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You are here: Home / Archives for Numbers and Proportions

Numbers and Proportions

Highlights from the January Post Archives on GeometryCode.com

Thursday, 1 January 2026 by Bruce Rawles

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)

  • Intention and Symbols
    Published January 2009
    Villarceau circles: torus plane intersection yields vesica piscis cross-section
  • QR code for GeometryCode.com
    Published January 2011
    QRcode for www.GeometryCode.com
  • ZomeTool sculpture at Denver Art Museum
    Published January 2012
    ZomeTool-DenverArtMuseum2
  • Sacred Geometry bulletin for January 2013
    Published January 2013
    mirrored pentagon grid
  • Local Author event in Castle Rock, CO 18Jan2014
    Published January 2014
    Local Author Showcase - Castle Rock, CO
  • GeometryCode bulletin for January 2015
    Published January 2015
    7thTermFibonacciGarden-MichaelSchneider
  • The Dance of Venus
    Published January 2016
    The Dance Of Venus poster by John Martineau
  • Multiplication Tables on Circular String Art lead to Mandelbrot and classic curves
    Published January 2017
    Mathologer - String Art Mandelbrot
  • Assorted 3D model geometry links of interest
    Published January 2019
    HexaSticks 3D model
  • 3D Geodesic Animation inspiration
    Published January 2020
  • Geometric Lamps … and Coloring Pages by HYBYCOZO / COZO
    Published January 2022

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!

Filed Under: 2012 Bulletins, 2013 Bulletins, 2014 Bulletins, 2015 Bulletins, 2D Geometries, 3D Geometries, coloring books for adults, Fibonacci Numbers, Fractal Geometry, golden ratio, polyhedra, sacred geometry animation, sacred geometry architecture, sacred geometry art, sacred geometry books, sacred geometry calendars, sacred geometry metaphysics, sacred geometry news, sacred geometry toys, sacred geometry videos

A few German sacred geometry books available for shipping costs only

Sunday, 2 November 2025 by Bruce Rawles

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:

  • Mandalas der Heiligen Geometrie (MDHG) das universale Design der Natur (SGDS German edition)
  • Bewusst malen – Heilige Geometrie

(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…

Filed Under: 2D Geometries, 3D Geometries, coloring books for adults, Fractal Geometry, golden ratio, Platonic Solids, polyhedra, sacred geometry art, sacred geometry books

Vandorn Hinnant – 45 Years of Dreaming with Open Eyes – Exhibition Catalogue

Friday, 4 April 2025 by Bruce Rawles

For many years, I’ve admired the geometric art of geometer colleague and artist Vandorn Hinnant; this month’s post features a pdf file of a retrospective of four and a half decades of his geometry-inspired artistry. Enjoy! Be sure to visit his websites LightWeavings.com and VandornHinnant.com for much more info.

Vandorn Hinnant – 45 Years of Dreaming with Open Eyes – Exhibition Catalogue

Download the pdf here.

The Title/Cover page provides the details of the live exhibit which runs from March 20 through June 28, 2025 at the Mary G. Hardin Center for the Cultural Arts, 501 Broad Street, Gadsden, Alabama.

Here are just a few examples of his inspired/inspiring art:

Vandorn Hinnant: "Times Golden Arrow” a.k.a. "Golden Rectangle Mandala” ©2010
Vandorn Hinnant: “Times Golden Arrow” a.k.a. “Golden Rectangle Mandala” ©2010
Vandorn Hinnant: The Truest Eye a.k.a. Root Three Spiraling Fractal Trigons ©1997
Vandorn Hinnant: The Truest Eye a.k.a. Root Three Spiraling Fractal Trigons ©1997
Vandorn Hinnant: Ark ©1997
Vandorn Hinnant: Ark ©1997
Vandorn Hinnant: Before We Knew - From The MOTHER of Everything ©2002
Vandorn Hinnant: Before We Knew – From The MOTHER of Everything ©2002

Filed Under: 2D Geometries, golden ratio, sacred geometry art, sacred geometry news

Encyclopedia of Polyhedra By George W. Hart … and other geometric gems

Sunday, 16 March 2025 by Bruce Rawles

Apples and Oranges by George W. Hart https://www.georgehart.com/Applorng.html
Apples and Oranges by George W. Hart
https://www.georgehart.com/Applorng.html

It’s time to revisit the work of other polyhedra enthusiasts. Of particular note (having crossed my digital path again recently) is George W. Hart who has shared some excellent resources along these lines (faces, vertices, etc.) for decades. Here’s one gold mine to explore: Virtual Polyhedra: The Encyclopedia of Polyhedra By George W. Hart. Here is part of the table of contents and highly recommended if you want to explore the genres and categories of polyhedra; lots of great imagery, rotatable 3D models (rotate them with your mouse), facts and details:

  • Platonic Solids (Regular Convex Polyhedra) Background List of models
  • Kepler-Poinsot Polyhedra (Regular NonConvex Polyhedra)Background List of models
  • Archimedean Polyhedra (Semi-Regular Convex Polyhedra) Background List of models
  • Prisms and Anti-Prisms Background List of models
  • Archimedean Duals Background List of models
  • Quasi-Regular Polyhedra Background List of Models
  • Johnson Solids (the remaining convex polyhedra with regular faces) Background List of models
  • Pyramids, Dipyramids, and Trapezohedra Background List of models
  • Compound Polyhedra — Introduction Background List of models
  • Stellated Polyhedra — Introduction Background List of models
  • Compounds of Cubes Background List of models
  • Convex Deltahedra Background List of models
  • Zonohedra Background List of models
  • Uniform Polyhedra Background List of models
  • Uniform Compounds of Uniform Polyhedra Background List of models
  • Stellations of the Icosahedron Background List of models
  • Stellations of the Rhombic Triacontahedron Background List of models

Perhaps the most well-known and ubiquitous polyhedral shapes have fold-up models (and related patterns) for the first 3 categories above in Sacred Geometry Design Sourcebook (SGDS pages 196-220) and also on this website for the 5 Platonic Solids and 13 Archimedean Solids as well as Illustrations of Platonic And Archimedean Solids, Math Tables for Platonic And Archimedean Solids (and much more about polyhedra on this website.

compound of five cubes by George W. Hart
compound of five cubes by George W. Hart

If you exhaust this generous resource (or somehow overlook a few gems) here are more of his contributions – the videos are particularly helpful if you want to replicate his constructions:

  • The Pavilion of Polyhedreality
  • George’s Instagram page with some excellent 3D geometric sculptures
  • Mathematical Impressions: The Golden Ratio (short video)
  • Hyperboloids (short video)
  • Little Zonohedral Library (short video; my favorite of these which also features our mutual friend Russell Towle who showed me his 3D zonohedral models (such as Rhombic Spirallohedra)
    years ago in Dutch Flat, California)
  • Ceci n’est pas une lampe (fun, clever, short video)
  • Birdland (another clever short video)
  • Seven Slide-Together Constructions (another interesting short video)
  • … and MANY more!
Stained Glass Ball By George W. Hart https://www.georgehart.com/stained_glass.html
Stained Glass Ball By George W. Hart
https://www.georgehart.com/stained_glass.html

Filed Under: 2D Geometries, 3D Geometries, applications, Archimedean Solids, golden ratio, Platonic Solids, polyhedra, sacred geometry art

Geometric gemstones

Friday, 6 December 2024 by Bruce Rawles

triangular geometric gemstone: Hashnu trioptic blastRecently a friend (Rick Fuller) showed me his gemstone grinding equipment including two grinders; one with a caliper-style readout to set grinding angles and another with a digital readout. My experience in this area is negligible, but the designs of cut gemstones seem profoundly connected to 3-dimensional solid geometry and all the laws, theorems, and insights gleaned over centuries about geometric proportions and more. Geometry is everywhere, and the geometric (polygonal) facets of countless gemstones are an excellent example.

Thinking about both the microscopic and macroscopic geometries involved and how this might be optimized in this endeavor, I did a quick online search for “gemstones cut to angles that match atomic lattice structure” and found this interesting AI-generated response:

Gemstones and Lattice Structure

Based on the search results, Corundum (Al2O3) is a crystalline mineral that exhibits a hexagonal crystal system. Its lattice parameters are well-defined, and the angles between its crystal axes are fixed at 90° and 120°.

In the context of gemstone cutting, Corundum is notable for its unique properties. Rubies and sapphires, which are varieties of Corundum, can be cut to specific angles to enhance their optical properties and brilliance. The cutting angles are designed to match the atomic lattice structure of the mineral, taking into account the hexagonal symmetry of the crystal.

For example, the refractive indices and birefringence of Corundum can be optimized by cutting the gemstone at specific angles, such as:

  • The “ideal cut” for rubies and sapphires, which involves cutting the stone with a pavilion angle of around 40-41° and a crown angle of around 25-30°. This angle combination minimizes internal reflections and maximizes brilliance.
  • The “step-cut” or “tabby cut,” which involves cutting the stone with a series of flat, parallel surfaces. This cut style is often used for sapphires and takes advantage of Corundum’s hexagonal symmetry to reduce birefringence and improve color distribution.

While other gemstones may not have the same level of precision in their cutting angles, Corundum’s unique properties make it an ideal candidate for cutting to angles that match its atomic lattice structure. This attention to detail can result in more vibrant colors, higher brilliance, and a more pleasing visual appearance.

Searching for related topics led to these quasi-random resources:

  • Hashnu Stones & Gems (the site listed in the book my friend showed me; the first to photos are from his book)
  • Gem Stone Geometry (Ceylon Natural Gems)
  • Gemstone Encylopedia (315 Gemstones from Actinolite to Zunyite)
  • Geometric Shape Gemstones (Etsy)
  • Solid geometry (Wolfram Mathworld)
  • Euclidean geometry (an excellent and extensive article)

… and since we’re on the topic of geometric jewelry, check out Sacred geometry jewelry by artist David Weitzman (Ka-Gold) … Related Content: Flower of Life, Seed of Life, Golden Spiral, Fruit of Life, Vesica Piscis, Star tetrahedron “Merkaba”, Fibonacci Whirling Squares Spiral “Phi”, Metatron’s Cube, Chambered Nautilus, Labyrinth, Torus Tube, Tetractys, Sri Yantra mandala, Tree of Life, Hexagram (Star of David).

pentagonal geometric gemstone

Filed Under: 3D Geometries, Numbers and Proportions, polyhedra, sacred geometry jewelry

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Sacred geometry Jewelry Sacred geometry jewelry by artist David Weitzman Ka-Gold Related Content: Flower of Life, Seed of Life, Golden Spiral, Fruit of Life, Vesica Piscis, Star tetrahedron "Merkaba", Fibonacci Whirling Squares Spiral "Phi", Metatron's Cube, Chambered Nautilus, Labyrinth, Torus Tube, Tetractys, Sri Yantra mandala, Tree of Life, Hexagram (Star of David).
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