Trihyperboloid

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 R[z] denotes the real part of z. 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 ln256=8ln2. 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 (2024), who showed that

where I_1=1/6 and I_2=1/3 are the volumes of the two tetrahedra with common face (0,0,1), (0,1,0), and (1,0,0) and apices (0,0,0) and (1,1,1) and

Plugging in the values for I_1, I_2, and I_3 then gives the expected result

(OEIS A257872).

See also

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References

Referenced on Wolfram|Alpha

Cite this as:

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

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