Developable roller".
• Steve Mathias. "Sphericon series".
• "Polysphericons". h-its.org. Heidelberg Institute for Theoretical Studies. September 2019.

Bibliography

• Hirsch, David (2020). "The Polycons: The Sphericon (or Tetracon) has Found its Family". Journal of Mathematics and the Arts. 14 (4): 345–359. arXiv:1901.10677. doi:10.1080/17513472.2020.1711651. S2CID 119152692.
• Seaton, Katherine; Hirsch, David (2020). "Platonicons: The Platonic Solids Start Rolling" (PDF). Bridges 2020 Conference Proceedings.