**1.**My first book was Polyhedra Models by M. Wenninger.

It is one of the books that I open most often and is quite worn out by now. Unfortunately the templates do not seem to have the correct x, y ratio.

**2.**Another book gave me a lot of inspiration was Symmetry Orbits by H. Verheyen. I found the theory difficult (I missed some proof somewhere, but that is most probably due to a lack of some deeper algebra knowledge from my side) and the book contains quite some little mistakes with wrong texts at or wrong references to pictures, but that keeps you alert as well All in all I learned a lot from that book.

**3.**The third book that I value a lot is Polyhedra by Peter R. Cromwel. It gives a very good introduction to different topics within polyhedra. Unfortunately it doesn't use that symmetry group notations from Verheyen's book.

## Books about Polyhedra

- marcelteun
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### Books about Polyhedra

Within this thread I thought it would be nice to gather a list of inspirational books about polyhedra.

1. Adventures Among the Toroids by B.M.Stewart - there is almost limitless exploration possible from here.

2. Platonic and Archimedean Solids by Daud Sutton - small and very simple introduction to polyhedra but still inspirational.

Jim

- Dinogeorge
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### More books

The first book I read featuring polyhedron models was Cundy & Rollett'shedron wrote:Marcel's list contains some truly inspirational books. Let me add:

*Mathematical Models*, the first edition of which Bruce Chilton loaned me for a few weeks back in 1958 when I was in eighth grade. I picked up a copy of the second edition at the MIT student bookstore when I was an undergraduate, and a few years ago I discovered the third edition for sale on Amazon. Also on Amazon I managed to find a used copy of the first edition, so now I have copies of all three editions in my math library.

Two other inspiring books were H. S. M. Coxeter's

*Regular Polytopes*and Coxeter, DuVal, Flather and Petrie's

*The Fifty-Nine Icosahedra*. I read Bruce's copy of the latter at first, but later, when I was a freshman in college, I found it on sale at either the MIT or the Harvard bookstore, and also the second edition of the former, which went on sale then (1963). I also remember originally borrowing

*Regular Polytopes*from the Buffalo Public Library when I was still a high-school kid: I had to take a bus downtown to the main branch to get it, and I began reading it on the return bus. It was a winter evening with lots of wet snow in the streets. Funny how such things stick in your mind.

I vividly recall being awestruck by such figures as the Archimedean polyhedra, the Kepler-Poinsot polyhedra, and the compound polyhedra, which are now just "old hat" to me. By the time I finished high school, I had built a model of the final stellation of the icosahedron, which I still have on my shelf, much the worse for wear. I painted it in ten colors to show off the enneagrammatic faces. Some of the models I made for my high school math class may still be gathering dust in some closet there.

Also as an undergraduate I made a photocopy of Coxeter, Longuet-Higgins and Miller's landmark paper, "Uniform Polyhedra." Xerox machines were not generally available in the mid-1960s, so I used our office's wet copier when I was a summer employee at Buffalo's Roswell Park Memorial Hospital. I don't recall whether I originally got the paper via interlibrary loan or whether Roswell's own library had a copy. Now I have it as a PDF, which I can email to anyone who wants one.

Another book I find myself turning to more and more frequently these days is Coxeter's collection

*Twelve Geometric Essays*. For a long while, Coxeter carried on in polytope geometry almost single-handedly, as mathematics sped off in other directions. Now, more an more people are taking a longer look at some of the things that Coxeter spent his life studying.

As a grad student at the University of Toronto (which I applied to just because Coxeter was there), I provided Coxeter himself with a few computer-generated pictures for his sequel to

*Regular Polytopes*,

*Regular Complex Polytopes*, which is presently in its second edition. I can't tell you how flattering it was to appear in the index to

*Regular Complex Polytopes*alongside Gauss and other

*real*mathematicians. It was Coxeter, too, who showed me the manuscript for Magnus Wenninger's

*magnum opus Polyhedron Models*, which he was then reviewing, and who provided me with Magnus's address so that we could correspond. I recall being incredulous that someone actually went to the trouble of calculating (apparently without the benefit of a computer) and building paper models of

*all*the figures in "Uniform Polyhedra." I think Magnus and I have continued corresponding off and on for about 35 years now.

- robertw
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I have split this topic up, as it went off-topic. The posts that have moved are now in a new thread for discussion of cube compounds here: http://software3d.com/Forums/viewtopic.php?t=45. You'll have to subscribe to that thread again if you wish to be notified of new posts.

Book-wise, I think I remained unaware of many of the important books when I was young, and they are hard to come by here in Australia. I didn't know about Coxeter's "The 59 Icosahedra" until after I wrote my first stellation program (long before Stella, and using a hideously inefficient brute force method). I saw reference to the "fact" that there are 59 stellations of the icosahedron, but could never see why it should be the case. I put "fact" in quotes because it all depends on what criteria you use.

I was also unaware of Wenninger's "Polyhedron Models", which I probably would have gone crazy over had I discovered it. Instead, I only knew of other books, and none really showed nets for anything very complicated.

There was one book, the title of which I've forgotten, which included nets for the compound of 5 cubes. They suggested a very sturdy construction, first building a single cube, then covering each face with another part turning it into a compound of 2 cubes, and finally adding a third level to complete the model. I made one, and it's rock solid, with three layers of cardboard on the visible parts of one cube! I still have it:

Looks a bit amateur, with parts coloured by texta!

Anyone know what book that was? It seemed to be aimed at a young audience, with large print and not too many details maths-wise. At least that's how I remember it.

Rob.

Book-wise, I think I remained unaware of many of the important books when I was young, and they are hard to come by here in Australia. I didn't know about Coxeter's "The 59 Icosahedra" until after I wrote my first stellation program (long before Stella, and using a hideously inefficient brute force method). I saw reference to the "fact" that there are 59 stellations of the icosahedron, but could never see why it should be the case. I put "fact" in quotes because it all depends on what criteria you use.

I was also unaware of Wenninger's "Polyhedron Models", which I probably would have gone crazy over had I discovered it. Instead, I only knew of other books, and none really showed nets for anything very complicated.

There was one book, the title of which I've forgotten, which included nets for the compound of 5 cubes. They suggested a very sturdy construction, first building a single cube, then covering each face with another part turning it into a compound of 2 cubes, and finally adding a third level to complete the model. I made one, and it's rock solid, with three layers of cardboard on the visible parts of one cube! I still have it:

Looks a bit amateur, with parts coloured by texta!

Anyone know what book that was? It seemed to be aimed at a young audience, with large print and not too many details maths-wise. At least that's how I remember it.

Rob.

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Cundy & Rollett

This sounds like it might be Cundy & Rollett'srobertw wrote:There was one book, the title of which I've forgotten, which included nets for the compound of 5 cubes. They suggested a very sturdy construction, first building a single cube, then covering each face with another part turning it into a compound of 2 cubes, and finally adding a third level to complete the model. I made one, and it's rock solid, with three layers of cardboard on the visible parts of one cube! I still have it:

Looks a bit amateur, with parts coloured by texta!

Anyone know what book that was? It seemed to be aimed at a young audience, with large print and not too many details maths-wise. At least that's how I remember it.

Rob.

*Mathematical Models*, which describes the construction of the five cubes compound in just that way. (On the other hand, that book was not necessarily for young readers.)

Indeed, if you don't have internal braces of some kind or stiffen the paper somehow, the finished five-cubes paper model has an annoying tendency to "pop in" at its octavalent re-entrant vertices.

### Re: Cundy & Rollett

If you use double tabs and continue the tabs where the edges form pentagrams - do it pin-wheel style with 5 sets of double tabs, each holding two pairs of triangles (and thereby bracing each octavalent re-entrant vertex) - the model is so rigid you would not believe it!Dinogeorge wrote:Indeed, if you don't have internal braces of some kind or stiffen the paper somehow, the finished five-cubes paper model has an annoying tendency to "pop in" at its octavalent re-entrant vertices.

- robertw
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### Re: Cundy & Rollett

I'd believe it (although I guess strictly speaking, I don't believe that it would be so rigid that I wouldn't believe it! ). Just some small struts across internal parts often helps a great deal. See my model of the Great Dodecicosidodecahedron for example.oxenholme wrote:If you use double tabs and continue the tabs where the edges form pentagrams - do it pin-wheel style with 5 sets of double tabs, each holding two pairs of triangles (and thereby bracing each octavalent re-entrant vertex) - the model is so rigid you would not believe it!

However, for a 5 cube model I would not have any tabs at the edges you recommend, as those parts would be in the same colour and be connected in single nets, so I'd probably come up with another solution. But that's OK, almost any pairs of internal double tabs meeting at a vertex would probably do

Rob.

- robertw
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### Re: Cundy & Rollett

No, it was not that book. It contained larger nets that you could trace off and use, and I remember the book was wider than it was tall. I feel sure it must be barely known these days as I have not seen a reference to it anywhere since school (on the other hand, I can't remember what it was called!).Dinogeorge wrote:This sounds like it might be Cundy & Rollett'sMathematical Models, which describes the construction of the five cubes compound in just that way. (On the other hand, that book was not necessarily for young readers.)

I don't have

*Mathematical Models*, but have photocopies of a few relevant pages. I don't think they describe constructing the 5 cube model in this way. They seem to just describe nets for the external parts.

Yes, even earlier I had built a very poorly constructed version. Possibly using nets traced from Mathematical Models, which were tiny! I didn't use tabs, but rather used a mass of sticky tape on the outside of the model. It looked awful, and of course there was no support inside, so trying to push the tape down didn't work very well. I found its crumpled shell in a box again years later.Indeed, if you don't have internal braces of some kind or stiffen the paper somehow, the finished five-cubes paper model has an annoying tendency to "pop in" at its octavalent re-entrant vertices.

Rob.

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Re: Cundy & Rollett

You're right. I checked my copies of all three editions of C&R and none describes putting the five cubes together that way. They do, however, describe building therobertw wrote:I don't haveMathematical Models, but have photocopies of a few relevant pages. I don't think they describe constructing the 5 cube model in this way. They seem to just describe nets for the external parts.

*five tetrahedra*that way (by starting with a single tetrahedron and gluing parts to it), so I clearly misremembered which figure they were describing. I do, however, also recall trying unsuccessfully to build the five cubes by starting with one cube. Probably I was trying to use their five-tetrahedra technique on the five cubes.

Last edited by Dinogeorge on Sat Mar 01, 2008 9:21 pm, edited 1 time in total.

Williams; The geometrical foundation of natural structure. Written decades ago in a (then) trendy fixed-pitch typewriter font. But it is still a standard reference for the Archimedean polyhedra and their spacefillings.

Pearce; Structure in nature is a strategy for design. Get this out of a good library. More insights into spacefilling polyhedra, and beautiful photographs.