Surface of constant width

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In geometry, a surface of constant width is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes. One defines the width of the surface in a given direction to be the perpendicular distance between the parallels perpendicular to that direction.

More generally, any compact convex body D has one pair of parallel supporting planes in a given direction. A supporting plane is a plane that intersects the boundary of D but not the interior of D. One defines the width of the body as before. If the width of D is the same in all directions, then one says that the body is of constant width and calls its boundary a surface of constant width, and the body itself is referred to as a spheroform.

A sphere is obviously a surface of constant width. A nontrivial example is Meissner's tetrahedron. Contrary to common belief the Reuleaux tetrahedron is not a surface of constant width.

[edit] References

• Guilfoyle, Brendan & Klingenberg, Wilhelm (2007), On C²-smooth surfaces of constant width, arXiv:0704.3248 

[edit] External links

• Spheroforms
• T. Lachand-Robert & É. Oudet, "Bodies of constant width in arbitrary dimension"
• How Round is Your Circle? Solids of constant width