Trihyperboloid

Consider the solid enclosed by the three hyperboloids specified by the inequalities

			(1)
			(2)
			(3)

This work dubs this solid the "trihyperboloid."

The basic shape of the trihyperbolid is that of a stella octangula with a "web" hung across adjacent faces.

The surface area of the trihyperboloid is given by

			(4)
			(5)
			(6)
			(7)
			(8)

(OEIS A347903), where denotes the real part of . The surface area can be given as a complicated (but likely simplifyable) closed-form expression based on evaluation of the integral

(9)

in terms of natural logarithms, dilogarithms, and trigamma functions (E. Weisstein Sep. 15-20, 2021).

Knill (2017) proposed as a challenge to Harvard summer school students that they prove that the volume was equal to . The problem was solved by student Runze Li, who gave the solution in terms of the mysterious integral

(10)

A more straightforward analysis was given by Villarino and Várilly (2021), who showed that

(11)

where and are the volumes of the two tetrahedra with common face , , and and apices and and

			(12)
			(13)

Plugging in the values for , , and then gives the expected result

(14)

(OEIS A257872).