It is unlikely that anyone knows how many types of superstitious nonsense exist, for counting them would be an enormous task, with no compelling reason to do it, and only a slim hope of actually finding them all. However, a given person will be more likely than other people to know about a particular type — if it is related to things the first person finds of interest. For example, a physician will be more likely to be aware of homeopathy, and the faulty ideas upon which it is based, than would a randomly-selected college-educated adult.
It won’t take long, reading this blog, for anyone to figure out that I have a strong interest in geometry. Were it not for this, I likely would be unaware of another type of superstitious nonsense: “Sacred Geometry,” which I cannot bring myself to type without quotation marks. If you google that term, you’ll quickly find an amazing number of websites devoted to that topic, with many extraordinary claims about certain polygons and polyhedra, but you won’t find more than miniscule amounts of logical reasoning on any of these sites, mixed in with large portions of utter nonsense.
Geometry is inherently interesting, and many geometrical shapes and patterns are aesthetically pleasing. However, to search for their mystical or spiritual qualities is to do nothing more, nor less, than to waste one’s time.
I did not create, nor discover, the figure above, but found this picture at http://earthweareone.com/a-new-form-has-been-discovered-in-sacred-geometry-meet-the-chestahedron/. It is described there as a polyhedron with seven faces (well, they call them “sides,” but they clearly mean faces) of equal area — three faces with four sides each, and four faces with three sides each. Knowing that, I tried to figure out exactly what the back side of the figure would look like, but the text at that website isn’t particularly helpful in that regard, being filled with claims, allegedly related to this shape, regarding the human heart as an “organ of flow,” not a pump (What’s the difference?); “the earth in its foundational form [as] not a sphere but rather [with] its basis [being] a ‘kind of tetrahedron'” (What?); and a (claimed) special relationship between this polyhedron, and the oh-so-profoundly-mystical Platonic Solids. If none of that makes sense to you, you are not alone. It doesn’t make sense.
I often use software called Stella 4d (which you may try at http://www.software3d.com/Stella.php) to investigate the geometric properties of polyhedra. Based upon comments about this polyhedron written by Stella 4d‘s author, Robert Webb, I was able to create the rotating virtual model below, with Stella, to help me understand what all the faces of the green polyhedron above look like, included those on the back side, as the figure is shown in the picture above. This polyhedron is similar to an octahedron, with a single face augmented by a tetrahedron, and with three pairs of coplanar equilateral triangles then fused into rhombi. Here is that figure:
This isn’t the exact polyhedron in the first picture, but is isomorphic to it. Vertices and edges are moved a bit, changing the size and shapes of the four-sided faces, as well as the dihedral angles between the triangular faces, until all faces are equal in area. This process turns the three rhombic faces into kites.
On the above-linked “Earth We Are One” website, where the “sacred geometry” of this polyhedron is “explained” (and where I found the non-moving first picture here), this is called a “Chestahedron.” While Stella can help someone understand the geometrical properties of the Chestahedron, it offers no information whatsoever about the spiritual or mystical properties of this polyhedron, nor any other. There’s a good reason for this, though: the complete lack of evidence that any such properties exist, for the Chestahedron, or, indeed, for any geometrical figure.
As for the people, whom I’m calling the “Sacred Geometricians,” who are pouring so much time and effort into investigations of these alleged non-mathematical properties of hexagons, pentagons, enneagons, many polyhedra, and other geometrical figures, I have three things to say to them:
- This isn’t ancient Greece, and you aren’t in the Pythagorean Society.
- That part of the work of the Pythagoreans had no basis in reality in the first place, anyway. Geometry, together with religion and/or mysticism, as it turns out, can be mixed — the Pythagoreans were correct, on this one point — but such mixtures are invariably incoherent and illogical, revealing the efforts to create them as activities which are both pointless, and useless. Just because two things can be blended does not imply that they should be blended.
- Please stop. You give me a headache.
I’ve been interested in the subject since Carl Sagan talk about Kepler’s crazy model of the solar system by nested circumscribed sphere platonic solids, but I agree its tough to find “meaning” in static objects.
The best I can see is you have to care about symbols, not symbol like an algebraic variable, but symbol like a realization of perfection, and a reminder of what’s hidden from direct experience, so we can know all forms in the real world are just approximations to mathematical relations that exist “behind” forms we see in the world, and that gives a sense of “order” in a world that can otherwise seem crazy.
I guess technology is our modern “sacred geometry”, so as long as technical progress seems to be continuing, we’ll believe we have some sort of control over the chaos.
Here’s one video series I totally didn’t get, but felt bad, since the author Robert Williams since he wrote a book I have called “The geometic structure of nature” which was very good. But his “Catenatic Geometry” escaped me.
I have been tempted for some time to just write a formula where as you define the number of sides and it sets said object as the base and then creates triangles on the the sides of those and intersecting 4 sided objects on top of them that meet the criteria of having equal areas for all sides. Theoretically you should be able to make it work for any 2n+1 number of faces and a base with n sides for n >= 3.
Robert: the spiritual path is one that seeks ultimate truth. Math, and in it geometry, is one form of ultimate truth. The contemplation of geometry may be understood as the contemplation of truth, and therefore: sacred. The contemplation of geometry has been a part of Islamic and other meditation for over a thousand years. Though it be not your path, that does not equate to no path. Charlatans selling junk… That is another matter. They are always looking for new markets. Regards, Brent
4. reverently dedicated to some person, purpose, or object:
a morning hour sacred to study.
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Truth is not logic: it is True 🙂
You can’t understand Truth if you try to understand it with logic.
Logic divides, Truth unites.
“Truth” which is unsupported by logical evidence should be called something other than “truth.”
Your opening remarks state that geometry is inherently interesting, but I think that idea warrants some further thinking.
What about geometry is inherently interesting? There are many things which are “aesthetically pleasing!” What makes geometry special? What makes it special *to you*? Why doesn’t everyone find it so compelling and interesting?
I think the people who get into the wonky “sacred geometry” have more in common with you than you might think (or like!). While playing with Stella4d, exploring new polyhedra, you’re experiencing the same kind of awe at the beauty that they are, but they feel an urge to connect the discoveries in this awe to something more. There is this urge to find some *meaning* to the awe they find themselves compelled by. How can something so fascinating, “perfect” looking, be meaningless, nothing more than mathematical curiosities? they ask.
The “sacred geometricians” find themselves JUST as compelled to the fascinating world of geometry as you, but the difference between them and you ISN’T (fundamentally) their intellectual rigor. Remember, Euclid and Pythagoras themselves both explored geometry with such intensity BECAUSE they felt as though they had stumbled upon something pointing to the deeper nature of the world, and needed to explore further. Buckminster Fuller bordered on “sacred geometry” on a number of occasions. if you google Vector Equilibrium, and you’ll find endless “sacred geometry” stuff expanding on Fuller’s thought.
Copernicus. Kepler. Newton. On and on. Some of the greatest mathematical and scientific minds of all time could not shake the feeling that there was something in geometry was a doorway to the deepest truths of the universe. Sure, lots of people without the tools to rigorously explore and understand geometry say some profoundly nonsensical BS, (like the commenter above talking about “truth” and “logic,” LOL) but (I think) one must be sympathetic to the urge to find more meaning in fascinating geometry.
Sure, every time we see it actually applied to the world, “sacred geometry” ends up being downright silly, usually because of their lack of rigor, these days, but if we’re going to dismiss them ALL, we ourselves need to apply more rigor. How do we KNOW that geometry DOESN’T point to something “deeper”?
Anyway, enough of that. Here’s Wikipedia’s page talking about the chestahedron, https://en.wikipedia.org/wiki/Diminished_trapezohedron#Special_cases
And I definitely recommend picking up some books on Copernicus’s and Kepler’s work on geometry from your local library. I believe Kepler actually preceded Penrose’s work on penrose tiling! You’ll likely find more images in those books than is available online, which you’ll no doubt find fascinating.
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Thank you — you have given me much to think about with this comment!
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