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Highlights from the August Post Archives on GeometryCode.com

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Here’s the eighth installment of selected blog posts published in August in recent years on GeometryCode.com. Each entry below includes the post title, published month, and a sample image to entice you to review prior posts.

(This is part 8 of a 12-part series; the highlights from prior archive posts in this series are at the end of this post.)

GeometryCode.com — Highlights from August Posts (from 2011 through 2025)

Thanks for reviewing these highlights from GeometryCode.com’s archives!

Here are the prior archive posts in this highlight series:

Highlights from the April Post Archives on GeometryCode.com

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Here’s the fourth installment of selected blog posts published in April in recent years on GeometryCode.com. Each entry below includes the post title, published month, and a sample image to entice you to review prior posts.

(This is part 4 of a 12-part series; the highlights from prior archive posts in this series are at the end of this post.)

GeometryCode.com — Highlights from April Posts (from 2013 through 2025)

Thanks for reviewing these highlights from GeometryCode.com’s archives!

Here are the prior archive posts in this highlight series:

Details about contents of Sacred Geometry Design Sourcebook and instructions for creating the graphics therein

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I’m glad I reviewed some of my oldest posts, as this post (originally published January 30, 2007) needed resurrecting and updating! I will leave the details from 18 years ago mostly intact, so if you want to dive in, scroll down to the “HOW-TO” INFO STARTS HERE heading for lots of “how to” info on many of the pages to make optimum use of my first book, Sacred Geometry Design Sourcebook!

Here’s another post from my blog archives that details the contents of specific pages of my Sacred Geometry Design Sourcebook, as well as instructions on how to replicate many of the images in the book.

30Jan2007 09:17 AM Filed in: Geometry-Math
Today’s post is an email conversation I had with a visitor to the intent.com website – our first website, which has changed hands several times since the 1990s! – that I probably could have written a decade ago, so this might be useful to those of you who already have purchased a copy of the book (and thanks to all of you who have supported me in this way over the years!) There are a few other goodies on my site map that might also be of interest.

Hi Julian:

Many thanks for your inquiry. I got so involved with the reply that I’ve decided to post it on my blog for today, as well as in this email; I’m grateful that you asked the question, so now I can answer it for the world.

On 30 Jan 2007, at 6:54 AM, Julian wrote:

Dear Bruce:

I was reading your web pages about your book SACRED GEOMETRY DESIGN SOURCEBOOK, and I wondered if you could tell me whether the book provides the reader with details of how to draw the 2-D patterns for themselves or whether they are just templates without such instruction.

The simplest answer is “yes and no”, depending on which of the 1300+ images you’re referring to.

The detailed answer is – here goes! – that I cover some of the philosophy and underlying math, concepts, and archetypal ideas at the beginning of the book, and provide a
generous assortment of references in the back of the book. For the remainder (majority of the 256 pages), I give the images as much room as possible so that not only can the patterns be photocopied easily (I went with spiral binding just for that reason), so that the image quality would be as high as possible for an 8.5″ by 11″ format.

“HOW-TO” INFO STARTS HERE:

Some of the illustrations give step-by-step procedures (in graphical form, assuming some basic familiarity with how to use a compass and straightedge, such as:

page 44 (showing how “unit cells” for the tiling patterns can be used to create an enormously expanded variety of additional patterns by recognizing how the space-filling shapes can be varied; this applies to the “unit cell” examples on pages 16-43,

page 45 (showing how each of the patterns on pages 16-43 can be embellished with fractal or other inscribed detail for each of the polygons for an infinite (literally!) variety of possible variations, which also apply to the majority of the remaining images in the book; page 237 gives a 3D example of this),

pages 46-84 have either explicit (most of these pages) or simple to observe implicit “unit cells” which show how these can be created in a great variety of ways,

page 84 (Pentagon Rotation Grid) gives the 73.2% proportion crucial to the exact construction of this pattern,

page 85 (Genesis of the Seed of Life) shows the step-by-step “compass only” construction of this important and universal pattern,

page 86-95 (variations on Seed of Life and Flower of Life) show how once the Seed of Life is constructed, many other patterns can be easily derived – Flower of Life, Hexagonal Grid, 2nd Harmonic Overlay which is used in Mika Feinberg’s beautiful LightSOURCE screensaver animation (no longer available); see my links (resources) page, Tree of Life, Fruit of Life, Heart and Ankh matrices, recursions, Metatron’s Cube, etc.,

page 95: since the Dodecahedron is the most complicated shape to derive from Metatron’s cube, the top center illustration on this page shows which vertices are used to create the “dodecahedron 2D shadow” with small circles highlighted in the larger image,

page 96: The general instructions for creating Nested Inscribed Polygons appear on this page,

pages 100-101-102: other examples of the crucial proportions needed to create these images either by hand or with a computer graphics program; numerous pages provide these instructions in the text without detracting from the space given to the images,

pages 104-105, 110-113, 123-124, 128, 130, 138-139, 144-145, 155, 163, 166, 171, 176-180, 182-187: all have instructions and details on how to create the images,

page 146: very detailed step-by-step instructions for inscribing a pentagon within a circle,

page 147: very detailed step-by-step instructions for inscribing a pentagon starting from one side of a given length,

page 156: very detailed step-by-step instructions for creating a golden rectangle (including “whirling squares and more),

page 156: very detailed step-by-step instructions for dividing a line by the golden ratio,

page 157: very detailed mathematical information about golden ratio progressions and powers, illustrated graphically,

page 188: shows how the Parthenon at the Acropolis in Athens, Greece, incorporates the golden ratio,

page 189: an amazing amount of data on this page about the Great Pyramid at Giza, Egypt showing phi (golden ratio) and pi proportions, and the proportions of the so-called “King’s Chamber” although the so-called “sarcophagus” (granite box) within is too large to fit through the only passage leading into that chamber, which violates the generally accepted funerary rite theory,

page 190: the classic “Measure of Man” (Vitruvian Man) by Leonardo daVinci, copied around the world, showing the golden ratio proportions in the human body,

page 191: the only known CAD drawing adaptation (to my knowledge) of Leonardo’s “Ideal Church” sketch,

page 192: detailed specifics about the Shoemaker’s Knife of Archimedes, giving several variations all showing the mathematical principle,

page 194: details of the geometry of the classic 1991 Barbury Castle, England crop circle formation,

page 195 and page 52: the details of the geometry encoded in the Sri Yantra (a classic Hindu mandala) and the cross section of the Great Pyramid at Giza, Egypt (also the “squaring the circle” conundrum), both with a 1-Phi-Square Root of Phi triangle which has a 51 degree, 51 minute slope,

page 196: numerous details common to the 5 Platonic and 13 Archimedean Solids,

pages 197-214; numerous details specific to the 5 Platonic and 13 Archimedean Solids, including how the insphere/intersphere/circumsphere/side length proportions are calculated,

pages 215-224: similar details specific to the 4 Kepler-Poinsot solids, the Star Tetrahedron (a.k.a. Stella Octangula) and related polyhedra,

pages 225-229: numerous graphics showing how the Platonic Solids relate to each other in a myriad of fascinating (awe-inspiring, really!) ways,

pages 230-235: numerous relationships between 3D polyhedra with 5-sided symmetry and the “shadow” they cast on a decagon (10-sided polygon), with construction details,

page 236: step-by-step instructions on creating an accurate drawing of the Icosahedron and Dodecahedron starting from a Golden Rectangle,

pages 237-255: generous appendices for hands-on explorers of all ages and levels of experience, including tables and charts of regular polygon angles, apothem, radius and side ratios and areas, radius ratios by coordination number for Ionic Chemical Bonding (which relates to properties of materials at the molecular and planetary levels), Fibonacci Numbers, Perfect Right Triangles (when I put the book together, I wasn’t aware of the Phi-1/Phi-Square Root of 3 Right Triangle that Mike Green of British Columbia introduced me to), Prime Numbers, extensive tables with all sorts of data on the Platonic and Archimedean Solids (useful for a variety of purposes, including model construction and computer simulation and animation, a map of planet Earth showing superimposed Platonic Solid Vertex Latitudes and (example) Longitudes, a fun stereogram with 6 Small Stellated Dodecahedra (there, I gave the clue  a short bio of myself, 3 pages of bibliography (more on my books and links pages and blog), and unique graphical index to all the illustrations in the book. Whew! I’d almost forgotten how much I packed into this labor of love over a decade ago!

Many of the 1300+ images are somewhat self-explanatory graphically (especially if you have created the basic shapes like the Seed of Life, Golden Rectangle, and a few others by hand with compass and straight-edge, which I highly recommend for anyone, as mentor Keith Critchlow so aptly reminded me when reviewing my original manuscript)…

… and of course, if you find re-creating one of the patterns challenging, I’d be happy to answer other questions via email so that I can share with other enthusiasts; thanks for asking!

Rithmomachia Game – David Joyner’s paper

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Here is a paper (87 pages in compiled LaTeX PDF format) of interest for math nerds from a long-time friend and cousin-in-law,* David Joyner. His stratospheric mathematical fluency and expertise (a retired math professor) leave me on terra firma, admiring his flights into esoteric math realms where my familiarity is negligible. Still, I’m trusting that it will resonate with the global math community. You can read more from David directly via his blog, Yet Another Mathblog

David’s book (Rithmomachia: is recreational arithmetic – a trimmed-down version of the paper) is also available on Amazon here.

David describes the Rithmomachia Game in detail in his document with this overview:

“The basis is in a Pythagorean philosophy of mathematics (where numbers and Platonic shapes have spiritual significance.)”

* I’m not sure if “cousin-in-law is the correct term. David & I met at my Aunt Gayle and Uncle Joe’s cabin at Lake Kirkwood, California, when we were in our late teens or thereabouts. David & I shared a beloved “Uncle Joe”; Joe was David’s uncle by blood, and my uncle by my aunt’s marriage. Can you tell I’m not a genealogist, either?

When David & I met, I had written up and hand-illustrated a few pages on the Golden Ratio, mostly gleaned from the two books below, both gifts to me in my adolescent years, both mentioning the golden ratio in different contexts – man-made architecture and natural geometric forms, which led me to explore what became known to many as “sacred geometry”:

David pursued his interest in math and became a professor! Happily, we reconnected some years ago, after he discovered this (GeometryCode.com) blog.

Here’s David’s treatise: The Recreational Arithmetic of Rithmomachia. Enjoy!

Rithmomachia gameboard: David Joyner
game board graphic: The Recreational Arithmetic of Rithmomachia by David Joyner