Where do these circles come from? 

I clicked on this video expecting a documentary video about a weird quirk in Minecraft's world generation that I hadn't known about, and instead I learned about geometry. Amazing.

This is really interesting! As someone who's made dozens of spheres in minecraft, this is something that I noticed but I never actually stopped to think about

Just watching the animation... it starts to look like drops of water creating waves on the sphere. So satisfying

I love how circles have relations with rectangles and triangles. The sine function shows this as well

If I had to guess, I'd say the rate of rippling would be proportional to the Hamming distance between the pattern and the nearby patterns since that's kind of like how the contour changes, but maybe it has something else to do with the sphere...

when you said “the sphere is the same size, the cubes are just shrinking” *you changed my view of that forever.*

Woah, seeing the ripples begin to interfere with each other as the sphere grows bigger through time is wild. You can actually see wave physics happening in real time as an artifact of geometry

Something about watching the huge rippling sphere towards the end gives me a crushing sense of scope and insignificance.

The Ripple frequency is related to the distance between repeating planes on the sphere. This is because a new ripple is spawned each time the cubes pass through another plane (thinking of the sphere as expanding). For the axis planes, this is clearly 1 cube length. For the the points between two axes, this is 1/sqrt(2) of a cube length. For the triple axis points, it is 1/sqrt(3) of an axis point. All of this math is heavily studied in X-ray diffraction, as it also dictates the angles for diffraction. The terms you want to look up are “Miller Index” and “Reciprocal Lattice”.

For thousands of years mathematicians concerned themselves with "squaring the circle" and tragically failed because calculus hadn't been invented yet. They knew intuitively it had to be possible, but the method was just out of their reach. Henry Segerman: "Let's have the computer sphere these cubes and see what happens."

0:45 when it zooms out, the colored sides of the cubes start blending in our eyes and making different colors, just thought it was neat to see on a macro scale.

Mesmerizing! Thank you for the demonstration

I love when a math video leaves you with more questions than you started with. Truly one of the joys of math is constantly realizing how much more math there is

Oh I see a little better now. It's like waves traveling from sources, and whenever the divisions are fine enough so that the interference of the 2 sources create a stationary wave (flat region) halfway between these sources, this point becomes a source. This happens since this region will now be flat, and thus whenever scaling is applied, the way to fill flat regions is with circles, and thus this point becomes a new source.

Great summarization and interesting topic!

The rgb colors blending together the further the zoom is so weird yet it's the coolest thing I've ever seen!

I really wish I could keep watching this video but my eyes stopped me after watching that vibrant-colored sphere for a minute.

In a museum about crystals, i have seen a physical experiment about it. You grow a big alum single crystal. Then you polish a perfect sphere out of it. Then you put it back into a crystal growing solution for a little while. You get a set of clear circular windows of different sizes on the sphere following a mathematical pattern. Almost looks fractal, and the main axes of the crystal are easy to identify. The rest of the sphere looks matte or rough. My amateur explanation: All those tangent planes that are parallel to any symmetry in the crystal structure inhibit crystal growth, the crystal prefers to grow by extending more irregular areas. You only let it grow for a short while. You used cubes. You could ose tetrahedrons, or other tilings of space. You would get different patterns of circles. The pattern reveals the crystal structure at the atomic level. This works for any crystals you can grow and polish into a sphere, and it is much more hands-on than x-ray crystallography, you get pretty objects that you can study hands-on. These are amazing for a museum display. You could 3D print some up at a scale where the tiling of space is visible, and next to it you have the ACTUAL thing showing it on a truly molecular scale. I am quite sure there should be a direct link between the images you get in x-ray crystallography, the cirles in your simulation, and the circular windows appearing on the crystal sphere.

i stoop listening half way through tbh cause the visual was just so mesmerizing

This is so satisfying yet mesmerising and hypnotising

So, if these circles appear where tangent planes are well approximated by a subset of a cubic honeycomb, the rate that each ripple occurs at must be related to some measure of 'average thickness' of each layer in that direction. That would explain why the coordinate axis circles are so much slower to appear.

Seeing my favorite youtubers work together even for something simple like this is awesome. Henry Segerman, Code Parade, Inigo Quilez for Shadertoy

I was just thinking about this the other day! Thank you!

More than the basic process of your video of ''how a sphere is based in a cubic world'', I've loved watching it thinking how it could show us how PIXELS work and it was mesmerising !!! Thanks a lot to amaze me, been a great time ! Have a nice day sir

It would be cool to see if dithering could be applied to make the pattern less prominent while preserving the shading. Like how dithering can help remove banding in video/images with low bit depth.

The circles correspond to points on the sphere where the normal vector takes the form of (h,k,l)/A for integers h,k,l, and A^2=h^2+k^2+l^2. I'm sure there's a proper name for these, but I'll call h,k,l the indices for the respective axes, and assume that they are non-negative, and co-prime. We see empirically that h,k,l are small values (typ, 0,1, maybe 2) but I'll come back to this. The tangent plane takes a 'diagonal' cut of the array of cubelets. We can translate it one cublet in any axis, and the distance from the original plane plane will be h/A, k/A, or l/A corresponding to the direction slipped. I believe that the distance to the nearest distinct plane will be just 1/A, given the GCF of indices is 1 from their being co-prime. We can think of this 1/A as being the layer thickness, which is largest when the indices are small. The thinner the layer, the more frequent the steps happen. How visible the steps are will relate to the layer thickness (1/A) relative to the feature size of the face texture, or max(h,k,l). The steps vanish quickly as indices get larger, with the largest index value hitting both the thickness and feature size. Obviously the largest steps are at the axes, e.g: (1,0,0)/1. Next largest will be simple diagonals at (1,1,0)/sqrt(2) then (1,1,1)/sqrt(3). We can also see circles for (2,1,0)/sqrt(5), (2,1,1)/sqrt(6), and (2,2,1)/3. The denominators of these expressions are the relative rates of ripples for these locations. (3,1,0)/sqrt(10) and a few others are just barely visible.

This was so insightful! Thank you for making this video.

This is an interesting pattern to see, reminds me of me when I joined a private SMP with my friends and I decided to make my base a floating iron sphere. There's a helpful chart that illustrates how to make proper "circle out of blocks" for minecraft. The construction process went like this: 1) I pillar'd up to the center point of the sphere. 2) From the center point, I'd make 2 circles of equal radius, one aligned to the ground and one perpendicular from the ground. 3) This is where I pondered on how to make the actual sphere and figured why don't I just start making circles aligned with ground level from the bottom of the perpendicular sphere with increasing/decreasing radius as I go up and... this works. I went to view the structure from outside to see the circles not only from the cardinal directions but in-between as well. It's very cool made my base stand out that there is a floating iron sphere in the sky.

Oh my god I have noticed this pattern for _years,_ I always thought it was just a simple matter of circular cross-sections in all planar directions which happen to be visible in a few select locations. wonderful explanation for an age old question

I love these types of videos. Informative & educational, no filler, no "remember to like and subscribe". You've earned my subscription, without the need to beg :) keep it up!

You are already so close to discussing the origin of the speed of the circles! Now I haven't studied this effect for longer than the video, but I figure the following is happening: the repeating cubes 'crystal' structure contains a number of planes, most notably the XY, YZ and XZ plane. 1 layer of that plane is a single cube thick in the Z,X, and Y directions. Within the same layer of the plane, you keep expanding until the plane interacts with the sphere, creating a circle. However, there are more planes in the cube crystal than those 3. Using vector notation, when planes are layered in the X, Y, and Z direction, you can say they are layered in the [1,0,0], [0,1,0] and [0,0,1] directions. But there also exists a layered plane in the [1,1,0], [1,-11,0], [1,0,1], [1,0,-1] [0,1,1] and [0,1,-1] directions. These diagonal planes are are the ones with the stripe patterns that you discuss at 2:18. And the [1,1,1],[1,1,-1],[1,-1,1],[1,-1,-1] directions are the 3 side patterns you discuss at 2:43. But how does the speed derive from this? Well, since we are working with discrete cubes, each plane of cubes also has a layer thickness in those directions, and that is where the speed comes from! The layers in the [1,0,0], [0,1,0] and [0,0,1] directions are 1 cube thick, so the diameter of the sphere must grow with 1 cube for a new layer on that plane to manifest the circles on. But the layers in the [1,1,0], [1,-11,0], [1,0,1], [1,0,-1] [0,1,1] and [0,1,-1] directions are actually 1/sqrt(2) thick, so the sphere only has to grow 1/sqrt(2) cube in diameter for the next layer to fit and thus next circle to appear. It's late, so I can't work through the algebra of calculating layer thickness for planes in 3 dimensions, but I believe this is where your answer will originate from: different layer thickness for different planes meaning new layers in those planes fit at different sphere growths.

I love that if you run lines between the Center points of the circles you get hexagons. Idk if it’s related, but it makes me think of all the things in nature that are derived from hexagonal patterns, and how that’s derived from spherical atoms in a crystal lattice. Beautiful 🙂

I’m surprised to see a high quality no-nonsense video coming from such a small channel. Great work!

This is a great video, I've always wondered why there were these circles in these simulations and I'm glad you've shown me how. It's provided me with a newfound liking for geometry in simulation but also cubes themselves

Thanks for the visualization!

Cool video! I am currently working on a spherical bubble dynamics problem using computational fluid dynamics on a hexagonal (cubical) grid and I am observing the same. Very interesting thought that you've shared. Thanks!

Interestingly, I feel this 3 coloured cube sphere can be used to make a colour wheel

I think the reason for the circles is obvious: At any point on the sphere, you have a plane which is tangent to the sphere that is represented by some arrangement of cubes. since the cubes are not infinitely small, there's distance between them, which can be represented by using multiple parallel planes. when a plane intersects a sphere, the point of intersection is a circle. This means that at any point on the sphere, since there is a corresponding plane that passes through that point, there's also a circle at that point in the voxelized version. for planes that line up with the coordinate grid, the distance between the cubes is greater, and the circles easier to see. as for the ripples occurring at different speeds, I think that has a similar explanation. Sine the planes at the top and sides are parallels to the cubes' faces, they must travel further into the sphere before reaching the next line of cubes that approximate the same plane. at the diagonals, this distance is shorter, and so the ripples can occur more frequently. (Sorry if I didn't explain that very well. Interdimensional explained it with fewer words in the comments here.)

I what's a math major, did some grad school, but only recently have I suddenly have been flooded with videos about splines and interpolations and it's been wonderful.

No idea what you said, but I was fascinated, well done!!

Finally someone explained this! What a wonderful video

I used to see patterns similar to these when working on my Master's research. They were apparent when trying to produce Point Spread Functions of certain systems using a Fourier Transform.

Please just leave a recording of this simulation up. It is very satisfying.

I could just watch an hour long video of this effect taking place, its that satisfying to me

It's one of those things I noticed, wondered about but never thought deeply enough. So I added it to my watch later to find out. Thank you for contributing to Shadertoy - i am building a neural language model for shadercode and currently building my dataset from all public shaders on Shadertoy

That is crazy, i realized the other day exactly this with my 3d resin printer. The layers are so thin, there are circles appearing in places they should not appear. I mean, the layers are going always in one direction, I was confused when I suddenly saw some circle shapes appearing in the other direction. Does anyone notice that before?

I found your shader and edited it to work in isf format for my VJ software months ago and have had a bit of fun mixing it with other shaders and live visuals and am just now finding this video where the shader was born. I feel like I was personally reverse engineered somehow. Cool cubes / sphere also, thanks.

Wow, this was simple yet fascinating!

The ripple frequency as a function of its relative complexity made me think of harmonics in the musical sense.

3:27 Ok, so to make a good sphere in Minecraft i'll need like a hundred thousands blocks of diameter

I was so hypnotized by the animation that I had to watch the video again to hear what the guy was saying

Great video quality. And i dont know how u think of this idea but giving insights on digital things like what an shader is structured. How it works in an example what creates this that phenomenon. Stuff that can catch eyes of the new. Blood that wants to create digital art pieces either for learning more about how their favorite digital experinces are made. Or for the understanding of new aspects and how u can utilize them. And generaly just inspiring stuff like how procedural animation whould work in a 3d space. How the gpus work with diffrent processes. (No iam not saying anything to make it like u need to do this. But theres allot i chouldnt even comprehend how it works and this video catched my eyes i am looking forward till next time )

This is equivalent to the 2D effect of "banding" so you could get rid of it by introducing "noise" (surface bumps) proportional to the... "bit depth" of the bands, i.e., the size of a cube.

It reminds me of harmonics of a square wave... I know it doesn't really explain anything but it inspires me somehow

the sub-circles that appear smaller at the odd angles remind me of when you're driving past orchards/crops on the highway and the gaps between trees line up at odd angles, except smaller. not sure if the effects are related at all but its neat :)

I did not know something I see everyday like this could make for a good topic. fascinating.

Is there a circle centered at each 3-vector of integers, with prominence/speed a function of their magnitude?

Because layers of circles are the closest approximation to a perfectly round curved surface, so you are going to see them more an more places as the accuracy goes up.

I could watch that cube sphere for hours, mesmerizing especially with the Moire effect at the end 👀

Beautiful animation!

I'm fascinated by the small R cases, like where radius is between 4 and 5. Where exactly can we claim the first circles (of each kind) appear, and how does their shape transform to larger R cases?

3:50 I bet this how an actual planet made out if Lego would looks from Space

I recognized the Lego globe from my last trip to Carlsbad, CA and immediately clicked the video. Best 4 min spent being mesmerized by a simulation!

A small and simple system can make something amazing and much more complex at the macro level. It reminds me of the game of life in a way.

Still not sure what's going on here but what I can say is that these are really pretty cubic spheres. It looks like drops of water every time more cubes are added.

First guess, there's a 'ripple center' for every ratio of red:green:blue, with the 'speed' of the ripples being tied to the ratio or angle of that trio. There might also be something about how often a tangent plane at that spot 'lines up' with a full lattice grid. I feel like there's an interesting way to generate or characterize a sphere like this...

This is so good. Thankyou!

Same effect apears in 2 dimentions on roof tiles or when you drive by a field with plants on grid. There are many ways to alighn your eyesite with a grid and all possiple alighnments are on a pythagoren tringle of integer grid points. Here as the sphere grows, more "integer" 3d orthogonal triangles are possible, and more circles seem to emerge on the surface.

These circles myke life hard when trying to compute curvature from an isosurface in such a cubic lattice. The areas where circles appear typically have the highest error.

3:38 *Void Termina Leaked Design*

i dont understand anything out from this video, i just watch how satisfying the sphere looks overtime

Really well explained.

My eyes: lego My brain: Minecraft

3:55 I challenge you to create this same sphere in Minecraft

This really reminds me of the spherical harmonics which have different bands of basis functions that represents signals from low frequency to high frequency.

Thankyou for this very useful information

You’re seeing the laws of physics visualised Because you’ve forced an aspect of our universe onto a simple block based world You see the mathematical harmony of the universe expressed in an unexpected way

*90% of people who have clicked on this video thought this was minecraft.*

I remember passing by the lego land globe and having the same exact question, thanks for the video!

The rate depends on the grid cubes that inter sect the tangent plane. At some angles you get "more planes" parallel to the tangent plane formed from the centers of the cubes. Put another way, if you partition the grid with a plane, there will exist a smallest distance to the next parallel plane with a different patition. That distance is different for some angles, and is largest for the cardinal planes.

Best minecraft sircle tutorial ever helps out alot thank you 👍

I watched till the end for maximum enjoyment. Anyways, crystallographers out there can give insight. I recognized the first few as directions where a crystal forms planes on their own.

So cool. Love it !

gets so hypnotizing in the end :o

My guess would be that in the limit, the centres of the circles are the projection of the integer lattice to the sphere along rays through its centre, assuming that the sphere has radius 1 and is centred on a lattice point. The patterns we're picking up on are the regular patterns corresponding to approximations to planes, each with a normal that has integer coordinates, and the circles are where those patterns start to differ from the corresponding plane. I'm still not 100% certain how to turn that into a rigorous argument that doesn't just beg the question (there's a question of perception also baked in here which is a little difficult to tackle), but I think it agrees with what we're seeing. It might be fun to try superimposing a perfect sphere that is marked up by circles that are the intersections of planes with integer normals up to some limiting magnitude in the integer lattice, and comparing that with the blocky sphere.

My eyes are hurting, but I love it, great explanation and visualization

The circles form along tangent planes. In order of emergence there are 6 at the center of the cube faces, 12 at the center of each edge, and 8 at each corner of the cube. Afterwards they form between faces and edges, faces and corners, edges and corners, more between faces and edges, between face-corner and edge, and so on infinitely. It gets complicated quickly but, the next circle is guaranteed to form along the greatest distance between each new center. Patterns from faces tesselate “squarely”. Patterns from edges radiate “rectangularly”. Patterns from corners tessellate “triangluarly”. Some patterns resemble other geometric shapes at various iterations. This is observable after 3:00. It helps to take a look at it while paused or to playback at 0.25 speed. It looks like this took a lot of work. Fun. Thanks! Pause and go to 3:05. Three circles form an equilateral triangle around the corner. Those same three are from a square around one of the faces and a rectangle around an edge. Lots more fun to be had here. 😅 Of course now you have to try this with tetrahedrons. 😁

Could watch this for a long time

You could characterise the circles like crystal cleavage planes. (1,0,0) for example at the coordinate sides and (1,1,1) at the diagonal. The other circles appear once other cleavage planes with whole numbers become large enough to become visible, for example (1,1,0) or (2,1,0) etc

This is honestly so interesting

The animation especially of the later stages of the sphere would make a nice screensaver

The first part was mesmerizing

Man answered a question that I wrapped my head around for literally my whole childhood…

I could watch that thing expand all day

I’ll need to watch this again a few times. And also not at midnight.

Reminds me of constructive and destructive interference patterns I’m curious if mathematical point sources where placed at the cardinal directions of a sphere, would similar interference patterns exist?

Very interesting. I think each point of the sphere defines a tangent plane.. the plane can be described be three number (x;y;z), according the same direction plane touching the 3 points : (x;0;0) (0;y;0) (0;0;z). When x,y, z are integer, the approximation of the plane in cubes make a repeating pattern... which in turn visually gives a specific ratio of colors according to the regular pattern of faces we're seeing. for instance (1;0;0) is red, (1,1;0) is yellow. On the sphere, if you select one of the integer planes, the points around the center where the points ceased to respect the average pattern, are circles .

Never found a better video for me. was just thinking about this the other day when playing modded minecraft looking at an AE2 crater lol

Thank you. I was curious about that whenever i used worldedit in minecraft.

i almost zoned out because how hypnotizing that cubic ball grow is