The Geometry of the Chestahedron

Make your own chestahedron with this template!

The chestahedron is a unique volume constructed of four equilateral triangles and three “kite” shapes (quadrilaterals). The chestahedron is unique in that the areas of the triangles and quadrilaterals are equal. The chestahedron is the first seven-sided volume with faces of equal area. You can examine one of the ways that the chestahedron is formed out of the transformation of a tetrahedron in this video.

The exact geometry of the chestahedron is known: all the angles in each face and all the dihedral angles (the angles formed when two faces meet). In the video, the chestahedron is formed when the “petals” of the small tetrahedron open up so that their dihedral angle formed with the base triangle is 94.83092. At this point, the area of the quadrilaterals is exactly equal to the area of the equilateral triangles.

The mathematics behind the chestahedron was formulated separately by Dr. Ron Milito as well as Dr. Karl Maret, M.D., and has been subsequently verified independently.

Dr. Ronald Milito has a doctorate in Biophysics, has studied human anatomy in a medical school setting, has done research on cancer detection, and for many years was a teacher of science and mathematics at the college, high school, and middle school levels. Dr. Milito’s solutions will be presented here soon.

Dr. Karl Maret has used the program Stella 4D to create a digital chestahedron, and has kindly provided images that show the face angles and dihedral angles:

1. Loll & David CRYER

TRUELY AMAZING!! You have discovered the key to the universe!!
My husband and i have been studying 7 for a few years and your discovery fits in perfectly to our beliefs to how intriguing 7 is to every aspect of the universe. Thankyou for your unbelievable discovery to the human race!!

Regards
David & Loll CRYER
Perth, Western Australia

2. P. Petran

Fascinating stuff.. One thing I can’t seem to wrap my head around are the values shown for the internal angles at the top of the kites..

i.e. How can these angles be at 71.7501 degrees when 71.7501 degrees x 5 = 358.7505 degrees, and not the 360 degrees required for a full ciircle?

What happened to the missing 1.2495 degrees?

*cheers
~Pete

• Bruce R. Gilson

Whenever you have a polyhedron, the sum of the angles at each vertex must be less than 360 degrees. If it were exactly 360 degrees, it would be flat, not a polyhedron vertex.

For example, in a cube there are three squares at each vertex, making 3 x 90 = 270 degrees, not 360.

In fact the amounts by which these totals are less than 360 degrees satisfy a rule: their sum over all vertices of the polyhedron must always be exactly 720 degrees.

• P. Petran

Yes of course, if all 5 pentagram kites were connected at the top of the chestahedron it would flatten out to a 2-D form @360 degrees.

My issue is with the internal angles as shown on the faces of the kites at the top of the chestahedron, which must be EXACTLY 72.000 degrees each, or they would not be consistent with the internal angles of the pentagram used to construct them.

The 3 kites that connect at the top must have a combined rotational angle of 3 x 72.000 degrees (216.000 degrees total) and NOT 3 x 71.7501 degrees (215.2503 total) as seems to be shown in the diagrams.

• Bruce R. Gilson

I assume that “the pentagram used to construct them” was only a first step; the angles probably needed to be slightly altered when the actual polyhedron was assembled to make all the faces equal.

I admit I haven’t worked out the detailed geometry, but I assume that Chester did and found those angles.

• P. Petran

re. the description of the last drawing:
“The image above shows the chestahedron as a two-dimensional net with the angles within each face shown.”

If those internal angles at the top of the faces of the 3 kites have to be manipulated to fit the form @71.7501 degrees, they no longer represent the 2-D angles of a true pentagram do they? In one of his demonstrations didn’t Frank say something along the lines of “you can’t make it fit, either it fits or it doesn’t”?

I’m thinking it has to be a mistake or a typo, and the real angles should have always read 72.00 degrees.

• Bruce R. Gilson

“If those internal angles at the top of the faces of the 3 kites have to be manipulated to fit the form @71.7501 degrees, they no longer represent the 2-D angles of a true pentagram do they?”

So they don’t, so what? He started with a pentagram, constructed the net he showed, then found that to make all the pieces fit together with equal face areas, he needed to alter the kites slightly to the dimensions shown.

I think you’re wrong in saying “I’m thinking it has to be a mistake or a typo, and the real angles should have always read 72.00 degrees,” and that the dimensions (lengths and angles) must be assumed to be correct as given. But until Frank Chester himself posts a response on here, we don’t know whether you are right or I am.

• P. Petran

A reference to the kite shapes from pg. 4 of the Seth Miller .pdf article @:

“””What is fascinating about the Chestahedron is that its two-dimensional,
‘unfolded’ version is found to relate exactly and unexpectedly to a perfect five-pointed star! It turns out
that five of the Chestahedron’s kite shapes, placed with their points together, makes the star pentagon.”””

But clearly this is not the case… the 3 kite shapes of the chestahedron do NOT fit exactly into a the shape of a 5 pointed star.
They’re close, but not exact. It follows too that if the top angles are a bit off, then all of the angles of the kites are a bit off.

Hopefully Frank or someone connected to him will read this and clear things up for us?

*cheers,
Pete

• Bruce R. Gilson

Your quote is a claim by Seth Miller, not by Frank Chester. Perhaps Miller saw a net without the angles being measured to four decimal places, and jumped to a conclusion, which is not exactly true, but only correct to the nearest degree. I see no claim by Frank Chester that the kites exactly match the shapes found in a pentagram. In fach, Chester does not, in this article, mention a pentagram at all, so it is not even clear that he started with a pentagram. In fact, this article refers to transforming a tetrahedron as the basis for the construction.

3. Bruce R. Gilson

Further comment: I see no mention of the pentagram in other places such as the Wikipedia treatment of the chestahedron. It would seem that the “exact” fit to a five-pointed star is an artifice of Miller’s measurements.

Of course, there is one possibility that could make you right. The measurements in this article are taken from a model constructed, using Stella, by Dr. Karl Maret. Possibly his construction of the model is inaccurate. The precise angles are the angles in Maret’s model, not taken from a verified construction according to Chester’s specifications.

4. P. Petran

“I see no claim by Frank Chester that the kites exactly match the shapes found in a pentagram.”

Ok so you don’t consider Miller’s interview to be a valid source of information. How about Frank’s own words?

@ 14:30 He shows the origin of the 2-D geometry of the kites and the very precise circles and pentagrams from which they come.

@18:50 “the top of a chestahedron comes from a 5-pointed star”

What you won’t find in the lecture is a reference to altering the internal angles of the 5-pointed stars to make them fit.

(I’m not trying to invalidate his work, just trying to figure out this nagging discrepancy.)

• Bruce R. Gilson

“[Y]ou don’t consider Miller’s interview to be a valid source of information.”
It’s Miller’s conclusion, not Frank Chester’s. That’s the difference.
“[T]he top of a chestahedron comes from a 5-pointed star”
Of course, “comes from” could, as you assume, mean “is a direct copy of.” But if it has the meaning I usually give for those words, all he is saying is that the ultimate source is a 5-pointed star. It certainly admits the possibility that the angles and lengths could be altered.
Of course, it is just us discussing this, with no way to tell who is right. Unless we hear from Frank Chester, this whole thread id totally speculative.

• Bruce R. Gilson

Actually, it is quite possible that both of us are partially right.

Take a regular octahedron. Replace a pair of opposite vertices by two that are spaced closer together or further apart. The resulting octahedron will not have Oh symmetry but only D4h, but it will still be true that all eight faces are congruent (and thus equal in area!) though they will no longer be equilateral triangles.

Possibly, the requirement that all seven faces of the chestahedron are equal in area, three are congruent kite-quadrilaterals, and the other four are congruent triangles does not determine the polyhedron’s geometry completely. It may well be that making the triangles equilateral requires the kites to be slightly altered from the shape they have in a pentagram, while one could also make a chestahedron with the kites precisely as they are in a pentagram, but then the triangles might not all be equilateral. The triangle/triangle edges might slightly differ from the triangle/kite edges.

This is only a conjecture. I have not the time to actually construct such a polyhedron, which would settle this conjecture. But given these possibilities, I wonder whether further discussion would be useful.

5. Bruce R. Gilson

I asked for help on another forum. And it seems my original thoughts were right. If one constructs a polyhedron according to the specifications that there be seven faces, three kites and four triangles, all equal in area, with the polyhedron having threefold symmetry, and all triangles equilateral, the angles 71.7501, 36.4998, and (twice) 125.875 come out as forced. So the close approximation to a pentagram is just that — a close approximation.

• P. Petran

Thanks for clearing that up.. so it would seem that each of the internal angles of the pentagram kites need to be forced/altered slightly in order to satisfy the requirement of all 7 faces having equal areas?

Isn’t this er, “cheating” in a way? Making it fit? I would think it was just as important to maintain the integrity of the angles used to dilineate the pentagram form with 5 equal divisions of a circle @ 72 degrees each, and preserve the corresponding golden mean ratios they represent, rather than worry about the slgiht difference in surface areas.

• Bruce R. Gilson

It depends on what you think is important. To me, this polyhedron’s special characteristics are that it has three kite-quadrilaterals and four equilateral triangles, all equal in area, and C3v symmetry. The very close approximation of the kite-shaped faces to the kites obtained from the pentagram is just an interesting coincidence. If you want to construct a polyhedron differing slightly from this one, with the angles and ratios corresponding exactly to those from the pentagram, you can, but it’s different, and LESS interesting in my eyes because the equality of the areas of the faces matters to me more than the close approximation to the pentagram (which I hadn’t even noticed until you brought it up). Your polyhedron would, of course, be indistinguishable from this one except by very precise measurement. But mathematically, it would not have identical properties.