This Puzzle Game Proves That Math Is Beautiful! - Engare

[@daanroelofs119]

Tyler out here almost reinventing Fourier transformations

[@wansichen3743]

technically if you have infinite many of those rotating things you can make any shape, just compute the distance to some center for ever point, plot the distance as a function of angle, make it periodic and perform fast fourier transform to get the fourier components, make the fourier component the size of the rods and the sign the direction of rotation, you can even draw tyler with this (3 blue 1 brown have a video on this iirc)

[@Oosedamoose]

@@wansichen3743I think it would be cool if someone made one of Tyler’s face lol

[@CielMC]

​@@wansichen3743​So, little nickpicks 1. Yeah, that's kinda the point of Fourier transformation, any curve can be turned into these constantly spinning arms stacked to each other 2. FFT is just a fast way to do FT, you don't "apply fft", you "use the fft algorithm to apply ft"

[@Aliosar22]

@@wansichen3743 Yep, the drawings were in the follow up video to the Fourier transform called "But what is a Fourier series?". I still remember, that video came out in the same month I had to write a test about working with signals and with that about fft.

[@maxweinstein1537]

@@wansichen3743 I believe that’s a Mathologer video, not 3b1b. Very cool video, highly recommend

[@considerthehumbleworm]

As long as the two speeds of the arms are rational, the resulting shape will always meet up with itself eventually. The arms can be whatever length you want. You could have as many arms as you wanted even

[@tehalfgroove4774]

Whats the name of the mathemtical process that this game is based on? What you mentioned about rational numbers bringing something back to its original place reminds me a lot of the beginning of a musical measure in a polyrhythm, where all the beats of time signatures sound at once.

[@detaggable9271]

@@tehalfgroove4774fourier transformations & fourier series, iirc

[@edim356]

​@@tehalfgroove4774Fourier transforms. They get even more cool though. With infinitely many rotating arms, there is a combination of lengths and speeds that will draw whatever image you want

[@gisopolis77]

And if they are integers (like in the game), it will always meet up with itself exactly after the inner arm performs one full rotation

[@alansmithee419]

@@edim356 Only closed-loop images are possible. Even simple images that break this will never work, such as two concentric circles. Though of course if you have multiple objects, you can always make multiple of these arm systems.

[@anonymouse4496]

Fourier transforms allow you to use a series of rotating lines to draw any shape with infinite lines, or approximate any shape with finite lines. The equation is surprisingly compact for something so complicated.

[@Eta_Carinae__]

Fourier Series is probably more appropriate for your meaning. The transform itself is a _transformation_ of your coordinates, that takes an image represented by a bunch of points as coordinates, and instead represents it with a bunch of sinusoids as coordinates.

[@Lilguybondo]

The game breaking at the end was golden

[@dylandepetro4187]

I mean the dude has a tendency of breaking any game that can be broken. Look at peglin, or that card game he has been playing.

[@NickiRusin]

​@@dylandepetro4187balotro can become way more broken, the stuff he's done is entry level

[@atticus9300]

12:20 😊

[@erikm8373]

I wonder if, when checking your answer, the game actually looks at the shape, or if it's based entirely on where and when you make the initial point.

[@cytrussToo]

If I had to guess it's probably easier from a programming standpoint to just check the timing and position (since you can add a defined field in which this works )

[@Crazyclay78YT]

look at the last one lmfao

[@cattynerd1859]

I'm pretty sure the coloring process is based on a shader, meaning it applies effects to what's there instead of calculating what's going on and picking from a set number of colorings. That last instance was just the shader wigging out from an unexpected edge case.

[@apppples]

I was thinking it could represent the openings in the shape as a graph and then find a coloring, but maybe it only supports up to some small n-colorable graph coloring? idk, i don't remember how hard it is to find a coloring of a graph

[@ryuuguu01]

Since if you only use integers for the settings ( and don't get a rounding error like the last one) you will always have a closed curve the shader could just be a flood fill with some symmetry checking. It would be fun to program.

[@tehcrashzord]

7:14 got me so good, well done editors!

[@willj3402]

love the last of us

[@IcelyPuzzles]

2:42 don't worry I got you, I also love 2 sideways balloons tied together with a big balloon in the middle! totally know what it is :)

[@SquidLikesTalking]

Chess Balloons Advanced

[@Khaim.m]

Chants Battle Advanced

[@gec101]

seriously what is it though I have no idea

[@SquidLikesTalking]

@@gec101 Go watch his channel, my favorite puzzle youtuber with Aliensrock

[@alphaofficial6466]

crossover battle advanced

[@ragequazar]

God, what an incredible game. I'm going to keep a close eye on the creator, because between this and Tandis, he has a monopoly on the "most satisfying possible math puzzle game" genre.

[@Phil_Chicken]

Wow! The editing and choices made for this video are really well put together. From 1 : Knowing your audience and what they like, it shows that you and your team have good knowledge of your content and how to make it. To 2 : The fade away to just simplistic and relaxing shape making, the way that is is done and how you fade out of it all at the end really shows that you and your team are conscious of the effect of good story telling and how to do it. Then 3 : The self awareness of Tyler to let the video breathe by itself and the editing taking advantage of that; it shows how much you have acquired knowledge of this job, it shows that you want to give a great story to your audience, it shows the beautiful art of story telling and finally it shows your growth. Thank you Tyler for giving us these videos, they help. And thanks to both: Noarsy and Ben Ranger for this experience.

[@sameez8701]

At this point, your videos are a part of my daily routine. The 12 hour shift in when you release them is much better

[@NaugOminis]

Such a cozy video to watch and relax to in the morning. Thank you Tyler and editors. P.S. the music is perfect props to whose decision it was to use it.

[@Gr33nBean21]

I find it really cool that the numbers being used for the slide puzzles are the arabic indic numerals system :)

[@CasualMitosisCollective]

Yeah, and the editors probably figured it out at some point in the vid.

[@sheersternfeld1914]

Yeah!

[@hirbodarya945]

well its actually persian numbers because its made by an Iranian developer

[@CasualMitosisCollective]

@@hirbodarya945 oh. Explains why it looked slightly off.

[@Gr33nBean21]

@@hirbodarya945 Now that you say it, they're using the persian numbers and not the arabic ones. The 5 is different

[@K_Hammond]

"You're not here to see puzzles, you're here to see what funny shapes I could make." That is SO facts. Also, if anyone out there knows a sandbox/simulator for this type of thing, please share it with the class.

[@reverse_engineered]

This is creating spirographs. There are many programs available that let you draw your own.

[@romajimamulo]

One example is GIMP, one of the built in filters make spirograph images

[@Keldor314]

The game in fact has a sandbox mode. Once you reach some point in the progression, the center part of the mosque in the level selection screen unlocks, and it has a selection of geometric sandbox toy environments for you to mess with, including some that never made it into the puzzle part of the game.

[@K_Hammond]

Oh, that's cool. I wanted to find simulations but I didn't know what the pattern-creating method was. Thanks!

[@Anastas1786]

Regarding the unicursal "stars" in the last half: Every star is made up of a repeating "sub-shape", and it looks like the speed of the dark blue "pen arm" on the end determines the number of sides or loops in the sub-shape while the speed of the white middle arm dictates how many times the sub-shape must repeat to complete the star.

[@zaywayokay]

the ending was so meditative. thank you for adding that in

[@ShadowKestrel]

loved that note at 2:45 great work as ever from Tyler and the editors!

[@donkulator]

would be super cool to have one of these that was basically a clock with the outermost one rotating once a minute, the middle one once every hour, and the center one making a rotation every day. i wonder what shape that would create

[@taiyoqun]

Put some light source behind the hands and photosensitive paint on the face of the clock or magnets and iron powder and you get to see the pattern as it goes along.

[@thatrandomrecorder914]

The blue one is the shapes symmetry, and the white one is how many shapes there are! Negatives make it more rounded and vice versa. Assuming white is x and blue is y, you can find and ordered pair for each pattern! (Yes I'm a math nerd, my freshman year algebra teacher is my dad's friend)

[@RiotGearEpsilon]

Engare deserved more attention than it got - I'm glad to see it getting more play.

[@dwarian5252]

0:22 the editors really saved him this time

[@Rubbly]

Really cool game! Makes you think about how much of art is math and vice versa, then lets you just play around at the end!

[@deloptin545]

was hoping this was a chants of sennaar episode, stuck on level 2 and wanted your help lol

[@jacknicoara3889]

hopefully we get that vid soon cause it seems like itll make a great series

[@DynablocksKunzite]

I think we're all hoping for a chants of sennaar episode soon.

[@calamari9436]

After watching the first 5 minutes of that video I bought the game and beat it lol its so good

[@avishaishitrit6371]

​@@calamari9436 same! Its amazing, finished it yesterday's evening

[@Aliensrock]

Tomorrow! Those vids take a while to edit lol

[@RavenDravenek]

They make Spirograph kits so you can make your own spiral art if you want something like this for your wall

[@fieqahazman3383]

this is the most wholesome and satisfying gameplay i ever watched. subscribed ❤

[@noki1307]

On the puzzle at 4:54 those symbols do match up with the Arabic numeral system which makes sense with the sides on the shapes. Very cool!

[@a_fine_boi5677]

What's weird is 4 is a letter, and 6 is a different letter aswell

[@zamasoom.]

@@a_fine_boi5677you are getting ٤ confused with ع

[@a_fine_boi5677]

@@zamasoom. No, in the game 4 is م but it's ٤, and 6 is ء but it's ٦

[@MHonari]

@@a_fine_boi5677These are how those digits are written in Persian.

[@talitameyer2581]

With the negative to positive scale question, the symbols, as you'd probably guess, were arabic numbers. It is intriguing this game!

[@NikoCubeRoot]

-3 -2 -1 0 1 2 3

[@HannahKossen]

What I find fascinating is that you always get a closed loop. There has to be some algorithm that finds the closest point to where you clicked while having a rational solution. Edit: nevermind it's just because the rotational periods of the arms are perfect ratios of each other.

[@alexholker1309]

Yep, if you take the rotational periods of each arm then the pattern will repeat after the lowest common multiple of all the rotational periods. If the ratio is irrational - such as Pi or sqrt(2) - then there is no lowest common multiple, and it will not form a perfectly closed loop.

[@forgotn42]

2:43 Ah, yes. A most classic shape. Sid from Ice Age.

[@YoshiTheGamer64]

Tyler always amazes me with his puzzle skills

[@chickenmaster3879]

The editors are just flexing in this one. Love the bit at the beginning

[@Zowiezo101]

Thank you for showing us some shapes. It was very satisfying to watch!

[@johnpaulboudreaux9569]

can't believe they made spirograph: the game

[@WorBlux]

This would have been really cool as a screensaver back in the day...

[@MohamedAyman-kh2ej]

7:26 The slider has the numbers in Arabic from -10 to 10

[@mcb187]

quick, give this man a spirograph!

[@bwax3135]

This could be an interesting way to think about groups under composition like D3. I wonder if there's a way to create a setup where the shapes have very controlled inputs, but various compositions of inputs create other inputs. Like putting one shape through another's pattern or something.

[@SalvatorTorres]

Damn, first time i liked tyler gameplay more than tyler commentary, but still i gotta thanks you man for giving me a chance to see this beautiful game❤ Keep it up, for the funny man on the screen or for the funny game on the screen, i think I found a subscription that'll last untile the very end😊

[@jamhopsey]

this is a next level video

[@Chocomint_Queen]

Tyler out here learning why spirograph kids were spirograph kids I spent hours drawing cool spiral shapes as a kid

[@ZooD333]

Get yourself a spirograph kids toy Tyler, it's a physical tool to make these patterns! Also Devin on the channel Make Anything made an amazing double pendulum glow in the dark wall art piece years ago that I think you would appreciate.

[@spellingchamp01]

I won't go into the math behind it, but you were onto something when talking about generalizing to any curve, as long as you are allowed up to infinite rotating arms.

[@Temperans]

Technically this is how etch-a-sketch works. You have two nobs and the position of which determines what gets drawn. In this case the position is just constrained to a double pendulum and their speed.

[@reverse_engineered]

Which makes it a simple form of spirograph.

[@cobracrystal_]

The game actually has a sandbox part where you can change rotations and lengths of 4 sliders!

[@Luck_612]

The 800 possible shapes is lower than the actual number, you could put the dot on starting points and lines

[@AveragealienWhocosplays-gj2tg]

the lines and colors intertwine so well in these games

[@skiller5034]

Not to mention there are multiple points on the small one you could select that would produce slightly different results, even with the same rotation settings.

[@jozimastar95]

teachers after asking somethink in exam that they never teached us : 2:44

[@reverse_engineered]

The cool colored look seems to be a combination of a couple of different strokes for the curve and filling in a bounding hull of the curve. First, the background. With a single closed curve, you can start from a point within the curve (given the setup of the curve, we always know where the center is), then project a ray from the center outwards, and look for the further distance away that the ray intersects the curve. This defines your bounding hull, which could be approximated with a polygon. Now fill that polygon and you're done. Second, the stroke. When we watch it unfold, it's drawn with a simple black stroke of a relatively narrow width. If we make a couple of different strokes of different widths in different colors, you get the affect seen here. Overlay the stroke over the filled hull and you get the final image. For the shimmer, each stoke/fill can be seen as a bitmap. Simply use that bitmap as a mask over an animated gradient and you get the cool shimmer effect. It's entirely possible this is implemented differently using shaders - I don't have any experience programming shaders. But back in the day before shaders, this is basically what we would do. It's also how you might do it with a vector editor like InkScape or Adobe Illustrator, where everything is a piecewise curve with a stroke, and if it's a closed curve, then it can also have a fill.

[@dahn57]

This is literally computerised spirograph, and I've no doubt that a computer could easily replicate any alterations you might put in for length and rotation of the arms.

[@Giguv05]

Yup Here's some lines of code you can use to do it in python import matplotlib.pyplot as plt import numpy as np plt.subplots(subplot_kw={ 'projection': 'polar'}) a=7 b=4 x = np.arange(0, 2 * np.pi, 0.01) y = np.exp(1j*x)+0.4*np.exp(1j*x*(1+a))+0.3*np.exp(1j*x*(a*b+1)) plt.plot(np.angle(y), np.abs(y)) plt.show()

[@karnature9391]

Really enoying this puzzle content ! The music of this game is so mesmerizing. Leads into the mystery that the game proposes .

[@ThatNerdGuy0]

I woke up just in time for the video I love the new upload schedule, now instead of breaking my sleep schedule it fixes it

[@ryanjohnson3861]

so one really cool thing about the shoes made with the lines is that through some complicated maths its possible to control the shape to such a high degree you can draw any thing that you want, I can' t remember where but I saw one set of rotating lines draw the Mona Lisa

[@coryman125]

I remember a flash game like the sandbox part of this on Kongregate, though the name escapes me. You could add and remove arms, change the speed and direction they rotate, and toggle drawing on them. Was very fun to play with!

[@crazyabe4571]

there were like 3-4 of 'em actually, Flash spirograph, SpiroStudio, and a couple others- unfortunately with the murder of flash half of 'em were lost and the others don't work anymore even with "ruffle" the defective flash emulator.

[@coryman125]

@@crazyabe4571 Yeeaah :( They're probably around on Flashpoint actually, I've never thought to look

[@phantomskull8135]

Hey Tyler, just wanted to let you know, in case you didn't, there is actually a thing that can do these kinds of things. It's called a spirograph, and they are super awesome and relaxing.

[@Dr_Nobody2962-YouTube]

This is why I love math because it dose so much.

[@SophW-s2f]

2:54 beehive

[@victorzarrabeitia3875]

The intro is so funny

[@solomond9417]

No one is talking about that legendary opening. I love the idea that Tyler is just trapped in the void in between recording sessions lol

[@evixivityreal]

i would LOVE a game like this where you can create the sticks and make WHATEVER you like

[@Corn_Fiend]

i dont know if this is a hint for future puzzles or if this is something that hasnt been said yet but there is a pattern to the shapes made and no matter which combination it is it will always be unique. the white slider indicates how many copies there are of each individual shape and the blue slider indicates how many sides each shape is going to have, i dont know how to determine positives and negatives but if anyone was curious like i was here is part of the answer

[@aba3220]

Love this! Would love to see more of Tandis!!

[@huskykid0295]

He already said why he wont be playing more in the previous video.

[@gap-lo9zj]

7:14 I love this reference to The Last Of Us. Tyler as Ellie is perfect

[@macdjord]

@Aliensrock: Actually, there are 4 variables in the sandbox mode: speed of the middle bar, speed of the outer bar, where on the outer bar you click, and the phase angle between the two bars. (To explain that last one: imagine the bars are both set to speed 1. If you click the top of the outer bar when all 3 bars are pointing straight up, you'll get one shape. But if you click it when the first two bars are pointing straight up and the last straight down, you'll get another. It's not just a matter of time; at speed 1-1, the bars are all in synch, so starting from all-up you'll *never* get up-up-down. To get up-up-down, you'd have to set the outer bar to speed 0 for a half-turn, then restart it.)

[@ladyravendale1]

I remember a flash game on Kongregate that I used to love playing, where it was this but you could use arbitrary colors, sizes, speeds, and linkages. Sadly I was never able to find it again, especially now with the death of flash.

[@crazyabe4571]

there were like 3-4 of 'em actually, Flash spirograph, SpiroStudio, and a couple others- unfortunately with the murder of flash half of 'em were lost and the others don't work anymore even with "ruffle" the defective flash emulator.

[@thelostsoul9446]

Thank you so much for this.

[@maxalon2479]

12:37 LOL TF JUST HAPPENED 😂

[@dhruvemital]

I was just going to say this is just fourier transformations visualized beautifully!

[@abuazamazim]

4:15 the symbols below the designated shape is arabic numbers ١=1 ٢=2 ٣=3 ٤=4 ٥=5 ٦=6 ٧=7 ٨=8 ٩=9 ١٠=10

[@crazywavybaby]

THE INTRO IS SO UNHINGED YOUR EDITORS ARE INSANE

[@bony_ttv]

And thus Tyler describes one of my favorite math playthings, the spirograph.

[@polymerkiki]

It's so cool this game is. So mesmerizing even if it's just visual show of algebra. Reminds me of high school when they were telling stories about al-Khwarizmi, the father of algebra. I remember I was so curious when my teacher were doing the storytelling. I can't imagine how much more excited I'd be if they show this game as well in class.

[@foRonable]

you can generalize coloring it in! The lines are just black dots, so you can go line by line, filling color in from the first black dot, and switching color after every black dot. If you run to the edge of the screen, just backtrack a little, and uncolor the part that was filled before. done!

[@MathewSan_]

Great video 👍

[@TheSheep1]

0:21 we were all thinking the same Tyler…

[@sykes1024]

[Tom Haverford intensifies] "There's just something about these shapes."

[@bananapajama2753]

Day 36 of sacrifing my sleep to support tyler After skipping a day due to getting good sleep, im back and we have this beauty of a game Still an interesting puzzle game

[@JamesTDG]

I hope the game's sandbox gets expanded

[@analander9222]

Holy hell Fourier transformation is now a game?

[@niczic123]

I LOVE WEIRD MOVING OBJECTS AND POINTS THAT DRAW LINES LET'S GOOOOOOOOOOO

[@MrBlinky10101]

That final shape was INSANE

[@electromika]

i love the arabic numerals being used here

[@MandrakeGuy]

id like to see more tandis, it looked interesting

[@Dark_Slayer3000]

Aliensrock is my favorite educational maths content creator ❤

[@sirzorg5728]

Fourier proved that any shape can be made with by tracing the end where only the length and rotation speeds of the arms changes.

[@roseroy5041]

This was very satisfying to watch. Thank you for sharing such great game.

[@6thearchivist]

i am absolutely here for the pretty shapes and i love it

[@eduardpeeterlemming]

Aliensrock a day keeps sadness away

[@blu_skyu]

Those rotating bars are made from fourier series. 3B1B made a video on it. You actually can make anything with enough bars! It's really cool!

[@OlujaDoTokija]

Please give us more Tandis Tyler!

[@BleachWizz]

12:38 - did you found a figure it can't fill in? there's too much going on in that middle circle xD

[@Nevir202]

7:13 maybe the best editor moment ever lol

[@Stilyxx]

I really like this game and watching you play it, but I can't wait for your next videos on Chants of Sennaar! I actually bought and finished it over the week-end after seeing your first video on it, and I'm looking forward to seeing your approach to it

[@sai_ai__8446]

Just a reminder that you do not have to place the dot at the end

[@Lordike3]

Liked based on the first 3 seconds of editor magic. Praise.

[@emiaugacha9614]

This was very relaxing to watch :D

[@cameodamaneo]

That is the best intro I have seen on any video.

[@konvitalik]

the “and if you don’t know what that is, fuck you!” was incredible

[@sophiesmidnightsquad]

40 minutes ago! I really love all the unique puzzle games you find! Thanks!😊

[@izza4673]

1:11 Technically, that’s just the moon, earth and sun