Extending Ulam Beyond Primes

Originally made 5 June 2017
Produced with HitFilm Express
-
Unfortunately I don't have the picture I mentioned in the video anymore, but I'm sure if you understood this video then you can find out how to generate the lists yourself (and maybe even more than I did here...)

[@dcterr1]

Another very intriguing video! I have fairly good programming skills, so I'm sure I can write code to perform the constructions you suggest near the end of the video, which I will try to do. Who knows where this may lead!

[@sehr.geheim]

this is amazing, I was extremely shocked when I saw the video had only 350 views. Definitely gonna post this in my math whatsapp group

[@Israel2.3.2]

I enjoyed this.Reminded me of the early days just staring at tables of numbers trying to understand the cosmos. 3:43 The reason that successive box/fence entries have differences of 4 in the even case and fail to have differences of 4 in the odd case is an easy consequence of prime factorization and properties of finite difference equations. Even squares have the form 4×1, 4×4, 4×9, 4×16, etc. After division this yields box/fence sequence in the even case as 4×1, 4×2, 4×3, 4×4, etc. Hence we have a sequence whose differences are constant with value 4. In order for a sequence to have constant difference 4 it is necessary that the nth term have the linear form 4n + c for some constant c. Forming box/fence in the odd case yields (2n-1)×(2n-1)÷n for the general term. Equating this with the desired form 4n + c and simplifying yields n(c+4) = 1 But n is a variable quantity while c is constant so this identity cannot be satisfied in general, hence we cannot have divisibility by 4 in the odd case. 3:58 The modified sequence replaces odd squares with odd squares shifted down by 1. If a number is odd it is of the form 4n + 1 or 4n + 3. So the odd squares have the forms 16nn + 8n + 1 and 16nn + 24n + 9 That is 4kn + 1 and 4sn + 9 for integers k = 4n + 2 and s = 4n + 6 Looking closely at these forms shows that odd squares are always 1 more than a multiple of 4. Damn that took too long to type. This fact about odd squares is an elementary but important result in elementary number theory and is easily proved using the language of congruences. 5:05 the constant differences of 8 given the slope 2 sequence 13, 58, 135, etc. can be shown by noticing that these values are displaced from the corners of the square in a regular manner, namely 13 = 4×4 - 3×1, 58 = 8×8 - 3×2, 135 = 12×12 - 3×3, etc. The general form of the nth value is 4n×4n - 3×n Forming the box/fence quotient gives the sequence 13÷2, 58÷4, 135÷6, etc Or 4×2 - 3÷2, 8×2 - 3÷2, 12×2 - 3÷2, etc. A sequence which has the general term 4n×2 - 3÷2 Or 8n - 1.5 For the general term in the box/fence sequence. Taking finite differences yields the constant difference 8 as desired. Now for your interpolation. If in our general term 4n×4n - 3n we substitute 4n + 2 for 4n then divide by 2n + 1 we get the box/fence sequence of the intermediate terms which have the form (16nn + 13n + 2.5) / (2n + 1) = 8n + 2.5 This lies between 8n - 1.5 and 8(n+1) - 1.5 in our interpolated box/fence sequence. Taking differences yields 4 as a constant difference. You asked whether sequences of larger slope will exhibit constant differences. The answer is yes. Every such sequence corresponding to slope m will have the form 2mn×2mn - (m+1)n for the nth term To get box/frame sequence we divide by frame mn which yields 4mn - (m+1)/m This is a linear sequence in n so that its constant difference is 4m. For slope 3 constant difference is 12, for slope 4 its 16, etc. One can derive your interpolation formulas with similar substitutions.

[@maxwellgrossman]

I may have a way to find the exact parametric equation for those shapes traced in the spiral, I’ll get back if I find anything

[@maxwellgrossman]

Ok cool so here it is, its a pretty messy equation, but it has a lot of floor function stuff going on so you could break it into piecewise segments and evaluate them each individually for some cleaner expressions, but for now here's the parametric equation for the collection of curves you're getting there www.desmos.com/calculator/pycor396sv lmk what you think lol :)

[@RickyMud]

@@maxwellgrossman that’s bad ass I wish I could use Desmos the way you do

[@maxwellgrossman]

@@RickyMud Lol thank u dawg, I bet you easily will be able to, because for me at least I think it's just a matter of experience that got me to where I am.

[@user-rm5rz2zh8k]

@@maxwellgrossman awsm dude! Finally a commenter on youtube who comments related to the video and boy is it helpful!

[@msthurnell]

It’s a problem of trying to sort out what is an artefact of the square grid. Obviously Ulam had square grid paper to doodle on as opposed to circular triangular or hexagonal grid. Using this kind of grid each successive square shell adds four squares connected by only a point, 4x0+4=4, 4x2+4=12, 4x4+4=20 and so on using the sequence. 0,2,4,6,8…. Where the added 4 represents a square connected to the next level in by only one point not lines. Commencing the Ulam-like spiral by using one diagonal pawn capture-like move produces the same result as the traditional spiral except the same numbers line up horizontally and vertically instead of on diagonals. If one continues take a diagonal diversion at the end of each completed circuit then interesting patterns emerge. If you decide when each square circuit is complete you will exit sideways then keep going around in the reverse direction until the completion of that circuit (play reversi) then again you will get some interesting diagonal alignment of the even square numbers, odd squares scattered symmetrically and some strings of primes. It would seem that whatever process you use to number the 100 squares of 10x10 grids some interesting alignments happen. There is probably a finite number of ways to cover 100 squares with a continuous string of numbers 1-100. There’s a way to get a four chevron design of of and even numbers but I’ve temporarily forgotten how. It’s in my notes. Google hasn’t forgotten me tho and is now reminding me about a workshop I ran on this topic in 2013

[@chrisg3030]

I experimented with Ulam type number spirals on a hexagonal lattice instead of a square one. If you start with 7 at the centre, every subsequent odd number n lies on the same straight row of hexagons as n²

[@etc.3062]

amazing video!!! can’t believe you did so much on just pencil and paper

[@CertifiedSkank]

This is so brilliant because anybody could have figured this out, but they didn’t. You did,

[@osiand9328]

Amazing video! This deserves way more than 3k views

[@chomastiarnoldo1892]

Why do few views? Excellently done!

[@luker.6967]

This is awesome!

[@olbluelips]

Beautiful!

[@nothj]

You deserve more.

[@ricardospengler2477]

It looks like a pyramid, and the center is the top of it!

[@DustinGunnells]

You are AWESOME!

[@ivanlott3515]

here's just a mark that i was the 475'th subscriber. i *know* this channel is gonna get big. these videos are a lot like 3 blue 1 brown, but more relaxed and better shows the development of concepts to better understand the problems.

[@TheOneMaddin]

I think the distribution of square numbers in the Ulam spiral is pretty well understood.

[@swift3564]

8:10 Looks like kirby

[@pezcadron6364]

Ok, now i can't unsee it

[@jursamaj]

Pick any starting square A. Now pick any adjacent square B (horizontal, vertical, or diagonal) where either B-A≧4 or B is in the next fence. Now form the quadratic 4N^2+(B-A-4)N+A. For N=0,1,2,…, this quadratic will give you the sequence following the ray A→B. The 4N^2 is why your box/fence numbers increase by 4. Indeed, forming a ray that skips thru the lattice at any angle will yield a sequence following a quadratic pattern.

[@richarddeese1991]

Thanks. As to the program, I think I'd create an array table, which contains 3 parameters: the number in each box, and its x & y coordinates. You should be able to use that to join the idea of drawing lines between the centers of various boxes with the numbers in those boxes. It should also allow you to change the rules governing which boxes are connected. For extra credit, try a program that prompts you on which relation(s) you want to connect, then carries it out. Let me know how it goes! tavi.

[@user-ex8dk3ic3x]

I did the Ulam Spiral with the Fibonacci sequence interesting results :)

[@ThePinkPhink]

we really seem to be working on same ideas.. technically all this visualizations might be done very easy in AutoCAD using the LISP language. i have made some routines that generates this sort of progressions (also l-system patterns, kolakoski sequence patterns and so on), and i guess its good idea to co-operate.

[@dcterr1]

I believe that some of the results you show in this video are very easy to prove. For instance, intersecting the continuous Ulam spiral with almost any ray starting at the origin is easily shown to yield a quadratic sequence.

[@graemeirvine7060]

Is there such a thing as a negative Ulam spiral that starts off to the left? Would it make any difference?

[@AshWedFalcon]

I think that 3-d placement of numbers could get us more interesting insights. I would use a double cone, half for positive and half for negative integers. Above the center (number 0), I would put a first "ring" with 1 and 2, then a second one with 3, 4 and 5, increasing ring size by one. Of course, I would use a spiral and not rings, so all real numbers could fit in-between...

[@mateuszkarbowniczek4518]

Python is easy to learn programing language that you can use for that purpose.

[@dvfantail]

Seconded!

[@mattikemppinen6750]

I've used Matlab for exploring this because it has builtin tools to transform numeric data into images, I guess some Python addition (maybe NumPy?) has similar tools.

[@ZaqZiemba]

Great video! And I’m confident that there are packages to create Ulam point spirals in python... but thankfully I don’t have to look due to the gentleman who created the GitHub program. Oh well!

[@ghastlymicrowave3207]

I wonder how effective it would be if you refered to a single point on a coordinate plane as a single number in code using this method. Probably not very but it's a interesting thought.

[@robertmiller5258]

Can you extend it to the complex plane?

[@carlowood9834]

That's what I also thought.. that is, I think there is something when you look at matrix as complex numbers (vertical being imaginary). However, not using the Ulam spiral, but one of the other representations where you see the primes lay on parabola.

[@hillaryclinton2415]

What happens when you map the primes?

[@JusSomeGuyOnInternet]

there are other videos on KZbin about the ulam grid that answer your question. give it a search

[@PULSAR-yb2nb]

nice

[@MrRyanroberson1]

3:40 ah yes you discovered (2x)^2/x = 4x, amd that the consecutive difference is 4. And also 4x + 4 + 2/x is not 4x

[@feridunabi7723]

i strongly recommend you check out the video on the ulam spiral by 3blue1brown

[@jursamaj]

I don't recall any 3b1b video on the Ulam spiral. Any idea which one?

[@feridunabi7723]

@@jursamaj kzbin.info/www/bejne/e3yWY52lbM5ogrM

[@jursamaj]

@@feridunabi7723 I kinda suspected that was the video you meant. Of course, it doesn't involve the Ulam spiral at all, so you're just wrong. It's a good video, but not related to this one.

[@feridunabi7723]

@@jursamaj are you a moron

[@thomasstewart9752]

7:58 It's kirby.

[@Nellak2011]

I actually know how to program a solution to that. I was just learning about how to program in Javascript using a framework called P5. P5 is a framework for artists, and it would make short work of this drawing problem.

[@thevikingwarrior]

Now play with logarithms!

[@WildAlchemicalSpirit]

I wonder if the Egyptians or Mesoamerican people had an understanding of this... maybe they used it when they were building the pyramids... 🤔

[@carlowood9834]

Wait, did I it right that you study math? :( Try drawing squares around 0...n^2-1 .. And then just forget about the spiral/image and work with bloody quadratic polynomials.

[@olehkinash9562]

I am surprise that you write points on paper instead of using some software. That is uncommon these days. Just use python to draw it or some other programing language

[@mattikemppinen6750]

why choose when you can do both?

[@NonTwinBrothers]

Noooo he don't have the programming skills