# Building Regular Polygons for Polyhedra

Polyhedra in SuperMag will only be stable if their faces are
stable. Otherwise, they will collapse. I don't know what the
exact conditions are that a regular polyhedron's face must
satisfy for the polyhedron to be stable, but it seems sufficient
to guarantee that all vertices of the face will always have the
same distance from some point, which need not necessarily be the
center of the polygon (of course, in two dimensions it would always
be, but we're building three-dimensional constructions).

On this page I'll show how to build all stable polygons that are
needed to construct all Platonic
and Archimedean solids. Note
that I'll only discuss polygons which use one bar per edge. If you
are interested in building larger polygons, see my
page on scaling polyhedra.

## Triangle

Triangles are stable without any support:

## Square

Short squares are best stabilized with a long bar in a diagonal.
Long squares can be stabilized with a short 4-star:

## Pentagon

Pentagons can be stabilized with a 5-star. Note that you cannot
stabilize a long pentagon with a short 5-star because the bars are
too short.

## Hexagon

Hexagons are best stabilized with a 6-star, the center of which
is the center of the triangle:

## Heptagon

I have not yet needed a heptagon, but I figured out how to stabilize
them, anyway. A short heptagon can be stabilized with a long 7-star:

For a long heptagon, a 7-star can be used as well, but each segment must
be made out of a long bar directly connected to a short bar:

## Octagon

For an octagon, the same techniques can be used as for stabilizing
heptagons:

## Nonagon

I haven't figured out yet how to build a stable nonagon, but
then, I haven't had any need for one yet.

## Decagon

The decagon is the first regular polygon (or rather the one with the
lowest number of vertices) that cannot be stabilized with a star, at
least as far as I can tell. The problem is that there are too many
bars that have to connect to the sphere in the center. One solution I
have found to this problem is that instead of using a single sphere
as the center, one can use a pentagon and then connect each vertex of the
pentagon to two neighboring vertices of the decagon. This technique
works for short as well as for long decagons: