Sacred Geometry people are my math heretics— The mad with power inquisition version of me would want them burned at the stake. As it is? I just seethe at their “incorrect appreciation of geometry”
They worship wrong.
I bought a book of geometric constructions — (not reading about it too carefully...) Just cracked it open. It cheerfully gives instructions for constructing a triangle, square, pentagon, hexagon, and heptagon…
A footnote begrudgingly admits the heptagon is only 98.25% correct. 1/
I almost thought that footnote wouldn’t even be there! This book is that glib. The footnote contains the most interesting fact presented in the entire book.
I must compulsively check all these constructions to see if they are “real” they probably are— but still.
I wonder how the heptagon approximation was created— I get the desire to just know how to make a serviceable regular solid— but to talk of heptagons but not unconstructability ? heresy I say!
@futurebird
I know that book. Ain't nothing wrong with approximations if you cop to it. That heptagon is similar to one Albrecht Durer draws and he doesn't admit to its inaccuracy.
You can (and should) construct a heptagon by origami methods.
http://origametry.net/papers/heptagon.pdf
I wonder how the heptagon approximation was created
💯 I'm wondering too. Answering this would probably require to extend approximation theory, i.e. continued fractions, to arbitrary algebraic fields. Not sure if that should be done ... (or has been done).
@helge @futurebird as a computer-programming-poisoned mind, I'd probably approximate anything trickier than a triangle by repeated bisection of angles using the binary representation of the fraction. You can get that as close as you like if you're patient.
That includes the pentagon, which has an exact construction but I can never remember how to do it.
Nooooooooooooo! That's inefficient.
If my memory serves me right, it's probably one of the worst possible ways to do it. All numbers of the form n/2^k can be constructed ... Now, one needs to dig through the approximation theory, on why this converges slowly in the number of steps.
But... I like constructions because they are theoretically exact!
But this does kind of expose why including an approximation for a heptagon alongside the traditional exact methods for the other regular polygons (even with a "this is just an approximation" footnote) is... weird and heretical.
@futurebird @helge
I'm not sure I can back this up, but... maybe philosophically speaking *counts* can be exact but *measurements* can't, so lengths and angles aren't really a thing, and even exact compass-and-straightedge constructions are based on axioms that don't make sense in the real world.
So I'm very relaxed about absolutely anything being an approximation, however heretical that is 😀
@helge @futurebird I doubt Heron of Alexandria (or whomever he copied his homework from) had access to any of those though.
[monthly mention that I regret losing the bookmark to a math professor's YT channel that was dedicated to doing math proofs known to ancient Greeks using only the techniques that were known to them]
@helge @futurebird wait, I think it's this guy?
https://www.youtube.com/@njwildberger/playlists
Weird how YT search gave me nothing for years and now he just pops up again.
Mind you, I never got around to really sitting down to listen to his lectures (until now) so I'm not sure how good his material is - I just loved the idea of old, now overlooked math techniques being kept alive the same way some people keep medieval craftsmanship alive or something.
This channel aims to explain a lot of interesting mathematics to a broad audience, to introduce exciting new research directions, and to fix some of the logical weaknesses that beset the subject.. You'll find playlists on Rational Trigonometry (much simpler, more powerful), Linear Algebra, Algebraic Topology, History of Mathematics, Universal Hyperbolic Geometry (a complete new treatment of this subject), the Foundations of Mathematics (it needs fixing), the Sociology of Pure Maths, Playing Go, Mathematics and Music, Differential Geometry, The Algebra of Boole and even an elementary introduction to K-6 mathematics. I (N J Wildberger) am a professional mathematician, BSc U. Toronto 1979, PhD Yale 1984 and Honorary Prof at UNSW, Sydney with over 50 papers, one or two books, and a love of teaching. And hundreds of math videos. Also, please check out the Wild Egg Maths channel, where Members have access to dozens of Research Maths videos over a wide range of topics.
@helge @futurebird So I just did a pre-emptive milkshake duck check and I find:
"DIVINE PROPORTIONS : Rational Trigonometry to Universal Geometry by N J Wildberger"
https://web.maths.unsw.edu.au/~norman/
Please tell me that that's not the same as sacred geometry? 😅 I mean he's still a maths professor so his stuff is still subject to peer review right? Right?
(also could find no information regarding milkshake-duckness so probably ok?)
No, this just seems to be normal math with an over excited title-- that's the whole problem with the "sacred" crowd-- they are hard to distinguish!
@futurebird @helge Yeah, it makes me wonder how many potential readers he lost because of the title.
The wikipedia page about the book says he's a finitist and came up with these proofs basically just to get rid of limits and actual infinity, so still a little quirky. But that's more of a philosophy-of-maths thing that one may agree or disagree with - plus the proofs apparently were still novel and correct, so something good has come out of it either way :)
https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry
More on the math behind this: http://youtu.be/oYlB5lUGlbwCatch David on the Numberphile podcast: https://youtu.be/9y1BGvnTyQAMore links & stuff in full descr...
@futurebird … think i found the opposite yesterday … NOT-a-heresy, but an imaginary number! 😁 … and not just some hypothetical, in a math class; in a electronics formula! something I could solder together in the real world! 🥰
found this in radio stuff, re: “Impedence” … matching the feed-line to the antenna. 🤷♀️
🤔 “Reactance” component seems like “Imaginary Resistance” … might as well just say it’s measured in “Imaginary Ohms” 😂👍 cool! thanks.
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