I just updated this sacred geometry calendar which has images from the fascinating geometric Martineau Solar System models. Here are details about the 12 images which are highly accurate geometric models of pairs of planetary orbits in our solar system. They were inspired by the remarkable work of John Martineau (publisher of Wooden Books), who wrote “A Little Book of Coincidence in the Solar System.” Tiled images of planetary surfaces and related images form the backgrounds. Images are two or more orbits defined by simple 2D (circles, triangles, squares, pentagons, hexagons, octagons) or 3D geometries (e.g. tetrahedron, icosahedron, dodecahedron) typically with 99.9% or greater accuracy. The work picks up where Johannes Kepler’s mystical exploration left off.
I printed and published the first copies of Sacred Geometry Design Sourcebook (SDGS) in December 1997. It took a year of intensive graphics work using Claris CAD – does anyone remember that Apple application? – to convert and extend numerous hand-drawn geometric illustrations into digital form. My wife (Nancy Bolton-Rawles) took an evening class and I started with an evening a week – while she was in class – learning how to use this early CAD program for something other than video routing switcher control panels and making these geometric archetypes with far more accuracy and resolution than I could possibly achieve by straightedge and compass (the classic geometer’s tools) or any other physical mechanical means. Those evenings quickly overflowed into seemingly every conceivable spare moment. I realized that these CAD drawings (plus a few other unique images such as a stereogram, some unusual tables, charts, spreadsheets and even a cameo illustration of a sphere by Nancy (who was taking art classes during that time) merited a book. There are over 1300 geometric CAD illustrations counting all the variations I placed in the margins of many of the larger full page (8.5″ x 11″) images.
It’s amazing to reflect on the journey of explorations that have branched off (fractals, anyone?) from this first book by countless colleagues. I am profoundly grateful to one and all for the links, shared discoveries, and inestimable support in 2 dimensions, 3 dimensions, 4 dimensions and beyond over the past two decades … and before and after that, too – since we’re all interconnected beyond time and space! It’s been wonderful fun to get emails, postal mails of unexpected geometrically-inspired gifts, and other correspondence via Facebook (GeometryCode.com and Geometry Code) plus other social media, etc. Thanks, everyone, for your generous and steady contributions and support over the past two decades! 🙂
If you are new to this website (or would like to peruse some of these topics again), you might want to start with this introduction to sacred geometry and then explore many years of bulletins, posts, and articles about related geometric coloring books – of which SGDS is a prime example! – and other sacred geometry art, books, calendars, food, audios and interviews, jewelry, music, news, physics, spaces, toys and videos, the golden ratio, Fibonacci numbers, other interesting numbers and proportions, the Platonic Solids and Archimedean Solids, other polyhedra, plus even some interconnections with the ancient, timeless Hermetic Laws and their relationship to modern physics … and metaphysics … which led to my second book, The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation.
To commemorate this 20th anniversary month (December 2017) of SGDS, here is an illustration that I added later, but didn’t make it into the book: a fold-up pattern for the Rhombic Triacontahedron, also known as Kepler’s Solid, which is an example of a zonohedron.
Here are the pages and posts on this website that refer to SGDS; enjoy! 🙂
Last, but not least, I’m copying the content from a personal blog post (from a decade ago) which provides details on many of the images in SGDS, including how to draw many of the 2-D patterns (which took the form of an email Q and A response):
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 in 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), the image quality would be as high as possible for an 8.5″ by 11″ format.
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 embellished with fractal or other inscribed detail for each of the polygons for an infinite (literally!) variety of possible variations, (which actually also applies 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, so 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; see my 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,
page 100-102: other examples of the crucial proportions needed to create these image 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 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 away , a short bio of myself, 3 pages of bibliography (more on my 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 are finding re-creating one of the patterns challenging, I’d be happy to answer other questions via email that I can share with other enthusiasts on my blog (which I’ll do with this reply; thanks for asking! 🙂
I’m also working on a number of related projects that will complement the book with video “hands on” procedures, etc. Stay tuned!
A new iOS app called Inspirit provides (hexagonal symmetry) kaleidoscopic animation imagery. It appears that there will be options to show either a single hexagon (1 image cropped, reflected and rotated in traditional kaliedoscopic fashion), as well as a fully tiled image (with the hexagonal “cell” translated vertically and horizontally to file a honeycomb (hexagonal) grid. Example video here:
(Thanks to Luke G. for alerting us to this new item.)
Thanks to geometer colleague and dream researcher, Ed Kellogg for alerting me to this very interesting (and fun) video “What Is Reality?” from Quantum Gravity Research (check out the lovely video loop of a slowly rotating polyhedron made of tetrahedra on their website’s home page) – in addition to meshing interconnections with our faithful omnipresent proportion, the Golden Ratio – also talks about higher dimensional geometric polytopes, Planck’s time and space constants, meaning, self-representing symbols, pixelation, geometric codes, non-local information … of course, consciousness – that ‘elephant in the room’ that mystics (not just in India) have been riding for millennia. 🙂
“The (8-dimensional) E8 Lattice … to generate that 3D quasicrystal, the substructure at the pixelated fabric of reality, we project this 8D crystal to 4D, and then we convert that to 3D. … just like the basic shape of the 3D cubic lattice is the cube, the cell shape of the E8 lattice is an 8D shape with 240 vertices. We call it the Gosset Polytope. When the Gosset Polytope is projected to 4D, it becomes two identical shapes of different sizes. The ratio of their sizes is … 0.618 (the Golden Ratio.)”
The video – after mercilessly exploding a cubic lattice made of Zometool, tsk, tsk! 🙂 – then references the appearance of the Golden Ratio in an equilateral triangle inscribed in a circle which we have documented on this website for many years.
“The Golden Ratio may be the fundamental constant of nature. … It is weirdly ubiquitous in the universe, appearing everywhere from the quantum to celestial scales. … it appears in black holes. The golden ratio is the precise point where a black hole’s modified specific heat changes from positive to negative. Ø = (M^4) / (J^2) … and it is part of the equation for the lower bound on black hole entropy. The golden ratio even relates the loop quantum gravity parameter to black hole entropy. Ø = 2 ^ (π𝛾) … Why does this support the claim that the golden ratio is the fundamental constant of nature? Because a theory of everything must unite general relativity with quantum mechanics and a black hole is where these two theories converge at their limits.”
“Meaning is subjective and requires choice.” … Seems like physics is treading on metaphysical turf again. Fun! 🙂
To dive a bit deeper into the projected polyhedral shape (which evidently resolves to an icosidodecahedron), check out this cool video – featuring Klee Irwin of Quantum Gravity Research – (“The 20-Group Twist”.)
For further detail, check out this video: “Quantum Gravity Research: an Overview Presented by Klee Irwin.”
“I felt glad to see this model incorporates E8 theory. I met Garrett Lisi when I attended his presentation on E8 theory (“An Exceptionally Simple Theory of Everything”) when I went to the 2010 Joint Mathematics Conference in San Francisco years back and felt quite impressed, but had heard nothing since. (See attached paper and check out this video link: https://www.youtube.com/watch?v=-xHw9zcCvRQ ).
I really enjoyed QGR’s ”What Is Reality” video, as it brings together a whole set of variable’s I’ve had an interest in – sacred geometry, consciousness, the golden ratio, the universe represented in a pixilated interface, derived from a non-local beyond time and space information matrix code, etc. all integrated into one congruent and entertaining presentation.
I especially found it quite cool that the golden ratio pops up as the ratio between the two sizes of the projection of the 8D Gosset polytope (from E8 theory) onto 4D – I’d never heard of that.
… check out this paper also: “Quantum Walk on a Spin Network and the
Golden Ratio as the Fundamental Constant of Nature,” http://www.quantumgravityresearch.org/wp-content/uploads/2017/04/quantum-walk-spin-31.pdf
One possibly important insight that occurred to me. The narrator talked about tetrahedrons as fundamental units (“physical reality pixels”), with each of them having different states. This reminded me of Donald Hoffman’s Conscious Agents, CAs, which he represented in a first rough approximation as triangular shapes/processes, functioning analogously in a manner somewhat similar to Turing machines:
From “Objects of Consciousness” (https://www.academia.edu/8227575/Objects_of_consciousness),
Even this simple model of interacting hierarchies of CAs leads to the identification of “a wavefunction ψ of the free particle with a harmonic function g of a space-time Markov chain of interacting conscious agents.”As a next step, bringing in another dimension of processing for more complex CAs, one would add another point to create higher dimensional tetrahedral CAs (“TCAs”), as it would add one more fundamental point, and three more communication channels might lead to the emergence of many more equations from the interaction of hierarchies of more complex TCAs. If so, what might that additional “point” represent, and what might the communication channels ? Something I find quite fun to think and to speculate about!
This has already given me ideas about what an upgrade of the “Beyond the Matrix: Conscious Realism and Lucid Being” workshop that I just gave at the Anaheim IASD conference might additionally include, in version 2.0. <g>”
This is an impromptu video conversation about Quantum Field Theory and Modern Physics in response to a recent post inspired by an email from researcher and geometer colleague, Frank van den Bovenkamp. Bruce Rawles and Frank converse about this fascinating and arcane world, weaving some of his new ideas ideas in with metaphysics and other related topics. Be sure to read the prior inter-related post: Golden Ratio and Quantum Field Theory – Frank van den Bovenkamp.
Here are some links mentioned in the conversation and related sites :
- A Quantum Theory of Microvita
- Shrii P.R. Sarkar’s discourses on microvita and cosmology – a selection
- Frank van den Bovenkamp’s YouTube Channel
- video: A Path Integral based on Synchronized Quantum Field Fluctuations
- Fractal Heart – Hunting the Hidden Dimension (PBS documentary excerpt)