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2023 Martineau Solar System Calendars available now

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Once again this perennial calendar is available! These timeless images portray amazingly accurate geometric representations of the planetary orbits in our very own solar system – now available in printed calendar form with dates for 2023!

Martineau Solar System Calendars

Martineau solar system calendar -12 months

These 12 images are highly accurate geometric models of our solar system; details here. They were inspired by the remarkable work of John Martineau (publisher of Wooden Books), who wrote “A Little Book of Coincidence”. 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. A newly revised calendar (which is updated every year) with images from the Martineau Solar System is again available for 2022.

The Golden Icosahedron and other interesting Golden Ratio appearances

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Golden Icosahedron

How did I miss this very cool, interesting polyhedron? Glad to have just discovered this fascinating shape (with lovely rotating animation) on this post by “RobertLovesPi” here, with this commentary:

The Platonic Icosahedron has twenty faces which are equilateral triangles. In the Golden Icosahedron, twelve of those triangles (the yellow ones) have been replaced by acute, isosceles triangles with a leg:base ratio which is the Golden Ratio.

Here are a few other appearances of the golden ratio (a.k.a. golden proportion a.k.a. golden number a.k.a. golden section a.k.a. extreme and mean ratio a.k.a. divine proportion a.k.a. Phi (ø) or Tau (τ) proportion a.k.a. sacred cut,  etc., and other synonyms for the ubiquitous (1 + √5)/2 ≈ 1.6180339887… number):

This unexpected golden ratio appearance first made my acquaintance at a sacred geometry conference hosted by Luke Gatto a few decades ago in Boulder, Colorado, and I shared it many years later here in 2011:

How to graphically derive the Golden Ratio using an equilateral triangle inscribed in a circle

Inscribed Triangle Phi

and resurfaces here on a Quora post – and I’m copying the entire account verbatim; thanks, Johann!

Johann Holzel

Author has 4K answers and 1.1M answer views

Some ratio has to satisfy a/b = (a+b)/a. The fact that this ratio should have such a simple continued fraction, involves √5 and therefore pentagons, has a special relation between its negation and inverse, etc., these must have all been surprising developments when each was first discovered (in many cases, by ancient Greek geometers—although apparently it, or √5, was also the first number proven to be irrational, which was even more surprising).

But nowadays, you learn most of those things together, and are immediately shown how they fit together, so there’s not much room for surprise.

And most appearances in nature make sense once you think about them. For example, there are cases where the most irrational logarithmic spiral is obviously the best to use, and the fact that it can be closely approximated by the arithmetically-simple Fibonacci spiral makes it unsurprising that evolution would hit on it multiple times.

(In fact, most claimed appearances in nature or in art are either not even close to ϕ at all, or are just boring cases of an artist who heard great art is supposed to be full of golden ratios so they followed what they thought they were supposed to do.)

So the most unexpected places are probably the ones that come from amateurs. And probably the most famous of these is the equilateral triangle inscribed in a circle, just because of how deceptively simple it is.

A 1983 article in American Mathematical Monthly consisted of basically this picture, accompanied by the word “Behold!”

A and B are the midpoints of two sides; extend the line from A to B out to the circle and the intersection is C.

The ratio of AB to BC is obviously somewhere around 1.6-ish just from visual inspection. So is the ratio of AC to AB.

But in fact, they’re identical, which means they’re both exactly ϕ, the golden ratio.

Why?

The golden ratio doesn’t just show up in triangles unless you put it there (e.g., the golden triangle is specifically designed as the isosceles triangle with the self-similar bisection), and that isn’t the case here.

This was discovered by George Odom, an amateur mathematician, who was in correspondence with famous geometer H.S.M. Coxeter. Coxeter wanted to see how many readers could work out why ϕ shows up here.

To get the answer, you have to notice that this chord looks like it can be used to construct a regular pentagon. Then you can complete the construction, and prove it’s exact via the chord theorem. And then there’s no mystery left—unlike triangles, pentagons are full of the golden ratio. (Coxeter just gave the constructed pentagon as the answer.)

That’s deeper than the kind of thing you’d expect from what looks like a simple recreational mathematics problem. And even after you get it, it’s still almost magical that something so with no obvious pentagons, or self-similar folds, or anything like that, gives you ϕ, so simply.

 

 

High Resolution Imaging Science Experiment (HiRISE) camera photos

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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!

Mars: Martian Dust Devil Trails (NASA)

Mars: dunes (NASA)

Mars: dunes (NASA)

Mars: dark dunes (NASA)

Mars: carbon dioxide Ice - late summer (NASA)

Mars: dunes (NASA)

Mars: MRO orbiter banded wind ridges iapygia (NASA)

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.

Are Physics, Geometry and Science pointing to (Non-Dual) Metaphysics?

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I just finished listening to an extended (almost 3 and half hour) fascinating podcast interview of cognitive scientist and author Donald Hoffman by Lex Fridman speaking on “Reality is an Illusion – How Evolution Hid the Truth” – (Here is a YouTube clip, evidently excerpted from the longer podcast.) If nerdy math and physics is over your head (as much of this was for me) you might want to skip to the last half hour or so where Hoffman delves into a more philosophical discussion with Fridman where more arcane subjects like Amplituhedron, Permutation matrices, Birkhoff polytopes and cosmological polytopes (new or minimally explored concepts to me) give way to a conversation that seems to lead one toward non-dual metaphysics (with which I’m much more familiar.) Here’s a related TEDtalk by Hoffman: Do we see reality as it is? I don’t pretend to understand much of the in-depth new math and physics that Hoffman talks about but I particularly appreciate his willingness to consider the idea that matter derives from consciousness and not necessarily the other way around! Here’s a quote that hopefully, I haven’t taken out of context: “The notion of self can be an artifact of projection…” (at about 2:16:39 into the podcast).

While doing some very preliminary perusals of some of the links above, I found a few interesting geometric links, including:

mirror-faced icosahedra descend on a city

 

 

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

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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:

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.

The Geometry Code: Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation - front cover artwork

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.

The Geometry Code book - Sample Chapter - Cover Page 1