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Highlights from the February Post Archives on GeometryCode.com

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Welcome to our curated recap of some of the blog posts published in February across recent years on GeometryCode.com. Each entry below includes the post title, published date, and a graphic image from the post, presented as an enticement for the curious to review prior posts. (This is part 2 of a 12-part series; here are the Highlights from the January Post Archives on GeometryCode.com)

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

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

In case you missed it, here are the Highlights from the January Post Archives on GeometryCode.com

Highlights from the January Post Archives on GeometryCode.com

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I occasionally encounter ICYMI (In Case You Missed It) anthologies of archives from publications that have been around a while. Since that’s true for GeometryCode.com, I thought I would do the same for this site. I’m sure there are plenty of new (or newish) visitors who have probably never seen many of the prior posts from this website, as well as seasoned readers who may have missed or perhaps forgotten some of the content we’ve featured over the years… since 2009 (and before that for a decade or so on other websites such as our former online home, intent.com. (Thanks to the “Wayback Machine” for maintaining archives of our original, primitive (embarrassing by today’s standards) work in our early web explorations. The format and underlying architecture have changed significantly, but the content of “Sacred Geometry Home Page” hasn’t changed much since this snapshot from Jan. 15, 1998!)

With a bit of clerical assistance from AI, I’m presenting a retrospective (by month, starting with January) of some of the 222 prior posts from GeometryCode that might merit revisiting. However, I will still add new finds as they cross my path, so this quasi-sabbatical won’t be devoid of new content. If you (or someone you know) has something that would enhance this website, please let me know! Thanks for your many years of support while I spend a few months (probably a dozen) re-exploring perennial topics via prior posts with evergreen concepts, ideas, and imagery.

I am using chronological order (rather than the more conventional reverse chronological order), putting recent posts at the bottom to give buried content some breathing room for new or renewed exposure.

Welcome to our curated recap of some of the blog posts published in January across recent years on GeometryCode.com. Each entry below includes the post title, published date, and a graphic image from its content, presented as enticements for the curious to review prior content.

GeometryCode.com — Highlights from January Posts (from 2009 through 2025)

These last few summaries are courtesy of ChatGPT. Each entry below includes the post title, published date, and a multi-paragraph summary of its content.

  • Cedar Sprig Symbolism: Freedom or Fragmentation?
    Published January 2023

    In this thoughtful January post, Bruce Rawles reflects on participating in a local New Year’s Day Peace Hike in Yachats, Oregon. He uses the symbolism of cedar sprigs as a metaphor for both individual and collective identity. He connects the fractal geometry of the cedar — with its self-similar branching patterns — to how we might view unity and fragmentation in human thought and social life.

    The post weaves imagery from the hike with contemplative commentary on how symbol and form interact: the geometric complexity of cedar sprigs becomes a lens to examine inclusion, peace, and the fractal nature of experience. Rawles draws philosophical connections between natural patterns and inner psychological landscapes, prompting readers to consider whether they view life through lenses of separation or wholeness. This piece blends personal narrative with meditative geometry, exemplifying the blog’s broader mission of uniting mathematical form with lived experience.

  • 2024 Martineau Solar System Calendars Available Now
    Published January 2024 (note that calendars are available again for 2026!)

    This post announces the release of the 2024 Martineau Solar System Geometry wall calendars — a perennial favorite among readers interested in precise geometric representations of planetary orbit relationships. The calendar series is built from ray-traced images that model the intricate spatial relationships among the planets, combining astronomical accuracy with sacred geometry aesthetics. These calendars serve both as functional datekeepers and as contemplative art pieces that invite reflection on the harmony of celestial mechanics.

    Beyond simply promoting the calendar’s availability, the post elaborates on how each month’s artwork is grounded in long-observed geometric patterns linking planetary motion to geometric forms. It draws a line from scientific modeling to artistic expression, inviting readers to contemplate how geometry underlies both the heavens and artistic representation. The narrative suggests that such resources — whether calendars, prints, or patterns — function as mirrors to natural law and can enrich both intellectual and aesthetic appreciation.

    Martineau solar system calendar -12 months

  • M. C. Escher’s Geometry and Illusory Perspectives Revisited
    Published January 2025

    This engaging post reflects on the work of Dutch graphic artist M. C. Escher and the ways that his art embodies rich geometric ideas. The author revisits classic Escher works — including impossible constructions, symmetry patterns, tessellations, and recursive spaces — and ties them into broader themes about how the human mind perceives space and structure. Escher’s art serves as a springboard for discussing how geometry, often thought of as abstract and formal, can be visually vibrant and conceptually playful.

    Throughout the text, readers are encouraged to appreciate how Escher’s imagery not only plays with visual paradoxes but invites a willingness to consider multiple perspectives simultaneously — a theme with deep resonance in non-Euclidean geometry and spatial reasoning. There’s also a conversation about how such visual explorations can inspire geometric creativity and reframe the way we look at patterns, proportion, and dimensionality in both art and mathematics. The author hints that future posts will build on these provocations, particularly around non-Euclidean geometry and related artistic interpretations.

    3 interlocking Escher-inspired reptile tiles

  • Thanks for exploring these highlights from GeometryCode.com’s January archives!

Hyperbolic Tessellation Generator … M. C. Escher might have loved this!

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I recently found this post on Facebook:

FaceBook post: Hyperbolic tesselations made with Python code generated by Gemini 3
FaceBook post: Hyperbolic tesselations made with Python code generated by Gemini 3

It immediately reminded me of some similar images that inspired me decades ago by M.C. Escher, such as “Circle Limit I‘ (a woodcut dated November 1958) and this image from a post earlier this year:

Fellow geometer, David Joyner, recently suggested exploring Python and Blender to make more accurate geometric images with AI and/or automation tools, so that may be my next dive into geometric experimentation, dusting off my decades-neglected programming skills; various projects and dabblings with FORTRAN, Pascal, BASIC, C, HyperTalk, Lingo, HTML, CSS, PHP, JavaScript, POVRay, MegaPOV, and a few others.

Meanwhile, to whet my interest, I googled and easily found what appears to be the source code for the first image. My to-do list now has an attempt to replicate these geometric hyperbolic tessellations and try a few of my own using this resource on GitHub. It appears that I may need to complete some preliminary Python tutorials first; I’ll try using the W3Schools Python tutorial. If you have other resources you would recommend, contact me.

Here are a couple of examples from the GitHub source page:

A few German sacred geometry books available for shipping costs only

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Update 2 November 2025: We’ve been back in Oregon for a few weeks, and I still have a handful of German books to find homes for, so I’m re-dating this post in case anyone would like to purchase a book for only my shipping costs!

Update 21 July 2024: We’ve been in Arizona for exactly a year today, and I still have a handful of German books to find homes for, so I’m re-dating this post in case anyone would like to purchase a book for only my shipping costs!

(Originally published 25Jun2023):

We’re making good progress toward moving to Arizona (mid-July 2023), so I’m anticipating having more time to do in-depth posts after our move. Meanwhile, I’d still like to lighten our load and not move these books yet another time… I’m offering (until mid-July, while supplies last) – limited supply; limited offer) asking only for my shipping costs (US only) or free if you happen to be on the central Oregon coast and can pick them up here in Yachats before we move! in the Tucson, AZ, Portland, OR metro area, and can pick them up.

German sacred geometry books are available for shipping only (for a limited time!)

I have a handful of both of my German-translated books:

(If you live in Germany, you’ll probably want to order the Bewusst malen – Heilige Geometrie book, still in print, directly from the publisher, Silberschnur Verlag, since the shipping from the USA to Germany is likely more than buying it locally.)

If you speak German or know someone who does and has the least interest in geometry, I am offering special FREE books (very limited supply) of both of these books if you cover my postage costs. If you live near Yachats, Oregon, Green Valley, Arizona, Hillsboro, Oregon, you can pick them up here and skip the postage expense. For details on these limited-time, limited-supply books, contact me.

Note that you don’t have to be able to read or understand German to enjoy either of these books, as they are both lavishly illustrated – FAR more imagery than text. Also, each copy of the out-of-print MDHG book has a unique set of die-cut fold-up models of the 5 Platonic Solids (included behind the back cover) that make a neat mobile if you add your own thread.

MDHG Platonic Solid fold-up mobile

Please contact me before mid-July 2023 if you are interested in this special offer!

Mandalas der Heiligen Geometrie (MDHG): German edition of Sacred Geometry Design SourceBook by Bruce Rawles - GeometryCode.com

sacred geometry coloring book (German): Bewusst malen - Heilige Geometrie by Bruce Rawles

Here are a few more interesting geometry-related links (videos) that have crossed my path recently; several short introductory videos that add some useful and interesting additions to commonly cited facts:

  • Golden Spiral Symbol – What Does It Mean?
  • What Makes the Golden Spiral Symbol Special?
  • What is Sacred Geometry? (evidently using an AI-generated voice/script?)
  • How Does the Golden Spiral Relate to Sacred Geometry?
  • What is the Meaning of the Golden Spiral?
  • What are Some Examples of the Golden Spiral?
  • What is the Fibonacci Sequence?
  • … and several others…

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!