Exploring Four-Space

$210 of $12,000 goal

Raised by 4 people in 42 months
I've been working with geometric figures in higher-dimensional spaces since I was a high-school kid 50+ years ago. I discovered many intricate and fascinating figures worthy of sharing with the rest of the world. So I am presently writing a picture book about them for eventual self-publication. About 34 chapters are in various stages of completion. In the meantime, it takes time and resources to produce such a volume, and I'm retired. Time I have, resources are short. So any crowdfunding support would be most appreciated. If you'd like to see the kinds of things I write about, please download the PDF at


for an example. It will be (after some editing) a chapter of the book, which I am calling The Polytope Book. I have posted the Foreword to this book as a PDF, which I wrote several years ago and now needs updating, at my website here:


The Foreword explains the purpose of the book in much greater detail than can be provided in this campaign description.

As an example of the kinds of pictures I intend to include in The Polytope Book, here is a chart of the 18 non-prismatic convex uniform polyhedra:


Please feel free to copy the chart as you wish. As more funding drifts into this website, I'll post more information about the contents of The Polytope Book at my polychora website.

Some people may remember me from my Marvel Comics Index series of the 1970s and 1980s; others may remember me from my various dinosaur works from 1978 through 2009. So you know I am quite able to self-publish my work when it is ready.

Some of the crowdfunding I may generate here will go toward funding a larger polychora website that will feature numerous movies of the three-dimensional sections of various four-dimensional figures, such as this one:


which takes a look at a 4D star I discovered that I call a Great Swirlprism. We can't see all four dimensions of the star at the same time, but we can present the figure as a three dimensional object that changes through time, the so-called "fourth" dimension. The Great Swirlprism has 120 identical pentagrammatic retroprisms as its cells, and 120 corners, or "points." Notice how those points appear suddenly as groups of disjoint expanding polyhedra that eventually comverge into a single figure, then diverge as the star passes through our three-space. All these terms and phenomena, and many more, will be explained in The Polytope Book (or you can look them up with various search engines online).

Incidentally, if you're interested in dinosaurs, here is a link to my Dinosaur Genera List, a table of all published dinosaur names from the 18th century to date:


I’ve always been a compiler of interesting things. Here is my current table of the solar system’s numerous planetary (but not asteroidal) satellites. Additions and corrections always welcome:


Do not forget to read the Updates. I bring lots more neat stuff to the table from day to day in those posts.
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Greetings! Attached is our holiday geometric e-card for 2016. It’s a little busy, what with the portrait of our nine-year-old dog, and the figure is rather simple as such things go, but we’ve been a little busy ourselves this year, so it will just have to do. I was doodling around with the Stella4D program searching for an e-card figure when it occurred to me that one may pleasingly combine, without too much trouble, regular five-pointed stars and six-pointed “stars” (actually, each is a regular compound of two equilateral triangles: a figure known to Judaism as a Mogen David) as faces of the same polyhedron. Thus, voila! A neatly “inclusive” holiday star-polyhedron.

Here is the URL for an endless GIF movie of the polyhedron rotating:


The Mogen David is the simplest nondegenerate regular compound polytope. There is a Mogen David analogue, which I call a “diplosimplex,” in every space of dimension two or greater. It is the simplest-possible nondegenerate symmetric compound in its space. In three-space, the diplosimplex comprises two regular tetrahedra (which are three-dimensional simplices) vertically inscribed in (having the same eight corners as) a cube, a familiar figure discovered by Johannes Kepler and called by him a Stella Octangula. In dimensions four and higher, neither the core nor the case of the two intersecting simplices forms a regular polytope, so the higher diplosimplices are not conventionally described as regular compounds. But because they always have twice the number of symmetries of either component simplex, I call them “super-regular” compound polytopes. The Stella4D program easily constructs the four-dimensional diplosimplex, which has ten corners.

Polyhedron model-makers will notice the resemblance of the “stars and Mogens” figure to the uniform snub ditrigonary icosidodecahedron, Bowers acronym “seside.” In the latter figure, the isosceles triangles are all made equilateral. This forces the equilateral triangles in the Mogen David face pairs to rotate a little bit out of exact anti-alignment and to change their sizes so that their edges have the same length as the edges of the pentagrams. The Mogen Davids thus transform into nonregular syncopated uniform compounds of two equilateral triangles. One can use the Stella4D program to make the “stars and Mogens” polyhedron quickly by faceting the Archimedean truncated icosahedron.

G.O. (and Andrea and Prince Petey-Boy)
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In update 15 there is a typo in the URL for the chart of Kepler-Poinsot solids. It should read


For some reason I cannot get to that update to edit it directly.
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I had to leave off writing about geometry for a bit and instead updated my table of solar system satellites. New information has come in from the New Horizons mission, and the recent discovery of the first satellite of Makemake finally compelled me to add data on all known satellites of transneptunian dwarf planets. The new total is 193 solar system satellites in the table. As usual, additions and corrections are most gratefully accepted. The PDF is at


Download as many copies as you like.

While you’re on the web, visit my eBay store:


No math or geometry books there yet, but lots of aerospace mags, some paleontology, and science fiction and comic-related stuff. For the well-heeled, there’s my complete run of Turok Son of Stone.
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Another interlude

Space Log was a space-industry publication covering artificial earth satellites and other spacecraft of the Space Age during the twentieth century. I managed to assemble a complete run of the title via eBay and other sources a few years ago, and I wrote up this adventure in a short, nine-page autobiographical essay, available as a PDF here:


To view all the pages of the very first Space Log (April 1960), go here and click on the individual pages:

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$210 of $12,000 goal

Raised by 4 people in 42 months
Created June 6, 2015
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42 months ago
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