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.
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
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.
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.
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
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. 🙏🥳
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.
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.
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.
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.
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.
This video inspired a five-hour geometry study adventure. I'm currently doing light experiments with sunlight and prisms, to artfully blueprint the arcs of elliptical ratio of colrs of the rainbow on planed surfaces, and mark out the polynomials used on the image in graffitti, to point out the beauty of the math. I want to do a color rendering of flared white light in rainbow prism, basis cubic Hermite splines of flared light, according to the time of day... show that either nature becomes abstractly algebraic or abstractions can become natural beauty through algebra. I spent the day studying the R value (sunlight) as it intensifies and dims during daylight hours, and catch the differentiation of color spectra as the day progresses and the hue intensity of each of the seven color go (x = -1,1), individually, according to the composite curve of the solar elliptic. I'm awestruck at the interplay of white light and black reflective surface, along with pure brilliant white light (noon) versus fractionated light at sundown. I like to watch how these these four P values interact independently to accentuate the degree of contrast/hue intensity. We can compose a Bezier curve along Sunrise and Sunset, P0-P1, and create a trajectory of color/light intensification of each individual hue of the rainbow, according to time of day by calculating the Hermite interpolations. If I want to tell this whole mess as a narrative story, I'll tell it this way: I am P0; mmy worst nightmares and suicide is P-1 while my most amazing potential come true is P1. R, the path and intensity of sunlight, is my karma, my most probable course of life. My limit is NOT P1, my limits are in form of Q... the tangental points that I'm not aware of but ruminate in the back of my mind, always on like an autonomous program, silently effectuating the true trajectory of my life R while my mind P0 is programmed to look at life in terms of P-1 to P0 to P1. The variable trajectory of each rainbow color's intensity and luminosity within its P0 elliptical subset is my actual performance in each of the 7 areas of life, like harmonic notes... does my rainbow make music or make noise? It depends on the Q value... you can imagine Q tangents like aura borealis coming off of the rainbow, a faint reflection of the rainbow itself, because of the reflective properties of the background plane (like fabric textures).... the Q tangents represent the WASTE of ENERGY, wasteful emission of luminosity that dims the rainbow itself. LIGHT and GEOMETRY are a PHILOSOPHY of LIFE. Imagine what humanity can do if it stopped obsessing about the P-1 delimiter that it cannot control, we all go up and down succeed and fail rise and drop. The unseen, and unqualified vector is Q; questions are unanswered, so the unseen and unqualified gain victory like a military coup. Questions unanswered and violent silence paint the sun black and reduce the visibility of light R.... you're blind to your environment, trajectory flatlines, tangental strings stole the luminosity of your rainbow. Or you answer your questions, mind your Ps and Qs, and make the sun brighter by making your future greater. **poet, applause**
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.
You could have the 1st cog go around the outside of the stationary cog then have the second cog on the inside of the 1st and the third on the outside of the 2nd etc... or mix orders of inside/outside etc... & then do it all in 3D with balls.
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. :)
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.
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
"Once you start doing impossible things, it's hard to stop."
Love this!!
I have no idea where TH-cam is hiding you, but this was absolutely fascinating
I don't think there's a better video about spirographs out there. This deserve more views.
It’s cool being able to get to a channel and be able to claim you’ve been there before it blows up.
you should have submitted this to the 3blue1brown competition if you didnt, amazing video... very beautiful... took it even further than i imagined
“Mathematics is the perfect tool for problems you’ve created for yourself”
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.
this was beautiful, it made me more emotional than most movies
So the key to Dr. Strange's magic is math...
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
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
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 !
Truly a video showcasing the beauty of not only the spirals but mathematics, the true heart of what describes them
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.
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
Thanks a lot. A beautiful mathematic journey starting with a spirograph, something so magic,
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. 🙏🥳
Amazing! I was recommended this a year later. Please continue your adventures with Math!
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!
Math has been around. Math is and will be everywhere, waiting to be discovered.
we are not nerds anymore, now we are real grown scientists. Thank you for being you!
I like the progression from toys to math
mesmerizing, i dont even know what happened, felt like a ' through the looking glass ' experience for me, well done!!!
great job, evolution over time describes the manifold
I just stumbled across a spiralgraph and am fascinated with the math. I hope to Start learning and understanding soon
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.
Beautiful trefoil knot at 9:21!
This is amazing please do more stuff! I always loved those Spirograph toys, they were awesome!
Loved the way you took us on the same journey of discovery you took yourself! Nice video
This is probably one of my favourite entries! Holy... This is mindblowing and super fascinating!
Wow, this video is absolutely visually stunning! Love your choice of music, great work.
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😄
This is such a well made video. Really a lot of talent went into every aspect.
well produced, and most importantly well researched
Simply fantastic. Great job. Would love to see more.
wonderful work! please keep posting !
These graphics are incredible, and the beautiful, smoothly morphing spirals even more so! Awesome work
5 minutes in, with the addition of the second circle, I immediately started to think about Fourier transformation shenannigans :p
Very good explanation. Thank you ❤
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!!!
Stunning video, hope to see more!!
Amazing and interesting video😍😍😍
Keep making these videos👍👍👍
Full sumpport 😁😁😁
Amazing video this could explain so much about the movement of atoms
Both interseting and very beautifull stuff there. I love the esthetic of the vidéo !
this is some truly beautiful work
never saw an spirocraph that is so fucking cool!
Brilliant. Please make more videos.
This is absolutely incredible, thank you for making this
Really cool work! Thank you, this definitely raised my interest!
This thing needs more views
Wow! Beautiful! Thank you for bringing this to us.
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.
Glorious... Thank you
It's great
I don't understand why you are so underrated
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.
What an amazing video! Gave me so much inspiration that I immediately had to pause halfway through and start coding while watching this
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.
This video inspired a five-hour geometry study adventure. I'm currently doing light experiments with sunlight and prisms, to artfully blueprint the arcs of elliptical ratio of colrs of the rainbow on planed surfaces, and mark out the polynomials used on the image in graffitti, to point out the beauty of the math. I want to do a color rendering of flared white light in rainbow prism, basis cubic Hermite splines of flared light, according to the time of day... show that either nature becomes abstractly algebraic or abstractions can become natural beauty through algebra. I spent the day studying the R value (sunlight) as it intensifies and dims during daylight hours, and catch the differentiation of color spectra as the day progresses and the hue intensity of each of the seven color go (x = -1,1), individually, according to the composite curve of the solar elliptic. I'm awestruck at the interplay of white light and black reflective surface, along with pure brilliant white light (noon) versus fractionated light at sundown. I like to watch how these these four P values interact independently to accentuate the degree of contrast/hue intensity. We can compose a Bezier curve along Sunrise and Sunset, P0-P1, and create a trajectory of color/light intensification of each individual hue of the rainbow, according to time of day by calculating the Hermite interpolations. If I want to tell this whole mess as a narrative story, I'll tell it this way: I am P0; mmy worst nightmares and suicide is P-1 while my most amazing potential come true is P1. R, the path and intensity of sunlight, is my karma, my most probable course of life. My limit is NOT P1, my limits are in form of Q... the tangental points that I'm not aware of but ruminate in the back of my mind, always on like an autonomous program, silently effectuating the true trajectory of my life R while my mind P0 is programmed to look at life in terms of P-1 to P0 to P1. The variable trajectory of each rainbow color's intensity and luminosity within its P0 elliptical subset is my actual performance in each of the 7 areas of life, like harmonic notes... does my rainbow make music or make noise? It depends on the Q value... you can imagine Q tangents like aura borealis coming off of the rainbow, a faint reflection of the rainbow itself, because of the reflective properties of the background plane (like fabric textures).... the Q tangents represent the WASTE of ENERGY, wasteful emission of luminosity that dims the rainbow itself. LIGHT and GEOMETRY are a PHILOSOPHY of LIFE. Imagine what humanity can do if it stopped obsessing about the P-1 delimiter that it cannot control, we all go up and down succeed and fail rise and drop. The unseen, and unqualified vector is Q; questions are unanswered, so the unseen and unqualified gain victory like a military coup. Questions unanswered and violent silence paint the sun black and reduce the visibility of light R.... you're blind to your environment, trajectory flatlines, tangental strings stole the luminosity of your rainbow. Or you answer your questions, mind your Ps and Qs, and make the sun brighter by making your future greater. **poet, applause**
Wow! This video was really well made. Super interesting. Thanks for sharing with us all :D
Excited to think about it and take a look thanks you! First time here! subscribed!
Amazing work dude! Part 2 is a must
what a beautiful video... really amazing - you're a talent.
This is a phenomenal video!
Absolutely amazing bro
The format was really nice, good work!
Very good job. Thank you
I loved my Spirograph as a kid and was recently surprised to find a Spirograph phone app
I really enjoyed this video, you are amazing!
This one was simply stunning... Loved it
Good job! You might want to look on the Wild Gears drawing tools. It’s more flexible yet complicated than a Spirograph.
im working on a program thatd be a bit beefier and do more circles pretty much. wait for updates!
This is a great video! Thank you for this, I might try this out to discover even more beauty in the world of spirals!
Awesome job, man. Congrats!
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!
congrats on getting recommended
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
reallly nice video; got me in the mood of coding some spirograph like stuff myself
hoping to see more stuff from you !
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!
3:59 the stuff of light that appears insides mugs!
Красавчик!
i am in strong agreement 🤭
amazing video!
I made a little desmos project to generate those last year, plenty of fun!
Thank you. This was really interesting to see
Beautiful fuorier series
Stunning !
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
Wonderful
this video was very well made - great work!!
You could have the 1st cog go around the outside of the stationary cog then have the second cog on the inside of the 1st and the third on the outside of the 2nd etc... or mix orders of inside/outside etc... & then do it all in 3D with balls.
Very cool and informational!
Subbed. Keep it up
Fourier analysis and Julia sets in one video. Nifty.
It's very beautiful