Fractal Geometry

High Resolution Imaging Science Experiment (HiRISE) camera photos

Sunday, 31 July 2022 by Bruce Rawles

I suppose one might say that these amazing images are more about geography than geometry, but I imagine that you might also notice that the abstract and seemingly amorphous shapes in these stunning images have some rather interesting geometric attributes, too! Some exhibit fractal patterns one might see flying over USA’s desert southwest and all show the profound intricacies of natural forces sculpting these desolate landscapes. These are all from NASA’s High-Resolution Imaging Science Experiment (HiRISE) camera on the surface of Mars. There are plenty of details on the NASA website that I won’t replicate here.

Of course, if you haven’t already checked out images from the newly celebrated James Web Space Telescope, check out those, too!

Thanks to fellow geometer, Dave Van Dyke – whose art, animation, and music I’ve featured on my personal blog – for alerting me to these artistic images from our cool, neighboring planet.

Reflections on the Sacred: Geometry, Mind, Non-duality

Sunday, 5 June 2022 by Bruce Rawles

Over the years, I’ve been asked by many: “What Makes Sacred Geometry sacred?” to which I usually respond with something like this: Either everything is sacred … or nothing is sacred. While that terse response may seem to be evading the question, I’ve realized more and more deeply that it’s true, and if we gradually (with help from the Inner Inspiration we all can employ) shift our perception from the ephemeral to the eternal then our view of everyone and everything becomes kinder, more compassionate, tolerant and forgiving of the minutiae of the fleeting phantasms of form and the cosmically temporary. If truth doesn’t change, our task must be to seek and find what is true and cannot change; the laws of mathematics and geometry seem worthy candidates for possible means of finding the unalterably true. However, for something to be unequivocally true, it must apply to everyone and everything. The discipline of the scientific mind requires that we keep looking if we find exceptions to our postulates and theorems and persist until we find the safety, security, and serenity of what can never change. William Shakespeare penned lovely prose about the ideal of looking ever inward to the mind for what doesn’t alter or change when it (the mind) finds the alterable:

Sonnet 116: Let me not to the marriage of true minds
BY WILLIAM SHAKESPEARE

Let me not to the marriage of true minds
Admit impediments. Love is not love
Which alters when it alteration finds,
Or bends with the remover to remove.
O no! it is an ever-fixed mark
That looks on tempests and is never shaken;
It is the star to every wand’ring bark,
Whose worth’s unknown, although his height be taken.
Love’s not Time’s fool, though rosy lips and cheeks
Within his bending sickle’s compass come;
Love alters not with his brief hours and weeks,
But bears it out even to the edge of doom.
If this be error and upon me prov’d,
I never writ, nor no man ever lov’d.

Seeking what is true for all, we realize we are all in the “same boat” either expressing kindness or in one way or another asking for it. Seen from this perspective, our true kinship – our real relationship – to everyone and everything merits kindness in response to expressions or requests for the same. That seems like a very serviceable and practical definition of sacredness – generalized to encompass all, sacredness reflects the perfection beyond the forms our senses and the mind’s machinery of interpretation seem to establish as real. We rarely question the vast number of assumptions we make every second about the world that seems vulgar and profane or at the very least disappointing in its impermanence.

A few dictionary definitions of the word sacred (most closely aligned with how I think of the word) are “revered, inviolable, inviolate, unimpeachable, invulnerable, untouchable, inalienable, protected, defended, secure, safe, and unthreatened.” These adjectives would suggest that nothing in this material world – where every form that appears eventually disappears – would qualify for sacredness; no exemption for the changed, changing, and changeable from decay, dissolution, and death. Yet while the laws of physics (particularly the Second Law of Thermodynamics) would suggest that nothing in spacetime is forever, we certainly can use our experiences in the world of space and time to guide us (even if largely as counterexamples) to find what is endless.

Our experience of sacredness – which I would equate with the purely innocent eternal nature of our being and all that remains in mind after we have completely forgiven ourselves and everyone for everything – depends completely on the thought system with which one views everything! If we look for what unites us, what is intrinsically the same for all – regardless of all the polarizing temporary attributes that seem to define our ephemeral personas and certainly our bodies – we find the sacred, innocent undifferentiated awareness from which all creation derives and to which we all belong.

What prompted this metaphysical musing? My computer calendar showed me that a decade ago today, I published my second book in print form, followed by a Kindle edition a few months later. I was also reflecting fondly on some related anniversaries:

10 years ago today, The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation became available in print
15 years ago (2007), I “rediscovered” the pure non-dual metaphysics of A Course In Miracles by way of Gary Renard‘s first book: The Disappearance of the Universe: Straight Talk about Illusions, Past Lives, Religion, Sex, Politics, and the Miracles of Forgiveness and shortly thereafter started ACIMblog.com and started a weekly study group (now online)
25 years ago (1997) I launched my first book, Sacred Geometry Design Sourcebook – Universal Dimensional Patterns
30 years ago (1992), having “rediscovered” what has become known as sacred geometry via fellow mystical explorer and contemporary, Gregg Braden at a lecture in my mother’s living room in Livermore, California a year before – we (my now wife, Nancy, and I) accompanied Gregg (21 of us) to Egypt to explore the Great Pyramid of Giza and numerous sites up and down the Nile River
50 years ago (1972) I graduated from Livermore High School

Thanks to all of you for many decades of support, collaboration, and shared enthusiasm for finding what is true! Here are some excerpts from the June 2012 bulletin that provide details about my second book, which combined interests in both sacred geometry and non-dual metaphysics. The book “The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation became available 10 years ago today: on June 5, 2012.

	Learn more about The Geometry Code book
	Free sample chapter from The Geometry Code book with free bulletin subscription
	Preface by the author from The Geometry Code book
	Table of Contents from The Geometry Code book
	Acknowledgments for The Geometry Code book
	Endorsements for The Geometry Code book

About the book: The Geometry Code

While I was completing work on Sacred Geometry Design Sourcebook, (completed in 1997), I was already working on another book, which was ready on June 5th, 2012. Rather than reinvent the wheel, I think the Preface I wrote explains how the book came about and gives a good “big picture” of what it is about. I’ve uploaded the Table of Contents, Endorsements from friends and colleagues, Acknowledgments, a sample chapter, and more.

SMAs in SMA (SMA-ll stellated dodecahedra in San Miguel de Allende)

Thursday, 5 May 2022 by Bruce Rawles

¡Feliz Cinco de Mayo! Last week, my wife Nancy and I enjoyed a long-overdue, much-needed getaway/vacation and chose San Miguel de Allende, Mexico, a place we’d heard great things about for decades from numerous sources. We very much enjoyed our immersion into the culture, cuisine, and art that this historic city is known for. In addition to world-class dining, lots of “heart decor” and so much more, I was amazed at how many 3D geometric stars were present in this lovely city, both in numerous shops and even in permanent street art! Since artsy stars, wings, and hearts are here in abundance, there are plenty of stellated polyhedra, but the predominant variety seemed to be one of my favorite shapes, the small stellated dodecahedron (one of the four Kepler–Poinsot polyhedra) which I’ve featured in numerous places on this website. I mentioned (in my broken Spanish, resurrected from high school classes a half-century ago) to a few of the shopkeepers that a foldup version that would fit in a tourist’s suitcase might afford more sales, since shipping these lovely fabrications is probably a bit expensive.

If you want to make your own (I suggest trying in cardboard first) here’s a foldup pattern from page 219 of Sacred Geometry Design Sourcebook; you’ll want to make 6 copies of this page. Join the two edges labeled 1 and also join the two edges labeled 2. If you fold together just one copy of this page, you’ll get a double pentagonal pyramid, which is another interesting shape. However, if you fold up the 2 pentagonal pyramid halves, crease the middle shared edge, and fold the other direction, you’ll have 1/6 of what you need to make this classic shape. MC Escher fans will probably recall this classic image.

Small Stellated Dodecahedron fold-up pattern from page 219 of Sacred Geometry Design Sourcebook - Universal Dimensional Patterns by Bruce Rawles — Small Stellated Dodecahedron fold-up pattern from page 219 of Sacred Geometry Design Sourcebook – Universal Dimensional Patterns by Bruce Rawles

I’ve included an assortment of photos from this trip below, including a few other familiar geometries you’ll probably recognize; enjoy! For many more photos of our San Miguel de Allende, Mexico trip this past week, visit my personal Facebook page.

Small Stellated Dodecahedra hanging from the ceiling of the gazebo in the centro park adjacent to the Parrochia church, San Miguel de Allende, Mexico — Small Stellated Dodecahedra hanging from the ceiling of the gazebo in the Centro park adjacent to the Parrochia church, San Miguel de Allende, Mexico

cluster of brass Small Stellated Dodecahedra; San Miguel de Allende, Mexico

Small Stellated Dodecahedra and many other colorful artsy decorative items in another street vendor shop; San Miguel de Allende, Mexico

Small Stellated Dodecahedron -hanging in a restaurant interior courtyard; San Miguel de Allende, Mexico

street vendor photo: metal small stellated dodecahedra and other art objects; San Miguel de Allende, Mexico

metal stellated polyhedra decorating trees in an interior courtyard; San Miguel de Allende, Mexico — metal stellated polyhedra (one of many polyhedra) decorating trees in an interior courtyard; San Miguel de Allende, Mexico

metal stellated polyhedra; San Miguel de Allende, Mexico

interior of a metal stellated polyhedron; San Miguel de Allende, Mexico

Small Stellated Dodecahedra and other geometric lamps; street vendor lamp shop; San Miguel de Allende, Mexico

large metal small stellated dodecahedron on street corner 1 block south of Parrochia church; San Miguel de Allende, Mexico — daytime photo of a large metal small stellated dodecahedron on the street corner one block south of Parrochia church, San Miguel de Allende, Mexico …

… the same large metal small stellated dodecahedron casting light on the street corner at night; San Miguel de Allende, Mexico

courtyard art: large stellated cube; San Miguel de Allende, Mexico

reflecting sphere art in hotel lobby; San Miguel de Allende, Mexico — reflecting sphere art in a hotel lobby; San Miguel de Allende, Mexico

Other examples of small stellated dodecahedra from this website:

sculpture by Gregory Hoag of I-Connect (formerly Metaforms).

zometool stellated dodecahedron — zometool construction (the outer blue shape is a small stellated dodecahedron)

Fractal flotilla? Topological temple? Self-similar shrine? Growing options for fractal art

Monday, 28 June 2021 by Bruce Rawles

$IMception: fractal flotilla?$

This intriguing image is among those featured in this article about a Kickstarter campaign – by Raksha Singla, creator of IMception, a new project which aims to add a newcomer to the existing field of fractal generation software:

“This exciting new program on Kickstarter is a gateway for an eternal journey into the world of fractal illustrations. It is so simple to use that anyone can instantly become a confident artist.”

Here’s a related article with some current options for making your own fractal artistry: Ultimate Guide to Fractal Generators: The Best Fractal Software in 2021 (Free & Paid)

Here are numerous prior posts on the GeometryCode website that reference fractals.

Sketches in Code: Randomness and Patterns in Science and Art – Livermore, CA exhibit

Wednesday, 28 April 2021 by Bruce Rawles

While visiting my hometown of Livermore, California this month, I was fortunate to see a very well done exhibit in the lobby of the Bankhead Theater entitled “Sketches in Code: Randomness and Patterns in Science and Art” which featured several interactive exhibits as well as numerous examples of 2D and 3D art, many of which will be familiar to those of us who have studied these realms for decades. Here is the detail from the Livermore Arts website:

Science, math, computers and art meet nature Explore how the intersection of mathematics, science, art and computers can be used to visualize and create artistic pieces which also describe and explain nature. As you tour this exhibit, we hope you see that, while art has often been described as an opposing discipline to mathematics, science and computers, in many ways they are very similar. In the end, all are creative processes that involve a journey of discovery. This exhibit is co-sponsored by Livermore Arts and Quest Science Center Bankhead Theater Gallery Feb 12–May 2 Open Fridays 2-6pm; Sat and Sun 1-5pm Limited entry, masks required. Family-friendly interactive exhibits with Covid protocols in place. Join us for this (exhibit) of science-meets-art.

If you’re reading this before May 2, 2021, you may still have time to see the exhibit in person if you’re in the San Francisco Bay area; recommended for the entire spectrum of those completely unfamiliar with the subject to those well-versed. Here are some cell phone photos of the exhibit items (most self-explanatory):

The first two photos are of a particularly nice 3D printed Sierpinski Tetrahedron (fractal tetrahedron) at the top of this article, with the second photo being a fun interior view with a pentagram-like interior contour:

If you’d like to make a Sierpinski Tetrahedron – a great geometry classroom or home school project! – here’s a fun sped-up how to video: (which also demos the same idea for a Menger Sponge.

$examples of fractals$

$fractal examples$

$fractal branching$