Drawing Hypocycloids (synthwave enumeration)
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- Опубликовано: 29 сен 2024
- This synthwave enumeration shows the creation of 32 different hypocycloids, which are curves that are obtained by following a point that lies on circle rolling inside another circle. If the ratio of the radii of the two circles is a rational number, then the described curve will close after a certain number of rotations of the inner circle (can you figure out how many rotations in terms of the ratio?). Can you determine how many cusps the curve will have in terms of the ratio? What do you think will happen if the ratio is irrational? Have you every played with spirograph?
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This animation was inspired by a fun article from Carol Schumacher and two undergraduates from the November 2021 issue of Math Horizons (where they draw generalized hypocycloids using tangle toys instead of circles). See maa.tandfonlin... for more.
#manim #hypocycloid #spirograph #circle #rollingcircles #rational #count #mathvideo #math #mtbos #animation #iteachmath #mathematics #enumeration #synthwave
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Pacific by Vyra | / vyramusic
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creativecommon...
Wanted to share this interactive desmos graph I made on hypotrochoids: www.desmos.com/calculator/spirograph
(variable settings are in the "4 Parameters" folder)
Very cool. Thanks for sharing! Having interactive version is awesome for playing around :)
i have a question. how do you remove all the top and bottom lines?
@@tesseract7586 What lines are you referring to? Grid lines or something?
Just posting cause I think it helps with the algorithm somehow.
Thanks for trying! :)
Cool. As I was watching I kept thinking about the the size of the biggest circle that would fit inside the finished diagram. There must be a relationship there...
I’ve been working on some nested hypocycloids filling the circles inside:) fun to play with. Thanks!
The radius of that circle would appear to be distance of the midpoint of the hypocycloid arc from the centre of the main circle, and that in turn has to do with how close the outer radius is to half the inner radius. For the straight line equivalents to these hypocycloids that's probably fairly easy to calculate. For the arcs, I'm not so sure. Are they circular arcs or something else?
@@phodd If the outer radius is R, then the inner radius is r=p/q*R where p/q is the listed fraction (or the reciprocal of it actually). So the inner circle that is created has a radius of .5*(R-2r). I don't think the curves are actually parts of circles.... but I guess I don't know off the top of my head :)
The numerator of the characteristic fractions is the number of spikes the resulting star has.
What about the denominator, though? It has some relation to how far around the circle you go with each step but that seems to be proportional not to the denominator but to the difference between numerator and denominator.
The denominator should be the number of times the inner circle has to travel around the outer circle before the curve closes up. :)
@@MathVisualProofs Oh, of course ^^'
Wow! Cool! 😂
Thanks!
100/49
Am I imagining this or did the channel used to be called micro visual proofs
Definitely. Might change it back. Has suggestion to try a different name. Intro to videos still says it. What are your thoughts?
@@MathVisualProofs I think "mathematical" has too many syllables so I prefer the previous name. Also, I feel "micro visual proofs" sounds almost more playful and less strict and rigorous than "mathematical". After all, your videos seem to emphasise the less common and more inspiring visual approaches to proofs rather than more common ones.
@@Mr_Happy_Face thanks! I was probably gonna change it back. I was seeing if having math in the title helped with analytics at all :)
Interesting patterns!
Definitely! :)