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Re: Balance point of a solid

  • To: mathgroup at smc.vnet.net
  • Subject: [mg113613] Re: Balance point of a solid
  • From: Andreas <aagas at ix.netcom.com>
  • Date: Thu, 4 Nov 2010 04:02:02 -0500 (EST)
  • References: <iam332$jvn$1@smc.vnet.net> <iar4jk$gmi$1@smc.vnet.net>

Ray,

Thanks for looking at this.

> This is a top-of-the head response, so it may be completely off, but
> why not think of the solid as two pieces: a flat-topped triangular
> column, height = min(h1,h2,h3), and a tetrahedral cap. The centroid
> of each piece is the simple average of its vertices. The centroid of
> the whole thing is the weighted average of the two centroids, with
> the weights being the volumes of the pieces.

The centroid of the triangular column you described clearly falls
directly below its center of mass, but I don't think the centroid of
the tetrahedral cap would fall directly below it's center of mass.  In
the simplest analogy, if you divide a right triangle into equal areas
with a line that intersects either of the sides forming the 90 degree
angle (and thus parallel to the other leg forming the right angle, it
won't intersect at the center of the side.

That said, I very much take your point about dividing the problem into
more manageable pieces.  Something I may need to do as I try to extend
it into n dimensions.

> 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.

Does this explanation make sense?

Best,
A





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