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Ever wonder what the extension of Reuleaux's triangle might be in 3D? If we interpret the 2D version as the intersection of 3 circles, then it would be tempting to investigate the intersection of 4 spheres. The resulting intersection is the shape, which resembles a somewhat bloated tetrahedron. Unfortunately, the above shape does not quite have a constant breadth. In the figure below, take any point on edge 1 (except the two endpoints, i.e. vertices)), its distance to any point on curve 2 (again except the two endpoints) is slightly greater than the radii of the spherical faces. A 3D shape of constant A constant breadth 3D shape can be constructed by rotating a Reuleaux's triangle about one of its axes (line from any vertex through the midpoint of the opposing side). The resulting shape, as shown below, resembles an acorn. As a bonus, we also throw in the rotated Reuleaux's pentagon, also a 3D shape of constant breadth. |
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