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.