There was someone's comment here with a code that worked in the browser, but it was deleted by TH-cam (TH-cam didn't even send the link to verify it!) editor.p5js.org/drawliphant/sketches/lrlHI7jd_ Thanks to David Oliphant for implementation!
@@minnarew pascal's snail is the more accurate term here. It just so happens that a cardioid is a special case and that's the one he showed in the video.
I recently bought one of these toys because it felt like there would be something to them. I love that you got so into it and documented your discoveries. Your enthusiasm makes me want to try work some things out too now! Thank you for making this :)
Glad to hear it! If you interested in the topic, i recommend to check out Joe Freedman's Amazing Cycloid Drawing Machine on TH-cam. It's a mechanical representation of how this spirals could be drawn.
The Spirograph I remember as a kid had pins that you would use to pin the fixed wheel down to a piece of corrugated cardboard. That allowed more creativity and flexibility than later jigs, since I could pin down parts in ways the makers did not plan for.
Will do! I'll post the link in community page as soon as i comment the code and make it less messy 😉. One thing, i don't think that it'll be as beautiful as in video, because final animations was rendered in After effects and it was NOT a real time render due to lot of post processing.
okay - THIS is the reason i'm excited for the future. I know that's a big thing to say, but i just think about all the people who can learn about literally anything they want/dream about in the palm of your hand. It just makes me happy, having found your amazing video and thinking about my younger self, and how much my life may be different if i was exposed to all the world's information so young... if that makes any sense at all? Probably not hah
The constraint on the second and the third circle reminds me of when we are calculating the equation of motion of the end point of a double pendulum, which is a nightmare
I think I found a real cool set of numbers using your tool. It is a decagon with a lot of interesting features. A hollow rectangular shape in the center, and "angled" lines internally. It makes me think I'm looking at a 3D object. Probably can't share a link on TH-cam, so if someone wants to see I'll write down the numbers.
I have spent a lot of time doing similar maths but never implemented it in such a pretty way. Thank you for this. The reason I was investigating this was to understand why you get a straight line when the rolling circle is half the radius of the larger circle. Keep up the amazing work
The ratio of the radii is 2/1 so that the number of nodes is 2 (the two ends of the line). If you move the pen position inside the rolling circle you get an elipse with the limit of a circle when the pen at the centre of the rolling circle.
We used to play with these in the 80's... they were made of different colored clear plastic and they came in a ton of sizes. We played with them ALL the time! Thank you for bringing my childhood drawings to life. 🙏🥳
Why I get reminded of Spiral manga by Ito Junji with your spiral obsession anyways you can be a great teacher with all that graphics and the structure of the video was fantastic ❤️
I wonder how this relates to fourier transform and how you can make pretty pictures with circles spinning at suitable frequencies. Are these the same thing? Or can this be a kind of generalisation of that?
This is actually a special case of a simple Fourier series function. If you draw a function in the complex plane that is of the form z(t)=(1+1/k)*exp(i*2pi*t)+1/k*exp(i*2pi*2*k*t), you get the epitrochoid two circle trace with parameter k. You would get the three-circle trace by accounting for a third term in the series sum.
A few points: At 3:30 placing the drawing point outside the wheel is exactly equivalent to another arrangement with a larger outer wheel and a normal drawing point on an inner wheel. For instance, your example appears to be R=7, r=5, d=8. But using R=7, r=2, d=5/4 yields the same figure, with suitable scaling. 3:40 Positioning the drawing wheel outside the base wheel: The spirographs we bought in the 70s in America did this. Instead of your wooden panel with multiple cutouts, there were several full wheels plus some annuli, any of which could be pinned down. Some sets had some additional pieces, like a bar with round ends, and even a 3-way splitter and other pieces that snapped together. All the pieces in the sets were plastic.
Thank you so much for making this. This is amazingly put together and the best video I have seen so far made for this challenge! Please make more videos!!!
I sometimes make spirograph drawing programs in javascript to amuse myself at work. I appreciate your in depth analysis of what causes the different shape categories. I've observed the same types of shapes, but mostly through trial and error changing rotation speed and gear count.
This channel IS amazingly good. These equations deserves a derivation in another video. Your quality is way too high and represents the work of an artist and mathematician. This work will get better because all the quality is in here. I am a math student and reader. I can safely say that your work is spot on. This leads us to differential geometry and the Frenet Serret equations for unicursal curves. This IS needed.
something that i realised near the end of watching this, i'm very sure i've seen many of those shapes on old osciliscopes using sine waves in the x and y direction, i don't have the ability to check it out but i think it could be fantastic!
I programmed only two-circle version myself and stoped, while you pushed it further more to make it really beautiful, nice video! Some interesting Russian accent
Epicycloids are fascinating... I also spent a summer playing with them and wrote a small drawing application (and an article on medium) but I added some randomness into the mix, like: the direction and speed of the rotation and the number of circles to see what I get. I have to say I prefer your visuals here - they better describe how each factor controls the overall shape. Awesome job!
One think I remember learning as a kid when using the Spirograph that led to some interesting insight into the geometry and mathematics, is that some shapes have a peculiar difficulty in drawing them: The gear would naturally pull away from the edge of the ring, spoiling the shape. When reaching the critical point in the curve, I would have to carefully turn the wheel with a finger and keep it pushed against the edge, rather than letting the force of the pen do it. Basically, the force of the pen near the center of the gear would point nearly directly away from the outer ring; the force of the pen matches the direction of the line at that point.
The actual Spirograph toy is somewhat different from that (home made?) laser cut toy you showed It does both hypo- and epi- trocoids. The full set has two fixed rings of different sizes, a flat bar, _many_ more drawing gears, and a few odd shapes including a triangle with rounded edges and a cross shape with teeth only on the extreme edges, and a pointy American Football shape. The "fixed" rings have pen holes as well so you could reverse the process and use them as drawing gears too, though the regular drawing gears have pen holes but not _pin_ holes, they could be fixed in place if you put the pins along the edge of the (too large) pen holes and used several of them. The transparent plastic allows you to see what you are drawing. There are _many_ more pen holes, arranged in a spiral so they are separated from each other even though close to the same radius. This allows fine tuning of the drawing's size.
Mine was more like his, a rectangular frame which had an elliptical hole in the center lined with teeth and gears with holes in them. There was an insert that fit into the large hole and had two smaller holes(one was another ellipse and the other was a circle).
Очень интересное видео, потрясающая графика, очень приятный монтаж, а акцент не страшен) Бодрый старт для канала, просто супер! Желаю удачи и буду ждать новых видео)
Thanks for this inspiring video, the beautiful aesthetics of this inspired me to take a look at some program i'd already written (an additive synthesizer, which is basically just a finite sum of sine terms according to magnitude, frequency and phase input vectors) and expanded it to allow a secondary output with an alternate phase vector - by setting the phase difference between the outputs to pi/2 this gives a perfect sin/cos output pair for X-Y graphing. Then the spirograph examples are just a special case for limited number of terms, fixing the frequency ratios to be rational, and some limitations on the relative magnitudes of terms. You can actually get some really cool "fake 3d" rotations in the plotting if you play around with shifting the phase offsets between X and Y to things other than pi/2, and it being a synthesizer also means you can listen to what specific graphic patterns "sound" like (turns out most "beautiful" sine chord ratios are far from harmonic), or have a live FFT view of the partials in the spectrum.
Absolutely gorgeous... What about an extrapolated version in 3D space with rolling spheres ? Hard to tell how it could look like. I guess number of parameters might increase significantly.
You'd probably need a raytracer to render it, but it would be interesting. If you don't mind writing code to define your scenes, POVRay has an "isosurface" object type where you can supply a formula and it'll render the surface directly. The Windows version comes with a built-in text editor with syntax highlighting for the POVRay Scene Description Language. Unfortunately, the Linux version seems to be command-line only and expects you to use other programs to write your scenes.
I tried making something exactly like this 4:50 in roblox but failed. And for some strange reason Fourier series were the inspiration, because of how I interpreted it I guess. I wish so bad to make a physical object do that in Roblox 😔
There was someone's comment here with a code that worked in the browser, but it was deleted by TH-cam (TH-cam didn't even send the link to verify it!)
editor.p5js.org/drawliphant/sketches/lrlHI7jd_
Thanks to David Oliphant for implementation!
you called a cardioid Pascal's snail????????????? what?
@@minnarew pascal's snail is the more accurate term here. It just so happens that a cardioid is a special case and that's the one he showed in the video.
Thanks for the shoutout! feel free to edit the values yourself for all those variables at the top of the code, and try out different ratios!
@@Drawliphant Well done!
@@max_sparbot well now im upset that cardioid microphones arent called snail microphones instead
"Mathematics is the perfect tool for solving problems you've created for yourself."
I've never felt so attacked by something I 100% agree with x)
hahahaha, indeed. :)
Absolutely incredible graphics
Woah! I didn’t expect you here!
Omg, Big Cactus himself! Thanks mate, i'm really like your videos.
@@sortofschool thanks! you should really make some more love the style
@@sortofschool Thanks but how did you spend a whole summer? Didnt you get bored and frustrated after a day or so?
dream = cring
I recently bought one of these toys because it felt like there would be something to them. I love that you got so into it and documented your discoveries. Your enthusiasm makes me want to try work some things out too now! Thank you for making this :)
Glad to hear it! If you interested in the topic, i recommend to check out Joe Freedman's Amazing Cycloid Drawing Machine on TH-cam. It's a mechanical representation of how this spirals could be drawn.
I have no idea where TH-cam is hiding you, but this was absolutely fascinating
It’s cool being able to get to a channel and be able to claim you’ve been there before it blows up.
"Once you start doing impossible things, it's hard to stop."
Love this!!
I don't think there's a better video about spirographs out there. This deserve more views.
you should have submitted this to the 3blue1brown competition if you didnt, amazing video... very beautiful... took it even further than i imagined
The Spirograph I remember as a kid had pins that you would use to pin the fixed wheel down to a piece of corrugated cardboard. That allowed more creativity and flexibility than later jigs, since I could pin down parts in ways the makers did not plan for.
It was a brilliant toy.
Take a moment to acknowledge the incredible editing, music and script of this video. Never thought a video about spirals would evoke such emotion.
“Mathematics is the perfect tool for problems you’ve created for yourself”
Please make this open source. Or even payed software. I would love to play with this. It seems like endless fun, and I'm sure I'm not alone.
Will do! I'll post the link in community page as soon as i comment the code and make it less messy 😉.
One thing, i don't think that it'll be as beautiful as in video, because final animations was rendered in After effects and it was NOT a real time render due to lot of post processing.
@@sortofschool Aaah, okay :)
We shall see. It will be fun either way. Thank you, keep up the good work
th-cam.com/video/bqRvLR3PLf0/w-d-xo.html
This video can help
@@spiralofinspiration3653 Done!
github.com/sokolov-teach/spirograph
@@sortofschool gbu
okay - THIS is the reason i'm excited for the future. I know that's a big thing to say, but i just think about all the people who can learn about literally anything they want/dream about in the palm of your hand. It just makes me happy, having found your amazing video and thinking about my younger self, and how much my life may be different if i was exposed to all the world's information so young... if that makes any sense at all? Probably not hah
this was beautiful, it made me more emotional than most movies
Damn this video is crazy good even though it wasn't made by a big channel. Subscribed!
Dude, the aesthetic in this video is INCREDIBLE :o props to you, that is some incredible work !
The constraint on the second and the third circle reminds me of when we are calculating the equation of motion of the end point of a double pendulum, which is a nightmare
Truly a video showcasing the beauty of not only the spirals but mathematics, the true heart of what describes them
So the key to Dr. Strange's magic is math...
I think I found a real cool set of numbers using your tool.
It is a decagon with a lot of interesting features. A hollow rectangular shape in the center, and "angled" lines internally.
It makes me think I'm looking at a 3D object.
Probably can't share a link on TH-cam, so if someone wants to see I'll write down the numbers.
Please do it sir, we would love to see😄
I have spent a lot of time doing similar maths but never implemented it in such a pretty way. Thank you for this.
The reason I was investigating this was to understand why you get a straight line when the rolling circle is half the radius of the larger circle.
Keep up the amazing work
The ratio of the radii is 2/1 so that the number of nodes is 2 (the two ends of the line). If you move the pen position inside the rolling circle you get an elipse with the limit of a circle when the pen at the centre of the rolling circle.
Thanks a lot. A beautiful mathematic journey starting with a spirograph, something so magic,
I just stumbled across a spiralgraph and am fascinated with the math. I hope to Start learning and understanding soon
I'm quite a bit older than you, and my childhood spirograph was much more elaborate, and DID let you put the drawing wheel outside the ring!
mesmerizing, i dont even know what happened, felt like a ' through the looking glass ' experience for me, well done!!!
Amazing! I was recommended this a year later. Please continue your adventures with Math!
We used to play with these in the 80's... they were made of different colored clear plastic and they came in a ton of sizes. We played with them ALL the time! Thank you for bringing my childhood drawings to life. 🙏🥳
Why I get reminded of Spiral manga by Ito Junji with your spiral obsession anyways you can be a great teacher with all that graphics and the structure of the video was fantastic ❤️
This is amazing please do more stuff! I always loved those Spirograph toys, they were awesome!
Beautiful trefoil knot at 9:21!
Wow, this video is absolutely visually stunning! Love your choice of music, great work.
Loved the way you took us on the same journey of discovery you took yourself! Nice video
I wonder how this relates to fourier transform and how you can make pretty pictures with circles spinning at suitable frequencies. Are these the same thing? Or can this be a kind of generalisation of that?
This is actually a special case of a simple Fourier series function. If you draw a function in the complex plane that is of the form z(t)=(1+1/k)*exp(i*2pi*t)+1/k*exp(i*2pi*2*k*t), you get the epitrochoid two circle trace with parameter k. You would get the three-circle trace by accounting for a third term in the series sum.
I think they're the same, and this is just a bandlimited version that cuts off after two or three terms.
@@kupariseppo7566hi
It’s a special kind of Fourier series.
This is probably one of my favourite entries! Holy... This is mindblowing and super fascinating!
I like the progression from toys to math
A few points:
At 3:30 placing the drawing point outside the wheel is exactly equivalent to another arrangement with a larger outer wheel and a normal drawing point on an inner wheel. For instance, your example appears to be R=7, r=5, d=8. But using R=7, r=2, d=5/4 yields the same figure, with suitable scaling.
3:40 Positioning the drawing wheel outside the base wheel: The spirographs we bought in the 70s in America did this. Instead of your wooden panel with multiple cutouts, there were several full wheels plus some annuli, any of which could be pinned down. Some sets had some additional pieces, like a bar with round ends, and even a 3-way splitter and other pieces that snapped together. All the pieces in the sets were plastic.
This is such a well made video. Really a lot of talent went into every aspect.
this is some truly beautiful work
These graphics are incredible, and the beautiful, smoothly morphing spirals even more so! Awesome work
Both interseting and very beautifull stuff there. I love the esthetic of the vidéo !
Amazing and interesting video😍😍😍
Keep making these videos👍👍👍
Full sumpport 😁😁😁
Thank you so much for making this. This is amazingly put together and the best video I have seen so far made for this challenge! Please make more videos!!!
well produced, and most importantly well researched
It's great
I don't understand why you are so underrated
great job, evolution over time describes the manifold
Stunning video, hope to see more!!
Amazing video this could explain so much about the movement of atoms
I just found this video through 3b1b and it really does look beautiful!
I sometimes make spirograph drawing programs in javascript to amuse myself at work. I appreciate your in depth analysis of what causes the different shape categories. I've observed the same types of shapes, but mostly through trial and error changing rotation speed and gear count.
wonderful work! please keep posting !
Good job! You might want to look on the Wild Gears drawing tools. It’s more flexible yet complicated than a Spirograph.
Simply fantastic. Great job. Would love to see more.
Wow! Beautiful! Thank you for bringing this to us.
This is absolutely incredible, thank you for making this
Brilliant. Please make more videos.
Amazing work dude! Part 2 is a must
This channel IS amazingly good. These equations deserves a derivation in another video. Your quality is way too high and represents the work of an artist and mathematician.
This work will get better because all the quality is in here. I am a math student and reader. I can safely say that your work is spot on. This leads us to differential geometry and the Frenet Serret equations for unicursal curves.
This IS needed.
something that i realised near the end of watching this, i'm very sure i've seen many of those shapes on old osciliscopes using sine waves in the x and y direction, i don't have the ability to check it out but i think it could be fantastic!
Math has been around. Math is and will be everywhere, waiting to be discovered.
Awesome job, man. Congrats!
I programmed only two-circle version myself and stoped, while you pushed it further more to make it really beautiful, nice video!
Some interesting Russian accent
What an amazing video! Gave me so much inspiration that I immediately had to pause halfway through and start coding while watching this
Absolutely amazing bro
Really cool work! Thank you, this definitely raised my interest!
reallly nice video; got me in the mood of coding some spirograph like stuff myself
hoping to see more stuff from you !
I loved my Spirograph as a kid and was recently surprised to find a Spirograph phone app
Epicycloids are fascinating... I also spent a summer playing with them and wrote a small drawing application (and an article on medium) but I added some randomness into the mix, like: the direction and speed of the rotation and the number of circles to see what I get. I have to say I prefer your visuals here - they better describe how each factor controls the overall shape. Awesome job!
The format was really nice, good work!
Excited to think about it and take a look thanks you! First time here! subscribed!
we are not nerds anymore, now we are real grown scientists. Thank you for being you!
5 minutes in, with the addition of the second circle, I immediately started to think about Fourier transformation shenannigans :p
Wow! This video was really well made. Super interesting. Thanks for sharing with us all :D
This is a phenomenal video!
I really enjoyed this video, you are amazing!
I would love to see some further experiments. For instance with shapes of constant width (i.e. a reuleux triangle)
This thing needs more views
don't know how you preformed at the SoME1 but this video is my personal favourite.
I made a little desmos project to generate those last year, plenty of fun!
This is a great video! Thank you for this, I might try this out to discover even more beauty in the world of spirals!
Very good explanation. Thank you ❤
Such a good video, nice work!
what a beautiful video... really amazing - you're a talent.
One think I remember learning as a kid when using the Spirograph that led to some interesting insight into the geometry and mathematics, is that some shapes have a peculiar difficulty in drawing them: The gear would naturally pull away from the edge of the ring, spoiling the shape. When reaching the critical point in the curve, I would have to carefully turn the wheel with a finger and keep it pushed against the edge, rather than letting the force of the pen do it. Basically, the force of the pen near the center of the gear would point nearly directly away from the outer ring; the force of the pen matches the direction of the line at that point.
Lmao tf
never saw an spirocraph that is so fucking cool!
Glorious... Thank you
whoa how is the gem of a video not gotten more views
Rly nice content bruh, subbed instantly
The actual Spirograph toy is somewhat different from that (home made?) laser cut toy you showed It does both hypo- and epi- trocoids. The full set has two fixed rings of different sizes, a flat bar, _many_ more drawing gears, and a few odd shapes including a triangle with rounded edges and a cross shape with teeth only on the extreme edges, and a pointy American Football shape. The "fixed" rings have pen holes as well so you could reverse the process and use them as drawing gears too, though the regular drawing gears have pen holes but not _pin_ holes, they could be fixed in place if you put the pins along the edge of the (too large) pen holes and used several of them.
The transparent plastic allows you to see what you are drawing.
There are _many_ more pen holes, arranged in a spiral so they are separated from each other even though close to the same radius. This allows fine tuning of the drawing's size.
Mine was more like his, a rectangular frame which had an elliptical hole in the center lined with teeth and gears with holes in them. There was an insert that fit into the large hole and had two smaller holes(one was another ellipse and the other was a circle).
Очень интересное видео, потрясающая графика, очень приятный монтаж, а акцент не страшен)
Бодрый старт для канала, просто супер!
Желаю удачи и буду ждать новых видео)
Awesome work!
This one was simply stunning... Loved it
this video was very well made - great work!!
Stunning !
Thanks for this inspiring video, the beautiful aesthetics of this inspired me to take a look at some program i'd already written (an additive synthesizer, which is basically just a finite sum of sine terms according to magnitude, frequency and phase input vectors) and expanded it to allow a secondary output with an alternate phase vector - by setting the phase difference between the outputs to pi/2 this gives a perfect sin/cos output pair for X-Y graphing. Then the spirograph examples are just a special case for limited number of terms, fixing the frequency ratios to be rational, and some limitations on the relative magnitudes of terms.
You can actually get some really cool "fake 3d" rotations in the plotting if you play around with shifting the phase offsets between X and Y to things other than pi/2, and it being a synthesizer also means you can listen to what specific graphic patterns "sound" like (turns out most "beautiful" sine chord ratios are far from harmonic), or have a live FFT view of the partials in the spectrum.
Absolutely gorgeous... What about an extrapolated version in 3D space with rolling spheres ? Hard to tell how it could look like. I guess number of parameters might increase significantly.
You'd probably need a raytracer to render it, but it would be interesting. If you don't mind writing code to define your scenes, POVRay has an "isosurface" object type where you can supply a formula and it'll render the surface directly. The Windows version comes with a built-in text editor with syntax highlighting for the POVRay Scene Description Language. Unfortunately, the Linux version seems to be command-line only and expects you to use other programs to write your scenes.
I tried making something exactly like this 4:50 in roblox but failed. And for some strange reason Fourier series were the inspiration, because of how I interpreted it I guess. I wish so bad to make a physical object do that in Roblox 😔
Beautiful fuorier series
Very good job. Thank you
amazing video!