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Pythagorean Theoremtheorem for Rightright-Corner Hyperbolic Simplicescorner hyperbolic simplices?

My answer to the "Favorite Equations"equations" question was the Pythagorean Theoremtheorem for Rightright-Corner Tetrahedracorner tetrahedra:

where A$A$, B$B$, C$C$ are the areas of the "leg-faces" and D$D$ is the area of the "hypotenuse-face".

Edit to add a couple of references
"The Lawslaws of Cosinescosines for Nonnon-Euclidean Tetrahedra"tetrahedra" (PDFpdf file) (by me) derives a Law of Cosines for which each non-Euclidean Pythagorean Theorem is a special case. I do not know if these results exist elsewhere in the literature. (BTW: There's a lot of unnecessary equation manipulation shown; I was using this as an opportunity to learn LaTeX. :)

A passage in above jumps into a discussion of "pseudofaces" by making passing reference to the Euclidean case. You can read about Euclidean pseudofaces and how I find them useful here:
  
"Heron-like Resultsresults for [Euclidean] Tetrahedral Volume"tetrahedral volume" (by me).

Here, $L^\star$ is half of Catalan's Constantconstant, and $H^\star$ is also known in the literature. (It is, for instance, the maximum of the Clausen function $\mathrm{Cl}_2$.)

If there's going to be a Pythagorean Theoremtheorem for Hyperbolic Simpliceshyperbolic simplices, then it must apply to this case, ideally relating these values in the non-Euclidean Pythagorean tradition:

Pythagorean Theorem for Right-Corner Hyperbolic Simplices?

My answer to the "Favorite Equations" question was the Pythagorean Theorem for Right-Corner Tetrahedra:

where A, B, C are the areas of the "leg-faces" and D is the area of the "hypotenuse-face".

Edit to add a couple of references
"The Laws of Cosines for Non-Euclidean Tetrahedra" (PDF) (by me) derives a Law of Cosines for which each non-Euclidean Pythagorean Theorem is a special case. I do not know if these results exist elsewhere in the literature. (BTW: There's a lot of unnecessary equation manipulation shown; I was using this as an opportunity to learn LaTeX. :)

A passage in above jumps into a discussion of "pseudofaces" by making passing reference to the Euclidean case. You can read about Euclidean pseudofaces and how I find them useful here:
  "Heron-like Results for [Euclidean] Tetrahedral Volume" (by me).

Here, $L^\star$ is half of Catalan's Constant, and $H^\star$ is also known in the literature. (It is, for instance, the maximum of the Clausen function $\mathrm{Cl}_2$.)

If there's going to be a Pythagorean Theorem for Hyperbolic Simplices, then it must apply to this case, ideally relating these values in the non-Euclidean Pythagorean tradition:

Pythagorean theorem for right-corner hyperbolic simplices?

My answer to the "Favorite equations" question was the Pythagorean theorem for right-corner tetrahedra:

where $A$, $B$, $C$ are the areas of the "leg-faces" and $D$ is the area of the "hypotenuse-face".

Edit to add a couple of references
"The laws of cosines for non-Euclidean tetrahedra" (pdf file) (by me) derives a Law of Cosines for which each non-Euclidean Pythagorean Theorem is a special case. I do not know if these results exist elsewhere in the literature. (BTW: There's a lot of unnecessary equation manipulation shown; I was using this as an opportunity to learn LaTeX. :)

A passage in above jumps into a discussion of "pseudofaces" by making passing reference to the Euclidean case. You can read about Euclidean pseudofaces and how I find them useful here: 
"Heron-like results for [Euclidean] tetrahedral volume" (by me).

Here, $L^\star$ is half of Catalan's constant, and $H^\star$ is also known in the literature. (It is, for instance, the maximum of the Clausen function $\mathrm{Cl}_2$.)

If there's going to be a Pythagorean theorem for hyperbolic simplices, then it must apply to this case, ideally relating these values in the non-Euclidean Pythagorean tradition:

replaced http://mathoverflow.net/ with https://mathoverflow.net/
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My answer to the "Favorite Equations" questionMy answer to the "Favorite Equations" question was the Pythagorean Theorem for Right-Corner Tetrahedra:

... where $Li_2$ is the dilogarithmdilogarithm.

where the right-hand side is symmetric in the four parameters $L_i$ (representing the volumes of the four legs of the simplex), which are all set equal to $L^\star$. (When the formula is populated with infinitesimal quantities, it should collapse to the Euclidean sum-of-squares relation.) However, while the dilogarithm has many interesting properties, the connection between $H^\star$ and $L^\star$ is not obvious (to me). In a separate MO posta separate MO post, I note a hypergeometric series "similar" to a series for $H^\star$ (what I call "$T(1/2)$" there) that has a direct relation to Catalan's constant (and therefore $L^\star$), but this hasn't yet provided appropriate insights into relating $H^\star$ to $L^\star$ directly.

My answer to the "Favorite Equations" question was the Pythagorean Theorem for Right-Corner Tetrahedra:

... where $Li_2$ is the dilogarithm.

where the right-hand side is symmetric in the four parameters $L_i$ (representing the volumes of the four legs of the simplex), which are all set equal to $L^\star$. (When the formula is populated with infinitesimal quantities, it should collapse to the Euclidean sum-of-squares relation.) However, while the dilogarithm has many interesting properties, the connection between $H^\star$ and $L^\star$ is not obvious (to me). In a separate MO post, I note a hypergeometric series "similar" to a series for $H^\star$ (what I call "$T(1/2)$" there) that has a direct relation to Catalan's constant (and therefore $L^\star$), but this hasn't yet provided appropriate insights into relating $H^\star$ to $L^\star$ directly.

My answer to the "Favorite Equations" question was the Pythagorean Theorem for Right-Corner Tetrahedra:

... where $Li_2$ is the dilogarithm.

where the right-hand side is symmetric in the four parameters $L_i$ (representing the volumes of the four legs of the simplex), which are all set equal to $L^\star$. (When the formula is populated with infinitesimal quantities, it should collapse to the Euclidean sum-of-squares relation.) However, while the dilogarithm has many interesting properties, the connection between $H^\star$ and $L^\star$ is not obvious (to me). In a separate MO post, I note a hypergeometric series "similar" to a series for $H^\star$ (what I call "$T(1/2)$" there) that has a direct relation to Catalan's constant (and therefore $L^\star$), but this hasn't yet provided appropriate insights into relating $H^\star$ to $L^\star$ directly.

$$L=3 \int_{\rm{acos}\sqrt{x}}^{\rm{acos}\sqrt{\frac{1}{3}}} \rm{atanh}\sqrt{\frac{3\cos^2t-1}{1-\cos^2t}} \\, \mathrm{d}t$$$$L=3 \int_{\rm{acos}\sqrt{x}}^{\rm{acos}\sqrt{\frac{1}{3}}} \rm{atanh}\sqrt{\frac{3\cos^2t-1}{1-\cos^2t}} \, \mathrm{d}t$$

$$H=6\int_{\rm{acos}x}^{\rm{acos}\frac{1}{3}} \rm{atanh}\sqrt{\frac{3\cos t-1}{1-\cos t}} \\, \mathrm{d}t$$$$H=6\int_{\rm{acos}x}^{\rm{acos}\frac{1}{3}} \rm{atanh}\sqrt{\frac{3\cos t-1}{1-\cos t}} \, \mathrm{d}t$$

$$H = \frac{4}{2!\\;3!} M^3 + \frac{18}{3!\\;5!}M^5 - \frac{ 918 }{4!\\;7!}M^7 + \frac{24786}{5!\\;9!}M^9 - \frac{ 6018759 }{8 \cdot 6!\\;11!} M^{11} - \frac{ 8233607961 }{80\cdot 7!\\;13!} M^{13} - \cdots $$$$H = \frac{4}{2!\;3!} M^3 + \frac{18}{3!\;5!}M^5 - \frac{ 918 }{4!\;7!}M^7 + \frac{24786}{5!\;9!}M^9 - \frac{ 6018759 }{8 \cdot 6!\;11!} M^{11} - \frac{ 8233607961 }{80\cdot 7!\;13!} M^{13} - \cdots $$ where $M := (6L)^{1/3}$. This isn't the relation I'm seeking, but observe that, for infinitesimal $L$, we have $H \approx 2 L$; that is, $H^2 \approx 4L^2 = L^2 + L^2 + L^2 + L^2$, which is the corresponding Pythagorean relation for Euclidean $4$-simplices. Even so, despite my best efforts of playing with these series, I have yet to get any insights into the nature of a non-infinitesimal connection.

$$L=3 \int_{\rm{acos}\sqrt{x}}^{\rm{acos}\sqrt{\frac{1}{3}}} \rm{atanh}\sqrt{\frac{3\cos^2t-1}{1-\cos^2t}} \\, \mathrm{d}t$$

$$H=6\int_{\rm{acos}x}^{\rm{acos}\frac{1}{3}} \rm{atanh}\sqrt{\frac{3\cos t-1}{1-\cos t}} \\, \mathrm{d}t$$

$$H = \frac{4}{2!\\;3!} M^3 + \frac{18}{3!\\;5!}M^5 - \frac{ 918 }{4!\\;7!}M^7 + \frac{24786}{5!\\;9!}M^9 - \frac{ 6018759 }{8 \cdot 6!\\;11!} M^{11} - \frac{ 8233607961 }{80\cdot 7!\\;13!} M^{13} - \cdots $$ where $M := (6L)^{1/3}$. This isn't the relation I'm seeking, but observe that, for infinitesimal $L$, we have $H \approx 2 L$; that is, $H^2 \approx 4L^2 = L^2 + L^2 + L^2 + L^2$, which is the corresponding Pythagorean relation for Euclidean $4$-simplices. Even so, despite my best efforts of playing with these series, I have yet to get any insights into the nature of a non-infinitesimal connection.

$$L=3 \int_{\rm{acos}\sqrt{x}}^{\rm{acos}\sqrt{\frac{1}{3}}} \rm{atanh}\sqrt{\frac{3\cos^2t-1}{1-\cos^2t}} \, \mathrm{d}t$$

$$H=6\int_{\rm{acos}x}^{\rm{acos}\frac{1}{3}} \rm{atanh}\sqrt{\frac{3\cos t-1}{1-\cos t}} \, \mathrm{d}t$$

$$H = \frac{4}{2!\;3!} M^3 + \frac{18}{3!\;5!}M^5 - \frac{ 918 }{4!\;7!}M^7 + \frac{24786}{5!\;9!}M^9 - \frac{ 6018759 }{8 \cdot 6!\;11!} M^{11} - \frac{ 8233607961 }{80\cdot 7!\;13!} M^{13} - \cdots $$ where $M := (6L)^{1/3}$. This isn't the relation I'm seeking, but observe that, for infinitesimal $L$, we have $H \approx 2 L$; that is, $H^2 \approx 4L^2 = L^2 + L^2 + L^2 + L^2$, which is the corresponding Pythagorean relation for Euclidean $4$-simplices. Even so, despite my best efforts of playing with these series, I have yet to get any insights into the nature of a non-infinitesimal connection.

Added range of parameter $x$ in volume formulas
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Added strikingly-similar leg/hypotenuse volume formulas; made minor edits to discussion.
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added discussion of special case
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