You Won’t Believe How These Shapes Roll! New Discovery in Math

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Trajectoids Nature Article www.nature.com/articles/s4158...
Make your own trajectoid colab.research.google.com/dri...
Images and Footage courtesy of Shamini Bundell and Yaroslav Sobolev.
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Jonathan Koppelman, Michael Seydel, Cy 'kkm' K'Nelson, Thorsten Auth, Chris Flynn, Tim Barnard, Izzy Ca, Tate Lyles, Richard O McEwen Jr, Scott Ready, John H. Austin, Jr., Brian Wilkins, David Johnston, Thomas Krause, Lynn Shackelford, Ave Eva Thornton, Andrew Pann, Anne Tan, Jeffrey Dutt, Joseph Lamoree, Francisco, Marc-Antoine, Chris Davis, Thomas Urech, chuck zegar, David Tuman, Ben Mitchell, Tyler Simms, James Mahoney, Jim Felich, Fabio Manzini, Jeremy, Sam Richardson, Robin High, KiYun Roe, DONALD McLeod, Ron Hochsprung, Aria Bend, James Matheson, Kevin Anderson, Alexander230, Tim Ludwig, Alexander Del Toro Barba, Justin Smith, A. Duncan, Mark Littlehale, Tony T Flores, Dagmawi Elehu, Jeffrey Smith, Alex Hackman, bpatb, Joel Becane, Paul Barclay, 12tone, John Lakeman, Jana Christine Saout, Jeff Schwarz, Yana Chernobilsky, Louis Mashado, Michael Dean, Chris Amaris, Matt G, Dag-Erling Smørgrav, John Shioli, Todd Loreman, Susan Jones, Bryan Andre, Miles Freeman, Bunny Lushington, JRM, Motty Porat, Michael Tardibuono, Yaw Mintah, Carlos Escolar, Anthony Docimo, robert lalonde, Julian Nagel, Cassandra Durnord, Antony Birch, Paul Bunbury, David Shlapak, Kent Arimura, Phillip Rhodes, Michael Nugent, James N Smith, Roland Gibson, Piotr Klos, Joe McTee, Oleg Dats, John Spalding, Simon J. Dodd, Tang Chun, Michelle, William Toffey, Michel Speiser, James Horsley, Brian Williams, Craig Tumblison, Cameron Tacklind, 之元 丁, Kevin Chi, Lance Ahmu, Markus Lindström, Steve Watson, Midnight Skeptic, Potch, Thomas P Taft, Indrajeet Sagar, Markus Herrmann, Gil Chesterton, Alipasha Sadri, Pablo de Caffe, Colin Byrne, Nick H, Jesper de Jong, Sofia Fredriksson, Phat Hoang, Spuddy, Sascha Bohemia, tesseract, Stephen Britt, KG, Hansjuerg Widmer, John Sigwald, O C, Carlos Gonzalez, Thomas Kägi, James Palermo, Chris Teubert, Fran, Wolfgang Ripken, Jeremy Bowkett, Vincent Karpinski, Nicolas Frias, Louis M, ROBERT C PAYNE, Moose Thompson, Rick DeWitt, Pedro Paulo Vezza Campos, S, Garrett Chomka, Rebecca Lashua, Pat Gunn, George Fletcher, RobF, Vincent Seguin, Michael Brunolli, Shawn, Israel Shirk, Jesse Clark, Steven Wheeler, Philip Freeman, Armin Quast, Jareth Arnold, Simon Barker, Lou, amcnea and Simon Dargaville.
Creator - Jade Tan-Holmes
Script - Joshua Daniel
Animations - Andrew Brown
3D printing - Stanley Lee
Music - epidemicsound.com
Chapters:
0:00-0:47 What's a trajectoid?
0:47-1:42 The basic idea
1:42-8:54 Cool math
8:54-10:40 Trajectoid Show-n-Tell
10:40-11:57 Applications of trajectoids

[@Kaldrin]

Math is always about "this looks fun let's try" turning into "wait this is actually very useful"

[@pumbaa667]

My favourite example are Prime numbers. 2'500 years of research just for the fun of it. And finally a real life usecase appears with asymmetrical encryption ^^

[@valiakosilla2413]

Not always. If you do not believe me ask from my wife. I have used probably ten thousand hours to all kind of mathematic hobbies with very small useful results.

[@aloysiuskurnia7643]

I beg to differ! To me maths is about "this looks fun let's try it whether if it is useful or not!" leaving a lot of tools, possibly unuseful at the moment, scattered all around the place. Whether other branches of science accidentally stumble upon our tools and finding it useful is up to them, not us :^)

[@sk8rdman]

Not always. Many mathematical breakthroughs were made in the pursuit of a specific practical goal. I'm sure Newton was a mathematically curious guy, but that alone was not why he invented calculus. He was very interested in understanding planetary motion, and he invented calculus in the pursuit of a rigorous mathematical model that helped explain his observations. It's a similar story with Leibniz. He independently invented his own system of notation for what we now know as calculus, because he needed it to understand and design his calculating machines. It wasn't mere curiosity that motivated these men. They invented calculus because they needed it to solve other (very different) problems that they were working on. The pursuit of mathematical curiosity is great, and it's also great when we find our discoveries have unexpected applications, but it would be a mistake to say that that's how it always does or should work. In fact, understanding the specific sort of problem that motivated a mathematical discovery can often help provide context and intrinsic motivation towards better understanding the math ourselves.

[@johnchessant3012]

I really like clever ideas like changing from 1 to 2 periods that suddenly makes trajectoids a lot less rare!

[@Sau1Bage2-im4mc]

Im very much not lying but i had the same idea while watching the video before she said it

[@IceMetalPunk]

It goes from "infinitely rare" to "guaranteed" just by doubling and rotating. Sometimes math is very cool 😁

[@vigilantcosmicpenguin8721]

Feels like a hack.

[@flameofthephoenix8395]

What's most interesting is they said it was infinitely rare then showed that all infinite trajectories have a copy that makes the trajectoids showing that it is at least 50% of all trajectories.

[@dominiklukacs7677]

"So here is a trajectoid of my heartbeat" *Immediately stops*

[@valiakosilla2413]

I thought also that I would be somewhat worried if my heart was powered with trajectoid.

[@Ath3nx_2]

@@valiakosilla2413 same💀

[@jeremyrixon150]

Hey KZbin Algorithm! Roll as many lumpy shaped objects as you have in this direction. We want people to follow the lines to UpAndAtom!

[@upandatom]

:)

[@Th3OneWhoWaits]

😀

[@freddyvejen743]

yes we do!

[@CMHE]

Are you calling me a lumpy shaped object?

[@firstname405]

​@CMHE we are all lumpy shaped objects 💖

[@gardenlizard1586]

So working out the shape of the rock in rock and roll.😊

[@GlennHanna8]

I agree with you on not being an expert at something yet being a good explainer by breaking things down. There's a joy in learning and understanding something that seemed difficult at first and then sharing all the parts that made it come together and make sense. Even mentioning the thoughts or ideas that might lead us the wrong way naturally and say "don't think of it that way like I kept doing... think of it this way instead" is very helpful.

[@ShawnHCorey]

If a trajectoid doesn't complete the path ending in the same orientation, it will repeat the path at a different angle. If that angle is a rational number, it will eventually come back to the initial orientation and then repeat itself. If the angle is an irrational number, it will never repeat itself; the angle of its path will always be different from any before. PS: I should clarify. If the angle measured in degrees is rational, it will eventually repeat itself at the initial orientation. If the angle is measured in radians, then if angle/2π = a/b where a and b are integers, it will eventually repeat itself in the initial orientation. I made this clarification because mathematicians like to measure angles in radians.

[@glennac]

LOL! 😂 “Rightway up”. I see what you did with the globe Jade❣️😜

[@Splarkszter]

Woah. You explain soooo well. I love the neat practical examples and everything. THANK YOU!!!!!!!!!!

[@upandatom]

Thanks!

[@silverharloe]

10:30 so one could say you really put your heart into this video?

[@upandatom]

>_

[@JasmineLi]

O-O

[@raman_14264]

Literally you talent is insane being a teacher, and also sense of humor. Your videos literally make maths a fun subject. Can't wait for your next video

[@jawaduddin4244]

I saw your short on this and wrote an article on my engineering blog about trajectoids a little while back - thank you for bringing this back!

[@mingmerci6103]

I absolutely love this. This made me smile way more than it should

[@upandatom]

math is wonderful :)

[@AsianDinner]

I've been feeling very stupid lately, but I discovered your videos recently and I love how you present information in such a fun and approachable way. Thank you for your hard work, you deserve the million!

[@LetsGetIntoItMedia]

I've been looking forward to this one! I remember commenting something about the physical practicalities of these shapes, so it was cool to see you explore those and highlight some issues here! Great video as always, Jade 🤩

[@iamsushi1056]

This new field of physical geometry that’s coming up with things like gombocs and trajectoids is so cool.

[@Porcuspine360]

It’s crazy how breedable she is

[@ShaneH42]

Geometry makes my brain not want to brain but your demos really helped. Great video

[@me0101001000]

I see this having interesting applications in materials science, too. There are lots of people in materials science who work on something called "Advanced Materials", which involves creating new materials from existing ones which have incredible new properties. I can see these trajectoids being used to inspire or even create new crystal coordinations with very interesting structures and properties. I'm excited to see this eventually trickle into my field!

[@piyush197sharma]

I am following this channel for 2 years and realised that it was very brilliant part for me she cleared my all doubts about quantum physics and quantum biology thank you very very much

[@BooleanDisorder]

You have such a vivid and clear way to explain things. Thank you! 😎

[@Nicks721]

Absolutely incredible, you explain everything very smoothly (unlike the lines of some trajectoids you showed... the trajectoid of the line represanting the smoothness of your explanations will roll forever!!!)

[@vladimir10]

Awesome vid! Really missed your content, happy to see you again!

[@pluggedingaming8169]

Having two periods, the same pattern mirrored on each side of the ball, it equally splits it perfectly in half and ensures that the path you want is still followed.

[@sosanzehra1227]

Jade,I was eagerly waiting for your video and it was such a cool one!

[@dj_laundry_list]

What if I said that the Fourier transform decomposes a hilbert space into orthogonal basis vectors?

[@cogwheel42]

5:30 - Yes, I actually did! You did such a good job introducing the topic I anticipated the punch line :)

[@debtanaysarkar9744]

Yayyyy, you're back again, awesome video once again

[@GermanSausagesAreTheWurst]

Your videos just keep getting better and better. I am in awe of not only your mathematical ability, but also your video production.

[@Hossak]

Hitting it out of the park as usual. Nice work Jade!!!

[@abdul-kabiralegbe5660]

I'm happy to see this channel grow. Over 700k!

[@gameon6252]

Thank God you had stan with you jade , he is the MVP. Was always excited to know about trajectoids thanks Jade🎉

[@powertomato]

I started following this channel because I noticed familiar topics from my university classes and on each one I was thinking "I wish my professor was this good at explaining it". I really think this kind of breaking things up to its most basic concepts opens it up to a much broader audience and leads to a deeper understanding. Math and logic in school are often very dry and driven by purpose. That's like teaching art to learn brush techniques, but never stopping to appreciate how beautiful the paintings are. Thanks for showing the beautiful side of math.

[@MemphiStig]

I always enjoy you explaining stuff and introducing me to things I've never heard of. Your videos are always well done, informative and fun.

[@JaiMadhusuthan]

Really great video as always. You are the best at explaining high level concepts you are my go to person for content like this!

[@LarkyLuna]

This feels like a Fourier series but you're embedding the periods into a sphere instead of a complex circle I wonder if you can relate the two in any way or reduce fourier into a special case of trajectoids Very interesting math!

[@DerpMuse]

Not quite fourier. These trajectoids have an identity of 4pi.

[@AbdullahCumhur]

I once saw your short video about this topic and I tried to recreate it myself but it didn't work, so after watching this video I'll try again lol

[@gazman1238]

Really fascinating. Thanks for the vid. These trajectoids seen to relate to a sphere the way a cam relates to a circle.

[@adamengelhart5159]

This is the most fun math thing I've seen since the monotile from last year! Thanks!

[@user-ro9fk7xc9y]

Sad you stopped making videos.

[@tagbon]

Great video. Always fun and educational - thanks! 🥰

[@dhonantarogundul1737]

Since Fourier Transform is also closely related to periodic things, I wonder if there is some kind of homomorphism going on between the trajectoid and the Fourier Transform.

[@orterves]

Everything about this is awesome

[@aianyoung]

Thank you for continuing to make excellent videos on complex subjects in an easily digestible way. I've greatly enjoyed watching your channel for the past few years!

[@punditgi]

Jade is back and totally awesome! ❤🎉😊

[@stephenpuryear]

Jade I am so glad that you have persisted in making these videos. I also really love the background that you frequently shoot in front of. That particular shade of blue is soothing but also eye catching, along withe the formulas on the black placards Lastly, I am convinced that as a species, we need to keep descending deeper into three or more dimensions as we seek "explanations" for how our world really works. Thanks for these videos!

[@faisalsheikh7846]

Plz make a video on Godel Incompleteness theorem

[@mimetype]

Isn't it incomplete?

[@NamanNahata-zx1xz]

​@@mimetype You did not just

[@mimetype]

@@NamanNahata-zx1xz Sorry :)

[@davidgillies620]

I saw this and immediately thought "parallel transport and spinors" and lo and behold, up pops the Bloch sphere. The angle doubling as applied to qubits is a dead giveaway. You see that everywhere, from light polarisation to quantum spin states.

[@vitaminluke5597]

"Trajectoid" sounds like an overly specific online political insult.

[@rpow6861]

all this kinda stuff is so damn cool!! i haven't got a clue about any of it but that doesn't stop me from loving it!!

[@kjh23gk]

You explained this brilliantly!

[@upandatom]

thank you!

[@JoshtMoody]

This is freaking fantastic. Thank you so much for making this video. When I watched this I was instantly reminded of the WW2 'mechanical computers' to get pretty accurate shelling. This is taking it up a level though. I love it. I am going to try to make one of these things to make a mechanical computer to model levitation of liquid rubidium in vacuum. I can cross-reference to some FEM modelling in COMSOL and then have some decent confidence in my prototype before I assemble and test it. Thanks again I am so jazzed.

[@kenjinks5465]

When tiling a plane, you can start with a square lattice and manipulate the boundaries of one cell to create different shapes that tile the plane. If you start with a sphere with an equator, you can design a path so that when you apply your shape half way around the equator, and the inverse of the shape on the other half of the equator, the two halves will always have the same area. So, this is a tiling on a sphere problem, but you only get two tiles on the sphere. I wonder if it can be broadened and start with three equators at 90deg to each other, and apply the manipulation to the edges and constrain the eight faces to have the same area, what properties that object may have. Since the ball now needs to rotate less than 180deg to produce one period, the period I would theorize to be more stable.

[@bungkiukissinquinker7563]

I was missing this lady's videos for a few days now. Happy to learn new stuffs again from her.

[@Watchmedothatfor.u]

I found this channel yesterday and i already love it ! Go physics,math and astronomy ❤❤🎉🎉

[@jakubr2769]

0:07 LITTLE PRINCE!

[@kam1470]

I see Jade, i watch! :D Congrats on 714K Subs. Its been 500K the last time i congratulated, so you went a long way in short period of time.

[@matteogiberti3297]

As usual... Amazing video and Amazing Jade!

[@chad6849]

Most underrated channel on YT. Note: due to content not because my daughter is named Jade as well.

[@upandatom]

>_

[@algorithminc.8850]

A really fun one ... thanks ... Cheers ...

[@vortonu3155]

In honor of the discovery, I will end this sentence with two periods..

[@aniksamiurrahman6365]

Now I have a new challenge to these mathematicians - discover at least one trajectoid solid whoich traces a completely aperiodic path.

[@benjaminwood8736]

Not possible (I don’t think) because the shape would need an infinite number of sides, if it had a finite number of sides, when you push the shape from on face to another from the same direction, it always goes to the same next face (otherwise it wouldn’t make periodic things either), since pushing from each face in every direction leads to limited options, it means that eventually you would have do the same thing twice, I think the total number of options is somewhere in the ballpark of ((number of faces attacked to current face) * (number of faces)!)

[@benjaminwood8736]

Actually thinking a bit more it should be around (the sum of ((the number of attached faces to current face) * (number of faces)!) for each face)

[@aniksamiurrahman6365]

@@benjaminwood8736 May be. On a second watch, a different question came to my mind. The mechanics of real world trajectoids should also be studied. Like, how their mass, volume, the driving force, and the smoothness of the surface relate to mobility. That may not be that costly of a research either. But a quite laborious one. I think I shud post this reply too as a OP comment.

[@MrHerhor67]

Like, a ball?

[@benjaminwood8736]

@@MrHerhor67 A ball makes a straight line, the force put on the trajectoid doesn't change

[@alex4833]

This is neat :). I was intrigued throughout and am interested in Brilliant's courses. Awesome video! Have a marvelous week, Jade! :). Take care.

[@CasshCoww]

I like how the papers in her background changed orders

[@puffinjuice]

I like how you demonstrated the ideas with a ball and clay snake. Clever!

[@robertsmith4129]

This is fascinating to me. I’ve studied a lot of math but have never really considered this concept.

[@user-tc9qy8hv4n]

Great video 😊 you make learning more fun.

[@mohammadrezamoalemnia9201]

I always learn from you and I love that!

[@angelsachse9610]

These shapes are quite peculiar! I believe that the biggest problem with getting them to roll smoothly is that the center of mass is rolling up and down, a problem also faced in the construction of similar shapes that I have taken an interest in, namely developable rollers. The problem could be somewhat mitigated by putting a spherical cavity at the center of the trajectoid, then filling it with a viscous liquid like molasses and a heavy metal sphere. Action Lab made a Video about such a contraption titled 'The World's Slowest Ball'.

[@captainchaos3667]

Love the animations in this. Must have been a challenge.

[@Amethyst_Friend]

The famous raised eyebrow of curiosity. Thanks Jade

[@Mrkino]

Thanks for another great video!

[@drunkuchiwa9361]

Great video !!!!!!!!!🙌

[@Geopholus]

I am quite sure that this inquiry into trajectoids, has an absolutely MIND BOGGLING & universal application to helping explain /( prove ?) not only that string theory fits with TOE, but how electron "paths" result in constructively reinforcing standing waves (which result in fundamental particles), . I have done lots of experiments with what I call the "baseball curve" the shape of the two flaps of leather (equal in size) that are used to cover a baseball, and now see that the 2 pi R 180 degree rotation trajetoids also fulfill the property, of tracing a path around a sphere, that when translated by the trajectory of it's own outline, end up equally distributing coverage of the sphere,... (hence a stable standing wave)

[@EyesOfByes]

5:44 No way they did that *Le Petit Prince* reference!! That's the most adorable meme I've ever seen!

[@vigilantcosmicpenguin8721]

I just think it's rude to the elephant to roll that shape over it so many times. But probably not as rude as digesting it as an anaconda.

[@christoforusbayurisanto1793]

Very insightful! Thank you

[@LeeCanPotato]

I look forward to signing things by rolling my own unique ball covered in ink.

[@ofgaut]

Such a great and easy to understand explanation.

[@raffimolero64]

You know it's a great explanation when it leaves you feeling like you could've discovered it yourself.

[@matthewhafner962]

When you got to the 2 period part, I went "oh yeah, antimatter." If you rotate a particle only 360 degrees, you get it's antiparticle. You have to rotate it 720 degrees to get the same particle again.

[@pisscow6395]

Such a cool video! I struggle a lot in understanding math and physics but this was so well explained and entertaining!

[@brromo]

The straight line trajectoid is a cylinder. The circle trajectoid is a cone

[@AndTecks]

my trajectoid landed me in jail :(

[@AlperenBozkurt-tx2bx]

How

[@Shaynes73]

So sorry

[@AndTecks]

lets just say, daddy Icarus flew too close to the sun. Don't make the same mistakes as me.

[@AndTecks]

@@AlperenBozkurt-tx2bx 42

[@AndTecks]

@@Shaynes73 ty love

[@marcfruchtman9473]

Interesting. Thanks for mentioning the Bloch Sphere.

[@MathIndy]

An electron requires 720 degrees to complete a single rotation. The two cycles of the trajectoid made me wonder if there is any mathematical connection between the two.

[@marcfruchtman9473]

@@MathIndyYes! Unfortunately, I am not a math head... I am sure Bohr could probably hash out the math. It might also just be that they share they same problem geometrically. And the quantum state is somehow related in that way... ( I have been trying to piece things together conceptually tho, and the 720 degrees relationship was very striking).

[@fnx427]

Should've named them "Pathtatoes".

[@Lladra]

Cool! thanks

[@merijein9418]

having a shape that can imitate EVERY line a person can draw is actually damn amazing. I love math, but my little peanut brain can just look at it and say "WOW!"

[@HiggsBosonandtheStrangeCharm]

....i had no interest in trojectoids until Jade presented them in this video.....she is such a great presenter.....such a pleasant voice......so bright.....i've fallen in love......with trojectoids that is........thank you Jade for making such wonderful videos......

[@ramonhamm3885]

Fascinating!

[@yaa40]

I'm been subscribed to you channel for who knows how long. This is the first time I realized - you have a jumping eyebrow! It's awesome. Oh, and your videos are great too :D

[@dkloke]

Heartbeat trajectoid is very cool, props to Stan! Has anyone thought about and tried having hollow trajectoids with one or more weighted balls (or trajectoids?) on the inside to gather and release the weight to overcome at least small loops and abrupt turns? These would have to be tuned for specific inclines and initial velocities. The path for the internal weights might have to be a tunnel, the path moving closer to the center then dropping towards the surface to give the kinetic kick to overcome the difficult transition. But not hard to realize with a 3D printer, although the inner weight (small ball bearing?) might have to be placed into the trajectoid during a pause in the printing process. Thank you for the video!

[@Acceleratedpayloads]

I think you can apply trajectioids to orbits, if you imagine a cone-like shape rolling across a paper. I dont have an intuition for it yet, but you can imagine projection of a cone cross section on to a paper, then you create a trjaectoid of that path. Then, deform the paper.

[@Acceleratedpayloads]

I think two trajectoids can model stable 3 body situations, if you have a fixed single body as the center point

[@mikhailgastby4680]

Fantastic. This shape made a new view to represent 3d objects and formal actions , great vídeo, and you run infinal vídeo Jade ? Hahahaja

[@kjbunnyboiler]

A Jade day is a great day👍👍

[@SaiGanesh314]

Can't really express how much I'm in love with this! I study at the Universitetet i Agder (UiA) in Norway and I think we have a 3D printer lying around somewhere in our Mechatronics Lab... now I want a trajectoid of my heartbeat and body silhouette drawn on the boundary xD 😂 Thanks for spreading your infectious passion for math into me. 3B1B and you have been strong forces for me to tinker about mathy-silly things that i daydream about ✨️ 💞

[@AdityaSharma-th1gl]

Beauty with Brain😊 A very rare combination

[@freddyjosereginomontalvo4667]

Awesome videos as always say 🌍🌟 You are awesome ❤ Take care

[@AgentOccam]

You could do a whole video on the qubits/quantum physics aspect. What exactly IS a qubit? What is it to exist in a "mix" of states? (That, and "Spooky action at a distance" are the two most confounding things in quantum physics.)

[@szboid]

Thanks Jade, It brightens my day and my mind when I watch one of your videos.

[@upandatom]

Wow, thank you!

[@Impatient_Ape]

So which set of orthogonal basis functions do you use to decompose the closed paths on the sphere? Is it still sines and cosines (Fourier)? Is it Legendre polynomials?

[@maxwellhavoc6996]

I can’t wait to try this for myself. I have a 3D printer too