Wikipedia:Reference desk/Archives/Mathematics/2016 January 25

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January 25

Extending the Steinmetz Solids to n dimensions...

If Vn is the (Hyper^n)volume in n dimensions of the intersections of all circles formed from the combinations of dj^2+dk^2=1 for any 1<j,k<n how does that volume Vn compare to the volume of the unit cube and the volume of the volume sphere in n dimension? (So in 4 dimensions, all points meeting w^2+x^2<=1, w^2+y^2<=1, w^2+z^2<=1, x^2+y^2<=1, x^2+z^2<=1 and y^2+z^2<=1, note this is *not* equal to the unit sphere as (sqrt(2),sqrt(2),sqrt(2),sqrt(2)) is on the edge of this volume and outside the unit sphere).Naraht (talk) 03:52, 25 January 2016 (UTC)

I get for the hypervolume for dimension n:

$2^{n}(n-1)\int _{0}^{\tfrac {\pi }{4}}\sin ^{n-2}\theta d\theta .$

Actually there are other ways to generalize to higher dimensions. For example in dimension 4 you could talk about the intersection of the 4 "hypercylinders" x²+y²+z²≤1, x²+y²+w²≤1, x²+z²+w²≤1, y²+z²+w²≤1. For any n and k, 1≤k≤n you could define the solid to be the intersection of the (n choose k) cylinders formed by taking coordinates k at a time and restricting to the k-ball. For k=1 this gives you the hypercube with side 2 and for k=n you get the n-ball, with the remaining sets a nested sequence between these two. --RDBury (talk) 09:36, 25 January 2016 (UTC)

Beautiful! Can the function be generalized with k as well?Naraht (talk) 19:22, 25 January 2016 (UTC)

I've been working on the k=3 case but it appears that the computation rapidly gets more difficult as k increases. For k=2 you end up integrating over a segment of a circle, which is where the limits 0 to π/4 come in. For k=3 you need to integrate over a spherical triangle which is tricky. For k=4 you need to integrate over a hyperspherical tetrahedron which is even worse. Of course there may be a simpler way that I'm missing; I'm trying to use spherical coordinates but maybe there's a different coordinate system that will work better. --RDBury (talk) 01:15, 26 January 2016 (UTC)

Want name of book

In one of Calvin Clawson's books on mathematics, chapter 12 deals with certain aspects of prime numbers, such as Goldbach's conjecture. Is the name of this book Mathematical Mysteries: The Beauty and Magic of Numbers?Bh12 (talk) 21:53, 25 January 2016 (UTC)

You can see the table of contents for that book at it's page on amazon. Chapter 12 is called "Goldbach's Conjecture", so it sounds like that's probably the book you're thinking of. Staecker (talk) 23:08, 25 January 2016 (UTC)

Resolved