Oloid

An oloid is the convex hull of two circles of equal radius in perpendicular planes, with the center of each circle lying on the other (Schatz 1975, p. 122; Bäsel 2018). Its boundary is an octic surface (Trott 2004, pp. 1194-1196). The oloid can roll smoothly on a plane.

The oloid is illustrated above. For comparison, a two-circle roller is formed from two equal disks in perpendicular planes whose centers are separated by times their radius. It also rolls smoothly (Nishihara).

An oloid can also be generated by the motion of a diagonal of Schatz's invertible cube. When one of the six tetrahedra in the articulated ring is fixed, the opposite diagonal sweeps out the oloid surface (Paul Schatz Stiftung).

For generating circles of radius , the surface area is

(1)

the same as that of a sphere of radius . Dirnböck and Stachel (1997, p. 117) computed the enclosed volume by numerical integration. It has the closed form

			(2)
			(3)

(OEIS A215447), where and are the complete elliptic integral of the first kind and elliptic integral of the second kind, respectively, with elliptic modulus (Bäsel 2018).

Bäsel (2018) gives the mean tangent diameter (mean width) as

			(4)
			(5)

where is defined by

			(6)
			(7)

Define the nome

			(8)
			(9)

Then

(10)

where is the elliptic gamma function, denotes the imaginary part, and is the principal branch of the natural logarithm. The equivalent rapidly converging series is

(11)

(E. Weisstein, Sep. 23, 2026).