The Plotting of Beautiful Curves (Euler Spirals and Sierpiński Triangles) - Numberphile
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- เผยแพร่เมื่อ 15 มิ.ย. 2024
- This the first or two parts with Matt Henderson and his plotter... Part 2 (featuring Pi) is here: • Plotting Pi and Search...
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Matt Henderson on Twitter (he posts lovely animations there): / matthen2
Matt Henderson Numberphile Playlist: bit.ly/MattHendersonPlaylist
Euler Spiral World Map: • A Strange Map Projecti...
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Part 2 (featuring Pi) is here: th-cam.com/video/tkC1HHuuk7c/w-d-xo.html
He's boring
Do you have a link to the Mathematica code for the turtle function, so I can show my son?
The most impressive part of this whole video is taking the paper off the plotter mid-print, showing it off, and then putting it back on the plotter and being able to continue the print with everything still lined up properly...
I'm guessing there's something analogous to "drawing pin holes" so that he can just attach the paper in the same fashion as before: I would be croggled if it actually uses old tech like drawing pins ;-)
@@PhilBoswell doesn't it just use the _POWER OF HIS MIND?_
Im sure the paper was bucked against jigs on a couple adjacent sides
I still love you though
haha, ikr!
1:41 - "a can of hyperbolic paraboloids" - that brought me back to my calculus class where my professor kept referring to that shape as a pringle
I could watch that machine draw all day… sooooooo satisfying
Hey, you're the Navier-Stokes enthusiast!
Seriously though, Tom, when's your next turn to guest host a Numberphile video?
Imagine if it could do tattoos! If you have any space left, I'm sure you'd have plenty of ideas for the machine haha
For those without Mathematica, Python has a built-in turtle graphics module.
Yeah, you can basically find a turtle library/function for any language these days, this is a classic :)
@@BrianBlock Thanks Seymour Papert. You gave generations of kids some serious fun while they were learning through osmosis with the Logo Turtle and Language.
Mathematica comes free on the Raspberry Pi...
@@flyingphysics9664 is it fully functional Mathematica or is there limitations? Does it have access to knowledge base?
I used that Python turtle module myself. I got it to draw the Sierpiński triangle myself. I picked up a recursive code on line.
I remember programming the Turtle at school in the 1980s. We had a physical Turtle robot and we could get it to draw big images on the large sheets of paper on the floor.
We had something similar that didn't draw anything. It looked like a small roomba that you could program with forwards, lefts and rights
We had something similar to "Turtle" on our Apple II-Es when I took a basic/introductory computer skills workshop for a one-marking period elective back in eighth grade back in 1985/'86 where we would input some simple geometric instructions, and the cursor ("turtle") would draw triangles, squares, pentagons, hexagons, stars, etc.
Dang, guess I missed out on the cool Turtle lessons as a student in the 2010s. We just programmed Turtle using the Java Virtual Machine.
I participated in a summer camp with the turtle robot. The instructor laid a course out on the floor and we each programmed our solution. One person thought the movement units were feet instead of inches, so on their attempt, the turtle barely moved. The teacher announced that the solution looked correct other than the scaling.
This is kind of amusing. I make Minecraft datapacks, usually based on fun math concepts. One of my main inspirations is this channel, and sometimes I'll try to recreate the processes in Numberphile videos in Minecraft. But this time, I posted a datapack about a topic just before you! My latest video was the Sierpinski Arrowhead Curve, which was generated with the same replacement method, and I'm working on a larger video about Lindenmayer (replacement) systems.
Finally a Numberphile video with a plot. 😉
ba dum tss
I definitely watch for the plot
You've got 42 likes which is the best number of likes.
Video is boring but the peanut gallery is on point! :))
lol
Takes me back to the early 1970s when I was an undergraduate, tying up the (admittedly not much used) Hewlett-Packard XY plotter on a timesharing DECSystem 10 drawing dragon curves...
I plotted dragon curves around the same time! On the plotter we used, you had to issue each drawing command twice to get perfect corners. That was because the pen decelerated at the very end of a command and that was easier than coming to an abrupt stop. I used just one command for each segment of dragon curves because perfect corners made it look like an incomplete grid, not a curve.
I also drew a 31-gon and all its diagonals.
Videos with Henderson are always great. Looking forward to part II.
Amazing at 7:47 - very similar to the way DNA packs itself when condensing "coils of coils". Even the little ball-ish nodes look like the histone proteins that it coils around.
Like a fractal, nature loves fractals :)
I thought the same! Wonder if there's a connection.
Yeah! It brought the same idea to my mind!
@@matdex connection is probably just an optimal packing formation - every (human) cell has around 6 feet of DNA that it needs to store inside a tiny nucleus. Fun fact: with ~10 trillion cells in your body, thats 10 billion miles of DNA you're carrying right now.
@@chinobambino5252 No wonder I weigh so much...
Those spirals of spirals are beautiful! They remind me of how the continued fraction expantion of some real number x can be used to give more and more accurate rational approximations to x.
Next step up: for the turtle: an automated combine harvester let loose in a very large corn field to produce a Sierpinski triangle - that would certainly upstage the usual crop circle. :-)
I imagine that such a field would have to be super flat as I'd think any irregularities in the topography would likely throw off the combine-plotter
@@shruggzdastr8-facedclown Some of the modern combines have impressive technology for very accurately locating themselves in real time. Makes for a very expensive turtle though. :-)
@@shruggzdastr8-facedclown
As long as the slopes are within the machines' physical limits, no problem.
GPS, especially with local transponders & on-board physical sensors in the control-loop, are centimetre-accurate. And, you can remote-input driving instruction into the really expensive ones. :)
Can anyone hack a combine?
I would recommend using a Hilbert Curve ruleset instead, since fields tend to be made up of parallel rows and more closely resemble a square (or rectangle, which can be thought of as a series of (overlapping) squares.)
Honestly, this kind of video is the core reason I like this channel
Man, I remember Turtle from computer class waaay back in 1989. Then last year I discovered Python comes with a simple Turtle implementation. So now I guess I only have to build a plotter lol!
Really cool this program is still around.
Matt Henderson Numberphiles are definitely my new favourite Numberphiles
You'll love the second part of this one!
I agree - great topics and very interesting. Plus his accent is very relaxing to listen to ☺️
@@numberphile second part?) That's awesome!
beautiful mathematics
Ok, but why does the substitution trick work? I can kind of see that it replicates the nested symmetry of the shape, but it would be really nice to see a proof of it. Numberphile has recently been stopping _just_ short of the proper maths itself, which is a bit of a shame.
It's not a trick so much as it's a rule. It's an example of Lindenmeyer systems (L-systems) that use rules like these to generate structures with some self-similarity or of a recursive nature. In addition to Forward and Turn (+/-) rules, there are also Scale (multiply/divide length), Scale (multiply/divide angle), Push/Pop (for generating branches), Trim (ends a branch), and in 3D you also have additional rules to deal with line thickness.
The rules in this example are specifically designed to create self-similarity, which is not a guaranteed result of any combination of L-system rules.
Love it when I see my area of research in a numberphile video! Lindenmeyer systems which are what the guest used to generate a sierpinski triangle! Personally I'm using them to generate 3D trees!!
Watching these videos is so relaxing
The spirals of spirals reminded me of how if you zoom out far enough in space, you see that galaxies are grouped into clusters and superclusters of galaxies.
Yay pretty maths drawings!
I absolutely love this and all of Matt's videos. Cheers!
Matt was amazing in this video. The articulation, the body language, the work. More videos please.😊
To me, this is pure art and I really mean that, especially what he shows in part 2
I love how in this video everything is regular and orderly, even when it seems chaotic it leads to something orderly. And the next video is just straight random chaos.
Would be cool to expand on lindenmayer systems a lot more
and show how they can mimic treelike fractals.
An L-system i found is
A -> - C++A
B -> B - - C+
C -> D
D -> AB
you start with AB and + & - is a 45° turn.
C -> D seems like a redundant step, you could replace it with C -> AB
@@ideallyyours Try it, see if there's a difference skipping the D.
Don't skip D-day.
I think in the limit they look the same with or without the D.
But with D it's much easier to draw by hand on squared paper. On that the diagonal lines are longer by sqrt 2, but that doesn't change the original scaling much.
@@Brontalo Maybe you found an elegant way to "time" when rules are applied by adding a holding step C -> D, so that different instances of C/D are substituted which could give a more organic and less layered look.
Don't think I've been this early to a video before.
One of the most satisfying things I've ever printed is the Sierpinski Pyramid. Never had to take it's 'pen' off the paper the entire way up the object.
Thank you Numberphile! After seeing this video I made a version of it on Scratch. Pretty fun to show how it works to my little student and how math can be beautiful without being useful.
The turtle gave me a great sense of nostalgia
When I was in high school, I used to waste my French classes plotting the Dragon Curve on a paper like this. I would have pages of L's and R's marking left and right turns.
Turns out I am both retaking French classes, and bumping to fractal drawings again - almost 20 years later. Now I am focusing a bit more on my French, though.
A neat thing about these plots for rational numbers is that your turtle will either run around in circles or run off forever in a set direction, depending on the fraction you give and the base. It can never do things like spiral outwards or walk pseudo randomly from a rational number input ; the exact fraction simply affects how much dawdling and pattern making it does along the way.
I have been waiting for your new video! Thank you so much.
Part 2 of this one will knock your mathematical socks off!
@@numberphile 🧦👟 kick off! Yay!
Very fun to watch the machine work and the patterns to emerge.
It inspired me in many ways. Thank you very much for sharing this video.
turtle graphics is my favorite python module, i love it a lot, its so very simple, powerful and fun i even have a yt video i made using it that im actually pretty proud of
Would be cool to see a dragon curve made from Euler spirals.
I really like this guy's math visualization animations
oh hey! i remember doing stuff like this in Scratch lol
Thanks for the upload. Very interesting indeed.
If you follow the output of an input of the Zeta function, especially for the higher imaginary parts of the input, and especially between 0-1 for the real part, you get a lot of Euler spirals as well.
Remembered something like this years ago, i think it was the square squigle fractal vid.
It's amazing what you can do with recursive formal grammars. Douglas Hofstadter goes into great detail in this vein in _Gödel, Escher, Bach_ .
bloop, floop, gloop
Love this guy! Also I want a whole video of just that machine
I need this too. I also want the exact program he used so I can play around with it.
After a past video that called it "yooler spiral", this is the long-awaited redemption video.
I wanna see these run forever
Please post the full video of the machine drawing the curve somewhere. That was amazing
If he'd made the turtle a rabbit, that printer would have been much faster.
That’s cool! Going to try this out myself :)
Really appreciate when you don´t just show the pretty picture, but take the time to build up to it from the basic rules. All that complexity from two simple statements!
My favorite part of this video is the Pilot pen that the machine draws with
The turtle pattern reproducing itself in high iterations is amazing
I Wonder what it looks like in billions of iterations
I knew all those repressed LOGO memories would come in handy someday
I love this channel
The British artist Harold Cohen in the 70s had produced "Aaron" an expert system that produced important exhibitions (at the Tate Modern and elsewhere) producing large-scale artworks using a 'turtle' - but did not use pre-determined forms. Really interesting.
6:05 1.0456 is beautiful
thanks so much that was great
Did he just call Pringles hyperbolic paraboloids at 1:42? I'm gonna use that...
It is not a case of chaos....if we repeat it enough times and zoom out enough we can see that it essentially will be the Euler spiral nested on itself. we need theta to be an irrational number for a chaotic patterns with different degrees of chaos maximum being with the golden ratio i think
Love the spiral of spirals!!! For Sierpinski, what happens when you do other angle pairs except +/- pi/3?
Thanks!
I remember learning about logo in 3rd grade. Drawing stuff was so good and fun as hell.
More Matt please
I was in a course where we were studying computer programing and the system had turtle graphics package. Our mid term test was to program a clock that showed hours, minutes & seconds. I was the only one who programmed an analog clock. To get the hands moving I drew the first in on (B&W displays that time & age) and then off moved the angle and drew them on etc. Nice excercise but the teacher was not excited as the graphics were supposed to be the next part of the course.
Put that in an art gallery and it'll be better than most of the stuff there
That Euler spiral done to 1,000,000 iterations looks reminiscent of the dragon curve to me
love it !
euler spiral used in transportation engineer .like highway and road design
I appreciate that you wrote Sierpiński correctly with ń, it isn't much nor a big thing, but it warms me a bit (im used too see polish surnames without polish letters)
And overall, cool video!
man i love Turtle! so great to see it used like this :D
This reminds me of writing HPGL scripts back in the day, to run my serial plotter.
I remember me programming the Turtle in the 1997 in the beginning of high school in Poland.
Quarter of century ago.
It was the first and the last time i had something in common with programming.
I perfectly remember each and every command, just like it was yesterday, it was fascinating.
Gracias por tanto. Saludos from Bo
I want a pen plotter that doesn't cost an arm or a leg, is that even possible nowadays? We used to have an HP plotter (I want to say something like 7475?) but I don't know where that went and I'll bet USB won't touch it :-(
@11:32 "It's within the rules of Numberphile". Then again, so is the Parker's Square.😀
I'm a biochemist and one of the Euler spirals you showed at 8:23 looks similar to the super packaging of genomic DNA in eukaryotic cells
The spirals of higher order were a surprise! Though not too strange in retrospect.
Another fun thing about the Sierpinski gasket is that it's related to the exclusive-OR operation.
Love this cute thing ! You may be interested to try the angle list [1:0.99:100000] and Boom, a symmetric and beautiful pattern !
Has Part 2 been posted yet?
Must be a high quality pen on that plotter!
Please post the spirals source code!
5:08 I want to see animation, where theta is increased very slowly (n being constant).
Is that a V7 pilot pen ?
Where is the part 2 coming by the way :)?
Not first but still glad to have gotten a notification!
Andy Murray’s secret second brother ;) - amazing video!
These Euler spirals appear in the partial sums of the Riemann Zeta Function.
"A can of hyperbolic parabaloids." Amazing
that was epic
Part 2 pls Brady
That is fascinating! :)
I
Sounds like LISP, played with it on the BBC and had a copy on the Amiga too :-)
Is there a way to construct a fractal Euler-spiral-of-Euler-spirals and derive a theta value from that?
This reminds me of Logo - a programming language for telling the turtle what to do.
I am getting flashbacks to going mad on the first computer I owned, the C64, and programming fractals of various kinds I couldn't even print, nor save, just change a parameter and marvel at what might take another day to appear. I resorted to opcode subroutines to speed it up, but by 1995 or so, even Basic was now faster than that.
You’ve created some very valuable brown paper, if they’re for sale!
He seems to be using an AxiDraw (or clone). You can do these graphs in huge very easily with a polargraph (or makelangelo). Very simple to build and 1 m x 1 m size is easily achievable and very cheap to build.
Had plenty of hours of fun so far with it.
Good ol' turtles... i know the from a minecraft mod where you can program in lua :D
They can break and place blocks, move around and turn. It's a lot of fun.
im really curious about how you could derive the continuous version of the euler spiral from this discrete version.
for example, turning 1 degree every 1 unit moved, we could find some recurrence relation (difference equation hopefully?) describing this, and then look at how that relation changes for turning 0.5 degrees every 0.5 units moved, turning 0.25 degrees every 0.25 units moved. i may do this later
"[…] from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved."