~10 years ago I had a renewed interested in perspective drawing, and was struggling with an 'exercise' that utilized curvilinear perspective & oblique angles.
Anyways, I stumbled on some of Rafael's drawings and found their inclusion of the guide lines / measuring lines invaluable, but for the hell of it, I emailed him some questions.
He promptly responded and then dumped images and reference material on dropbox for me - very nice guy.
Personal fav is "Dürer" that demonstrates Albrecht Dürers method of projecting a spiral.
At first glance the "masterpieces" looked like a crap ton of lines and circles around a sketched shell or butterfly. You know, the kind you see when you tell Stable Diffusion to "sketch". Then I watched the video for a sec... WOAH!!! He is talking about a "formula" quite a bit... ctrl-f "formula" == 0 results.
Now I am trawling github to see if I can find some processing or similar libraries that I can play around with. Golden Ratio sketches, defining physical objects and shapes from purely mathematical constructions... I love rabbit holes!
Would be very pleasantly surprised if a drawing like "Blue Spikes Shell" is actually reducible to a readable mathematical construction.
A surface that exhibits very strong variability in curvature (like the spikes in the shell) is most likely a collage of different pieces. These would (at best) have only a locally valid mathematical formula. Achieving a visually convincing result through smooth patching introduces lots of ugly "numbers".
Most likely the "formula" concerns only some aspects of the drawing, like overall symmetry and relative sizes.
I believe that we still know so little about nature and its laws. Sometimes it seems that we are moving in the direction of technological development instead of developing our knowledge about nature and aligning ourselves with it.
Technically amazing - such fine detail, and of course precision. However ...
I recently heard the saying "a great artist knows when to stop" (it was Ben Afflek talking about AI art - I guess he was quoting someone?). I feel like in the case of these drawings, less detail would actually be better. More readable, perhaps.
Still, an impressive amount of effort for each one, given only straightedge and compass.
I think the detail IS the art, or at least the strongest element that the artist wishes to demonstrate.
Removing the details removes the reference to the formula(e), which is the soul around which the beatiful pictures are created. In this case, the art is the process, more-so than the end result - which is a by-product.
> he unfolds the significance of the Golden Ratio, showcasing its spiritual depth and presence within the natural order.
Yikes. The golden ratio has limited significance, nothing to do with spirituality, and little presence in nature [1]. Araujo's pictures look great, but in almost any of them you could replace the golden ratio with 1.6, 1.7, or 1.5, and get something no less beautiful.
I'm also a mathematician, and it doesn't bug me at all.
> nothing to do with spirituality
It clearly does have a lot to do with spirituality, for many people, as symbol. Much as the cross does for Christans. Neither of these symbols have spiritual meaning to me - a cross is just a cross, a spiral is just a spiral. I don't have much need for spiritual symbolism myself - when I meditate, I rather to focus on a simple sound or light source - but I'd consider it highly egotistical of myself if I was to start judging other people's use of symbols just because they don't understand maths, or whatever.
> fetishisation
I've yet to ever see this word used except to denigrate other people's beliefs, or as an attempt to make the user feel superior. I believe you did both here.
My advice, one mathematician to another: chill and let people have their symbols. Don't expect them to have a deep understanding of mathematics, much as we don't have a deep understanding of their need for spiritual symbolism. Nonetheless we can let each other be, and all get along.
Who knows, perhaps a fascination with this "sacred geometry" as they call it, might be a starting point for someone to have a genuine interest in mathematics.
In my experience, sacred geometry, and all these other terms like frequencies and energies and what not create highly ambiguous language. Esoteric concepts piggy-backing on physics’ success (or mathematics), trying to legitimize nonsense. Manganese-balancing rose quartz. Nuclear vibrations. Quantum uncertainties with at best questionable claims about determinism.
The only people I’ve ever seen healthily walk the boundaries were from philosophy/physics/math to spirituality (whatever that means for individuals), not the other way round.
So basically, the same as all human language then?
If you're saying that geometric terms should only be allowed to be used in a strictly defined mathematical sense, even by non mathematicians, that's a kind of gatekeeping that doesn't make any logical sense to me.
Firstly, it completely misunderstands how humans use language, how we play with it, make puns, reuse terms from other social groupings (maths) in new ways that have meaning to another social group (new age spirituality), how we bend and twist the meanings of words over time.
Secondly, it's impossible, and unwarranted, to try and police this, so I hope you're good at dealing with being ignored.
~10 years ago I had a renewed interested in perspective drawing, and was struggling with an 'exercise' that utilized curvilinear perspective & oblique angles. Anyways, I stumbled on some of Rafael's drawings and found their inclusion of the guide lines / measuring lines invaluable, but for the hell of it, I emailed him some questions. He promptly responded and then dumped images and reference material on dropbox for me - very nice guy.
Personal fav is "Dürer" that demonstrates Albrecht Dürers method of projecting a spiral.
https://www.rafael-araujo.com/product-page/d%C3%BCrer#
At first glance the "masterpieces" looked like a crap ton of lines and circles around a sketched shell or butterfly. You know, the kind you see when you tell Stable Diffusion to "sketch". Then I watched the video for a sec... WOAH!!! He is talking about a "formula" quite a bit... ctrl-f "formula" == 0 results.
Google: Rafael Araujo artist formula. Ahhhh! https://www.rafael-araujo.com/calculation
Now we are getting somewhere... and another that shows the process a bit better: https://hazelhomeartandantiques.blogspot.com/2015/04/the-cal...
Now I am trawling github to see if I can find some processing or similar libraries that I can play around with. Golden Ratio sketches, defining physical objects and shapes from purely mathematical constructions... I love rabbit holes!
Would be very pleasantly surprised if a drawing like "Blue Spikes Shell" is actually reducible to a readable mathematical construction.
A surface that exhibits very strong variability in curvature (like the spikes in the shell) is most likely a collage of different pieces. These would (at best) have only a locally valid mathematical formula. Achieving a visually convincing result through smooth patching introduces lots of ugly "numbers".
Most likely the "formula" concerns only some aspects of the drawing, like overall symmetry and relative sizes.
Inigo Quilez has some good videos generating art and scenes with just mathematical constructions:
https://www.youtube.com/c/InigoQuilez
This is a deep rabbit hole indeed :)
Does Donald Knuth have any metaprogramming for golden section/ratio?
About golden section in the universe there is a nice book by Scott Olsen - https://www.goodreads.com/book/show/316282.The_Golden_Sectio...
I believe that we still know so little about nature and its laws. Sometimes it seems that we are moving in the direction of technological development instead of developing our knowledge about nature and aligning ourselves with it.
Hi, id like to introduce you to eastern philosophy
https://x.com/rafaela31416
Is the Monarch one based on a geometric way they fly or something?
[flagged]
Technically amazing - such fine detail, and of course precision. However ...
I recently heard the saying "a great artist knows when to stop" (it was Ben Afflek talking about AI art - I guess he was quoting someone?). I feel like in the case of these drawings, less detail would actually be better. More readable, perhaps.
Still, an impressive amount of effort for each one, given only straightedge and compass.
I think the detail IS the art, or at least the strongest element that the artist wishes to demonstrate.
Removing the details removes the reference to the formula(e), which is the soul around which the beatiful pictures are created. In this case, the art is the process, more-so than the end result - which is a by-product.
I own a framed print of the first one (the orange butterflies) and it's a delight.
> he unfolds the significance of the Golden Ratio, showcasing its spiritual depth and presence within the natural order.
Yikes. The golden ratio has limited significance, nothing to do with spirituality, and little presence in nature [1]. Araujo's pictures look great, but in almost any of them you could replace the golden ratio with 1.6, 1.7, or 1.5, and get something no less beautiful.
The Wikipedia page is fairly good on this, especially the "Disputed observations" section: https://en.wikipedia.org/wiki/Golden_ratio#Disputed_observat...
As a mathematician, fetishisation of the golden ratio bugs me.
[1] The main place is spiral arrangements of leaves, petals, etc. Vi Hart explains why (watch all three parts): https://www.youtube.com/watch?v=ahXIMUkSXX0
I'm also a mathematician, and it doesn't bug me at all.
> nothing to do with spirituality
It clearly does have a lot to do with spirituality, for many people, as symbol. Much as the cross does for Christans. Neither of these symbols have spiritual meaning to me - a cross is just a cross, a spiral is just a spiral. I don't have much need for spiritual symbolism myself - when I meditate, I rather to focus on a simple sound or light source - but I'd consider it highly egotistical of myself if I was to start judging other people's use of symbols just because they don't understand maths, or whatever.
> fetishisation
I've yet to ever see this word used except to denigrate other people's beliefs, or as an attempt to make the user feel superior. I believe you did both here.
My advice, one mathematician to another: chill and let people have their symbols. Don't expect them to have a deep understanding of mathematics, much as we don't have a deep understanding of their need for spiritual symbolism. Nonetheless we can let each other be, and all get along.
Who knows, perhaps a fascination with this "sacred geometry" as they call it, might be a starting point for someone to have a genuine interest in mathematics.
In my experience, sacred geometry, and all these other terms like frequencies and energies and what not create highly ambiguous language. Esoteric concepts piggy-backing on physics’ success (or mathematics), trying to legitimize nonsense. Manganese-balancing rose quartz. Nuclear vibrations. Quantum uncertainties with at best questionable claims about determinism.
The only people I’ve ever seen healthily walk the boundaries were from philosophy/physics/math to spirituality (whatever that means for individuals), not the other way round.
> highly ambiguous language
So basically, the same as all human language then?
If you're saying that geometric terms should only be allowed to be used in a strictly defined mathematical sense, even by non mathematicians, that's a kind of gatekeeping that doesn't make any logical sense to me.
Firstly, it completely misunderstands how humans use language, how we play with it, make puns, reuse terms from other social groupings (maths) in new ways that have meaning to another social group (new age spirituality), how we bend and twist the meanings of words over time.
Secondly, it's impossible, and unwarranted, to try and police this, so I hope you're good at dealing with being ignored.
> As a mathematician, fetishisation of the golden ratio bugs me.
I know, but hear me out: it's a decent hook for teaching people about Geometry, Recursion, and Dynamic Programming.
Araujo seems to relate the golden ratio to the icosahedron, so I wonder if there is something to that??
(Or is it just that the largest rectangle that inscribes a regular hexagon has the same ratio?)