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Oloid - mklabgr - 08-15-2026
Oloid
Source: Wikipedia
Topic: Geometry / Developable surfaces
Discovered by: Paul Schatz
Year: 1929
The oloid is an unusual three-dimensional geometric solid discovered by Paul Schatz in 1929. It can be constructed from two congruent circles of radius $r$ lying in perpendicular planes, arranged so that the center of each circle lies on the circumference of the other; the distance between their centers is therefore exactly $r$. The oloid is the convex hull of these two circles. Interestingly, one third of each circle lies inside the resulting solid, so its boundary can equivalently be generated from two circular arcs, each subtending $4\pi/3$.
Despite its complicated appearance, the oloid has a remarkably simple surface area: $A=4\pi r^2$, exactly the same as that of a sphere of radius $r$. Its volume is considerably more complicated, involving complete elliptic integrals: $V=\frac{2}{3}\left[2E\left(\frac34\right)+K\left(\frac34\right)\right]r^3\approx3.0524184684r^3$. (Wikipedia)
Perhaps the oloid's most fascinating feature is the way it rolls. Its boundary is a developable surface, meaning it can be flattened onto a plane without stretching, and during one complete rolling motion every point of its surface eventually touches the ground. Unlike a sphere or cylinder, however, its center of mass does not travel along a straight line but follows a gently meandering path, although the vertical variation is quite small—only about $0.0429r$. At every instant the oloid contacts the ground along a line segment of constant length $\sqrt3,r$. This combination of simple construction, unusual motion, and elegant mathematical properties makes the oloid an especially striking example of how elementary geometric ingredients—just two circles—can produce surprisingly rich three-dimensional behavior.
Key takeaways
- The oloid is generated from two equal, perpendicular circles whose centers lie on each other's circumferences.
- Its surface area is $4\pi r^2$, identical to that of a sphere of radius $r$.
- When it rolls, its entire surface comes into contact with the plane.
- Its rolling motion is neither the simple rotation of a sphere nor that of a cylinder; its center follows a characteristic meandering trajectory.
ARTICLE / ARCHIVE [PDF]