Intention and Symbols

Today's post is an edited response to an email I just received, asking about the pentagram (or pentacle) and it's cultural associations. There is more on about this on my other site, and this blog site


Hello again. It's been a few years since I last emailed you and this time I have to ask you for some advice. ... I have been interested in sacred geometry since I visited your previous site. I think it was called 'Intent'. ... I felt a strong correlation between the traditions of ..., sacred geometry, alchemy and mystic symbolism.

I am currently designing a range of posters around the four seasons which is loosely based on I think it's the 'Ryder' tarot deck. I am drawn to using some of the symbolism but the one that I am having a bit of difficulty deciding on, is the use of the Pentacle or Pentagram. I think it is also referred to the 'morning star'.

The rub for me is not so much the symbols themselves but their association with Illuminati or satanic symbolism that is used as subliminals in some advertising. I might be barking up the wrong tree but I feel the miss use of these elements might be clouding the higher message that these symbolic elements and resultant graphic language hold for humanity. This is the message that I understand in your work. I really want to help people realise a higher potential and level of conciousness in themselves but the last thing I wish to do is propagate any luciferic conciousness.

I guess my confusion stems from not knowing enough, and maybe I shouldn't be meddling in things I don't understand. Could you please steer me in the right direction with regard to this issue.

Sincerely (name omitted)


To respond to your question about pentagrams, or pentacles, I always like to "back up" to a more fundamental question. What is the thought system, intention, perspective or attitude of the question? If it is an inclusive, all-encompassing, transpersonal one, then the issues around any malevolence dissapear, since the symbol, regardless of the form, because the content will be one of kindness and integrity. If the thought system is divisive, petty, personal, based on the fantasy of separation that inevitably occurs when we take sensory data as evidence of isolation or differences, then we will see whatever symbol appears to us from an egoic orientation.

In the specific case of pentagrams, or pentacles, synchronistically I've been reviewing the movie "The DaVinci Code" this week and the "Langdon" character played by Tom Hanks in the movie describes (I'm paraphrasing here) how the pentagram has been used by pagan cultures as a symbol for the divine feminine principle. I would suggest that any shape with 5-sided (pentagonal) symmetry intrinsically has metaphoric connections to the principle of interconnectedness, since the pentagram has the golden ratio (approximately 1.618) encoded in every proportion. There are numerous references on my website (particularly the tutorial page) about how this ratio represents the idea that there is no separation, with examples from the cross section of every DNA molecule (and also the shape of several of the molecular chains that are attached to each turn of its helical design, to spiral galaxies, Chambered Nautilus shells and countless other examples in nature, art and architecture, including the Great Pyramid at Giza, Egypt.

Like anything else, the motivation is the key. We can use a hammer to build or destroy a house, and our thoughts and the intentions they serve are no different in this regard. Historically, there have been associations with the pentagram and both benevolent and malevolent intentions. But the same can be said about any other shape, symbol or design. The swastika has been used by Nazi Germany for an emblem of world conquest, yet also by native Americans to represent healing.

The vesica piscis has been used by a major financial institution as their commercial (credit card) logo, on the cover of the Chalice Well in Glastonbury, Somerset, England and countless examples of medieval religious art to represent Christ.

(I also found this interesting geometric connection with the torus, while exploring this idea this morning):

One might suggest that the PRINCIPLE of Christ (that there IS no separation) is what is being depicted in this symbol. Returning to the original premise of this writing, perhaps one can (and I make every effort to do so! Happy recall that ANY symbol can be used in the service of truth (beyond our finite perceptions limited by space, time, matter and duality) or enslaved by the realm of apparent separation and its attendant motivations.

I hope this is helpful!

LightSOURCE Thanksgiving Sacred Geometry Sale

I just got this announcement from my colleagues at LightSOURCE who make brilliant, lovely DVDs, screensavers (including our collaboration on GeometryCode), etc., so I'm passing this along:

Announcing our
Annual Thanksgiving Sacred Geometry Sale, our way of expressing our gratitude to cherished friends and fellow appreciators of Sacred Geometry:

Come see our new Sacred Geometry website with new product additions including:
-           "Hanta Yo" - Native American Power Shields, DVD, with "The Shaman's Heart" Hemi-Sync 5.1 Surround Sound soundtrack
-          LightSOURCE videos for iPod and iPhone
-          Sacred Geometry Stickers and Note Cards, and more.  

Every product is on sale.  Order now through Thanksgiving and save 20-25% on all products plus receive a free Sacred Geometry Gift. 

Save 20% on all 3-D Sacred Geometry Forms and Pendants. 

Buy one Sacred Geometry Poster and receive another of your choice FREE. 

For an additional 5% savings (25% total) on all CD-ROMs, DVDs, digital downloads, fine art prints, T-shirts, Note Cards & Stickers, enter coupon code: 11272008 

Every physical (non-digital) order will receive a complimentary CD-ROM of Sacred Gaiametry, The Magic of Crop Circles. 

Each digital order (downloadable) order will receive a complimentary download of Hot Ice, featuring animated crystalline snowflakes. 

Click the link below to visit the Internet's Premiere Sacred Geometry website:


Orders must be placed before 12:00 AM Sunday, 11/30/2008, discounts will be subtracted from each product at checkout.  Sacred-Gaiametry CD-ROMs may be shipped separately from main order but shipment will not be delayed, download link for Hot Ice will be provided upon completion of the checkout process.
CodeName: MANA LLC
18624 Downing Rd
Kearney, MO

8-8-8 Crop Circle

Thanks to Asha and several other friends (including my Mom who usually sends me these links) for alerting me to this lovely 8-8-8 Crop Formation.

In addition to the overall "figure 8" shape, I particularly like the rhythmic expansion and contraction of the "bead circles" which
corresponds to the inward and outward pointing arrowheads. Neat! As with any symbol, it can either distract us with emphasis on the form (and its implied dualistic nature), or serve to bring our mind toward wholeness, peace and pure non-dualistic awareness... I recommend the latter thought system!

Pacific Domes

A long overdue thanks to local friend and colleauge Asha who has been graciously hosting our local Geometers group (we whimsically call ourselves the "Hedronists" as well) for many months. (Also thanks to local friend and colleauge Libby who offered similar gracious hosting for many months prior!) Asha also heads up Pacific Domes which is doing wonderful things around the (oblate spheroidal) globe with Buckminster Fuller's geodesic invention.

I had the pleasure of building and living in a dome (a Cathedralite Dome kit from Aptos, California) in Georgetown, California in the late 1970's. In addition to just being a great space to be in, since we left quite a bit of our floor plan open, the acoustics were quite interesting, too! A neighbor (and long time mentor, Floyd Edwards who started the Creative Dynamics program (Leadership and Human Relations) my ex-wife and I enjoyed teaching in nearby adult schools for several years) had made a custom battery-operated clock with his airbrushed art, which hung on the wall downstairs next to the woodstove. At the time, we had a slant-board in the upstairs loft (looking up at the top 5 triangles of the dome) for meditation. One day, I realized the loud "tick-tick" I kept hearing was the minuscule sound of this little clock bouncing around the interior surface of the 39' diameter dome... and converging back at a point where my head was... amazingly, one could barely hear the clock standing next to it!

Another fond memory of the dome was the day our generous friends and neighbors spent most of a day helping us raise the 60 plywood-over-beveled-2x4-triangles into place. After hoisting 5 pentagons (all made of triangles) atop a 5' riser wall, then linking them together with 5 hexagons (also made of similar trianges) so we could now remove the supports and (after raising the scaffolding to get to the 21' peak), we finally put the last top 5 triangles of the dome into place. The second to the last triangle was fitted with a sturdy eye bolt, through which a long rope was tied. I got to hop up onto the now-sturdy roof with a sledge hammer and sequentially tap the 3 corners until the last triangle fit perfectly snug and the whole structure (just like Bucky professed) went from a loose wobbly thing to rock 'solid' (not accounting for the vast emptiness of quantum space, of course! Happy To help visualize this 3/8 sphere dome, check out
Poly from Pedagoguery Software. and animate the 3-Frequency Icosahedral Geodesic Sphere (and imagine 5 pentagons (defaulting to red color) resting on a 'cylinder'. Here's a "bird's eye view" of the dome we built and lived in, using a screen snap from Poly with appropriate triangles grayed out and more dome resources.



Pi Day

Happy Pi (approx. 3.14) Day! Of course, both Pi and Phi are encoded in the Great Pyramid.

Hyperdimensional Geometry of Music

Thanks to Ed K. for this neat article (with animation) on the Hyperdimensional Geometry of Music

Blueberry Stars

This was such a great question, I thought it would make a perfect blog post du jour!

Hi Bruce ...What does the star mean in sacred geometry?

This might depend somewhat on the specific geometry involved... In particular, are we talking about 5 or 6 (or some other number) of points on the star? Hexagonal (6-fold) stars (a.k.a. hexagrams) relate to the Flower of Life, Seed of Life, Fruit of Life, 2D matrix which occurs in a myriad of forms in the various kingdoms of the natural world (basalt column cross-sections in the mineral kingdom, cloverleaf and cellular structures in the plant kingdom, honeycomb in the animal kingdom, etc. just to name a few.

I am wondering if you have any idea what the star on the bottom of each blueberry would mean?

A little gem I learned from
Michael Schneider is that the blossom of every edible fruit has 5-sided (pentagonal) geometry. The five-pointed star (pentagram) was the symbol allegedly drawn on the hands of the Pythagorean initiates (perhaps because the notion of interconnectedness leading to the awareness of all-encompassing oneness is just as "threatening" now as it was to the predominant culture then Happy There is also a wealth of other examples in my book:
as well as quite a few on
the tutorial on my website (I imagine you've already read that?)
and the
links on the resources page:

I am doing food research and it seems like the star would mean something.  Doesn’t it look like a star at the bottom of a blueberry?

You got it! It is indeed the guiding archetype for this form, as well as so many others in nature! Happy

GeometryCode 2007 Contest Winner!

Congratulations to Judy Cole of Healing Partnerships ( for winning our 2007 contest! Judy submitted a copy of her exquisite Lissajous oscilloscope patterns "Healix - The Music of the Spheres: Mirror One version 2) DVD and says she was inspired by Sacred Geometry Design Sourcebook. Judy wins a copy of the GeometryCode screensaver and eBooklet. Thanks, Judy! Happy

Here's a static image of some of her engaging animation:

Pasted Graphic

I have fond memories of staying after hours in my high school electronics lab in Livermore, California, hooking up every oscillator in the lab to the horizontal and vertical inputs of a 'scope and creating endless variations of patterns like these. Judy uses interesting (close to Phi-based) harmonic ratios and elegant transitions to show how many of the complex patterns we're familiar with are simply the communion of number within the mirage of space time.

Ed Kellogg article on Trans-Personal Dreaming Beyond Time and Space

Thanks to Ed Kellogg for this superb article; here are his notes about source and reference material...

Glad you enjoyed my abstract - feel free to post it - or selections thereof - on your blog.
You can find my abstract online here:
Scroll down about halfway below Rita Dwyer, who chaired the "Cosmic Dreaming" session.
You can also find the abstract for my dream healing workshop here:
Scroll down to Kellogg.
They recorded two of my sessions, "Cosmic Dream Connections" (#306) and "Exploring Inner Space "(#321.)

"Trans-Personal Dreaming Beyond Time and Space."

Short Abstract:
Mystics through the ages have taught that although humans have separate personalities, that at a deeper level we share the same greater Self. The validation of psi-dreaming supports this idea, and has demonstrated that through dreams we can connect with people, places, and times unrelated to our waking lives.

Summary: Some models of consciousness posit that although humans have separate personalities, that at the deepest level we all share the same greater Self. As a metaphor think of our waking selves as leaves upon a tree - although the leaves differ from one another, they all belong to the same Tree and share the same trunk and roots. In this model, telepathy would not involve transfer of information over a distance between separate minds, but of simply accessing information by going within to find what the ‘Greater You’ already knows. This "Tree of Life" model of creation has found support in the writings and experiences of mystics through the ages. In recent years, scientific research under controlled conditions has demonstrated that through psi-dreaming we can connect with people, places, and times unrelated to our waking lives.

How might this manifest in our dreams? I've noticed over the years that while many people often dream of themselves in ways that closely match their waking physical reality experiences, that these people report also dreams in which they become someone else, or even something else, and experience the world from a very different perspective than in their waking lives. In my own case, I've experienced myself in dreams as many different beings - age, sex, race, etc. Often not only does my dreambody differ from its waking physical reality counterpart, but so does my dream personality and memories. However, through all this, my essential sense of self somehow remains the same - "I" remain "me", my dreambody, personality and memories seem no more "me" than does the clothing I wear or the car that I drive. I’d estimate that in 80% of dreams my dream self differs significantly in some way from my waking physical reality self.

Although I believe that although some dreams of this type may have to do with personality "aspects", that for most of them this explanation proves inadequate. Instead I favor different interpretations, in that I experience while dreaming parallel selves, past lives, future lives, and even other dimensional lives, and that tuning into different locations and different beings throughout the Multiverse just seems a routine and normal activity for my dreaming Self. And after many years of such experiences, I've finally come up with an answer to the perennial "Why do we dream?" question that rings true for me. Dreams provide "food for the soul", reconnecting us to our greater Selves and allowing a therapeutic release from the constraints of a time-space bound existence that reminds us, if only unconsciously, of the illusion of separateness.

Richard Feather Anderson slide shows and workshop in Ashland, Oregon

(Alas; all RFA events in this post are CANCELLED; as of 23 July 2007; the workshop might be rescheduled around Nov. 3-4, 2007)

Richard Feather Anderson
will be giving a slideshow "The Eight Patterns of Nature" on July 25, another slideshow "Geometry That Matters: Power of Creation, Laws of Harmony" on July 27, and a weekend workshop "Exploring the Symbolic Language of Geometry" July 28-29, 2007 (Friday evening through Sunday) in Ashland, Oregon. I had the pleasure of meeting Richard while attending a recent (MARVELOUS!) evening talk on labyrinths in Medford, following on the heels of the local regional labyrinth gathering here a few weeks ago featuring Jean Houston. Based on the enthusiastic response, this will be a more intensive follow-on workshop exploring labyrinths, sacred geometry, regional geomancy and much more. Since Richard and I complement each other's work, I will be assisting at this weekend workshop giving short multimedia presentations on topics that emphasize 3D and higher dimensional geometries. If you are interested, or would like to reserve a place in upcoming slideshow or workshops, contact Bonnie Bayard at 541-773-4286. The short presentations I give this weekend will be "samplers" of expanded workshops I'll be conducting in the southern Oregon area (or your vicinity!); contact me for details.

Here's more details from Richard:

What is Sacred Geometry all about and why should we care?


The Eight Patterns of Nature

Is it really true that Nature needs only eight patterns to create and sustain all life? Learn how spirals, meanders, and other dynamic patterns work for plants, animals, and the human body, and how free-flowing water purifies itself. The world will never look the same after viewing these astounding images and acquiring the ability to recognize cosmic patterns in ordinary things!

“Richard is the first person to explain the golden mean in a way I could understand.” – Jillian Brorby, feng shui consultant

The Geometry That Matters:
Power of Creation, Laws of Harmony

A visual introduction to the themes of the weekend workshop.

View a multitude of awe-inspiring images and awaken to the staggering recognition of the order and harmony of the universe. Learn about the essential role geometry plays in the continual creation of our universe. It is one of the greatest reservoirs of Nature’s laws of harmony. Sacred Geometry has the unique ability to depict the essence of a thing (by revealing its true underlying vibrational quality), rather than merely represent it metaphorically. Understand why knowledge of sacred geometry is crucial to be able to create beauty and ecologically sustainable culture, and design environments in harmony with Nature so they feed the human soul.

“You gave the best presentation of the whole conference! I learned more about sacred geometry from you in one hour than from any other talk.” — Ursula Rabe, feng shui consultant/astrologer and fine artist, Grass Valley, CA


Exploring the Symbolic Language of Geometry

Geometry is the universe’s secret way to turn energy into matter, and guide the growth of all forms of life. It holds the real key to the identity of DNA, a molecule, or an organism, and enables Nature to implant intention and form harmonious, life-enhancing relationships.

Learn about the benefits of tapping into this body of knowledge, including the ability:
* To recognize spatial patterns more quickly and accurately
* To read the geometry of a building as a language that may reveal the symbolic message the original builders wanted the edifice to embody and convey.

This seminar will be a rare opportunity to re-discover geometry as a rich, universal, symbolic language, which illuminates the core teachings of the world’s spiritual traditions in a pure, non-dogmatic way. (This obviously bears little resemblance to high school geometry or math!) Richard's use of a hands-on, visual, kinesthetic, Mystery School approach restores geometry to its ancient sacred function as a portal into meditation, soul-work, and timeless wisdom.

Understand proportions, like the golden ratio, that guide the growth of organisms.
Relive the initiation into the laws of manifestation that our cells received during embryonic development.
Discover the intriguing geometrical symbols hidden within cosmological mandalas.
Glimpse how to apply the laws of harmony to create art and architecture in which all the parts relate harmoniously to each other and the whole like a well-tuned orchestra.
Join us for a visual tour of Chartres, the most glorious French Gothic Cathedral, and witness how to decode the hidden messages in its geometry, to gain clues about its potential mystical uses and cosmological story.

“You helped me get all the connections, including the surprise of how geometry applies to group process. When I’m with a teacher who loves what he’s doing, I understand it in my body.” – Gail Gardener, massage therapist

Evening Talks start promptly @ 7:00 pm; Q & A from 9-9:30 pm
Suggested Donation: $10-20 in advance or at door

Sat-Sun. Workshop Fee: $225 - $325 sliding scale. If paid by July 20, Friday night slide show is included in seminar fee. Please register by July 20 to help make workshop happen.

Locations and materials list will be sent when payment is received.

For Questions on Content: Contact Richard at 707.544.8203

For Registration Information: Call Bonnie at 541-773-4286 between 9 am - 8 pm.
Space is limited to maximize personal attention.

To Register: Call first to reserve space, then confirm by mailing your non-refundable check, made payable to Richard Anderson, to Bonnie Bayard, POB 1691, Medford, OR 97501. Please write on your check which programs you have registered for.

Update (11Mar2008) from Richard Feather Anderson:


___ Booklet — The Patterns of Life — Description of the eight patterns that Nature uses to create and sustain all life on planet Earth. Each pattern allows Nature to function efficiently, using the least amount of energy and matter, thus conserving its resources. Examples are given of how the patterns manifest in the forms and dynamic forces of Nature. Knowing these patterns enables you to see the world more fully, understand the basis of feng shui, and recognize the archetypal power of single-path labyrinths. 5 pp.
$5.00+ .80

To place an order, Please print out and use order form below to calculate sales tax and shipping charges.

Total PRICE of ordered items
(Do not include postage on this line.) > __________

Add HANDLING CHARGE > __$2.00 ___

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Make checks payable to: Richard Feather Anderson.
We accept checks from U.S. Banks or International Money Orders.
Sorry, no credit cards.

Please mail orders with check to:
Richard Feather Anderson, 1570 North St. #17, Santa Rosa, CA 95404

Information: 707.544.8203

Customer Info /Shipping Label:

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Phone: Day ________________ Eve ______________

Richard Feather Anderson
American School of Geomancy
1570 North St. #17
Santa Rosa, CA 95404


Toroidal animations

Thanks to Luke for alerting me to this lovely toroidal animation... see also Nassim's animation (by Bob Gray) for comparison.

Adrian Rossiter's Excellent Geometry Gallery

If you haven't already explored it, Adrian Rossiter has an excellent gallery of geometric models and animations, including polyhedra, tensegrity models, polyhedron clusters, 17 types of polyhedron symmetry, jitterbug, gear (very cool!) and torus animations, zonohedra, geodesics, string art, octet trusses, and more.

Website That Links e, Pi and Phi

Here's a website that links together e, pi and phi. Lots to explore here.

Euler's Beautiful Equation

Here's an article about Euler's Beautiful Equation (linking e, pi and i).

More from Crater Rock Museum

Here's another few images from Crater Rock Museum with examples of spiral and other interesting shapes caused by the underlying mineral/crystal geometries (I believe the third one was artificially grown bismuth crystals with a rectangular maze-like pattern.




Ron Quinn's Vastu website

Here is Ron Quinn's website on Vastu (which I'm just learning about) that evidently is an ancient Hindu sacred geometry tradition that incorporates architecture and much more. He is giving presentations in southern Oregon in summer 2007.

new website for Nassim Haramein

Colleague Nassim has a new website and DVD, Crossing The Event Horizon. I attended Nassim's presenation for IONS in Petaluma, Calif. in 2004, and was quite impressed. He is giving another there on 22May2007 and a similar one in Santa Barbara on 20May2007.

Kepler's Inner Planets Crystal by Bathsheba

Bathsheba Grossman has created another lovely crystal in her array of exquisite art... this one is a
3D model of Johannes Kepler's diagram of the inner planets, first published in his 1596 book Mysterium Cosmographicum. I'd love to see this first hand....


I created a POV-Ray model a while ago entitled "
Homage to Kepler" which incorporates both the inner and outer solar system models.


Even at smaller print sizes you can see both the inner and outer models fairly easily. If you want a really close view, the print is available all the way up to 36" x 27". This early POV-Ray model was a precursor to my
Martineau Solar System Series now being exhibited in at least 3 different venues on this planet.



New website updates!


MAJOR website changes today! has moved to and has a new steward/owner! We've had a number of offers over the years to buy our domain,
and recently 3 at one time! However, until now, none had the intent we wanted to go forward with that idea.
The new owner plans to create a website with new content for soon:
"a community site based on the ideas of exchanging information on wellness, helping the planet,
personal development, and other topics related to health, the environment, and personal expression."

Please explore our new websites! We'd love feedback. (See section 2 about our new email addresses for this.)

Also, PLEASE fix any links you might have on YOUR websites to I have redirect pages in place
which should be around for a year, but now's the time to fix the links... THANKS! Thanks particularly to those
of you who have linked to our websites over the years. I'll be adding even more links to our new sites soon,
so please continue to send suggestions for sites that complement the work we're doing.

Here's the result of the "cattle drive" (still in progress) of moving about 12 years of web content to 2 newer sites:

I've moved all of the geometric (and related) content (from to:
Hopefully this new website will be much more accessible and navigable; you might even discover content
that has been there for a dozen years you just never realized was there! Happy This new format (featuring
CSS and SSI) will make it MUCH easier for me to maintain and enhance, so look forward to much more frequent
updates and additions going forward...

I'm moving everything else, such as my favorite books, movies, music, webcasts/radio, etc. and all the
New Frontiers of the Gold Country event archives (from to:
This latter effort is still in progress (just barely started, but I'll finish it ASAP, and I wanted to let everyone know early.

Formation of Fibonacci Spirals In Nature

Thanks to Rich in Penn Valley, California for this article:
Scientists find clues to the formation of Fibonacci spirals in nature.

Updated page on Martineau Solar System Series & article in journal Symmetry

Another update on the "cattle drive" I'm doing this week : here's an update on my Martineau Solar System Series. I also just received a complimentary copy of Symmetry: Culture and Science, Vol 16, No. 2, 2005, (Journal of the Symmetrion, editor: György Darvas, Polyhedra, Part 3; postal address: Budapest, P. O. Box 994, H-1245 Hungary) which has an article I submitted a few years ago on this subject on pages 193-196. You can also get a calendar with all 12 of these images here, and another with my Homage to Kepler art as a bonus cover image here. Here's an image from one of the first exhibits of this work at South Yuba Club in Nevada City, California.


Sonic Wind Sculptures by Ross Barrable

Thanks to Tim A. in China (and North Carolina) for this link with a great quote from Galileo, inspiring text and very interesting (and lovely) sculptures; I'd enjoy hearing the wind through them... I 'resonate' to the text, so I'll post a link to my blog about them... Happy

A couple of minor corrections on that page:

Geometry literally means “the measure of the earth" ... adding 'sacred' just adds the meaning and significance of that word Happy

Also, on that same page, the author/artist refers to the square root of 3 as transcendental. In mathematical terms (to be technically correct), I believe that number is just irrational. However, Pi, (the circumference/diameter proportion of any circle) which is almost always "nearby" or present whenever the square root of three is involved geometrically (either with the vesica piscis made from two circles, or the diagonal of a cube which can be perfectly inscribed or circumscribed by a sphere containing an infinity of circles) IS a transcendental number. In my presentations, I often share the metaphor that an irrational number (like the square roots of 2, 3, or 5) is not resolvable by our conscious 'rational' minds; however a transcendental number (like Pi) is not only irrational, it also transcends this 'dimension', since it can't be expressed even as a polynomial fraction (a fraction made of whole number polynomials, or numbers raised to higher powers or 'dimensions' metaphorically.)

Here's a
link to the MathWorld reference on transcendental numbers. Along with Pi, Euler's number, e (the base of natural logarithms) is perhaps also commonly known to many.

Platonic Solids Info

Thanks to Dan Radin for sending me a link to his website on Platonic Solids which has animations, Origami and more.


For those of you interested in a generalized Fibonacci sequence (summing the prior n terms) here's a reference; thanks to Michael in Arizona.

Yi-Jing Algebra

Thanks to Ed Kellogg for finding this interesting cube / I Ching relationship.

Flower of Life grid at different sizes

Thanks to Andrew M. for alerting me to this lovely illustration that appears to be a hand-drawn Flower-Of-Life variation with overlapping scalings... neat!

Beautiful Symmetry-E8 computation

Although the math is a bit beyond my conscious fathoming, I find the idea that there is a collaboration exploring beauty and symmetry in higher (would you believe 248?) dimensions ... and finding it! ... to be a satisfying notion Happy Here is a link to "E8" research and a "string art" visualization.

Hedronists meeting video - now online

For those of you interested in creating community around things geometrical, here's a link to an hour (in 6 installments) of video of a recent meeting of fellow "Hedronists" (a whimsical name a handful of us kindred geometers who have been meeting quasi-regularly in Ashland, Oregon call ourselves). Our hour potluck meetings discuss common geometric interests as well as a variety of ideas for mutual support and creating a local and global community to not only promote our work, but to make a positive difference on a global scale.

Torus-sphere and Spider/insectogram crop circle connection

Thanks to Micheal S. for alerting me to this link which shows a Sun - Earth - Plasma - Blast image with our beloved "torus-sphere" geometry, plus (due to the asymmetry of the solar wind around our planet) a rather remarkable resemblance
to some of the "insectogram" (or spider) crop circles of a few years ago, such as these few quick finds:

Scroll down on
this page to find "Fish/Eye Formation at Silbury Hill, near Avebury, Wiltshire, 9 July 2005"
with a similar link (and overlapping content)
here: note the 2nd and 3rd images past this line of text:
"This one appears in 13th July last year at West Kenet Long Barrow". Scroll down on
this page to find
"The Star Nation peoples have sent similar messages in other formations"

Bethe Hagens UVG grid for Google Earth

Thanks to colleague Dan Shaw, of Vortex Maps, I discovered this lovely UVG-grid add-on (which superimposes the great circle outlines of the icosa- and dodeca- grid for Google Earth. Here is a map of our county (Jackson) in southern Oregon showing where the lines cross west of Crater Lake ...


and a more zoomed out view showing the nexus above the Giza longitude crowning the Nile Delta ...


When you zoom out, you'll see lovely images of the Earth surrounded by the grid.

Interesting article on Sacred Geometry & Feng Shui

Thanks to Larry G. for sending the link to this Interesting article on Sacred Geometry & Feng Shui.

Very cool spiral animated "music box" - gotta see/hear it!

Thanks to Jordan P. for alerting me to this very cool spiral animated "music box" - Check out all the variations! There are a whole series of animations you can select with different musical intervals and corresponding geometries that trigger the notes; pretty fun!

Islamic Art, Golden Rhombus, Eric Weisstein's Treasure Troves of Math, Dodeca/Tetra...

Thanks to Jim H. for pointing out a permanent link to the article on advanced math in art/architecture and Islamic Art that I mentioned in my March email bulletin (the one I listed had expired after a week.)

Thanks also to lucid dream researcher and forum host
Ed Kellogg for showing me an interesting book by Shafica Karragula (author of Breakthrough to Creativity: Your Higher Sense Perception which I read a bit of decades ago) and Viola Petitt Neal entitled "Through the Curtain" which contains references to the diamond and other geometric shapes as appropriate for meditation and spiritual resonance. This reminded me of the significance of the diamond (rhombus) shape and the appearance of the Golden Rhombus (the diamond shape you get when you connect the midpoints of the sides of a golden rectangle) in the Rhombic Triacontahedron, also known as Kepler's Solid; I have a fold-up pattern for it on my website here. The conversation Ed & I had about the diamond/rhombus shape also lead to talking about Eric Weisstein's fabulous magnum opus, Eric's Treasure Troves of Math (and Physics and ...) which evolved into his Encyclopedia of Mathematics and the online version, a collaboration with Wolfram which I seem to refer to constantly. Kudos to Eric and all the other contributors which make this such an exemplary gift to humankind's understanding!

Thanks to Luke G. and Michael H. for this neat visual progression from R.W. Gray about
dodecahedron/tetrahedron nesting relationships.

248 Dimensions?

Thanks to Iona for this interesting article about 248 Dimensions! Looks like string art and string theory combined Happy

Sacred Geometry - Polyhedra Models

I had an inquiry over the weekend from some friends wanting to know where to get models of polyhedra. In addition to some awesome screensavers, here are some lovely polyhedra & geometric models (in gold an bronze) you might want to explore! More on the way... stay tuned! Happy


Here's a press release for geometry enthusiasts from Robert Webb:
(looks like a great update for this interesting looking software; another enticement to get an Intel-based Mac, I suppose Happy

I just released version 4.0 of Stella, plus a new product called Stella4D. It includes everything from Great Stella, but also includes
support for 4D polytopes! Watch a tesseract (4D cube) undergoing 4D rotation. Watch 3D cross-sections of 4D polytopes animating as the slicing hyperplane cuts deeper and deeper. All 1849 known uniform 4D polychora are included, plus more! Happy

I attach a photo of a paper model I made of a 4D polytope using nets created and printed from Stella4D (the cantellated tesseract).

Home page:

Get the demo of Stella4D here:

See a bunch of screenshots, including animated ones, here:

Vedic Math, Fibonacci, Pi and synchronicities

While looking for references for the existing ISBN number (after discovering that the new system has 13 digits) for my Sacred Geometry Design Sourcebook, I found some interesting gems like this page with interesting references to Vedic Math, 1/89, Fibonacci, Pi and synchronicities.

Details about the contents of Sacred Geometry Design Sourcebook, instructions for creating the graphics therein, etc.

Today's post is an email conversation I had with a visitor to the website 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 Happy

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 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 photocopies 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 links page
at, 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 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 away Happy, 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 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! Happy

I'm also working on a number of related projects that will complement the book with video "hands on" procedures, etc. Stay tuned!

Stereogram animation of 24-cell

Here's a neat stereogram animation (thanks, once again to Michael in Arizona!) by Tony Smith which is a 4-dimensional shape; see my links page for more on polytopes.

Sri Yantra, Great Pyramid, Golden Ratio and Vesica Piscis

Here's an excerpted dialogue about the Sri Yantra and it's geometry from a recent email with colleague Sherri Silverman; one of those "might as well share with a wider audience" conversations... Happy


Hi Sherri:

Your message is timely and appreciated!

I have an interesting "synchronicity" story to tell about the Sri Yantra. Over a decade ago, when I was researching the 1-Phi-Square Root of Phi triangle (cross section of the
Great Pyramid and primary geometry in the Sri Yantra), I discovered a website that gave the exact proportions of the Sri Yantra and it's construction ( ), and I was replicating these on my computer using Claris CAD. That evening, I happened to be listening to a favorite radio program, New Dimensions, hosted by Michael & Justine Toms. Michael's guest just happened to be talking about - you guessed it! - The Sri Yantra! His guest, whose name I've forgotten, mentioned that all authentic replicas have a small gap between the inner and outer regions of the design, so I made sure there is a very tiny gap on my version of it on page 195 of my book.

I'm curious if the CD has music with frequency ratios of
1: Phi: Square Root of Phi...

BTW, another friend in British Columbia (Mike Green) recently alerted me to a right triangle I wasn't aware of that has Phi: 1/Phi: Square Root of 3 (hypotenuse) proportions; amazing that I'd never run across such an elegant triangle linking these mystical proportions! Seems curiously connected, somehow! (The square root of three is the proportion of the
Vesica Piscis, the diagonal of a cube & circumsphere to intersphere ratio of the tetrahedron, and the surface area to edge ratio of the tetrahedron.)

Dear Friends,

I will soon send out one of my illustrated newsletters with Vastu tips relating to the eastern sector of your home or business. In the meantime, I wanted to let you know about a FABULOUS NEW ITEM FOR YOUR CONSCIOUS LIFESTYLE:

I have found a wonderful Sri Yantra CD that converts the mathematical form of the auspicious Sri Yantra into sound. It creates a centered, harmonious, meditative experience. This is not Sanskrit chanting, nor is it on the website shopping cart yet. Call Transcendence Design at 505.984.0108 to order this special CD. Prices follow the explanation of the CD. If you do not remember the benfits of the Sri Yantra, look at and for info on the two- and three-dimensional forms of the Sri Yantra.

The SRI YANTRA consists of 9 triangles arranged with 4 pointing upward and 5 pointing downward, superimposed on top of one another. Dr. Jeffrey Thompson, DC, BFA, has translated this ancient mandala into a sonic mandala mathematically in two different ways, corresponding to part 1 and part 2 of the CD. The sound vibration patterns that are generated by the complexities of the Sri Yantra make this soundtrack a powerful tool for reaching deep states of consciousness.

In each part of this CD, Dr. Thompson has used the circle around the triangular patterns of the Sri Yantra as a central reference point to which all of the triangles relate in distinct proportionate ratios. On track 1 you will hear the Sri Yantra assemble itself sonically one triangle tone at a time, starting with the frequency for the circle and ending with all twenty-seven angles playing together. This produces the exact relationship patterns you see as you look at the Sri Yantra visually.

For Track 2 of the Sri Yantra CD, Dr. Thompson calculated the area of the circle and the area of each of the nine (9) triangles. He then related the area of each triangle as a percentage proportion to the circle that enclosed it, making a ratio. In both tracks, Dr. Thompson has played the complete Sri Yantra through a series of seven octaves. You will hear the Sri Yantra begin as a mix of complex low frequency vibrations, which continually double in frequency every four minutes and become higher and higher, leading one upward. Many people feel the Sri Yantra begin as a sound vibration in their body and eventually exit through the top of the head.

The sounds on this CD have been processed with special 3-D recording techniques, so headphones or good stereo separation give the best results. This CD is best used sitting with an erect spine, with eyes closed and lights dim. You may also wish to listen to it while gazing at a picture of the Sri Yantra.

The Sri Yantra CD is only one of many Transcendence Design conscious lifestyle products. You can get an advance peek at more of them at

I have added a lot of new material to the website, including an article from the Santa Fean Magazine this past July. You can read "Do you vastu?" at

All best wishes for a truly centered year of grace, love, and abundance,
Sherri Silverman

Great Pyramid models

This morning I received an email from a website visitor in the UK asking if I sold pyramid models. In keeping with my recent resolution to not answer the same question twice (by posting generic, FAQ-ish info online, what a concept! Happy, here is my reply:

I don't actually sell pyramids, although it is a topic of great interest to me (especially the Great Pyramid
which I had the good fortune to visit in November 1992 after two decades of anticipation; quite an amazing experience there!)

I have a fold-up pattern in my book on page 189 of my book, Sacred Geometry Design Sourcebook,
which is available here:

or, the foldup pattern (which has the correct proportions for the Great Pyramid at Giza, Egypt) is also
online on my website here:

If you use the model above, the important proportion (as close as you can get it) is:
diagonal length (also the radius of the circle used to make the foldup pattern) : 0.95105645
side length (the four base sides of the pyramid): 1.0

This will give the exact golden section (1.618034...) proportions for the pyramid faces (the two halves of each face
can be 'reassembled' to form a golden rectangle, as is shown on page 189.

In terms of someone to make you a model, any craftsperson that is good with math should be able to fashion you a model scaled to your requirements; especially for large items that would need to be shipped, I highly recommend finding someone local to your neighborhood/community (and local matierials) if possible for optimal environmental factors! Happy

Since I get inquiries like these frequently, I'll post my response (you will be anonymous) to my blog today:

I hope this is helpful! Happy