Testwiki:Reference desk/Archives/Mathematics/2022 February 7

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February 7

Height of Simplexes

I'm working my way through Simplex and have been unable to find exactly what the height of each simplex is. I'm defining the "height" of an n-simplex as the distance from one vertex to the center of the n-1 simplex made up of the other vertices. So the "Height" of a 2-simplex is sqrt(3)/2. and the "Height" of a 3-simplex is sqrt(6)/3 (according to Tetrahedron). I'm presuming that they tend to 0 as n goes to infinity, but I don't know if there is an easy formula.Naraht (talk) 09:37, 7 February 2022 (UTC)

That obviously depends on the choice of a vertex, as in 2D example you may have three different heights of a triangle, depending on a vertex chosen as an apex. --CiaPan (talk) 10:21, 7 February 2022 (UTC)

I think we are assuming that it is the "standard simplex", which will be symmetric with respect to choice of vertex. See a discussion here: https://math.stackexchange.com/questions/1697870/height-of-n-simplex showing that this height is $\sqrt{\frac{n+1}{2n}}$. Your answer is different I guess because you assumed the simplex edge length is 1? The typical construction of the standard simplex makes the edge length $\sqrt{2}$. Staecker (talk) 12:34, 7 February 2022 (UTC)

The formula $\sqrt{\frac{n+1}{2n}}$ works for edge length $1$; the height of a 1-dimensional simplex is the length of its single edge, so just put $n=1$ and indeed $1$ pops out.  --Lambiam 13:07, 7 February 2022 (UTC)

Yes Lambiam is right- I popped it out wring! Staecker (talk) 14:39, 7 February 2022 (UTC)

So sqrts of 2/2, 3/4 , 4/6, 5/8, 6/10 , so 1, sqrt(3)/2, sqrt(4/6)= sqrt(6/9) = sqrt(6)/3, sqrt(5/8) = sqrt(10)/4, sqrt(6/10)=sqrt(3/5)=sqrt(15)/5 etc. which as n-> inf approaches sqrt(1/2) = sqrt(2)/2. Interesting.

In contrast, the volume approaches $0$ more than exponentially fast; see the last paragraph of Template:Section link. The volume formula gives an alternative way to derive the height formula, using an obvious way of obtaining the volume of the regular $(n+1)$-simplex from that of its predecessor by integration over the height.  --Lambiam 21:10, 7 February 2022 (UTC)

Template:U Linear over (factorial * power). *ouch*.Naraht (talk) 22:41, 10 February 2022 (UTC)