The Hypnotic World of Degenerate Spirals
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The Hypnotic World of Degenerate Spirals
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Re: The Hypnotic World of Degenerate Spirals
#2So the degenerate part is taking a fixed sample of points no matter how big the spiral gets. The effect is like you're sampling at larger and larger scales, so you get to sample the structure of the spiral at larger and larger scales, and the diversity of patterns you see, is the visual representation of the spiral at these larger and larger scales through the lens of the sample you are taking. That's my explanation anyway.
Re: The Hypnotic World of Degenerate Spirals
#3Really wonderful.
Anyone who knows how to implement this with projectM, or GLSL?
Re: The Hypnotic World of Degenerate Spirals
#4Reminds me of moiré patterns.
https://en.m.wikipedia.org/wiki/Moiré_pattern
This is found in magic angle graphene, which will likely yield a Nobel Prize in a few years: https://www.quantamagazine.org/when-magic-is-seen-in-twisted...
I wonder if there is a connection to these spirals and their segmentation lengths…
Re: The Hypnotic World of Degenerate Spirals
#5>In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
I'm surprised by that definition and always thought of a spiral as a curve with a monotonic signed-curvature function.
So, for example, the Euler/Cornu Spiral has a point of inflection where the curvature changes sign at the point of inflection, but the curvature increases continuously all the way from -infinity to + infinity as you travel along the length of the curve. So under my definition the whole Euler Spiral would count as a spiral, even though it stops revolving/emanating from a point just under 1/4 turn after the inflection point.
If you split a curve into segments at its curvature minimum and maximum points (vertices in the differential geometry sense [0]) then each segment has monotonic curvature and I'd define those as spiral segments. Vertices and monotonic curvature segments are preserved under inversion, which is mathematically useful.
In contrast, inflection points with zero curvature are not preserved under inversion. So the Euler spiral can be transformed by a suitable inversion to a curve like the one defined by Wikipedia, that is a curve emanating out from, for example, the origin.
Edit: just spotted this in the Wikipedia article on spirals 1]:
> Spirals which do not fit into this scheme of the first 5 examples:
> A Cornu spiral has two asymptotic points.
> The spiral of Theodorus is a polygon.
> The Fibonacci Spiral consists of a sequence of circle arcs.
> The involute of a circle looks like an Archimedean, but is not:
The Cornu spiral I've covered.
The spiral of Theodorus doesn't have a monotonic curvature function - it's a polygon approximation of the Archimedes Spiral, which does.
The Fibonacci Spiral's curvature function is a monotonic step-function.
The involute of a circle is a log-aesthetic curve, all of which have monotonic curvature functions. (The logarithmic spiral and the Euler spiral are also log-aesthetic curves.)
Re: The Hypnotic World of Degenerate Spirals
#6If you like this, you might also like: Belousov-Zhabotinsky reactions
https://www.youtube.com/watch?v=PpyKSRo8Iec
https://www.youtube.com/watch?v=kw9wF-GNjqs
They can form very neat patterns.
Re: The Hypnotic World of Degenerate Spirals
#7Reminds me of moiré patterns. https://en.m.wikipedia.org/wiki/Moiré_pattern This is found in magic angle graphene, which will likely yield a Nobel Prize in a few years: https://www.quantamagazine.org/when-magic-is-seen-in-twisted... I wonder if there is a connection to these spirals and their segmentation lengths…
Yep, I think the effect is basically caused by aliasing, much like the moiré effect. Very cool!
Re: The Hypnotic World of Degenerate Spirals
#8>In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. I'm surprised by that definition and always thought of a spiral as a curve with a monotonic signed-curvature function. So, for example, the Euler/Cornu Spiral has a point of inflection where the curvature changes sign at the point of inflection, but the curvature increases continuously all the way fr…
Another curve to add to this (very nice) collection is the parallel curve of an Euler spiral. It's mathematically very similar to a circle involute, but with some nice and interesting properties of its own.
Re: The Hypnotic World of Degenerate Spirals
#9Reminds me of moiré patterns. https://en.m.wikipedia.org/wiki/Moiré_pattern This is found in magic angle graphene, which will likely yield a Nobel Prize in a few years: https://www.quantamagazine.org/when-magic-is-seen-in-twisted... I wonder if there is a connection to these spirals and their segmentation lengths…
Yep, I think the effect is basically caused by aliasing, much like the moiré effect. Very cool!
No aliasing is required. I was first introduced to moire patterns by a book with a bunch of line patterns, and a sheet of transparent plastic with other line patterns printed on it. Put the latter over the former, move it around, watch the weird effects.
Re: The Hypnotic World of Degenerate Spirals
#10So the degenerate part is taking a fixed sample of points no matter how big the spiral gets. The effect is like you're sampling at larger and larger scales, so you get to sample the structure of the spiral at larger and larger scales, and the diversity of patterns you see, is the visual representation of the spiral at these larger and larger scales through the lens of the sample you are taking. That's my explanation…
Ah, relief. Thank you.