Consider the solid enclosed by the three hyperboloids specified by the inequalities
(1)
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(2)
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(3)
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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)
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(5)
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(6)
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(7)
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(8)
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(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)
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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)
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A more straightforward analysis was given by Villarino and Várilly (2021), who showed that
(11)
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where and are the volumes of the two tetrahedra with common face , , and and apices and and
(12)
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(13)
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Plugging in the values for , , and then gives the expected result
(14)
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(OEIS A257872).