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Given $2$, $2D$ vectors we can calculate the angle inside these vectors using either dot or cross product. - (And presumably many other methods too)

Given $3$, $3D$ vectors, how would I calculate the solid angle inside them?

Given $4$, $4D$ vectors, how do I calculate the 4-angle inside of them?

And then finally, $5$, $5D$ vectors same question.

My problem is actually in 5 dimensions, so that's the one I'm most interested in. Though it would be lovely to see a pattern emerge from calculation to calculation.

Thank you in advance

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  • $\begingroup$ For three vectors in $\mathbf{R}^{3}$: Calculate the interior angles of the spherical triangle whose vertices are the given vectors; the solid angle (a.k.a., the area of the triangle on the unit sphere) is the sum of these angles minus $\pi$. Offhand I don't see a nice way to generalize to higher dimension, unfortunately. $\endgroup$ – Andrew D. Hwang Jul 5 '17 at 23:58
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    $\begingroup$ @AndrewD.Hwang There is a generalization but I don't know how to carry out the actual computation. Computing a 5-d angle is equivalent to computing the volume of some 4-simplex living on 4-sphere. For even $n$, the volume of the spherical $n$-polytope is related to the "angle sums" of its faces through a generalized Gram relations (for statement, see this). The contribution from 4-, 3- and 2- faces are not that hard to figure out. The problem is I don't know how to compute the contribution from vertices and edges. $\endgroup$ – achille hui Jul 6 '17 at 1:12
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    $\begingroup$ Wikipedia gives a formula in the form of an infinite series: en.wikipedia.org/wiki/… $\endgroup$ – Hans Lundmark Jul 6 '17 at 8:51
  • $\begingroup$ This Formula is for the total solid angle in a given dimension. Sadly not what I needed at the time. - Still don't have a solution to this $\endgroup$ – Ben Crossley Jun 20 '18 at 18:49

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