Polyhedra with pentagram faces

[Pat]

I am attempting to construct the solid figure (polyhedron) which has
four regular five-pointed stars (pentagrams) meeting at each corner
(vertex).

The figures with three and five pentagrams meeting at each vertex are
well known; they were described by Johann Kepler in 1619. Those two
were rediscovered and two related ones characterised by Louis Poinsot,
in 1809; so they're known as the Kepler-Poinsot polyhedra
http://en.wikipedia.org/wiki/Kepler%E2%80%93Poinsot_polyhedron.

However, I can't find any mention of a polyhedron with _four_
pentagrams meeting at each vertex. I see no reason why it shouldn't
exist. In almost all cases where the sum of the angles of the
polygonal faces around a vertex is less than 360 degrees, there is a
polyhedron of that type.

There are a few exceptions; for example a shape with two pentagons and
one hexagon meeting at each vertex cannot exist because it requires
each pentagonal face to have alternate pentagons and hexagons as
neighbours, and you cannot alternate two things around an odd number
of sides!

This limitation does not affect the shape I am trying to construct,
but I have my doubts of its existence, because I have never seen an
account or illustration of it. However, neither have I seen any
article that says: "this shape doesn't exist and the reason is…"

I have, naturally, tried to construct the shape mechanically, using
Google's Sketchup. Make a pyramid with a square base and angles of 36
degrees at the apex (this is the angle between the edges of a
pentagram). Construct the four pentagrams whose upper point is a face
of that pyramid. Erect more pyramids on the 16 other points of these
pentagrams, and build pentagrams on each of the other triangular faces
of these pyramids. And so on ad… well in the latest attempt I've got
up to 18 pentagrams and haven't hit …nauseam yet; but I have a
suspicion the last word may be …infinitum.

Naturally some of the vertices of the new pentagrams coincide with
vertices already constructed, but each new one seems to have two or
three points that don't coincide; so they form new vertices of the
conjectured polyhedron.

There is an appealing regularity to the partial form; it looks like
the central pentagon of each pentagram contains a cluster of five
pyramids, each tilted outward at an angle of about 13.6 degrees from
the pentagram plane. That has a certain beauty to it, but it's a lot
of pyramids and more are implied each time a new pentagram is placed.

Is the process diverging without limit; or will a closed solid
eventually be formed - possibly with hundreds or even thousands of
faces? (please don't let it be thousands!)

It would satisfy my curiosity much more quickly to find a proof in a
few lines that the polyhedron in question cannot exist. But I can find
nothing of this kind on the all-knowing web.

Can anyone help?

PF

[Jasen Betts]

On 2013-02-16, Pat <Pat> wrote:
> I am attempting to construct the solid figure (polyhedron) which has
> four regular five-pointed stars (pentagrams) meeting at each corner
> (vertex).

you light have better luck with hexagrams.

> The figures with three and five pentagrams meeting at each vertex are
> well known; they were described by Johann Kepler in 1619. Those two
> were rediscovered and two related ones characterised by Louis Poinsot,
> in 1809; so they're known as the Kepler-Poinsot polyhedra
> http://en.wikipedia.org/wiki/Kepler%E2%80%93Poinsot_polyhedron.

these polyhedra are all based on the icosahedron/docdcahedron pair

> However, I can't find any mention of a polyhedron with _four_
> pentagrams meeting at each vertex. I see no reason why it shouldn't
> exist. In almost all cases where the sum of the angles of the
> polygonal faces around a vertex is less than 360 degrees, there is a
> polyhedron of that type.

if it's a regular polyhedron it's vertices will be on a sphere.
locate the vertices, and then the edges and planes, and the
polyhedron will appear.

> This limitation does not affect the shape I am trying to construct,
> but I have my doubts of its existence, because I have never seen an
> account or illustration of it. However, neither have I seen any
> article that says: "this shape doesn't exist and the reason is…"

immagine what it looks like when viewed from above a vertex.

> Is the process diverging without limit; or will a closed solid
> eventually be formed - possibly with hundreds or even thousands of
> faces? (please don't let it be thousands!)

I can't see your sketchup, but the main consideration should be would
be is each set of equivalent points on a sphere and do these spheres
have a common centre.

> It would satisfy my curiosity much more quickly to find a proof in a
> few lines that the polyhedron in question cannot exist. But I can find
> nothing of this kind on the all-knowing web.

I suspect what you're trying is impossible.

--
⚂⚃ 100% natural