Making a physical Lissajous curve

[@razielhamalakh9813]

Look, Parker square ain't gonna die. This is Parker Lissajous. Resistance is futile.

[@spencernelson9868]

Or a Parker Missajous, as he called it at 6:14.

[@stephenbenner4353]

I had almost forgotten about the Parker square.

[@2nafish117]

Damn you beat me to the joke

[@lunasophia9002]

Stephen Benner Parker memory, clearly.

[@stephenbenner4353]

So when are we going to be able to get the tee shirts?

[@cftug]

"I've done mine in... Excel." Of course you did.

[@razielhamalakh9813]

Quite frankly, I expected python there.

[@kindlin]

So much easier in any real programming language... Basically everything is besides graphics.

[@SoftBreadSoft]

+tj22 I don't think so. One entry for each variable and use the plot function with whatever time step desired. Would be about 15 short lines, very easy, also about 2 minutes. What is faster is what you're familiar with, generally.

[@nanoblast5748]

geogebra has an excel tab in it, so that's that.

[@labibbidabibbadum]

Easier to to see what's happening line by line in Excel , but agreed far slower and more cumbersome. However, Excel is great when you're explaining stuff to kids or non-programmers. They can't hold the concept of variables and matrix operations in their heads, but can readily get "this multiplied by this gives this".

[@0fuxTaken]

"I'm gonna call this..." Digging your own grave, Parker... Parker Lissajous

[@Hades948]

He had to have seen this coming.

[@DanTheStripe]

The comments section is *exactly* what I expected. #ParkerLissajous

[@pas-giaw6055]

Missajous

[@andrewkovnat]

Ah yes, the Parker Lissajous!

[@MAProsper]

I was waiting for this.

[@sagarramchandani3139]

Ahhhh am a day late! Good job Sire!

[@mirageinthedesert5448]

Andrew Kovnat or a parker missajous

[@meganwhitney3911]

Last weekend I went to Tompkin’s Square Park, the squarest park in New York. As much as people kid about the Parker Square, it’s truly an example of someone who keeps trying in an imperfect world. Thanks for showing the grit and not just perfection

[@TheMasonX23]

+

[@livintolearn7053]

"I'm gonna call this..." Oh why are you so eager to have your day ruined?

[@H2CO3Szifon]

"3, that's good enough for π" - an engineer-minded mathematician

[@datpudding5338]

If it wasn't for his love for pi, one could probably call that Parker Pi

[@seanm7445]

Your really excelled in this video

[@dliessmgg]

Oh! I remember you. You're the guy who asked for Irish people to react to abstract art.

[@Ranthos1]

Classic parker curve

[@Rubrickety]

"Near-missajous" is pretty good. Sadly better than the obvious "Parkerjous".

[@wierdalien1]

Yes but never miss a chance to rub good natured salt on t' wound

[@dragoncurveenthusiast]

The speed ratio of your turntables seems to be approximately 55 to 32 or ~1.72 (as counted via the number of frames one half-turn takes in the sped-up clip at 5:29, assuming equal diameters of the turn tables)

[@tonksdude]

wouldn't it be easier to just count the teeth on the gears when he shows 'em?

[@Septimus_ii]

@@tonksdude That assumes that the motors are going at the same speed, and I don't think they are

[@tonksdude]

@@Septimus_ii Fair enough

[@rewrose2838]

Ah good ol'nerdery

[@TheDiggster13]

Clearly this is a Parker Curve

[@RFC3514]

That wouldn't encapsulate the fundamental notion of failure. I propose "Parker Circle".

[@Agent-ic1pe]

@@RFC3514 even better, a Parker Sphere, missing an entire dimension but eh close enough. Also at 10:33, Parker π = 3

[@DukeBG]

You tried to make a joke, but it didn't really work, it came out crooked and kinda just wrong. But what's more important - you gave it a go. That's right, you made a Parker Square of a joke.

[@mstmar]

I noticed that your excel demo had shapes that were too far to the left, indicating a likely mistake in the formulas. So i decided to do my own math and found one. Gamma should be: =ACOS(SQRT((G10-E10)^2+(F10-D10)^2)/$H$2) (there was an extra *2 in there) After that fix i tried to recreate your lissajous. i estimated every value needed: A-x = A-y = B-y = 0, B-x = 5, A-r = B-r = 1, speed = 2, stick length = 3.5, offset = 0. The result was surprisingly similar to the one drawn

[@setiv2]

I agree with that fix! I was trying to work through the math on some paper and decided to look at the comments.

[@descuddlebat]

If only this comment wasn't stuck under 20 people saying "Parker Square"

[@GraceSerenityK]

"I could have used this program specifically designed for making pretty math things, but I chose Excel because of course I did!"

[@HitchHawk]

Matt, please stay imperfect. Be our Parker Matt. Someone who is right, but not quite right.

[@kindlin]

At the risk of being too obvious, you do know his name is... Matt Parker?

[@Ienjoylotsofstuff]

I'm going to school for instrumentation. and i was messing around with a signal generator and oscilloscope along with some other electronic components and i saw these curves show up occasionally.. pretty cool to see a video about them i didn't know if they had a name

[@h0verman]

oh yeah, i didnt even realize this is basically what an oscilloscope does, but just with much more varied lines than sine waves

[@pamdemonia]

Yeah, in the early 90s I did some video editing and we would use the lissajous figures that resulted from plugging the input in at x and the output in at y to calibrate the video signal.

[@roderickwhitehead]

Lissajous figure creation and interpretation was "a thing" in Mech. Lab 2 back when I did my undergraduate in Mechanical Engineering.

[@michaelmoorrees3585]

All of the older analog oscilloscopes have an X-Y mode, that does this. It was one of the "scopes" basic functions. Usually, the x-axis is tied to an internal "timebase", that puts a sawtooth wave along the horizontal axis. The timebase is sync'd to the Y-axis (vertical) input. In the real old days lissajous was used quite a bit.

[@sagarramchandani3139]

R.C. Whitehead It is a thing even today, am a physics major and Experimented with them last year.

[@PopeGoliath]

"Three: good enough for pi." -Matt Parker 2018

[@veggiet2009]

I'm surprised you didn't try setting them to a "1 to 1" ratio to see what kind of circle-ish thing that would surface.

[@amiwatchesyt]

He is fearing having a Parker circle

[@bjornolson6527]

Yes, “Circling” the Square might cause hemorrhaging, in this instance.

[@massimookissed1023]

Just waiting for Mr Segerman to build a 4D version.

[@my_temporary_name]

The picture you get is 2D, so if you calculate all possible curves depending on periods of both turntables you get a 4D shape, although it would be more practical to look at the 3D shape in X, Y, (periodA/periodB) space.

[@DeclanMBrennan]

Using quaternions. :-)

[@xaytana]

Wouldn't you need four inputs, one for each dimension, for a 4D output?

[@99CreeperKing]

Parker Curve

[@RFC3514]

But it _is_ a curve. The whole concept of Parker geometry is that it isn't what it was supposed to be. So this should be called a "Parker circle".

[@quinn7894]

But the thing drawn on paper wasn't a perfect curve.

[@Blustride]

What you have built here, at least according to Wikipedia (which is always right about everything, ever) is a Pintograph, which is a type of Harmonograph. The way most people do this is to use two pendelums. With a pendelum, you can easily restrict the axis of motion for each wave generator to one direction, and more precisely position them. On the other hand, they don't keep a constant amplitude. I'm going to call your invention the Parker Pintograph.

[@Holobrine]

I think you could get an honest lissajou curve if you had the rods connected to the turntables through slots, and if you had a way to hold the rods orthogonal to each other.

[@AguaFluorida]

And that device would be called a Harmonograph ;)

[@Holobrine]

AguaFluorida Not quite. Every pic I found of a harmonograph on google uses weighted pendulums, mine wouldn’t.

[@raykent3211]

A scotch yoke extracts one axis from circular motion. Very simple. For a crank that Matt is using you just need the con rod to be infinitely long, which may be less practicable.

[@wierdalien1]

Not going for #parkerjou?

[@tibees]

awesome!

[@rubenssautter9242]

Would be interesting to see a 3D version of these curves. Could we express in a math equation any knot? Some knots resemble each other

[@guidodurante9495]

That explosive ending was one of the best subscription requests I've seen lately

[@TN-lw5dt]

You guys are all joking about Parker Lissajous but idk nearmissajous is a pretty fine name to me :p That Excel plot, on the other hand... I think we can safely call that a Parker Nearmissajous?

[@eris4734]

Parkearmissajous

[@TrippLilley]

It's a Parker Plot

[@SassInYourClass]

You’re missing the joke entirely. Search Parker Square on KZbin.

[@Markovisch]

Parker = Near/Almost, so no need for that

[@U014B]

6:15 Personally, I'd have gone with "More-or-less-ajous".

[@verdatum]

In the 11th grade, I got access to a pair of tone generators and an analog oscilloscope. I figured out how to set the thing up in parametric mode and I spent hours looking at these things. They're great fun.

[@ianward1848]

loving the quote "where, where have I seen that before". So to the moving midpoint pattern, I know not really part of the video, but , to quote a famous mathematician "where, where have I seen that before", I realized it is the Mandelbrot cardioid and as you increased the speed by one you got the next shape( the kidney one) and so forth .

[@Walthanar]

back in highschool (we're talking end of the '90s) I programmed in qbasic a little thingy that, given the two relative frequencies of the x and y components, plotted the lissajoux figures with slowly increasing offsets, and after that it projected on screen the figures in an animation-like sequence. It was quite beautiful as you could visualize it as a single string around a spinning invisible cylinder, and I was very proud of my work. I think I also got highest marks in programming for that little creature :)

[@lukeh9811]

Hey man, Just finished your book. when I read the bit about the domino computer I said to myself "I remember that from somewhere" and then I checked your face at the back of the book and there you were. Thanks for the awesome content and reads.

[@harbingerofwarx995]

You're really great at creating things that are close but not quite right.

[@DysnomiaATX]

Near-missjous. Give this man an award.

[@MortiePL]

I love how the green shape in the middle of your program sheet resembles cardioid. :)

[@goldenrainbowwonderland4986]

Very nice physical project. Great to see mathematics like this.

[@AlaskaSkidood]

What a lovely Parker curve machine!

[@lidular]

Cool video Matt. You have set yourself Up for a Lot of parker square jokes though...

[@will4not]

I love when Excel is brought out. It's like math that you can while you're running reports.

[@Mobin92]

Geogebra is such a great piece of software. Even for more mundane tasks. It's so easy to do geometry tasks with it.

[@pyramear5414]

Lissajous plots are pretty useful for measuring the relative phase of two signals of a particular integer frequency ratio. By using an oscilloscope and using one input channel as X, and one input channel as Y, you can determine the phase between the two signals by recording the x-y intersect points. While more modern equipment can typically measure phase of two signals of equal frequency, the Lissajous remains an effective method for measuring the phase of frequencies of integer ratio frequencies greater than 1.

[@SineEyed]

And on that note, I suggest you look into the very nifty chaotic attractor of Chua's circuit, as shown on an x-y oscilloscope plot. Assuming you haven't seen it already, of course. If you haven't though, it's really neat - look into it..

[@erbro]

In fact, when I was very young, like in the 70s, I had a subscription to a magazine called De Jonge Onderzoeker (the young researcher) and there was an article in it with a sort of pendulum. It had a fixed length in one direction but you could alter the length in which it moved in the direction perpendicular to that. At the bottom of it was a small flask with sand, a little hole that let the sand seep through. It made perfect Lissajous figures!

[@ZeratulRahl]

Ah! Finally a video about the Parkerjous figures!

[@topilinkala1594]

Proper plotter's that did the same line going forward and backwards were robust machines. The pens were as short as possible and the moving arms were anchored at both ends. About the parametric equations: One get very nice polynomials out of one of those. This is how I've described it previously: Here's one each order starting from 1, which is y = x, the order two one is y = x^2 - 2, the third order one is y = x^3 - 3x, the fourth order one is y = x^4 - 4x^2 + 2, the fifth order one is y = x^5 - 5x^3 + 5x, the sixth one is y = x^6 - 6x^4 + 9x^2 - 2, etc. These are polynomial functions that have the following properties: 1. All local maxima and minima values for y are 2 or -2 and their corresponding x-values are between -2 and 2. 2. Each odd function goes through points (-2, -2) and (2, 2) and even function through (-2, 2) and (2, 2). These two properties mean that the polynomial oscilates between -2 and 2 when x goes from -2 and 2. Outside the range the function goes monotoniously to (minus) infinity. 3. The coefficients of the polynoms are integers and the leading coefficient is 1. This is something I find beautiful. That there are integer coefficient polynomials with leading coefficient 1 that have all zeroes inside small range (-2, 2) and that at the same region the polynomial function don't exceed -2 or 2. They can be generated by the parametric pair x = 2cos(t), y = 2cos(nt), where n is the order of the polynomial. The pair x = cos(t), y = cos(nt) gives the same first two properties scaled down by 2 but the polynomials leading coefficient is not 1 which is IMHO ugly. So that's why I like the scaled up versions.

[@567secret]

It's pretty cool that it kind of graphs a surface.

[@fzigunov]

And this is why mathematicians don't build things in the real world.

[@dexteritymaster]

"Three good enough for Pi" Matt Parker 2018

[@mojosbigsticks]

Fascinating. Loved it. Didn't understand a word of it!

[@frankharr9466]

THAT IS AWESOME! What a great use of Excel! AAAAA! I want to see ratios with various irrational numbers.

[@PanthereaLeonis]

Pulling out weird "art" supplies, I felt like I was watching Jazza for a second.

[@RMoribayashi]

In the 1960's & '70's the Franklin Institute Science Museum in Philadelphia had an exhibit that drew Lissajous figures on a pair of 3 foot square tables using sand held in a large brass funnel. You could adjust one of the two lengths of the compound pendulums to get the various figures shown in this video. Back then a brass funnel 6" across wasn't considered either a theft risk or a safety risk. 80's kids proved them wrong and after several injuries and thefts (and countless repairs) the demonstration was removed. By the 1980's seeing rowdy kids climbing on the exhibit and swinging the funnels at each other didn't surprise me. I just couldn't figure out why the sand disappeared so fast. What was the fascination with play sand that drove so many kids to steal it.

[@stevieinselby]

I remember a program to draw Lissajous curves on a BBC Basic computer, long before I knew even what trigonometry was, let alone parametric curves. No idea how it worked, but it did draw pretty patterns!

[@xaytana]

What you made is a common geometric drawing machine. I forgot the specific name of it, and some people refer to it as a guilloché drawing machine; though that term will also bring up cycloidal drawing machines on google. Lissajous curves are different. They take the (x,y) point of circle A and (x,y) point of circle B, at a specific time interval with each circle having their own speed, and plot that intersection as a (x,y) point on a separate grid. There's no fixed length of the arms, and the arms have to stay parallel to which axis they represent; assuming circle A is to the side(s), it's arm would need to stay parallel to the X axis, as circle B would have to stay parallel to the Y axis, assuming circle B was to the top and/or bottom. This would be fixed by having two of each circle, circles A1 and A2 at the sides, and circles B1 and B2 at the top and bottom, and with hollow arms, the pen would always be in a position that is square to both axes. Only problem then would be tolerances. I remember seeing some mechanical etch-a-sketch machine that used pulleys on the two knobs, but this was years ago, and I don't remember if the pen was square to each axis or how tight tolerances were; but that is similar in concept but vastly different in implementation.

[@damon314]

That is some spicy excel work

[@Yupppi]

That missajou pattern was cool. Like modern architecture drawing. Or like some of those thermodynamic graphs that have way too many variables in the same picture. Was actually a bigger fan of the midpoint patterns than the actual drawings.

[@michaela.warren3335]

I was playing around with the Speed Ratio (sr) on Matt's Spreadsheet in a similar manner as he did in the video. A few notable observations when 10 < sr < 127: sr=29, the red circle creates a cool pattern that is completely different than its surrounding sr=28 and sr=30 sr=31.665525, the red circle was completely filled in. sr>31.665525, the cycles start to revert back to the initial position sr=61.83151, the green cycle is almost a straight line sr=63.33105 (twice that of the previous sr), the red and green cycles are approximately the same as when sr=2 sr=126.6621 (twice that of the previous sr), all cycles are approximately the same as when sr=1 N.B. These are approximations made by brute force and a lot of spare time.

[@roderickwhitehead]

Matt, this is bread and butter undergrad Kinematics stuff for Mechanical Engineers. Look up +"5-bar linkages" +"2 degrees of freedom". That's what you have synthesized. Now if only I could magically remember everything I learned in Differential Equations... then I'd be somewhere.

[@michaelharrison1093]

The midpoint figure created in the Excel sheet is a limacon Very similar to the math behind harmonic sequence components

[@billborrowed3939]

I can’t believe, he gets to spend his money on turning tables to create lines. This is the best job on the world.

[@LamiaZan]

Aha ha...So,What's going on? So cute! Awesome presentation! Congratulations!

[@Petch85]

If it is worth doing, it is worth overdoing :-) And now I have to open Python :-)

[@descuddlebat]

@@Peter_1986 Yeah but not free

[@PeeDub1021]

"I've got two turntables ..." Ohhhh, Beck was making lissajous curves!

[@marethyuthefirstshinigami162]

Oh hey, Julio Mulero is one of my teachers, my programming teacher to be precise

[@Poultryphile]

Brought to you by Parker Square Engineering, Our bridges "almost" don't fall down.

[@moradan81]

I love the theme song for Stand-up Maths. On a slightly different note: when you comissioned it, did you specifically ask the composer to make it end unsatisfactorily? Did you ask them: make it unresolved at the end, I want people to feel like something is missing, and keep waiting for it, and for it to never come? Is that how this theme was conceived? Honestly, I do love it, really.

[@domjanabi6006]

the way that pen perfectly followed its previous path...

[@kasanekona7178]

That spreadsheet is perfectly replicating my signature!

[@MrGustaphe]

"if you see a mistake, let me know" well, you did it in Excel to begin with.

[@Latchfpv]

Nearmissajou is obviously the best name.

[@gsurfer04]

I like how the near-missajous and ending music are sort of in time.

[@revengewrld999]

Parkerjous. hahahahaha favorite youtube channel.

[@bassmaestro1628]

What I find most interesting is how the 5/3 and 3/5 make the same pattern (with at 90* turn) but then 5/2 and 2/5 make such different pattens. 1:00

[@AHBelt]

It might be in-topic for me to mention, a few years ago I took an online course on Alfred Hitchcock, and it was said the first computer graphics in a film might be the Lissajous curve drawn over Kim Novak's eye in the opening titles of "Vertigo". They used an old military thing.

[@Lohoydo]

Reminds me of of what you get from a spirograph. One way to improve it would be to have longer bolts (with a nut to hold the bolts vertical) then use multiple slats from each bolt alternating so the pen is held upright better.

[@AmanManglik]

In the gifs the length of the "sticks" is not fixed, it's variable, and their spatial orientation is fixed. While in your construction it's the other way around. It's not the construction tolerances which are messing with your shapes, you are not creating lissajous figures. It might be interesting to find non-random-blob pattern configurations in your setup though :)

[@GoldenSyrup42]

I agree, I think a physical setup like this would probably work: i.imgur.com/jWBrNzT.png

[@Shadsuna]

Now THAT would be fun to see how tolerances and inaccuracies would play into the final result

[@danilooliveira6580]

Stronginthearm XXVII I like it, and its also not that hard to build. the problem is that inaccuracies would break everything. so you would need to find a way to perfectly sync both circles... maybe a chain ?

[@thief9001]

Could you just have a single input motor that drives both circles so when you turn on a switch, both of them start moving at the same time? You could set them up with different ratios to make them spin at different speeds, but always start at the same time.

[@Shadsuna]

My line was in response to Stronginthearm XXVII's proposal of crossed guide sleds and two pairs of rotors. The idea is the following: Even if you do your best to control for sources of inaccuracy (perhaps using a single drive motor with complex gearing to the turntables like thief9001 suggests), the tiniest of production tolerance-caused error margins will have a noticeable result, as the further off of a pure integer ratio you get, the more distortion you introduce into the final image. Whether you'd like to classify all the possible result sets dependent on the initial conditions and unpredictable perturbations due to device imperfections is a matter of preference; and it seems to go agaist the Way of the Parker to even take such nuances into consideration... After all, the result simply needs to be Good Enough (tm). And that's why we love Matt (math)! :)

[@CaveSpiderRider]

I think I saw something like this a while back where they had a pendulum-on-a-pendulum type set up, and had a bag of sand as the weight. Swinging with different lengths for each pendulum made the different speeds, the sand was released slowly to 'draw' on a surface, Bob's your uncle, they had a lissajous pattern.

[@CaveSpiderRider]

Didn't explain it very well, found the video: kzbin.info/www/bejne/q4HFq5uujrmcpa8 Also wasn't two pendulums, was a pendulum with a Y shape so movement was restricted in one axis.

[@C4ZWX]

You can also draw these curve using a Y-shaped pendulum. When the pendulum oscillates perpendicularly to the Y, all the pendulum oscillate. When it oscillates parallel to the Y, only the low part move. => different lenghts = different periods of oscillation => You can chose the ratio between the periods in order to obtain Lissajous curves.

[@SineEyed]

This is true. A while back I saw this demonstrated by the science educator and KZbinr @Bruce Yeany. In the video, he shows similar sort of stuff early on, but not the Y-pendulum as you described until later on. In his demonstration, he's set up a little LED rig and captures the oscillations by taking some long exposure photos, with pretty neat results tbh. I was inspired to try it out myself, so I set up my own little long exposure rig. But instead of using LEDs like he did, I modified a $1 laser pointer to stay on indefinitely and captured the photos of the path it traced as it swung around pointed down at the floor. I got pretty neat results as well. It's something that's fairly easy to set up and have a go at yourself. For that reason too, an experiment like this is a great choice to demonstrate science-y type stuff to others, especially kids. Simple to set up, simple to take down, and easily explained to non-nerdy people to possibly deepen their appreciation of science in general. Win-win. Here's a link to the video I mentioned. It's timestamped to where Bruce shows the Y-pendulum configuration: kzbin.info/www/bejne/bYOrkqGmZr2dqcU

[@coolbojy]

the x and y coordinates for the second circle, B_x and B_y, respectively, use the radius of circle A (cell $B$3) rather than the cell for the radius of circle B

[@jacobclaassen4709]

Have 2 spinning plates along each cardinal direction of the paper. Have the pen be fixed to a sliding joint that can slide along a bar in between each spinning plate, these 2 spinning plates will spin at the same speed. Now you have 4 spinning disks on each side of the paper. I don't really want to put any more effort into this problem but I think you get what I mean.

[@exintrovert6803]

So now you’ve made an Excel Spirograph. Love it! What other classic toys can you recreate in Excel?

[@frankdehobbit8989]

Excellent Parker Lissajous ;)

[@nikosaarinen3258]

Parker curve!

[@Oyashirosamananodesu]

I have absolutely no idea what is going on here, but it looks fun.

[@gartackpsdav4984]

In your excel plot of the mid-point, the patterns showing up at times reminds me of a video on the Mathologer channel called "Times Tables, Mandelbrot and the Heart of Mathmatics". He does times tables in a circle and gets the patterns.

[@jossc8280]

Ah, yes. The neat curves you inevitably find after playing around in Desmos for long enough

[@SithBowman]

This way of drawing patterns on the ends of sticks ends up reminding me of an old flash game I played when I was younger (about 8 or 9 years ago) called rotatix, except it was just a link of rods spinning at different speeds with the "pen" connected to the final rod

[@leonidlgLeonid]

I love this guy, he always blows my brain😀

[@BillySugger1965]

Argh Matt! Your tinkering set off my engineer’s OCD and I was about to bill you for a contribution to my therapy fees. But then you did your simulation in Excel, which restored my equilibrium. Phew!

[@Darxide23]

Congratulations. You built a Spirograph. :D

[@Czeckie]

"I'm gonna call this..." haha, I love you dude

[@KyleDB150]

if you put two turntables on opposite sides with a rod going over the paper, repeat that at 90 degrees, and put a sliding block where the rods cross, and attach the pen to that, then you can get the actual lissajous instead of the near-missajous

[@NineteenHand]

This is fascinating.

[@scicodude]

This looks a LOT like @3Blue1Brown 's video about the geometric interpretation of the Fourier Transform! They both involve going around the circle with different speeds, so I suppose that makes sense.

[@Blitz6804]

The Parker Lissajous... putting in a lot of effort and trying really hard, only to just barely miss.

[@Paraboemba]

Make those sticks shorter and connect each to another long stick which you connect to the pen. That extra joint enables you to straighten the long sticks in some glider. Its a crank shaft system.

[@ray73864]

This just reminds me of the old Spirograph that I had as a kid back in the '80s.

[@sa3270]

I did a pendulum thing similar to the previous comment once. Except instead of sand dripped colored water on paper. Pendulum method will still have distortion from various factors, but at least the distortion is centered at the origin.

[@mralmightypenguin]

you could make a true pattern with four turn tables. if you syncronize two and attach a bar across them that has a slit along its length, you will have a track moving sideways but not changing its angle. then put another one over it at right angle. to eliminate pen wobble, you could attach two levels of tracks, one guiding each end of the pen. some kind of lubrication between tracks and pen may be neccessary.

[@beningram1811]

As you were changing the values of the speed ratio, it was interesting to me to see that the number of loops in the midpoint pattern was equal to the speed ration -1 (as long as the speed ratio was an integer).