Spheres and Code Words - Numberphile

[@Ranzha_]

For anyone wondering, the Morse Code at the beginning and the end are the same: NUMBERPHILE

[@NoriMori1992]

Numderphile?

[@victork8708]

@@NoriMori1992 Nnmberphile*

[@hgjfkd12345]

"So imagine them being surrounded by a five dimensional sphere" Oh sure, just let me put on my five dimension seeing hat and let me get right on that XD

[@2Cerealbox]

To be fair, a sphere looks about the same in every dimension.

[@ernestoyepez5103]

@@2Cerealbox nop they are more of "starshape" in the fourth there is a numberphile video about it

[@RalphDratman]

@@2Cerealbox From the center of the sphere, I'd say yes. Everywhere you look there are points r units away. From the outside of the sphere, the story is not quite as simple.

[@johnnyforeigner1009]

It's okay just pretend that you can imagine it and act accordingly

[@RalphDratman]

@@harrywatts760 If you assume you are in the middle of the sphere, it has to be the easiest n-dimensional object to imagine. Of course you have some extra directions to peer out into, but everything looks exactly the same no matter where you look!

[@TheEspenjo]

Alfa, Bravo, Charlie, Delta, Echo, Foxtrot, Grime.

[@SherlockSage]

Hotel, India, Jim.

[@Jared7873]

I thought G became Golf at one time.

[@TheTriumfAnt]

@@Jared7873 It still is Golf. He replaced Juliet and Golf with Jim and Grime. That's his name :)

[@grieferjones2237]

Hotel, Trivago.

[@64156ful]

@@TheTriumfAnt but... it's james not jim

[@nathansauveur6704]

What I've learned from this is that in a parallel universe, 24th dimensional farmers would have a really bad time stacking apples on their market booth.

[@Dracopol]

A 24th-dimensional apple a year keeps the doctor away...

[@Innengelaender]

@@Dracopol The Doctor doesnt like 24th-dimensional apples?

[@borekworek69]

@@Innengelaender yes

[@FeaturingMaxAsMax]

It's really the opposite. What Dr. Grime perhaps didn't make clear enough is how absolutely wonderful the Leech lattice is. Outside of dimensions 8 and 24, the best *known* packings are most likely many times worse than the *actual* best packings. The true miracle of Vyazovska's result is that in dimensions 8 and 24 -- but no others, aside from 2 and 3 -- the best known packing is in fact the best packing, period. So ... farmers in 19 dimensions have absolutely no clue what is the best way to stack their apples. But farmers in 24 dimensions know exactly how to do it!

[@kindlin]

@@FeaturingMaxAsMax Exactly. The 24th dimension packers have it easy, all the apples fall into place just like our 3D spheres do. It's the guys with 19 dimensions that have weird gaps they can't fill no matter they try.

[@AageV]

4:43 Ah yes, casually imagine 5-dimensional spheres.

[@Triantalex]

??

[@quahntasy]

Love every video which has James Grime on it.

[@SJrad]

same with matt parker

[@nonachyourbusiness1164]

He really is the best

[@BigDBrian]

he should upload more on his own channel

[@stuartofblyth]

+ Quahntasy - Animating Universe Ah! That explains the "G - Grime" instead of "G - Golf" in the phonetic alphabet (0:44).

[@klaxoncow]

I prefer the videos that have James Grime in it, rather than on it. Because when he's on a video, he gets in the way of me seeing it properly. But when he's in the video, that's okay, as he is the thing I'm actually supposed to be looking at anyway, so he's allowed to occlude the background at that point.

[@gergelykiss]

Well done on the pronounciation of Fejes Tóth! (At 7:34) I have never heard of him, but I managed to find him by google based on your pronounciation. Full name: László Fejes Tóth - apparently he was a really big shot in discrete and combinatorial geometry, hung out a lot with Pál Erdős, and was a monster of a table tennis player. Also a capable gymnast. Cheers for the name drop!

[@xCorvus7x]

So one takeaway is that higher dimensions have an increasingly ridiculous amount of free space between spheres, even in the densest packing. The packing density of a sphere packing in infinitely many dimensions seems to be zero.

[@michael_betts]

Spheres in higher dimensions are described sometimes as pointy, as the tip of the sphere can fall away in more dimensions. If you slice a perpendicular line of length 2*radius halfway along the radius of a circle, you get a line which contains about 86.6% points inside the circle. In dimension 3 the corresponding square countains about 58.9% points inside the sphere, this continues decreasing in higher dimensions. Also, if you pick a point on a unit circle it has a 0% chance of all of the dimensions (ex. 1/2, sqrt (3)/2) being less than a half. But in dimension 100, that is easily possible, as there are more dimensions to add up to 1 (ex. All dimensions can be .1). You get spheres in high dimensions where a random point is very likely to have small values in all coordinates, but you still have huge spikes out to the points where one coordinate is 1 and all others are 0. In some sense almost all the stuff of a high dimensional sphere is contained very close to the center in all dimensions.

[@renemunkthalund3581]

What James brushed over a bit is how this fact affects the efficiency of error correction of longer bit words. Any insight on that?

[@DDranks]

I interpret it to mean that while there is some gains to have error-correcting longer and longer words, (you need less error-correcting bits per bits of the message) there are diminishing returns. Using longer words will always win chunking the message into multiple words, but at some point you might as well start chunking because of the diminishing returns and because that reduces algorithmic complexity in the real life case.

[@nowonmetube]

You mean just like Atoms?

[@chloelo6415]

I have studied code word in abstract algebra, but the way James Grime put it into sphere packing is really illuminating.

[@We1mann]

6:47 James "a bit of Pythagoras" Grime strikes again.

[@Grimace1996]

That dimension 8 stuff reminds me of chemistry with the electron shells and how 8 is a stable number for the number of electrons per shell with how they're distributed

[@alexholker1309]

There is a term for the minimum number of steps between two error-free messages: the Hamming distance. In the first cube the Hamming distance is 3 (000 to 111), while in the second cube the Hamming distance is 2 (000 to 011). You can always detect any number of errors less than the Hamming distance, and you can correct any number of errors less than half the Hamming distance (by assuming the received output was input as whichever message is closest).

[@maxblechman2665]

The Morse code spells numberphile

[@VoteScientist]

@Lanz Friszt Max is not a nerd, he's probably a Ham of a Fist.

[@maxblechman2665]

VoteScientist nope just another nerd. Sorry

[@Doktor_Vem]

Thank you! I was curious as to what it said, but I couldn't be arsed to translate it, but thankfully there are others in the community who can! :D

[@maxblechman2665]

Doktor Vem -. --- / .--. .-. --- -...

[@NoriMori1992]

Numderphile?

[@bretterry8356]

I see that "golf" in the phonetic alphabet has been replaced by "grime" and "juliet" with "jim." I wholeheartedly approve.

[@PC_Simo]

They’re finally honouring James ”Jim” Grime, as they should. I approve, as well. 👍🏻

[@MrVernechannel]

0:52 Initially I read MURDERPHILE

[@nowonmetube]

FBI OPEN UP

[@TrimutiusToo]

Spheres are kinda pointy in higher dimensions... Reminded me of that old Numberphile video... =)

[@jjweger]

After the start, I was hoping we were covering error correction encoding.

[@jamesdavis2027]

Check out Ben Eater's channel for some great videos on the subject

[@normalasylum]

The sister channel Numberphile did a few videos covering it in-depth. Just search Computerphile error correction.

[@jjweger]

Neither are Dr Grime though.

[@rmsgrey]

He did touch on the difference between error detection and error correction (if not in those terms) so there's a start...

[@TheDruidKing]

Then you corrected that erroneous view.

[@nymalous3428]

It's interesting that as the number of the dimensions increase the amount of empty space in them also increases (at least with regards to packing regular/irregular shapes). I wonder how insane it would make a person to actually be able to see into one or more of these higher dimensions...

[@CorwynGC]

Play a lot of Golf do you Grime?

[@KlaasDeSmedt]

Who is asking? Jim or Juliette?

[@wynautvideos4263]

Hehehehe Jim and golf were replaced

[@Atemu12]

@@KlaasDeSmedt James of course!

[@rykermcintyre2158]

Might be more of a computerphile topic, but I would love to see a video made about the mathematics of error detection and correction, and how bits are actually sent over a network! Cyclic redundancy check, NRZI, 4B/5B, etc.

[@pegy6384]

Nice to see Pete get extra use from his playing card image on the coin here. I had to pause to tell that the tails were, in fact, tails. And as always, I enjoy seeing Dr. Grime's work on here-a nice wrap-up for the trilogy.

[@msolec2000]

Dr. Juliet Golf always delivers!!! Also, the morse at the end says "numberphile".

[@m.h.6470]

3:43 - 111 is actually close to exactly 3 code words (011, 101 and 110), not 2 as mentioned in the video.

[@julienbongars4287]

I heard of something similar to this before but I've never seen it conceptualized in the way you demonstrated... Awesome!

[@victork8708]

Wow! I think this case shows some kind of relations between discreet logic and concept of dimensions through group theory (or maybe category theory idk not educated)

[@alphabravo6877]

Thumbs up for "rolling a die" a rare correct pronunciation nowadays.

[@SnoopJonson]

Time to watch my favorite youtube channel, Numderphile

[@legendariersgaming]

It's always cool seeing two different fields of mathematics mesh so nicely with each other! :)

[@tylernass6263]

I love how James can turn anything into geometry

[@TurkishSupremacy]

1:11 "A mistake might happen..." - secondary camera enters frame -

[@mobermeyer]

In the telecom industry, some are starting to use LDPC (Low-Density Parity Check) for error correction. I don't know much about how it works, but I find it interesting that the algorithm was more or less forgotten after it was created in the 60s until it was rediscovered in 1996. It'd be interesting to know more about how that algorithm works and how it is similar or different from these spatial density concepts.

[@nivolord]

Very interesting, you want to pack codes as closely together, given that no two codes can be closer to eachother than '3' changes. So with (0) = 11111, you can choose (1) = 00011, (2) = 10000, (3) = 01100, each in sequence are '3' steps from eachother. But they form a square, because (0) is '4' steps from (2), and (1) is '4' steps from (3), so the packing isn't that dense.

[@Pika250]

Allow the dimension of the Euclidean space in question, d, to tend to infinity and watch the densest possible packing of unit d-balls tend closer and closer to zero. The Euclidean distance between points (0 ... 0) and (1 ... 1), each with d coordinates, is the square root of d, which tends to infinity as d does. And yet the d-balls are of radius 1. The tending-to-infinite orthonormal basis, alongside the tending-to-infinite Euclidean distance between two spheres' centers, is the reason the d-balls don't look so packed in the higher-dimension lattices compared to the lower-dimension ones.

[@tiikoni8742]

Is the density always getting lower on higher dimensions? Or is there some dimensions where density is higher than in previous dimension?

[@XanderGouws]

I don't think that "Jim" and "Grime" belong on that list of the NATO phonetic alphabet haha

[@creatif7957]

Right. J=juliett G=golf

[@palopo-t3v]

His name is Jim Grime

[@WilliamDye-willdye]

Another use for research into error-correcting codes: the search for extraterrestrial life. If a given error correction technique is mathematically proven to be optimal in some regard, then we can better focus our search for interstellar communication.

[@willis936]

Wow this is a brilliant way of visualizing codewords. When I took Information theory there was no geometry involved. Fair enough, we only covered the statistics and compression encoding. I should really learn more about FEC.

[@pauldavies8554]

Just jumped to this vid from a 2012 one starring James, and the dude hasn't changed a *jot* in nearly 7 years!

[@SciencewithKatie]

Was that a cartoon version of you on the coin! 😂

[@MaxDiscere]

lol I didn't even geht that ^^

[@Scanlaid]

And five tails! Glad I went back to look closely

[@Dracopol]

He also replaced Golf and Juliett in the NATO alphabet code with his name: Grime and Jim. What an ego on him! :-)

[@pegy6384]

@@Dracopol Dr. Grime doesn't animate the video--that was a hat tip from Pete McPartlan, the animator.

[@ZomB1986]

I see a snake head on the tails side

[@Uejji]

It makes sense for an error-correcting space to be low density, doesn't it? If less space is used for data, it leaves more room for parity, which is certainly very important for error correction.

[@YingwuUsagiri]

This feels really comparable to the ack/nak story with the weather reports by Brailsford on Computerphile.

[@vylbird8014]

The transmit-three error correcting code is horrifically inefficient, but also very simple to understand, so it's the standard example when introducing the concept of error correcting codes.

[@LeoStaley]

Niels Schellekens it's almost identical to the "the perfect code" computerphile video.

[@nosuchthing8]

I'd like to understand the reasoning behind a thumbs down. Without these error correcting codes their thumbs down might not be registered. A self defeating claim, or a paradox of sorts.

[@l.ijspeert9040]

The only thing I am still wondering about is why we are looking for the best packing in context of sending messages. I think this wasn't clear from the video. The reason I could think of was to be more efficient with your code, but James showed that a better packing could break your error correcting capability. Thus rendering that code useless, as that was the whole point.

[@austynhughes134]

What a great Sunday morning notification! Nothing like kicking off the day with a little numberphile!

[@menachemsalomon]

I was not aware that a bytestream transmitted over the internet was encoded in this way. However, I remember reading that 8-bit bytes are encoded using 15 bits on media such as floppy disks and CDs. Perhaps this is more a Computerphile project, but it's the change in polarity (for magnetic media) or surface pits and lands (for optical media) that is used to represent a '1' bit, and the absence of change over a given area that represents a '0' bit. However, a fresh disk - no pits or polarity changes - does not represent an area containing all 0x00 bytes, because every nybble (4-bit) value is actually encoded in a way that requires at least 2 pits/flips.

[@mikesummers-smith4091]

During the 1944 Battle of Cassino, a German officer sent a radio query, either "Ist Abt im Kloster?" or "Ist Abt. im Kloster?" Historians still argue which. Abt = abbot Abt. ='Abteilung', detachment, section

[@RBuckminsterFuller]

I like the idea of lunch orders being sent using the same system as launch orders... very robust.

[@rick9870]

yes, very donald trumpish

[@Triumvirate888]

Wait. So as the number of dimensions increases, the empty space between shapes also increases? That means that shapes are getting more prickly and spiky, or full of holes like Swiss cheese. If it keeps following that pattern, then it might explain why Light is both a particle and a wave. If you think of light as spheres in a very high dimensional space, then each little sphere would touch every other sphere, but would fill 0% of the spatial area in any given "box" of space.

[@sk8rdman]

But I've also heard, I think from one of Matt Parker's videos, that messages are also often encoded into a sort of su-do-ku arrangement, so that any gaps in the code can be extrapolated from the other end. I suspect some combination of both of these strategies are used? The spheres to equate for incorrect bits, and su-do-ku to equate for missing or uncertain bits?

[@ferrocen]

Well done Mr. Grime! Nice Esteregg... Golf Juliett for James Grime :-)

[@gigglysamentz2021]

Packing spheres in 3D is important for studying crystals ^^

[@snookerkingexe]

7:35 ... I was like "Mhh ... how would you prove, if this is the densest packing?" ... a second later I learned a toad proved it and now I feel utterly useless :(

[@justdata3650]

Interesting... mathematicians have a very different language. Being in the computer industry for a very long time I have never heard anyone talk about parity, CRC, checksums and packets/blocks on the various transport layers in terms of multidimensional space.

[@BrianStDenis-pj1tq]

I'm pretty sure messages on the Internet, which use TCP/IP protocol, use a simple checksum to detect errors. Erroneous transmissions are either ignored or cause a retransmission, in TCP/IP (Internet) communication.

[@vylbird8014]

I don't see the geometric connection exactly. All your packing calculations are for spheres in Euclidean metric space, but error correction codes are implemented in Hamming metric. Is there a proof that packing is equivalent in some way?

[@pavelhoral]

I would add that the most obvious use of error-correcting codes is in CDs and DVDs where you don't want a few scratches on the surface to ruin the disc.

[@darreljones8645]

Did anyone else notice that, as the value of n increases, the density of the best (or best-known) packing of n-dimensional spheres DECREASES?

[@vinodkumar-wm3oq]

So does that mean that compressed files are more prone to be corrupted? I strongly think that's the case. Please reply if you know more on this.

[@nowonmetube]

Or not, because you have to send less data ¯\_(ツ)_/¯

[@Ivo--]

I learned about this in university but I never thought about them being spheres in N-dimensional space. Very cool.

[@danieldc8841]

This fundamental encoding in E8 is thought to be related to the fundamental particles in the extremely simple theory of everything. Would be interested to see a video on that.

[@tiberiu_nicolae]

Jim and Grimes in the new revised phonetical alphabet haha

[@aksela6912]

I assume the 8-dimensional version is used because we often divvy up information into 8-bit words, aka bytes? But as the density is so low in 8D, would error correction be more efficient if we used smaller words, say of 4 bits, aka nibbles?

[@sean..L]

This is actually a lot easier to conceptualize after watching 1blue3brown’s video on Hilbert-curves.

[@steelwarrior105]

Little did everyone know, Dr. Grimes is secretly a materials scientist/engineer

[@celtgunn9775]

And someone there is into Ham radios.

[@jordanedlinger8347]

Funny thing about being a scientist... we typically enjoy learning anything we don't already know, then figuring out if it's true because there are too many bullshitters out there...

[@SumTingWong886]

Please do a video on lattice polyhedra and Ehrhart polytopes! 🙏🏾🙏🏾🙏🏾

[@bernhardmelitamann6512]

I put all the percentage numbers into excel and it showed me a graph. no surprise here. Do these numbers converge to something or just converge to zero? I think that there is something interesting behind the density in different dimensions.

[@ceegers]

0:44 I see what you did there...

[@kassy6373]

Hi Numberphile! Does calling the Ackerman function with Graham´s number as arguments really horrify mathematicians and if it does could you make a video of that/them? I think that would be fun.

[@gz6616]

Is it true that by using a 8-dimension code words system, the size of the information gets inflated by 8 times, so there is a sacrifice of efficiency to accuracy?

[@1st_ProCactus]

Not EVERY message is transmitted this way. I've never used this idea and I can still send data from one MCU to another. Do Infrared remotes work like this ?

[@zeekfromthecreek]

If you used squares or cubes instead of circles or spheres, couldn't you get 100% density? Do the density limits only apply to cirlces or their higher-dimensional analogs?

[@Shadow81989]

I have definitely seen a very similar video on the same topic before, probably on Computerphile. It was quite some time ago though, so I'm not sure, but I think it might have been with Prof. Brailsford? Great video anyway! edit: Just noticed that it's even linked in the description - which nobody usually reads, lol!

[@flymypg]

I like that an odd-numbered video rounds out the series.

[@Yonsucker]

How is it relevant though to have the densest sphere packing in the reals if the code just happens on the natural endpoints. We don't necessarily need spheres that don't overlap in the purely rational/real points, as long as they don't overlap on the naturals, do we? Wouldn't this kind of optimal codeword distribution require a different notion of density to be optimal?

[@OnlyPenguian]

Excellent and very clear.

[@yusefdanielhassounharmouch1520]

Just a question, we have all integers, and then we remove all the ones that are divisible by 2, then by 3, then by 5, then by 7, and so on, first, how would it add up if all are removing an infinity (in terms of percents)? Second, would you be removing basically the inverse of the list everytime you do it (2 --> 1/2 infinity, then 3---> 1/3--->1/2 infinity so 1/6 infinity) or is there some hidden math worth talking about?

[@kazedcat]

Infinity is not a number and this is one of the case where infinity behave differently. Infinity-infinity is undefined. But in your problem the result is infinity.

[@umanggada8684]

Does this then mean that universe has a pretty high number of dimensions as to how sparsely populated it is? Just a thought. I have no idea how dense the universe is.

[@1997benjaminvh]

Does the sequence of densities converge to zero?

[@shrirammaiya9867]

Correct, it's quite obvious.

[@Sopel997]

Watching the latest videos about spheres I was wondering about a related problem: For N>12 what is the largest radius for a 3-d sphere such that we can put N such spheres tangent to a unit sphere?

[@thatoneguy9582]

alpha bravo charlie delta echo foxtrot *g r i m e*

[@LeoStaley]

This is almost identical to the "the perfect code" computerphile video. The first part at least.

[@jaythompson7149]

The sexiest maths guy on Numberphile, bar none. Love you James! J

[@3117master]

Wasn't this done on Computerphile

[@matrixstuff3512]

How long have those pictures been on the floor for?

[@varunmuhilviswanathan3234]

Can you make a video stating the reason of the irrationality of pi?

[@impguardwarhamer]

hey numberphile, could you do a video about the Haruhi problem? It sounds pretty interesting, since its a maths problem solved by the internet in a funny way

[@Jet-Pack]

6:08 why do you not just call the length "a" and then cancel the a squares out? It would be mathematically more correct, right?

[@felixbillington6151]

Can you use linear programming such as simplex to solve for the solutions which would equal each value??

[@ultimateman55]

Cliffhanger! So it seems that the density approaches zero as the dimensions go to infinity? But man, that sure seems to happen fast. And what about for dimension N=11? I think string theorists might be very curious about that one. And what is the idea here? As the number of dimensions increases it becomes harder and harder to fill up space with N-dimensional spheres?! More! Give us more!

[@Maymz-uf6bc]

Hmm, so why is the best packing less dense as you go up in dimension?

[@EMAngel2718]

I feel like it would've been helpful if you talked about the failure rates of the systems in given numbers of dimensions because this video made it sound like it doesn't make any sense to use higher ones

[@danielescotece7144]

Wait isn't there a video on computerphile about the same exact thing?

[@omri9325]

Is this a 4 part video trilogy?

[@barnowl2832]

Where do these codewords start being used? Are they being used when I'm sending the data in this comment to KZbin? What about when the data rate is pushing the limit of the connection like in a facetime call or something? Or are they only used when mistakes will be costly?

[@vylbird8014]

Nearest to you? Your ethernet cable. Gigabit ethernet uses four-dimensional trellis-coded error correction.

[@novafawks]

Ah yes, error correction! Professor Brailsford has a similar video explaining the "points" concept on the Computerphile channel - just look up "Computerphile Error Correction"

[@GhoshA]

A nice video, after a long time.

[@MarkTillotson]

Is this 24 dimensional result relevant to the binary Golay codes?

[@jimi02468]

I want a video about the explanation of how you generalize the idea of sphere packing into higher dimensions.

[@Math-bz8bw]

Please can you explain fractional derivatives and fractional integral step by step

[@grandexandi]

is the packing efficiency in 1 dimension 100%?