90,525,801,730 Cannon Balls - Numberphile

[@iabervon]

What's the smallest positive integer that has never been Matt Parker's favorite number?

[@numberphile]

Whatever it is will be his new favourite!

[@tracyh5751]

No numbers are boring.

[@ssdd9911]

1

[@dlevi67]

We can put a higher bound on that at 2, since Matt calls it "a sub-prime", which ain't nice. That doesn't leave many contenders.

[@timclark2880]

90,525,801,731

[@xaiano794]

"I don't think anyone else has ever bothered doing this" - Mathematicians in a nutshell.

[@JosephDalrymple]

Hahaha pretty much. The vast majority of a lot of these videos result in two expressions from me, almost simultaneously: (1) that's absolutely fascinating, and (2) why on earth would anybody even think of doing that? [now how, mind you: why] 😂 I ask myself the same questions every time I conclude my work and discover the answer to one of my engineering problems. It starts out the same and ends with the same questions.

[@Triantalex]

false..

[@davidross7467]

Eagerly awaiting the follow up video in which Professor Eisenbud constructs the 31,265-gon with just ruler and compass

[@numberphile]

Good Numberphile knowledge.

[@AlisterCountel]

David Ross Sadly, the 31,265-gon is not constructible. One may draw one with a relatively high degree of accuracy though, using just a compass, no straightedge needed!

[@baijokull]

@@AlisterCountel Sure it is, just bring me 90,525,801,730 cannonballs!

[@bledathehun328]

And Carlo Séguin to 3D print it.

[@abd.137]

@@AlisterCountel I see what you did there.

[@jekyllgaming99]

The 31,265-sided polygon, also known as the Triamyriahenachiliadihectahexacontakaipentagon.

[@Sylocat]

Or the Parker Circle.

[@hamiltonianpathondodecahed5236]

~Felt like the name of some chemical compound~ Sorry

[@wizard-pirate]

Thanks for answering the dangling question of "What is the actual name for the 31265-agon?" I like Parker's Polygon, if not just because it's easier to remember.

[@이규헌-s3c]

Googolplex, also known as 'Ten Triacontrectricentitriadecatriavecitriacentitriadecitriaxonitriacentitriadecitriayoctatriacentitriadecatriazeptitriacentitriadecatriaattitriacentitriadecatriafemtitriacentitriadecatriapicitriacentitriadecatriananitriacentitriadecatriamicritriacentitriadecitremillinilliduotrigintatrecentillion'. (Note: this might be inaccurate, because it also feels like 9.1×10^(10^99).)

[@Proto_Divi]

medial omnicircumfacetopental triakishecatonicosachoron mommy

[@nataliekanakova9496]

You can see it in his eyes how excited he is about his discovery :)

[@ericstoverink6579]

Because he thinks this will make us forget about Parker squares.

[@richardtickler8555]

@@ericstoverink6579 maybe it turns out to be a parker pyramid

[@Nemelis0]

@@ericstoverink6579 No Parker Squares are unforgettable, but this will tell people that there are 31263 more sides to Parker than just the 2 in a square

[@macswanton9622]

@The Idiot Reviewer If it was a hat it would barely fit

[@janschmeink9296]

it's so cute right

[@jonthecomposer]

The Parker Conjecture: The amount of non-useful numerical discoveries will always be greater than the rate at which Matt Parker can discover them.

[@peppybocan]

well, non-useful. What he does is - searching for solutions of non-linear diophantine equations, a tricky task to do!

[@Tehcarp]

a modern day Tristram shandy

[@Veggie13]

You can't compare a number to a rate, silly.

[@jonthecomposer]

@@Veggie13 I do what I want! You don't know me! Wait, wut? lol It's all in fun, man. I say it because it's ridiculous. Nothing more, nothing less. :)

[@randomdude9135]

@@Veggie13 I was about to type just that.😂.

[@johnchessant3012]

Fun fact: A hyper-pyramid with a cube "base" can always be arranged into a square, no matter the height of the hyper-pyramid. This follows from the amazing fact that 1^3 + 2^3 + ... + n^3 = (1 + 2 + ... + n)^2.

[@NAMEhzj]

Thats very cool!

[@TheRealFlenuan]

But then the question is: Which can be arranged into a cube?

[@alexanderehrentraut4493]

@@TheRealFlenuan I don't think there are any.

[@witherfly5811]

Just as any 3-D Pyramide can be arranged into a Line. So nothing special.

[@pleaseenteraname4824]

Withi Nah, this is cooler

[@codesimpson6010]

"And so that's when 4,900 stopped being my favorite number, and I upgraded to just over Ninety billion!" - Matt Parker, 2019

[@j.hawkins8779]

LOL😂

[@ramonalainmirandaquintana6515]

"Is not useful, but I love it". Denotes both great appreciation for mathematical beauty, and terrible parenting skills :)

[@Erichwanh]

Please pin this comment, hahaha

[@maxnullifidian]

It may not be useful yet, but who knows what the future holds? It may someday be found to answer some critical question that hasn't even been asked yet!

[@tqnohe]

Describes my son Michael.

[@beauwilliamson3628]

@@maxnullifidian Like: What's the coolest way to stack the 90,525,801,730 plasma torpedos I've stockpiled for my invasion of the Galactic Empire?

[@maxnullifidian]

@@beauwilliamson3628 See, it's useful already! LOL

[@bookslug2919]

I thought this was going to be good but it's just a load of balls

[@zippy-zappa-zeppo-zorba-etc]

*Clap* *Clap* *Clap*

[@Danilego]

@@zippy-zappa-zeppo-zorba-etc *MEME* *REVIEW*

[@Kris.G]

ba dum tssss

[@fafnir242]

I applaud you.

[@vipcesh]

Hey! I thought he was really ballsy!

[@KurtRichterCISSP]

The 31,265-gon is one of my favorite shapes.

[@eternalreign2313]

It looks more and more like a circle the further you move away from it. In fact, even if you're right next to it you probably can't tell where one of the corners is.

[@KurtRichterCISSP]

@@eternalreign2313 depends how big it is relative to me...

[@therabbits69]

@@KurtRichterCISSP People should figure out how big it actually would be IRL using the average canon ball.

[@markreynolds1436]

@@eternalreign2313 I don't believe it's round. It's probably a sphere or something mad.

[@bertblankenstein3738]

Maybe a 31415-agon could be used in calculating pi. Can a 31265-agon be constructed?

[@unclvinny]

Brady, your animator is so good! They were really working overtime on this one.

[@spaceatlantis3504]

Yeah, his name is Pete McPartlan

[@iggusify]

"Matt, seriously, I'm proud of you!" -- That's beautiful...

[@Mr.E-Bachs]

I love Matt’s self-aware pause, “with enough... spare time and a laptop.” And that is why we keep coming back for more.

[@asailijhijr]

Something that nobody was looking for but that is genuinely impressive, a Parker Pyramid.

[@sophiegrey9576]

Or a Parker Cone.

[@spinter1310]

“That’s not useful”, said every mathematician about his discoveries a few decades or centuries before it is integral to revolutionary technology.

[@Kylora2112]

Now: Matt: "I found this useless object and it's fun and cute!" 3019: "Matt's discovery lead to the Unified Theory Equation and is the key to intergalactic and interdimensional travel. Human civilization is now based on 90525801730."

[@KuraIthys]

Meanwhile, think of all the people that aren't mathematicians that randomly mess around with numbers sometimes (like me say) come up with something weird, look at it, have no idea what it is, and then forget they ever did it... Hopefully I've never 'discovered' anything new or useful, because if I have I've since forgotten again. XD For that matter, I ran into a youtuber recently that claimed to have 'invented' something that I had made a version of more than 15 years ago. They presented it as some great amazing breakthrough, and 90% of their audience agreed, meanwhile I did it randomly, looked at it and decided 'this is so obvious I'm sure there's been thousands of people before me that have come up with the same thing.', and then just ignored it and more or less forgot I did it until I saw someone else do it. So... Did I invent something unique and not realise the significance? Or was my assessment that it was a super-obvious solution that's probably been done a million times before correct and this guy on youtube is just full of himself? Either way, you never know. What seems trivial to you may turn out to have been an amazing discovery and/or invention, which is lost forever because not even it's creator remembers it, simply because it didn't seem particularly important or impressive to them. XD

[@wolfson109]

@@KuraIthys now I really want to know what it was.

[@Xeridanus]

@@wolfson109 Same

[@petros_adamopoulos]

Not this one, trust me.

[@cooloutcoexist]

Takes a lot of balls to search for this number.

[@clayz1]

Thats probably enough cannonballs to do the job. He’s a agon-er.

[@sb-jo2ch]

Natural number: exists Matt Parker: It's free real estate

[@HonkeyKongLive]

My favorite part of this video is actually that I had to do some thinking about how "pentagonal" and "hexagonal" numbers are formed because at first the shapes seem wrong since their visual isn't like "pizza slices" that meet in the shape's center but rather wedges that meet at the top. Had to look that up and saw that any -gon numbers are constructed by making a "1x1" version of the shape and then building new layers around it anchored at one corner, leading to this rather unusual appearance.

[@SpySappingMyKeyboard]

There's one! Well, I didn't look any further, but there's only one! Parker proof?

[@pleaseenteraname4824]

Proof by exhaustion, i.e. I was too tired to look any further

[@maximiliand2180]

Person: "Hey Matt what is your favorite number?" Matt: "Yes"

[@unitrader403]

Computer: Error, encountered -NaN, expected Number

[@imveryangryitsnotbutter]

I agree, what is a very cool number.

[@BeeCeeJay]

I think this is my favorite video on your channel. Matt’s curiosity, skill, and sheer joy are all turned up to 11. Perfection.

[@michaelzimmermann3388]

Usefulness lies in the eyes of the spectator. I love your discovery, it is nothing short of amazing

[@pipolwes000]

"4,900 is the only number which can be both a square and a square base pyramid" Counterexample: 1.

[@dcs_0]

Tut tut. You forgot 0

[@SpaceboyBilliards]

How is a single sphere a square

[@cemerson]

@@SpaceboyBilliards How are any number of spheres a square?

[@Tfin]

A sphere is a square because you're only considering the packing of that number of spheres. You have a square tray with sides of length x(n). How many spheres of diameter n fit on it? But 1 is trivial, and works for all numbers.

[@SpaceboyBilliards]

@@cemerson 4 spheres is a square because the outline of them is a square shape. 1 sphere has the outline of a circle.

[@raghavproach2444]

I wish I had such a maths teacher, I have fallen in love with maths all over again. Would love to see more such cool stuff from Matt. Also, someone said this isn't a useful number, I think it is - it is fascinating and exciting, such that it pulls you towards math.

[@adelarscheidt]

How exactly do you arrange spheres on a pentagon surface?... I mean what rule(s) do you follow?

[@gadrill4285]

1 sphere = 1 unit

[@Theraot]

You melt them and pour them in a pentagon mold

[@theznayx808]

I think the idea is maximum packing. If you check that scene where they showed all the layers of the pentagonal one, you see there's big blank spaces inside.

[@debblez]

For height 1: Pentagon size 1 (just one sphere) For height 2: Pentagon sizes 1+2(total 6 spheres) For height 3: Pentagon sizes 1+2+3 (total 16 spheres) And so on

[@ig2d]

I am wondering the same thing. Can this problem be clearly defined...

[@NUGGet-3562]

I love how proud he is of his numbers. "And I found it first!"

[@betabenja]

1919: I thought about it and proved it in concept for infinity cases using a clever logical argument 2019: I brute force found one solution, and I didn't even do the calculations. who knows, eh?

[@ElektrykFlaaj]

well, we use bruteforce to calculate something 1919 mathematicians couldnt do analytically

[@GeodesicBruh]

betabenja oof haha

[@gajbooks]

I think it's more of "Is there anything within the realm of sanity that works, and can I find it without huge effort, because it's not exactly a useful answer".

[@adamsbja]

@@ElektrykFlaaj If they'd added a 1920th mathematician I bet they could've figured it out.

[@ElektrykFlaaj]

@@adamsbja imagine the possibilities with 1921 mathematicians

[@TeodorMusic]

Did I miss something or why do you ignore triangle base pyramid?

[@pedrolmlkzk]

Tetraedron

[@philipyao5989]

Not sure with squares but the formula is n(n+1)(n+2)/6

[@poofishgaming5622]

For 3 it’s 10, 120, 1540,

[@matthewstuckenbruck5834]

He has it in the description

[@quarkonium3795]

@@poofishgaming5622 and 7140

[@timsullivan4566]

That's nice - I've always thought 90,525,801,730 deserved some recognition. His factors must be so proud!

[@levengli]

Discovery for the sake of discovery. The beauty of pure mathematics. Well done, Matt!

[@nathanderhake839]

0:25 WHAT DO YOU HAVE AGAINST COMPLEX NUMBERS???!!

[@Ken.-]

The computer was probably thinking, "I must be doing something really important!" ....And that's why the AI computers will kill all of the humans.

[@bookslug2919]

Nope, they will just make our lifespans more 'efficient'.

[@TheRealFlenuan]

Let's hope

[@akaelalias1113]

At 0:34 Matt says that his club of favourite numbers is finite, but at 0:27 the graphic shows that all real numbers are a subset of his favourites!

[@arkantyne7122]

"What's your favourite number Matt?" "Oh, err, somewhere between, err, 90 billion, or was it 80? Maybe 85..."

[@Galileosays]

For the tetrahedrons, triangle based pyramid, there are 4 cases: side 3, 8, 20, and 34 are equal to triangles with sides of 4, 15, 55 and 119.

[@krumuvecis]

Hey, what about a triangle? Seems you missed the simplest polygon

[@SSGranor]

10. You can do it with 10.

[@krumuvecis]

@@SSGranor Ok, that's one solution. Are there any others?

[@PhilBagels]

Yeah. Why did he skip over triangles? One of the ways you can actually stack cannonballs, as opposed to octagons and 31,265-gons. 10 is the first and easiest one to find. I suspect there are others, but I don't remember, and I don't have time to look for them now.

[@krumuvecis]

Apparently there are only 5 solutions: 1; 10; 120; 1540; 7140

[@ffggddss]

@@krumuvecis And don't forget 0. Fred

[@to2podemosaprender630]

I like this guy... He's always happy to find this numbers, I feel like all these numbers are going to be useful one day... I used to think like that..

[@SlyMaelstrom]

"So the people whose names you're seeing on screen at the moment... these are all the people that found the number before Matt Parker."

[@numberphile]

Ha ha.

[@blackraven114]

There's an interesting infinite family of solutions. For any "k-agon" where k=3a+2 for some integer a, you can construct both a k-agon with side length 3a^3-3a+1 and k-agon based pyramid of base length 3a^2-2. Both will use (3a^3-3a+1)*(9a^2(a^2-1)+2)/2 "cannonballs". The k=5 case gives the trivial solution of a side length of 1.

[@alonamaloh]

I found this too! (a couple of days after you :) ).

[@tetsuoumezawa5833]

0:37 remember when matt had hair

[@andrewkepert923]

h=hair, f=favourite number. d/dt (hf) = 0

[@johnchancey3941]

At 6:42, Brady sounds like a proud father whose child did something somewhat strange but surprisingly creative

[@julienhau999]

I cannot imagine how you can stack cannoballs for higher base polygons I thought you could only make triangular base pyramids (tetrahedron) and square base pyramids only PLEASE respond matt

[@Ellyerre]

They had a note in the video at 3:35 that said that those pyramids wouldn't necessarily stack in the real world.

[@kevinslater4126]

Of course you can! With enough time, expendable labor, and glue

[@reidtrevar]

My list of favorite Numberphile videos is large, but finite. This is my current favorite.

[@harrypounds456]

i thought this was a genuinely cool discovery, well done!

[@guepardiez]

I don't understand the point of this. Only the outer layer of those polygons is regular. How are the inner layers formed?

[@jokusekovaan]

Right, some things weren't explained at all. Is this something like the least surface area where the surface is 'stuck' like oranges in a cardboard box, and each round thing above the previous layer touch at least 3 things below in a stabile way?

[@Koisheep]

The video says such forms may not be possible in real life, they are made up

[@kevinslater4126]

There’s a previous video about stacking cannonballs

[@TheRealFlenuan]

“Is this what it took?” 😂😂😂

[@SAAAMTV]

This. is. absurdly. fascinating. I've the same enthusiasm as numbers as Matt, but none of the coding knowledge to realise these sorts of things. Thanks for doing these!

[@beeell8017]

Is there a relationship between the 1k and 5k hexagon stacks? That could predict the next size?

[@Vaaaaadim]

If so, I think the solutions would be a recurrence relation. In the past before I had looked for where the sum of integers from 1 to n equals a perfect square, for whatever reason. In other words solutions to the equation n*(n+1)/2 = k^2, for integers n and k. I remade a program to look for this after seeing your comment, and found the following solutions 0,1,8,49,288,1681,9800,57121,332928 and I noticed that this sequence of solutions fits the following recurrence relation: F(0) = 0, F(1) = 1, F(n) = 6*F(n-1) - F(n-2) + 2 and it seems this recurrence relation predicates the next value of this sequence being 1940449, and indeed sqrt(1940449*1940450/2) = 1372105 an integer. Seeing that something like this comes up with a recurrence relation like this is what would prompt me to think that the hexagon stacks thing would also do similarly, if such a relationship exists. Though there is a difference between this example and that which comes up in the video, as my thing has a quadratic on both sides, whereas the video has a quadratic on one side, and a cubic on the other.

[@blackraven114]

There are more solutions higher up but there doesn't seem to be any sort of relation. I ran things up to 10^22 and you end up with 1045 , 5985 , 123395663059845 , and 774611255177760.

[@lvl1969]

Turns out the equation is that of an elliptic curve. Siegel's theorem gives us that there are only a finite number of solutions. The recurrence formula you are looking for is that of point addition/duplication on elliptic curves, but they will eventually start producing rational numbers.

[@mystifiedoni377]

These animations are amazing! Really help you understand what's being talked about.

[@LukePalmer]

At 4:36 I can hear matt thinking "yep, this is who I have become"

[@stever1693]

How many cannonballs would you need to stack before they started to fuse together at the bottom?

[@only2ndplace]

"It's not useful, but I love it." - every mathematician about every field of math ever

[@SteveHodge]

You can generate some cannon ball numbers with the formula: 1/2 (27x^7 + 189x^6 + 513x^5 + 684x^4 + 474x^3 + 171x^2 + 30x + 2)

[@Anchor9Studios]

7:29 is that Adam Savage of Mythbusters? He watches Numberphile? I feel like I’m in great company with so many superstars :)

[@trequor]

Props to you for actually reading the patreon credits

[@MrVasile]

Lots of comments asking about the "triangle base" case. The formula for a layer is (n x (n + 1) / 2. I haven't figured out the formula for n layers, but just brute-force adding all previous layers works in a spreadsheet. Once you do this you have the total balls in an n-layer pyramid. You then need to back-calculate to figure out how many balls on a side for a flat triangle shape. To get that calculation, you solve (m x (m + 1)) / 2 = t where t is the total in the pyramid and m is the sides of the flat triangle. There are two solutions (quadratic equation) but you can toss the negative solution (only a positive number of balls or any units makes sense). This results in checking if (-1 + sqrt( 1 + (4 x 2 x t)) / 2 is a whole number. Again, since we are counting balls, only whole positive numbers make sense. Result (up to 100,000): Pyramid height : flat triangle side 3 : 4 8 : 15 34 : 119 19,628 : 1,587,766 23,380 : 2,064,115 70,524 : 10,813,194 85,542 : 14,444,951 You are welcome!

[@odineinmann5299]

My guess is that there only exists 17 integer solutions to this entire problem as it is a variant of an elliptical curve

[@quarkonium3795]

The last four are not correct. Whatever code you used hit a floating point error

[@AviMehra]

I was expecting a 31,265-gonal Parker square but I got a 31,265-gonal Parker pyramid

[@numberphile]

BTW, there is a new Numberphile podcast out too: kzbin.info/www/bejne/b6qUc3qso7msh6M

[@MrCyanGaming]

Yeah but matt, I think cannon balls can't be stacked in shapes greater than a hexagon right? like, I'm pretty sure you can only have them in triangles squares and hexagons

[@julienhau999]

Yeah i dont understand either how they are stacked for higher polygons

[@dfhgjhg]

It's the Parker's cannon ball stack

[@KurtRichterCISSP]

The foundation level is inside a frame made of adamantium.

[@Tahgtahv]

@@julienhau999 They are all stacked in the same way. It's successive layers, with sides of length n-1. That fact that this would be more or less impossible to pull off physically is irrelevant.

[@theznayx808]

If you look at the scene with all the individual pentagon layers you see big spaces so it's probably just maximal packing

[@Latchfpv]

And this is now one of my new favourite Matt Parker number videos.

[@arthurdequeiroz8393]

Greetings from Brazil , absolutely love this channel

[@Duey8808]

I really like your observation that the number has been waiting since the beginning of time for someone to attach some significance to it. That's a very interesting way of looking at it.

[@joe9832]

3:38 - Damn, I was really looking forward to stacking hundreds of cannonballs - in differently shaped polygons nonetheless - in the comfort of my home, sadly confined within the real world.

[@whatno5090]

Luckily, by Siegel's theorem, for each number of sides of your polygon there are only finitely many solutions (if any at all).

[@deidara_8598]

1:10 Looks like Gauss' formula applied to three dimentions! If you had a pyramid in 2 dimentions you'd suddently be looking at summing n length at the base with one lower until you get to 1, so naturally one should be able to make a function that applies to all dimentions.

[@tyzonemusic]

I don't know if you took this into account, but from the animations in the video it looks like your pentagons, hexagons, etc. always have a cannonball in the center. If you look at flat squares however, 3^2 will have a cannonball in the center while 4^2 will not. Was that taken into account when calculating the different values you could get?

[@dkamm65]

I finished Humble Pi and now I'm reading Things to Make and Do in the Fourth Dimension. I just read the chapter on cannonball numbers today, and I come home to find a video about them waiting for me. Funny coincidence.

[@jonathanwalther]

Damn, these animations! Well done!

[@xander1052]

I calculated for the 31265-gon shape, its diameter would be approximately 7,186 km, approximating the shape as a circle and the diameter of the cannonballs at 30cm. which makes it just small enough to do this in real life (in theory), given if you can make 90 billion + cannonballs. For the pyramid, assuming that each cannonball level is stacked directly over the last, with none of that real world stuff that happens when you stack cannonballs the pyramid is a mere 77.7m high, and assuming it is close in shape to a cone, the radius of the base should be 18,269.30385... cannonballs, or 5480.79m, so a cone with a diameter of over 10km if my maths is correct, would like someone to check that.

[@muralikonda6233]

“ And I found it first “ 😂😂😂😂

[@TanookiOshawott64]

I worked out the equation he is using! Given the number of sides on the polygon s with sides of length n the number of balls in: 1 layer=n(0.5s(n-1)+2-n) n 1 pyramid= Σ i(0.5s(i-1)+2-i) i=1

[@Answerisequal42]

Does this work with tetrahedrons too?

[@ameyaparanjpe6179]

Amazing content as always Numberphile

[@hoggyapproved3040]

What about triangles?

[@LeoStaley]

Easily the most important comment here, but jokes reign Supreme.

[@WildAnimalChannel]

Did you know 1^2+2^2+....+24^2 = 70^2 is used in things like string theory in 26 dimensions, and is related to the Leech Lattice in 24 dimensions?

[@sekundus9274]

Are this Parker-pyramids?

[@LaSDetta]

It's definitely a Parker Pyramid!

[@Jp4wt]

as long as its not again a parker square xD

[@gadrill4285]

@Parody Poops pretty sure it'd be a pyramid. definitely not a cylinder though

[@gadrill4285]

@Parody Poops well i wouldn't call it a circle either

[@kourii]

I don't know, but that was a spot of Parker grammar there, mate

[@davidharmeyer3093]

Just checked by rearranging my cannonballs in my room. 90,525,801,730 does indeed work.

[@quesoestbonne]

Matt's mum: I asked you to tidy your room, why are you playing on your computer? Matt: First I have to prove it's possible with the number of cannonballs I have

[@Casowsky]

"Really! Is this what it took?" Lmao

[@awardmathpu4231]

This(pentagonal based pyramid) did help in discovering the pattern that never repeats called (Penrose tiling).

[@rafbammens1032]

He what about triangle based piramides? Just asking? Greeting Raf.

[@matthewpeltzer948]

10. 1 + 3 + 6 (levels of a triangular pyramid) = 1 + 2 + 3 + 4 (rows of a flat triangle)

[@RichardPenner]

{m -> 1, n -> 1, s -> 3 }, # 1 - trivial {m -> 3, n -> 4, s -> 3 }, # 10 - stack triangles 3 high or make one triangle with a side of 4 {m -> 8, n -> 15, s -> 3 }, # 120 {m -> 20, n -> 55, s -> 3 }, # 1540 {m -> 34, n-> 119, s -> 3}, # 7140 I am told these are all the solutions when the number of sides is 3, but I haven't seen a proof yet.

[@GeneralYouri]

@@matthewpeltzer948 Besides the trivial solution and your given solution of 10, there are at least three more: 120, 1540, 7140. There are no more until at least 2^53.

[@canadiannuclearman]

That equation is the sum of squares. as in. 1^2+2^2+3^2+4^2+5^2.... up to N^2. in a 4x4 square matrix and ask how many squares there are then it is the above equation with N=4. sum is 30 squares. 16 squares of 1x1. 9 squares of 2x2. 4 squares of 3x3. 1 square of 4x4.

[@DjVortex-w]

How exactly do you arrange spheres into a regular polygon?

[@kourii]

Pedant.

[@Rabbit-the-One]

Proud of you too Matt.

[@tomkerruish2982]

What about 0 and 1? Don't they always work for every shape?

[@sebastianjost]

They're boring. It's about the non-trivial cases. But you are right :)

[@ragnkja]

Tom Kerruish Too trivial.

[@tomkerruish2982]

Agreed, they're trivial. But IMO they should have been mentioned. You always need to take care of such cases. E.g., when defining a field, it's always made explicit that 0 and 1 are unequal.

[@arw000]

Matt Parker is just a joy.

[@Monosekist]

I’ve discovered every number... Just not a use for each one.

[@nok400]

I really like how you have uped the production of these videos. Keep it up, I am loving it. - A fellow Tim

[@Mr.Funkenstein]

Anybody else notice that 31265 is also a beautiful euphemism for “There’s 12 months in a year”? (12 inside of 365) Or am I just weird?

[@OlliWilkman]

I discovered a second decagonal cannonball number: 368050005576 (unless someone else already found it). It is the the 6511th decagonal pyramid number and also the 303336th decagonal number. I tested all numbers up to the ten millionth decagonal pyramid number, which is around 1.3e21. After a bit of pen&paper, and a few lines of code in the Julia language (took me a couple of hours), it takes about 0.02 seconds to test every number up to the millionth decagon pyramid number. After that I have to switch to using arbitrary precision numbers and the code slows considerably. Testing up to the ten millionth took about 92 seconds. I'll leave it running for the billionth as I go to bed now.

[@OlliWilkman]

In more detail: I noticed on OEIS that all the decagon numbers lie on a line in the square spiral, so that in the spiral, their X coordinate is positive and their Y coordinate is zero (when the coordinate system is chosen in a particular way, but it's arbitrary and other choices will just interchange X and Y or the sign). I implemented an algorithm that gives the coordinates in the spiral, and then just fed it decagonal pyramid numbers and tested whether the condition "X is positive, Y is zero" is fulfilled. Edit: I'll add that the clever bit here is that computing the spiral coordinates is a constant-time algorithm, which then makes checking whether a given decagonal pyramid number is also a decagonal number constant-time as well.

[@OlliWilkman]

Update: did not find a third one decagon cannon ball number, when checking up to the billionth decagonal pyramid number (which is around 10^27). So at least up to there we only have 175 and 368050005576.

[@EpicMathTime]

Woah, that's a lot of damage!

[@namewarvergeben]

I started reading "Things to make and do in the fourth dimension" just this morning and came across the Cannonball Numbers. I recognised the thumbnail immediately. What are the odds that Numberphile would upload a video of Matt Parker talking about this topic just now? I mean, coincidences like this happen, and it could have been any of the other topics I've read in the first few chapters so far, but man. Freaky.

[@w0ttheh3ll]

most of those pyramids aren't actually proper stacks of spheres, you would need some kind of complex support structure to hold them in shape, defying the point of stacking in the first place

[@HanabiraKage]

Would they be called Parker Pyramids, then?

[@bulgaria9003]

It's been so long...since I have seen a big number...on this channel...to the man who's talking numbers

[@partygirl0101]

the biggest meme of numberphile is back at it again

[@htmlguy88]

a meme ? ahem.

[@partygirl0101]

@@htmlguy88 yes

[@QuoteVG]

no the biggest meme is -1/12

[@michelegiacomini8487]

There is actually a different way to calculate the solutions that gives all possible answers for a given number of sides. For example there are two larger solutions for octagons (the pyramids with bases 49785 and 91839 long can be rearranged in octagons of sides 6413415 and 16068720 respectively). With the number of sides fixed, the problem is equivalent to calculating integer points on an elliptic curve. The nice thing is that there are algorithms for that. The downside is that this method is much more computationally heavy. For example my laptop struggles to calculate the solutions for 21 sides.

[@Fogmeister]

I just realised who you remind me of... You’re a cross between Rimmer and Kryten! It is uncanny.

[@bookslug2919]

Now you've said it I can't un-see it!

[@venkateshbabu1504]

Sine cube plus cosine cube equals one and solve for angles and get ratio. Whenever you see sphere.

[@thomasturner6980]

What would win, nuke or 90 billion cannonballs

[@slowsatsuma3214]

90 billion nukes

[@xbzq]

Some research indicates that this is old news (some 5 years old) from "Things to Make and Do in the Fourth Dimension"

[@danielleanderson6371]

...which is Matt Parker's book. Numberphile has been trickling out factoids from that book for years now. It's the reason I actually stopped reading it; most of it I'd already seen in vastly superior video format.

[@feeish]

3:35 I feel like if they "work" mathematically but can't be stacked in the real world it's kind of cheating.

[@todork.3240]

At this point i can say, that this channel cannot get any nerdier, or can it. Thanks for the great video.