This Puzzle Game Proves That Math Is Beautiful! - Engare 

Tyler out here almost reinventing Fourier transformations

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

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.

The game breaking at the end was golden

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.

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.

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

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 :)

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.

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.

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

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

"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.

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!

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

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.

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.

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

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

0:22 the editors really saved him this time

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!

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.

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

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

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

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

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

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

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)

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.

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

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

the lines and colors intertwine so well in these games

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

Tyler always amazes me with his puzzle skills

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

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.

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.

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

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😊

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.

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

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

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

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

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.

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!

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

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

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

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

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

@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.)

This was very relaxing to watch :D

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

Thank you so much for this.

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

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.

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.

12:37 LOL TF JUST HAPPENED 😂

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.

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

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

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

That final shape was INSANE

id like to see more tandis, it looked interesting

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

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.

Great video 👍

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

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

It looks so cool! I don’t know what shape it was that u put as your top 10 fav 2d self ineptersecting shape!

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

The intro segments remain creative and fun, how do they do it day after day :P

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!

Holy hell Fourier transformation is now a game?

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

0:21 we were all thinking the same Tyler…

this is a next level video

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.

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

Thank you for the beautiful shapes Tyler!

can't believe they made spirograph: the game

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

Aliensrock is my favorite educational maths content creator ❤

Aliensrock a day keeps sadness away

7:13 maybe the best editor moment ever lol

I hope the game's sandbox gets expanded

quick, give this man a spirograph!

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

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

THE INTRO IS SO UNHINGED YOUR EDITORS ARE INSANE

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

Please give us more Tandis Tyler!

Anybody have a link for the time lapse song?

I am loving the new schedule lets me watch more vid!

3:19 it looks like the michelan man had a child with jabba the hutt lol

sometimes i wonder if i'm insane watching some guy on the internet solving such puzzles. then i remembered that i'm too dumb to solve them anyways

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!

More chants of sennaar when? Love that game