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

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

(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

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

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

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

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

(OEIS A257872).

See also

Hyperboloid, One-Sheeted Hyperboloid, Steinmetz Solid, Stella Octangula

Explore with Wolfram|Alpha

References

Knill, O. "Archimedes Revenge Solution." https://people.math.harvard.edu/~knill/teaching/summer2017/exhibits/revenge/.Sloane, N. J. A. Sequences A257872 and A347903 in "The On-Line Encyclopedia of Integer Sequences."Villarino, M. B. and Várilly, J. C. "Archimedes' Revenge." 6 Aug 2021. https://arxiv.org/abs/2108.05195. To appear in College Math. J.

Referenced on Wolfram|Alpha

Trihyperboloid

Cite this as:

Weisstein, Eric W. "Trihyperboloid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Trihyperboloid.html

Subject classifications