# 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[edit | edit source]

There are several known classes of developable rollers.

### Prime polysphericons[edit | edit source]

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 . 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.

### Polycons[edit | edit source]

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.

### Platonicons[edit | edit source]

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

## External links[edit | edit source]

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

## Bibliography[edit | edit source]

- 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*.