Non Round Wheels – Can it be?

The main mathematical news

2008-9: Computational solutions for drilling perfect squares holes and regular hexagonal holes.

2009: A prototype for bicycle using non-round wheels was developed.

2012: Computational solutions for drilling any odd-sided regular polygonal holes.

2016: Great Britain redesigned the 1₤ coin in order to ensure its width is constant.

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Additional Theorems / conjectures / Open questions

* The circle is not the only curve of constant width – so is Reuleaux triangle.

* Any regular odd-sided polygon can be used to construct a curve of constant width.

* For curves of constant width d:

  • The perimeter is πd.
  • Reuleaux triangles’ area is the smallest, the circle’s area is the greatest.

* A prototype for the Wankel engine was developed. It replaced pistons with a rotor shaped like a Reuleaux triangle.

* A Reuleaux tetrahedron can be modified to form surfaces of constant width (Meissner tetrahedra).

* A general solution for drilling regular even-sided polygons is yet to be found.

* Open question: Meissner tetrahedra have the minimal volume of all 3D surfaces of constant width.

To the MNS presentation
The main mathematical concepts / Principles

Plane and solid geometry (MSC2010#97G40)

*   Curve of constant width

*   Reuleaux triangle

*   Tangent

*   (Regular) polygons

*   Circle / disk

*   Arc / sector / radius

*   Tetrahedron

*   Meissner tetrahedral

*   Sphere

*   Axis of rotation

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