A New Computationally Efficient Method For Spacing Points Equally On A Sphere

[@jakeaustria5445]

For those who are bludgeoning this video, please notice that it is 7 years ago. Don't flaunt your egos here. I personally don't care if he thinks he invented it, because it literally happens to every mathematician at some point. Personally, I thought that I discovered the Collatz parity test. I even derived it when I slept on the bus and got trapped in a police station. I spent the night deriving the equation and when I searched at wiki, damn!

[@sykes1024]

I did the same thing with the cross correlation function. Built it up from a discrete case of computing conditional probability with unknown time delay with a small number of events and then figured out how to extend it to a continuous case. At the time, I only realized that it was a kind of convolution but without one of the functions being time reversed as with a convolution. Years later when I went searching for a way to quickly invert the order of an array so that I could use the built in convolution function to implement my computation, I stumbled upon the built in cross correlation function.

[@jakeaustria5445]

@@sykes1024 I haven't studied cross-correlation functions yet, but I know convolutions and Laplace transforms. That's a great way to learn. I commonly find myself facing problems that I do not know the answer to and I don't want to rely on the internet so I solve it myself. The n*(n+1)/2, I thought I discovered it when I was in elementary. That's a real shame. Even the inverse transform in statistics, I derived it myself only to learn that it is called the golden rule in wiki. The inverse transform is a tricky one to derive since dx/dy does not mean 1/(dy/dx) but the derivative of the inverse function. It's a great feeling when you learn something and discover it yourself even if it comes out that it is already created.

[@gabberwhacky]

Hah I also discovered n(n+1)/2 in elementary 👍

[@opticulus]

im confused. are we supposed to be flaunting our egos or not?

[@semplar2007]

are you trolling X)

[@DavidZMediaisAwesome]

here’s my solution: put them all on the same point. they will all be 0 units apart and will therefore be the same distance apart.

[@theevilcottonball]

Genius! My sphere has radius 0, so any configuration will do.

[@dizzypear]

The engineer's solution

[@joshuabarqueesimeth4530]

fallacy, they cannot be the same distance apart if there is no distance for it to define the points as being apart, zero means nothing, if the distance between them is nothing, there isnt really a distance is there?

[@skycrafter1509]

why did the youtube-algorithm randomly decide to show a bunch of people a 7 year old video

[@xanderlastname3281]

KZbin does that some times

[@themammoth67]

Ye

[@MelindaGreen]

Because it doesn't treat viewers as points on a sphere

[@CLOUDEE33]

Wait fax

[@CLOUDEE33]

Why 7 year old vid

[@christianherrera4729]

Babe babe!! Wake up!!! New computationally efficient method for equally spaced points on a sphere just dropped!!

[@tisaconundrum]

Babe! We've been asleep for 7 years!

[@xane256]

Babe: “Honey, he doesn’t even show the distribution of point-point distances”

[@punpcklbw]

Thanks for the research. The uniform placement of points on a sphere was used in quantizing normal vectors used by lighting in computer games. Quake 2 engine has 162 predefined vectors for encoding the normal of a vertex with a single byte (rather than 12 bytes that the brute-force way would take). They are placed in a pattern that resembles a subdivided octahedron. The method on the video could be used to generate any number of equally spaced vectors with ease.

[@jonatan01i]

"it's impossible to place n points on a sphere" So that's why electrons have trouble deciding who goes where around the nuclei

[@Ruktiet]

Lol no. You’re confusing classical models for atoms with quantum mechanical ones.

[@wanmaster11]

maybe your algorithm could be used as a starting point for a guess using calculus to speed up the process for applications requiring greater accuracy?

[@dinhero21]

Isn't that just Fibonacci Spheres?

[@dairop3220]

Looks like them

[@omg_look_behind_you]

Yeah. The uploader also discovered e^iπ - 1 = 0

[@chaosschnitzl7422]

This one modell you made looked very much like this typical pollen. I wonder if there are similar Structures in Nature and how they are organized.

[@RawrBag]

Fibonacci sequence

[@ehrenmurdick]

Nice algorithm. I don't think it would work for the satellite spacing problem though, orbits don't work that way. Orbits must always be geodesics and geosynchronous only works directly above the equator

[@matchamitminze]

Geostationary* orbits can’t have an incline from the equator - geosynchronous orbits can have any inclination, and an inclination of 0 from the equatorial plane is what yields a geostationary orbit. :)

[@phoenixamaranth]

I thought he meant the spacing of the mirrors on the satellite, not the spacing of the satellites

[@nel_tu_]

​@@phoenixamaranthbro didn't watch the whole video

[@CNLohr]

A major benefit of this would be for compressing uniform vectors, i.e. one could dedicate "10 bits" worth of data, and out of that 10 bits, it could get an 3-vector, approximated to the closest vector. I have needed this before and compromised. Now, the real question is can you generalize this to ℝ4, then you could compress quaternions quickly/efficiently.

[@hohuynhquocchuong4925]

Try my spacing: N is number of point counter = -(N-1)/2 ;+1;(N-1)/2; α = (√5 + 1)*π*counter = golden_ratio*2π*counter Z = 2*counter/N R = √(1-Z^2) X = R * cos(α) Y = R * cos(α) P(X,Y,Z) will be N point on unit sphere

[@alisadeghi3156]

Y=R*sin(a)

[@danielmilyutin9914]

I did this with solving physics-like optimization problem. Where points repell each other. And I added viscosity to stop their movement. It was quite ago.

[@Fluoman_]

I think I used your code!

[@danielmilyutin9914]

@@Fluoman_ impossible. I didn't publish it. It was a little hobby in scilab lang.

[@Fluoman_]

@@danielmilyutin9914 damn. Well, somebody had the same idea then.

[@danielmilyutin9914]

@@Fluoman_ Yep. It was quite on surface.

[@robbiekavanagh2802]

Could be good for generating low poly planets for games

[@noahdirksen3623]

high poly aswell, Sebastian Lague tried using an algorithm like this but the problem is that its difficult to predict where the points will end up, so unloading unused vertices is a hassle

[@sus-kupp]

Good methods for generating spherical meshes already exist because it is rarely necessary to fine tune the exact number of vertices in the mesh.

[@TheTurt1e9]

The difficult part of this approach is creating the tris as the points are in a helix sequence. I believe you have to use Delaunay’s Algorithm to produce the result you’re thinking of.

[@profdc9501]

Here's another method based on Geodesic Packings: Design of a spherical focal surface using close-packed relay optics Hui S. Son, Daniel L. Marks, Joonku Hahn, Jungsang Kim, and David J. Brady

[@nartoomeon9378]

I recall, a spiral you used, seems like a method to get perfect flat 2-dimensional net of a 2-sphere. I forget the name, but it is a spiral stripe between poles with infinite small width. Maybe, it has various width in different places, but if they infinitesimal, that not so important.

[@nartoomeon9378]

@@mariokapalka7364 Euler curve yes?

[@TashiRogo]

Ah, the confident infallibility of youth.

[@Troloze]

3:14 isn't it possible to determine a function that gets the most efficient function for each number of points? Or at least for intervals? seems like a better idea than to use a single function for all values.

[@lukepowers8122]

thats what a neural network does

[@TheAlison1456]

1:48 J J Thomson discovered the electron? The first time I ever hear of him is right here in a maths video totally unrelated to that.

[@tuskiomisham]

you seem to be confusing two concepts. One is equally spacing points on the sphere, and two is maximizing the average distance between points on a sphere. technically speaking 5 points on a sphere can be equally spaced. it's not hard just draw Pentagon on the sphere. likewise maximizing the average distance between points on a sphere isn't hard either. there are many algorithms to do this. I can see why mixing the two concepts will make this difficult to do though

[@sydniecollings1821]

But where is the formula???

[@IARRCSim]

Yes, I expected the description to have links to GitHub or part of the video to explain the algorithm in more detail. Too much of the video is discussing applications. Code from www.openprocessing.org/sketch/41142 appears to solve the problem in a similar way. There is also a research paper at arxiv.org/pdf/0912.4540.pdf explaining a similar solution.

[@KamalFansa]

@@IARRCSim great link thank you

[@Miguel_Noether]

The paper is finally out!!!

[@timonix2]

I have been trying to find approximate tiling for spheres for weeks before I found this. This seem like a valid approach. Although in my case the average error is less priorities than the max error and I am guessing that the errors at the poles are far greater than the errors at the equator

[@lucaballarati9694]

I Will be borrowing this for gamedev

[@furkanunsal5814]

and the paper?

[@AySz88]

Look for "A New Computationally Efficient Method for Spacing n Points on a Sphere"

[@thelocalsage]

How does this fair for numerical analysis like approximating surface integrals of spherical functions? currently lebedev quadrature is the norm in non-periodic quantum computational chemistry but those are preset numbers of points. my favorite exchange correlation functionals are sensitive to the integration grid so it’d be cool to find out in very applications what the coarsest grid you can get away with is.

[@rainytreecat3992]

Use it to unfold it into a generally polygonal or circular map that isn't stretched or squished

[@richardandrews573]

I came across this same problem when designing my IEC fusor.

[@arthurheuer]

Another great approximation is creating an ico sphere, with 5 or more subdivisions, in Blender, and measure some segment lengths with the MeasureIt plugin. That seems a lot easier; for most uses, it's also suitable, I imagine.

[@Rav0000000]

The downside is that you can only subsivide in factors of 20, one for each face of the icosahedron. Any odd numbers outside of this cannot be spaced evenly using an icosahedron. The benefit of his method is that it theoretically can have n = any whole number from 1 to infinity.

[@minecraftermad]

golden ratio tho?

[@cyberpersona6267]

Cool. How do you calculate the Error/Efficiency of placement?

[@noahdirksen3623]

Im assuming he finds the average distance between all points, and then compare outliers

[@AySz88]

If 2:58 is correct, it's just the smallest distance between points, normalized by an upper bound calculated from what would be the point's share of the sphere's surface area. The OP also linked to a paper today - look for Kogan, Jonathan (2017) "A New Computationally Efficient Method for Spacing n Points on a Sphere," (Very late reply, I understand, but the alg has chosen this video today)

[@ONRIPRESENCE]

4:24 kinda reminds me of a Mandelbulb fractal hehe

[@JJ-fr2ki]

Anyone figure out how to solve the problem for an N-dimensional sphere?

[@shiv-iwnl8188]

Is it possible to find the nearest point on this sphere to some point p without a lookup table and in constant time?

[@absence9443]

Yes, but it's tedious. You'd translate the point into 3d polar coords and then approximate its deviation from point placement rows and columns with which you get the nearest point

[@тасяколомейцева]

3:23 You simply selected the number 0.1 + 1.2n for this area. Perhaps it would be better to use the Euler function? The unevenness in your method depends precisely on the divisibility en.m.wikipedia.org/wiki/Euler%27s_totient_function I did the cool job❤

[@imsatoboi]

Some people in the comments seem to think that ‘math’ is invented and people hold some sort of copyright over it. Guys , i get that someone might’ve figured it out before others , and thats an astonishing feat , but people who figure the same thing out themselves are not in anway inferior. I failed at maths fr. So i can be completely wrong. But lets just take a breath and enjoy the beauty in the process and how us humans are soo freakin cool. Peace.

[@phoenixamaranth]

And most of them are ignorant of the point of his video and paper: that his method solves the problem in a faster way. He didn't claim to be the first to solve the problem. His point was he came up with a fast algorithm to solve the problem

[@thederpydude2088]

Does this algorithm not find a better way to space 5 points?

[@HappyMathDad]

Google must have started using a new method for efficiently spacing points on a sphere, for video recommendations.

[@culan_SCP]

"computationally efficient" also shows the laggiest simulation ive ever seen (no hate, i love the math and video)

[@decycle2912]

i dont see the reason of using this method over fibonacci spheres

[@omg_look_behind_you]

Apples and oranges

[@MetaBuddha]

tbh.. Solid Work, man 🎉

[@phpn99]

Have you ever heard of quaternions ?

[@TearonQ]

comment made 1 day ago video made 7 years ago hmm

[@automatescellulaires8543]

@@TearonQ Hi there. I guess youtube algorithm decided we should watch this now. I wish i could have seen it 7 years ago.

[@TearonQ]

@@automatescellulaires8543 lol

[@tuskiomisham]

​@@TearonQ dude just gave a 1 day old comment a heart. he's chillin.

[@user-pr6ed3ri2k]

0:23 I just tried it myself, and I managed to equally space the points with the numbers 1, 2, 3, 4, 8, 12, and 20?

[@user-pr6ed3ri2k]

0:50 oh yeah I forgot about 6, but the others should work right?

[@user-pr6ed3ri2k]

1:58 buckyball numbers

[@user-pr6ed3ri2k]

^(60 has a somewhat solution too)

[@matchamitminze]

Well sure, you got all of those points because (aside from 2 and 3) those numbers correlate to the number of vertices of Platonic solids. 4 vertices for a tetrahedron, 6 for an octohedron, 8 for a cube, 12 for an icosahedron, and 20 for a dodecahedron.

[@user-pr6ed3ri2k]

@@matchamitminze yeah that's precisely how I got them 14 should somewhat work too (rhombic dodecahedron) Idk what would happen if I included Archimedean solids since I have no clue what they do

[@randymess7665]

But n still can’t equal 5?

[@BoxEnjoyer]

Man, the people in the comments here are reminding me why I stay away from of a lot of the math community. There's just the general sense of elitism if you go outside of the mainstream parts.

[@jbb4019]

It’s cool numbers are all they have 😂😂😂😂😂😂

[@colonelgraff9198]

4:49 Darth Vader and Moff Tarkin would like a word with you

[@djprisma847]

Dont sceam at me !!! 2:01

[@philip2205]

Just turn down the volume yourself. For me, the volume was no problem.

[@anastasiaklyuch2746]

3 point is just as problematic as 5, since they don't divide the sphere in all dimentions. 2 create hemispheres, so that's fine, but 3 are bad.

[@AdrianBoyko]

1 point is maximally asymmetric

[@trolleyproblem4318]

How is this related with sphere subdivision? Easy solution is to start with some of Platonic.

[@alexwang982]

this solves my chem problems

[@BuZzKiiLLA]

Very interesting. I was curious about this question based on something I am doing. I came up with the same result by different means and simply in my head and recreated it in blender. You said something about the poles and stopping it before it gets to the poles? what does this mean. also just out of curiosity. why does n=1.2. Without any effort, i am answering many problems that are in the world today and they are all related to the same solution. I am good at seeing patterns in things. And the fact that n=1.2 was interesting to me but it might be nothing at all hence why I ask.

[@arnabghosh3457]

link the paper or the article

[@RobleViejo]

HEY! This is genius. Literally genius. I was wondering. Does someone got a low N grid placed over Earth's Surface? It would be VERY helpful for my research.

[@rider2fois]

Nice cooking lecture

[@kenjohnson6101]

Can this be generalized to N dimensions?

[@swig.]

This could have cool application in earth orbit satellite configuration for like telecommunications. Equal spread means most consistent signal on earth for most area.

[@_ahsiA]

i think high speed lower accuracy isn't how you should approach this problem(not like i'm competent though). Isn't it better to let some supercomputer spend a bit of time to calculate these extremely costly and hard to change once done things?

[@JasminUwU]

​@@_ahsiAyou wouldn't even need a supercomputer, we don't have a trillion satellites

[@dimm__]

yet

[@debblez]

imagine being so delusional you think you invented fibonacci spheres

[@miron__]

Dr Wolfram did allow it to be added to his "Mathematica" language, so it can't have been that delusional

[@debblez]

@@miron__ yes. they implemented fibonacci spheres. has nothing to do with this bozo

[@debblez]

@@miron__ wolfram adding fibonacci spheres has nothing to do with this guy

[@PokeNebula]

I looked at the mathematica documentation and changelog and i see no attribution to this author. Is this video just lying about contributing to Mathematica? If not, and and this algorithm is simply just a subset or similar algorithm to fibonacci spheres, why didnt the author mention them at all? And why is this video getting a bunch of views all of a sudden?

[@debblez]

@@PokeNebula smart idea. And yes this is just a blatant lie. Although I get the impression this guy wasnt trying to trick anybody, he’s just clueless

[@juha-petrityrkko3771]

How does the even distribution help with the satellites, as they are constantly moving? We would need to prove that they stay at least near this optimal distribution.

[@honkhonk8009]

idk he just saying shit ig lmfao

[@phoenixamaranth]

Even when orbiting the satellites need spacing for optimal line of sight and distribution. We setup satellite networks now that are all about being spaced over optimal distances from each other

[@Xayuap]

you can't comb a furry ball without curls

[@ciniss]

How do you create faces tho

[@honkhonk8009]

yo why am I getting recommended this lmfao I fucking hate discrete math type shit.

[@ctrlaltdebug]

This will be great for round earth maps in games. No more flat earth!

[@menjolno]

youtube didn;''t just recommend me once, it was three times already

[@ulysseyang9733]

This video is stolen, there is another video that is created by someone else named the same and it is longer and has a description.

[@Zahlenteufel1]

Link or didn't happen.

[@monoplexbob]

Interesting math? Yes. Actually applicable to the real world? probably not. For the use case of where to put satellites in orbit. The heuristic doesn't consider weights of satellites near the poles being useless because nobody lives there. Also several hours of computer time to find an iterative optimal solution is much cheaper than a single rocket launch. Again, cool math but not a problem that needed solving.

[@phoenixamaranth]

We have satellites that orbit over the poles already and line of site to a satellite doesn't mean you live directly under it. And it's pretty presumptuous to assume it wasn't a problem that needed solving. History is full of examples where edge case mathematics has turned out to have practical real world applications or advantages found years and decades later. His method is to improve calculation efficiency vs other known methods of solving the same problem. Embedded systems and near real-time applications always benefit from faster algorithms.

[@joshuawhitworth6456]

I figured out the math to Holographic Waves.... Perhaps you might find it useful... Math to Holographic Wave Lengths 1÷(1÷Y×Z+Y)=A,B,C,...÷(A+B+C+...)=1 These are the rules.... Solve for every whole number less than Y. Then add them together Y = any whole number Z = any whole number less than Y including zero. A,B, and C are the various numbers you end up with for each whole number less than Y. Here's an example. Keep in mind I cut the numbers short do to them being infinitely long.... 1÷(1÷3×0+3)= 0.33333 1÷(1÷3×1+3)= 0.30000 1÷(1÷3×2+3)= 0.27272 0.33333+0.30000+0.27272=0.90606 0.33333÷0.90606=0.36789 0.30000÷0.90606=0.33110 0.27272÷0.90606=0.30100 When you add them together they should equal one... 0.36789+0.33110+0.30100= 1

[@ziyadcodes]

sheeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeesh. I 100% understand this. How did I get here 😭. I was just trying to make a normal sphere 😭

[@resunoon9994]

Omg, amazing

[@dragonsagesummoner6071]

What language is that? It’s not Java or c#. It is python?

[@hgilbert]

First I thought it was Prolog. But just checked Wolfram's site. Looks like it's Mathematica. During Lockdown I was trying to teach myself but gave up. 1 month free trial was over.

[@ngc-fo5te]

It's Mathematica

[@BenRasmussen-c3u]

very cool, great job.

[@tb9k_]

This will be perfect for Project BOFA.

[@jowillll]

2017 is 7 years ago? 😦

[@phoenixamaranth]

Right? Doesn't that feel crazy?

[@joshuawhitworth6456]

This is awesome! Cudos.

[@Miguel_Noether]

Where is this guy now? 🤔

[@asheep7797]

Seemingly still alive.

[@anastasiaklyuch2746]

Not applicable with sattelites, since you don't need them over the whole poles or the whole ocean. It really depends on what is actually needed, but other uses are cool

[@phoenixamaranth]

Why does everyone keep missing that he was talking about the mirror spacing on the satellite not the spacing of satellites?

[@anastasiaklyuch2746]

@@phoenixamaranth I wasn't talking about the student satelite, but the optimal satellite placement at 4:50 plz watch more carefully before making such comments about everyone.

[@phoenixamaranth]

@@anastasiaklyuch2746 Fair enough, my bad. I will point out we do run satellites over poles, oceans, etc. We run vertical satellites that orbit from pole to pole

[@anastasiaklyuch2746]

@@phoenixamaranth Yeah, that does makes sence :) This also adds complexity of sattelite motion to how they are positioned, big stuff.

[@ofirshelly3606]

so helpful kogan master

[@Bestofchatgpt]

I wanna play with the code

[@squeakybunny2776]

Sorry, I can't trust someone who is comfortable with mathematica

Excelent !!

[@rabbitkinggod4602]

Looks like someone's math paper they went to turn in for a class by KZbin, but forgot to remove the video after getting and F in the class.

[@tomoki-v6o]

good for starlink

[@xaf15001]

God 2017 is 7 years ago. Fuck

[@AJMansfield1]

Wow you really need a pop filter on your mic.

[@patty4449]

Not very accurate, think simpler and outside of the plane... The optimum placement is simple if you split the concept to a dot with rays and the direct orientation sequence to pi... Takes way less calculation and works on non spherical objects as well

[@pablomirandaandrade3712]

GREAT!

[@VeteranVandal]

#blessed_by_the_algorithm

[@jonathankogan1825]

scholar.rose-hulman.edu/rhumj/vol18/iss2/5/

[@DeviRuto]

this is the kinda thing that genetic programming would be pretty good at

[@estantaya]

time to steal

[@Kyoz]

🤍