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Consider a Euclidean tetrahedron with lengths of edges
$$ l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34} $$ and dihedral angles $$ \alpha_{12}, \alpha_{13}, \alpha_{14}, \alpha_{23}, \alpha_{24}, \alpha_{34}. $$ Consider solid angles \begin{split} &\Omega_1=\alpha_{12}+\alpha_{13}+\alpha_{14}-\pi \\ &\Omega_2=\alpha_{12}+\alpha_{23}+\alpha_{24}-\pi \\ &\Omega_3=\alpha_{13}+\alpha_{23}+\alpha_{34}-\pi \\ &\Omega_4=\alpha_{14}+\alpha_{24}+\alpha_{34}-\pi \\ \end{split} and perimeters of faces \begin{split} &P_1=l_{23}+l_{34}+l_{24} \\ &P_2=l_{13}+l_{14}+l_{34} \\ &P_3=l_{12}+l_{14}+l_{24} \\ &P_4=l_{12}+l_{23}+l_{13}. \\ \end{split} Then the following cross-ratios are equal to each other: $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[P_1, P_2, P_3, P_4]. $$ Question: Is it known? I have found a proof of this statement (to appear soon), but it involves quite tricky algebraic geometry. It will be very interesting to me to see a more elementary approach.

Addition: Similar statements hold in spherical and hyperbolic geometry. For a spherical tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}]. $$ For a hyperbolic tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{P_1}, e^{P_2}, e^{P_3}, e^{P_4}]. $$

Addition 2: One can prove a more general statement, which I formulate in the spherical case (but it is true in other geometries after appropriate modifications). There exists a $PSL_2(C)-$ transformation, sending eight numbers

$$ 1, e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}, e^{i(\alpha_{12}+\alpha_{23}+\alpha_{34}+\alpha_{14})}, e^{i(\alpha_{12}+\alpha_{24}+\alpha_{34}+\alpha_{13})}, e^{i(\alpha_{13}+\alpha_{23}+\alpha_{24}+\alpha_{14})} $$ to $$ 1, e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}, e^{i(l_{12}+l_{23}+l_{34}+l_{14})}, e^{i(l_{12}+l_{24}+l_{34}+l_{13})}, e^{i(l_{13}+l_{23}+l_{24}+l_{14})}. $$ I know neither an elementary proof of this statement nor any interpretation of the coefficients of the $PSL_2(C)-$ transformation.

Consider a Euclidean tetrahedron with lengths of edges
$$ l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34} $$ and dihedral angles $$ \alpha_{12}, \alpha_{13}, \alpha_{14}, \alpha_{23}, \alpha_{24}, \alpha_{34}. $$ Consider solid angles \begin{split} &\Omega_1=\alpha_{12}+\alpha_{13}+\alpha_{14}-\pi \\ &\Omega_2=\alpha_{12}+\alpha_{23}+\alpha_{24}-\pi \\ &\Omega_3=\alpha_{13}+\alpha_{23}+\alpha_{34}-\pi \\ &\Omega_4=\alpha_{14}+\alpha_{24}+\alpha_{34}-\pi \\ \end{split} and perimeters of faces \begin{split} &P_1=l_{23}+l_{34}+l_{24} \\ &P_2=l_{13}+l_{14}+l_{34} \\ &P_3=l_{12}+l_{14}+l_{24} \\ &P_4=l_{12}+l_{23}+l_{13}. \\ \end{split} Then the following cross-ratios are equal to each other: $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[P_1, P_2, P_3, P_4]. $$ Question: Is it known? I have found a proof of this statement (see here), but it involves quite tricky algebraic geometry. It will be very interesting to me to see a more elementary approach.

Addition: Similar statements hold in spherical and hyperbolic geometry. For a spherical tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}]. $$ For a hyperbolic tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{P_1}, e^{P_2}, e^{P_3}, e^{P_4}]. $$

Addition 2: One can prove a more general statement, which I formulate in the spherical case (but it is true in other geometries after appropriate modifications). There exists a $PSL_2(\mathbb{C})-$ transformation, sending eight numbers

$$ 1, e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}, e^{i(\alpha_{12}+\alpha_{23}+\alpha_{34}+\alpha_{14})}, e^{i(\alpha_{12}+\alpha_{24}+\alpha_{34}+\alpha_{13})}, e^{i(\alpha_{13}+\alpha_{23}+\alpha_{24}+\alpha_{14})} $$ to $$ 1, e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}, e^{i(l_{12}+l_{23}+l_{34}+l_{14})}, e^{i(l_{12}+l_{24}+l_{34}+l_{13})}, e^{i(l_{13}+l_{23}+l_{24}+l_{14})}. $$ I know neither an elementary proof of this statement nor any interpretation of the coefficients of the $PSL_2(\mathbb{C})-$ transformation.

Consider a Euclidean tetrahedron with lengths of edges
$$ l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34} $$ and dihedral angles $$ \alpha_{12}, \alpha_{13}, \alpha_{14}, \alpha_{23}, \alpha_{24}, \alpha_{34}. $$ Consider solid angles \begin{split} &\Omega_1=\alpha_{12}+\alpha_{13}+\alpha_{14}-\pi \\ &\Omega_2=\alpha_{12}+\alpha_{23}+\alpha_{24}-\pi \\ &\Omega_3=\alpha_{13}+\alpha_{23}+\alpha_{34}-\pi \\ &\Omega_4=\alpha_{14}+\alpha_{24}+\alpha_{34}-\pi \\ \end{split} and perimeters of faces \begin{split} &P_1=l_{23}+l_{34}+l_{24} \\ &P_2=l_{14}+l_{24}+l_{12} \\ &P_3=l_{13}+l_{34}+l_{14} \\ &P_4=l_{12}+l_{23}+l_{13}. \\ \end{split} Then the following cross-ratios are equal to each other: $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[P_1, P_2, P_3, P_4]. $$ Question: Is it known? I have found a proof of this statement (to appear soon), but it involves quite tricky algebraic geometry. It will be very interesting to me to see a more elementary approach.

Addition: Similar statements hold in spherical and hyperbolic geometry. For a spherical tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}]. $$ For a hyperbolic tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{P_1}, e^{P_2}, e^{P_3}, e^{P_4}]. $$

Addition 2: One can prove a more general statement, which I formulate in the spherical case (but it is true in other geometries after appropriate modifications). There exists a $PSL_2(C)-$ transformation, sending eight numbers

$$ 1, e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}, e^{i(\alpha_{12}+\alpha_{23}+\alpha_{34}+\alpha_{14})}, e^{i(\alpha_{12}+\alpha_{24}+\alpha_{34}+\alpha_{13})}, e^{i(\alpha_{13}+\alpha_{23}+\alpha_{24}+\alpha_{14})} $$ to $$ 1, e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}, e^{i(l_{12}+l_{23}+l_{34}+l_{14})}, e^{i(l_{12}+l_{24}+l_{34}+l_{13})}, e^{i(l_{13}+l_{23}+l_{24}+l_{14})}. $$ I know neither an elementary proof of this statement nor any interpretation of the coefficients of the $PSL_2(C)-$ transformation.

Consider a Euclidean tetrahedron with lengths of edges
$$ l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34} $$ and dihedral angles $$ \alpha_{12}, \alpha_{13}, \alpha_{14}, \alpha_{23}, \alpha_{24}, \alpha_{34}. $$ Consider solid angles \begin{split} &\Omega_1=\alpha_{12}+\alpha_{13}+\alpha_{14}-\pi \\ &\Omega_2=\alpha_{12}+\alpha_{23}+\alpha_{24}-\pi \\ &\Omega_3=\alpha_{13}+\alpha_{23}+\alpha_{34}-\pi \\ &\Omega_4=\alpha_{14}+\alpha_{24}+\alpha_{34}-\pi \\ \end{split} and perimeters of faces \begin{split} &P_1=l_{23}+l_{34}+l_{24} \\ &P_2=l_{13}+l_{14}+l_{34} \\ &P_3=l_{12}+l_{14}+l_{24} \\ &P_4=l_{12}+l_{23}+l_{13}. \\ \end{split} Then the following cross-ratios are equal to each other: $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[P_1, P_2, P_3, P_4]. $$ Question: Is it known? I have found a proof of this statement (to appear soon), but it involves quite tricky algebraic geometry. It will be very interesting to me to see a more elementary approach.

Addition: Similar statements hold in spherical and hyperbolic geometry. For a spherical tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}]. $$ For a hyperbolic tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{P_1}, e^{P_2}, e^{P_3}, e^{P_4}]. $$

Addition 2: One can prove a more general statement, which I formulate in the spherical case (but it is true in other geometries after appropriate modifications). There exists a $PSL_2(C)-$ transformation, sending eight numbers

$$ 1, e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}, e^{i(\alpha_{12}+\alpha_{23}+\alpha_{34}+\alpha_{14})}, e^{i(\alpha_{12}+\alpha_{24}+\alpha_{34}+\alpha_{13})}, e^{i(\alpha_{13}+\alpha_{23}+\alpha_{24}+\alpha_{14})} $$ to $$ 1, e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}, e^{i(l_{12}+l_{23}+l_{34}+l_{14})}, e^{i(l_{12}+l_{24}+l_{34}+l_{13})}, e^{i(l_{13}+l_{23}+l_{24}+l_{14})}. $$ I know neither an elementary proof of this statement nor any interpretation of the coefficients of the $PSL_2(C)-$ transformation.

Consider a Euclidean tetrahedron with lengths of edges
$$ l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34} $$ and dihedral angles $$ \alpha_{12}, \alpha_{13}, \alpha_{14}, \alpha_{23}, \alpha_{24}, \alpha_{34}. $$ Consider solid angles \begin{split} &\Omega_1=\alpha_{12}+\alpha_{13}+\alpha_{14}-\pi \\ &\Omega_2=\alpha_{12}+\alpha_{23}+\alpha_{24}-\pi \\ &\Omega_3=\alpha_{13}+\alpha_{23}+\alpha_{34}-\pi \\ &\Omega_4=\alpha_{14}+\alpha_{24}+\alpha_{34}-\pi \\ \end{split} and perimeters of faces \begin{split} &P_1=l_{23}+l_{34}+l_{24} \\ &P_2=l_{14}+l_{24}+l_{12} \\ &P_3=l_{13}+l_{34}+l_{14} \\ &P_4=l_{12}+l_{23}+l_{13}. \\ \end{split} Then the following cross-ratios are equal to each other: $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[P_1, P_2, P_3, P_4]. $$ Question: Is it known? I have found a proof of this statement (to appear soon), but it involves quite tricky algebraic geometry. It will be very interesting to me to see a more elementary approach.

Addition: Similar statements hold in spherical and hyperbolic geometry. For a spherical tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}]. $$ For a hyperbolic tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{P_1}, e^{P_2}, e^{P_3}, e^{P_4}]. $$

Addition 2: One can prove a more general statement, which I formulate in the spherical case (but it is true in other geometries after appropriate modifications). There exists a $PSL_2(C)-$ transformation, sending eight numbers

$$ 1, e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}, e^{i(\alpha_{12}+\alpha_{23}+\alpha_{34}+\alpha_{14})}, e^{i(\alpha_{12}+\alpha_{24}+\alpha_{34}+\alpha_{13})}, e^{i(\alpha_{13}+\alpha_{23}+\alpha_{24}+\alpha_{14})} $$ to $$ 1, e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}, e^{i(l_{12}+l_{23}+l_{34}+l_{14})}, e^{i(l_{12}+l_{24}+l_{34}+l_{13})}, e^{i(l_{13}+l_{23}+l_{24}+l_{14})}. $$ I don't know neither an elementary proof of this statement nor any interpretation of the coefficients of the $PSL_2(C)-$ transformation.

Consider a Euclidean tetrahedron with lengths of edges
$$ l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34} $$ and dihedral angles $$ \alpha_{12}, \alpha_{13}, \alpha_{14}, \alpha_{23}, \alpha_{24}, \alpha_{34}. $$ Consider solid angles \begin{split} &\Omega_1=\alpha_{12}+\alpha_{13}+\alpha_{14}-\pi \\ &\Omega_2=\alpha_{12}+\alpha_{23}+\alpha_{24}-\pi \\ &\Omega_3=\alpha_{13}+\alpha_{23}+\alpha_{34}-\pi \\ &\Omega_4=\alpha_{14}+\alpha_{24}+\alpha_{34}-\pi \\ \end{split} and perimeters of faces \begin{split} &P_1=l_{23}+l_{34}+l_{24} \\ &P_2=l_{14}+l_{24}+l_{12} \\ &P_3=l_{13}+l_{34}+l_{14} \\ &P_4=l_{12}+l_{23}+l_{13}. \\ \end{split} Then the following cross-ratios are equal to each other: $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[P_1, P_2, P_3, P_4]. $$ Question: Is it known? I have found a proof of this statement (to appear soon), but it involves quite tricky algebraic geometry. It will be very interesting to me to see a more elementary approach.

Addition: Similar statements hold in spherical and hyperbolic geometry. For a spherical tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}]. $$ For a hyperbolic tetrahedron $$ [e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}]=[e^{P_1}, e^{P_2}, e^{P_3}, e^{P_4}]. $$

Addition 2: One can prove a more general statement, which I formulate in the spherical case (but it is true in other geometries after appropriate modifications). There exists a $PSL_2(C)-$ transformation, sending eight numbers

$$ 1, e^{i\Omega_1}, e^{i\Omega_2}, e^{i\Omega_3}, e^{i\Omega_4}, e^{i(\alpha_{12}+\alpha_{23}+\alpha_{34}+\alpha_{14})}, e^{i(\alpha_{12}+\alpha_{24}+\alpha_{34}+\alpha_{13})}, e^{i(\alpha_{13}+\alpha_{23}+\alpha_{24}+\alpha_{14})} $$ to $$ 1, e^{iP_1}, e^{iP_2}, e^{iP_3}, e^{iP_4}, e^{i(l_{12}+l_{23}+l_{34}+l_{14})}, e^{i(l_{12}+l_{24}+l_{34}+l_{13})}, e^{i(l_{13}+l_{23}+l_{24}+l_{14})}. $$ I know neither an elementary proof of this statement nor any interpretation of the coefficients of the $PSL_2(C)-$ transformation.