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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 breadth (as opposed to simply width in 2D) can be defined as one for which, when tangential planes on opposing faces or edges are drawn, are always apart by the same constant distance.
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.