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Trihyperboloid


Trihyperboloid

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

x^2+y^2-z^2<=1
(1)
y^2+z^2-x^2<=1
(2)
z^2+x^2-y^2<=1.
(3)

This work dubs this solid the "trihyperboloid."

TrihyperboloidStellaOctangula

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

S=24int_0^1int_(sqrt(1-y^2))^1sqrt(2+1/(x^2+y^2-1))dxdy
(4)
=48int_0^(pi/4)int_1^(sectheta)rsqrt(2+1/(r^2-1))drdtheta
(5)
=24sqrt(3)-2pi-24+12sqrt(2)int_0^(pi/4)R[tanh^(-1)(sqrt((csc^2theta+1)/2))]dtheta
(6)
=24sqrt(3)-2pi-24+12sqrt(2)int_(sqrt(3/2))^inftyR[(tanh^(-1)x)/((2x^2-1)sqrt(1/2(1-x^(-2))))]dx
(7)
=18.76939626...
(8)

(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

 int((u-1)lnu)/((1-2(u-1)^2)sqrt((u-1)^2-1))du
(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 ln256=8ln2. The problem was solved by student Runze Li, who gave the solution in terms of the mysterious integral

 I=1/2int_0^1[(z^2+1)(1/2pi-2tan^(-1)z)+z^2-1]dz,
(10)

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

 V=8(3I+I_1+I_2),
(11)

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

I=int_0^1int_y^1int_(1-x+y)^(sqrt(1+y^2-x^2))dzdxdy
(12)
=(ln2)/3-1/6.
(13)

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

 V=ln256=8ln2=5.54517744...
(14)

(OEIS A257872).


See also

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

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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.

Cite this as:

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

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