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The Roots of Sacred Geometry
You guys tired of geometry yet? No? Great! Neither am I! I found another neat thing I'd like to show you.
Back in 2002, Kurt Hofstetter showed how fundamental the golden ratio is by demonstrating a very simple construction using five circles:
I discovered today that he considerably undersells that construction. Not only does it contain the golden ratio, but it also contains the other two fundamental ratios of sacred geometry, √2 and √3. Consider this pentacle-like diagram:
In this diagram, notice the the dark circle in the bottom-center. If it has a radius of length 1, then the blue lines are of length √3, the green lines are of length √2, and the red line is of length ϕ. Thus it is almost trivial to construct any of the classical regular polygons—the triangle from √3, the square from √2, the pentagon from ϕ, or their multiples—within the dark circle. Indeed, the triangle is already present; the square and hexagon take a single additional circle each, while the pentagon requires two additional circles:
I haven't had much time today, but I'd like to explore this construction further...
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Axé!
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I have been working through "the Diagram - Harmonic Geometry" by Adam Tetlow. It is amazing how many ratios can come out of a square with some diagonal lines drawn through it, and the amazing frequency of 3-4-5 triangles that form themselves within it.
I had no idea how many things would develop out of collections of circles! I suppose it should not surprise any of us--The universe is dripping with linkages and order that pops up out of apparent chaos. ;-)
You and other readers may be interested in this article from 2013 about Chemistry by Number Theory by Boeyens:
https://repository.up.ac.za/bitstream/handle/2263/33292/Boeyens_Chemistry_2013.pdf?sequence=1&isAllowed=y
Even if your math skills are limited, it is well worth grazing through the PDF just to look at the pictures-- The diagram that fits the planets of our Solar System neatly onto two Fibonacci curves blew me away!
--Emmanuel G
PS-- to answer your question-- Of course we are not tired of geometry! ;-)