HDG-APfSFaRPGGaN - Presentation

Hyperdimensional Geometry: A Primer for Sci-Fi and RPG Geeks and Nerds

Presentation

This book (presented here as a series of posts on my blog first) is intended to help fans of science-fiction (or "speculative fiction", as it's known these days) and role-playing games to deal properly with geometrical objects with more than 3 dimensions (and less, too). I've seen many misconceptions when it comes to those, and I want to point them out and help avoiding them.

It all started over 40 years ago, in 1982 or so, when I was just a teenager in 8th grade. The 1980s were a great time for anyone who loved science fiction in Peru. We got several weekly collections those years, both at the newsstands and the supermarkets... but of course, a teenager couldn't afford to buy them all. I got Verne's and Wells' classics, like Voyage to the Center of the Earth or The War of the Worlds, as well as Conan Doyle's The Lost World, and more recent authors, like Asimov and Clarke, Niven, Simak, Ellison, Bradbury, Heinlein, Blish, Brown, Weinbaum, Sturgeon... novelizations of the Star Wars trilogy or Raiders of the Lost Ark... different short-story anthologies... issues of Starlog and other magazines...

And then there was Fantaciencia.

Fantaciencia, Enciclopedia de la Fantasía, Ciencia y Futuro was one of those collectable encyclopedias, published weekly or bi-weekly in 1982 as 16-page issues (plus a central spread to be collected separately) by the Spanish publisher ECG Ediciones. It was a translation from the Italian Grande Enciclopedia della Fantascienza, originally published by Editoriale del Drago between 1980 and 1981 as 80 issues, which made a total of 11 volumes (the first 8 volumes collecting 8 issues each from 1 to 64, the next 2 volumes collecting the central spreads of 40 issues each, and the last volume collecting issues 65 to 80 as a single encyclopedic dictionary). Unfortunately, Fantaciencia lacked most of the sci-fi stories included in the original Italian version (most likely, for lack of publishing rights for those stories in Spanish), and the collection didn't make it past issue 40.

Still, the issues that were published contained a lot of interesting articles and information about stories which deal with various subjects in sci-fi. That's where I first learned, among other things, about tesseracts and polytopes. About Heinlein's And he built a Crooked House and Abbott's Flatland. And more.

Then, some time later, I came across Carl Sagan's Cosmos at a local department store. I browsed it, but unfortunately the part I looked at where right the pages dealing with hypothetical being living in the atmosphere of a Jovian planet. That made me think the book wasn't serious and I bought Miller and Hartmann's The Grand Tour instead, which was published in the same collection. Then some time later an uncle lent me Clarke's 2010: Odyssey Two, and upon reading it, I realized the illustrations I had seen in Cosmos were nothing more than an artistic interpretation of the beings described in the novel. So, I went looking for Cosmos again, but it had ran out at the department store. Luckily, I managed to find a copy at an office and school supplies and stationery store (I also got Clarke's 2001: A Space Odyssey at a bookstore nearby, but that's another story). And just in time, since a couple weeks later, a local TV network announced they were showing the TV show Cosmos, which translated into an increase in the book's price. Reading it, I enjoyed it a lot. In Chapter 10, I ran into the tesseract again, and Flatland. And then there was Appendix 2, which talked about the Platonic solids and the various equations which defined them.

Then later, at my school's library, I ran into A.K. Dewdney's "Computer Recreations" column from the April, 1986 issue of Scientific American (by 1986 I was in 12th grade, having stayed at school to graduate from the German Abitur at the end of 1987) where he wrote about a computer program titled HYPERCUBE which allowed the visualization of a rotating hypercube on the computer screen. I managed to write such a program for my Atari 800XL computer by following his instructions (incidentally, I need to get that program transfered to my PC at some time to share it on the Internet). The school library also had several interesting science-fiction anthologies, including the collected short stories of H.G. Wells, and short stories by Arthur C. Clarke, among others.

 

 

Then in November 1986, I read on the Colombian edition of the magazine Muy Interesante (which was a reedition of the Spanish magazine of the same name, which in turn was a translation of the German magazine P.M.). It was about the possible shape of the football (or soccer ball, for the ones who don't know better) in the past, present and future. It talked about regular polyhedra (the so-called Platonic Solids), and also about semirregular ones (of which the current footbal is one, the truncated icosahedron or "Buckyball", after Buckminster Fuller, and its proposed replacement, the rhombicosidodecahedron), mentioning how Pythagoras had described 11 of the 13 existing ones, and describing how they can be obtained from the regular ones by successive truncation operations. Starting from the shapes of the regular polyhedra, with nothing more than pencil, paper an an eraser, I also managed to find 11 of the 13 existing ones (although I had no iea of their names, since the article only mentioned a few of them), and after counting their various faces, I managed to infer the general equations that applied to them by deriving them from the equations for the regular polyhedra described in Appendix B of Sagan's Cosmos. Later on, after already graduating from the university at the end of 1992, I also read Ian Stewart's Mathematical Recreations column in the May, 1996 issue of Scientific American, where he mentioned in passing the semirregular polyhedra, so I decided to write him a letter telling him how I had managed to find those 11 semirregular polyhedra and arrive at the general equations by myself, and asking him help on finding the missing two. I never mailed that letter; some time later, after examining the equations, I finally came across the snub cube and the snub dodecahedron, and noticed how they were chiral, although at the tiem, I had no idea that term existed; I just noticed how they could be skewed either to the left or to the right.

By the year 2001 (so, 25 years ago), thanks to having Internet access at home, I managed to find some interesting articles about polytopes. written around 1997 or so. My interest hadn't vanished, and over the following years I kept drafting different depictions of 4-dimensional and higher-dimensional polytopes with pen and paper. The years 2013 and 2014 were particularly fruitful, when I drew 2-dimensional and 3-dimensional nets of the unfolded n-cubes (from the 4-dimensional tesseract to the 7-dimensional hepteract). Then I progressed to the n-simplices and the n-orthoplexes. By the year 2024, I was trying to find a 2-dimensional net for the unfolded 24-cell (the 3-dimensional net was simpler), but I couldn't find a nicely symmetrical one like I had done with other polytopes before. In 2013 I had also been drawing some raster graphics to illustrate my book, since I had already come across those misconceptions I had mentioned earlier while designing a maze for an RPG a friend of mine was running. By 2024 I was redrawing them as vector graphics instead. Right now I'm no fully done with them, but I've made much progress, and I'll be working on them as I go ahead publishing each chapter. Currently, the book is divided in an Introduction 10 Chapters and 4 Appendices, and I think I've fleshed them out enough already to finally start writing the final versions of them.

This book is dedicated to my dad, who kept making sure I went to get some sleep back then when I was still up at 4am in early August 2024 (two years ago) still trying to find some nicely symmetrical 2-dimensional net of the unfolded 24-cell, shortly before passing away from a sudden heart attack at the end of September. I really miss you, dad. I really do.

Lima, Peru, August 11th, 2026.