The Pythagorean theorem in hyperbolic 2-space is fundamental, and I discovered the hedronometric 3-space analogue years ago (see here), but what about dimension 4 and beyond?

As discussed in the Heron-like Strategies for Non-Euclidean Tetrahedral Volume post, my investigations into the 4-space case stalls-out because hyperbolic volume is defined in terms of the thorny Derevnin-Mednykh integral.

In the simplest (but hardly simple) case of an “isosceles” right-corner simplex with right-corner leg-faces and a regular hypotenuse-face, one arrives at these formulas for leg volume, \(L\), and hypotenuse volume, \(H\):

$$\begin{align}L &:= \frac{3}{2}\int_{1/3}^{x} \frac{1}{\sqrt{u\left(1-u\right)}} \; \mathrm{atanh}\sqrt{\frac{3u-1}{1-u}} du \\[0.5em] H &:= 6 \int_{1/3}^{x} \frac{1}{\sqrt{1-u^2}} \; \mathrm{atanh}\sqrt{\frac{3u-1}{1-u}} du \end{align}$$

for \( 1/3 \le x \le 1/2 \). Surely, a Pythagorean relation of some kind exists (right?), but I’m at a loss for what it might be. Given the specific extreme values at \(x=1/2\)

$$L^\star := \frac{1}{2} \sum_{k=1}^{\infty} \frac{1}{k^2} \sin \frac{\pi k}{2} = 0.45798\dots \qquad H^\star := \sum_{k=1}^{\infty} \frac{1}{k^2} \sin \frac{\pi k}{3} = 1.01494 \dots$$

the relationship is all-but-guaranteed to be non-trivial. (Note: \(L^{\star}\) is half of Catalan’s constant; and \(H^\star\) is, among other things, the maximum value of the Clausen function, \( \mathrm{Cl}_2 \). Maybe mentioning that here will make this blog post show up in web searches, and people who know a little about polylogarithms can give me some insights.)

Here’s a TeX’d-up version of this blind alley: “What is the Pythagorean Theorem for Right-Corner Simplices in Hyperbolic 4-Space?”