Re: Balance point of a solid

*To*: mathgroup at smc.vnet.net*Subject*: [mg113634] Re: Balance point of a solid*From*: Ray Koopman <koopman at sfu.ca>*Date*: Fri, 5 Nov 2010 05:13:18 -0500 (EST)*References*: <iam332$jvn$1@smc.vnet.net> <iar4jk$gmi$1@smc.vnet.net> <iatt52$eib$1@smc.vnet.net>

On Nov 4, 2:07 am, Andreas <aa... at ix.netcom.com> wrote: >> [...] >> I'm not sure how you want to extend the solid to n dimensions. >> Can you be a little more specific? > > The base equilateral triangle holds the key. I want to extend that > to n dimensions, place trapezoids on the edges, and connect the > resulting tops. > > Always tricky trying to visualize in more than 3 dimensions, but I > think the triangle would become a regular simplex. From wikipedia: > > "...simplex (plural simplexes or simplices) is a generalization of > the notion of a triangle or tetrahedron to arbitrary dimension. > Specifically, an n-simplex is an n-dimensional polytope which is > the convex hull of its n + 1 vertices. For example, a 2-simplex is > a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a > pentachoron." > > If I follow this correctly, to extend what I have into 4 dimensions, > my "base" would become regular tetrahedron, in 5 dimensions a > regular pentachron and so on. Then I build up the multidimensional > solid with additional trapezoids on all the edges. I was hoping you would algebraicize it. Is this what you mean? Let A be an n+1 x n matrix whose rows contain the coordinates of n+1 mutually equidistant points. Then the coordinates of the vertices of your solid are contained in the rows of the 2(n+1) x n+1 matrix {{A,0}, where 0 is a column of zeros, {A,h}}, and h is a column of (positive) heights.