Imagining Numbers as Shapes - Numberphile 

I imagine zero as a circle, dont know why.

I like how even though they sped up the video, they still kept in the little sound effects the guy made while he was drawing his triangles.

I would like to congratulate Simon for being the first person too young to collect a pension to actually sit through a 60 minutes episode (or at least part of one) in decades. The closest I ever managed was watching a replay of parts of the 60 minutes 2 episode where they presented the forged Bush national guard documents as genuine.

I imagine numbers as atomic numbers of chemical elements, 1 = hydrogen through 118 = oganesson, e.g. 1234 = magnesium selenium. (Of course, I had to work very hard at acquainting myself with the periodic table first!.) Oh, and 0 = neutronium. (I think the PIN is, e.g., lithium thorium.)

Recommended Video : Ticon - Out Foxed. Yeah, triangles, of course !

But how does he do 8? This question NEEDS answering! Show no nnercy on his fanpage/webslte, through the power of 1,7 million Numberphile subs we can surely get this kid to comment.

Brady, you should meet with Oskar van Deventer one time!

This guy likes triangles. Illuminati confirmed

The number you flashed up at 4:30 is 650.

This video: "Mmmmh...mmmh....mmmm" "Unnnh....unnh...uhhh" "Daaht....dahh...duh"

Nice idea .

I used to do exactly this when I was little... Mostly because I'm so bad at memorizing times tables though. lol

ok here's the important question, how do you visualize 2 ?

From solid mathematical objects, that all start with 1, to pure artistic abstractions, which have no any solid center or any solid point as a base ! Straight line, what shape is it ? A cat, a tree, a smile ? What numders ? Give me them and I rebuild the World !!!..

And then someone asks him to represent the number 1 and he cracks.

Does that kid imagine all numbers like that? How would he visualise the number 4?

You should contact him and get him to host an episode.

Is this guy challenging the internet to figure out his PIN.... that's a bold move, Cotton

I misread the title as "imaginary numbers as shapes" and was waiting to have my mind blown.

Illuminatie confirmed

that kid is now a phD candidate at the age of 18 wow...

All I did was imagine the Vihart video on triangles and remember Vihart constantly saying "triangles triangles triangles".

We have a term for such kind of learning: *Visualization*. And it's amazing.

5:20 Holy s***! that power leg spread, Simon's got going on.

13*2*5*3= 0390?

how does he do 7?

Jacob Barnett is actually quite an awesome young man. Obviously gifted, but he's also very personable and outgoing. He's definitely got a bright future ahead of him.

Wow, that was really intriguing. But I was quite surprised that there was no mention of Daniel Tammet? His way of seeing/'feeling' numbers in his head is truly amazing.

This is cool and all, but WTF are you gonna do with 210?

Wow, counting in base-prime

How to do more than one instance of 2, having a factor of 4 (or 8, 16, ...)? How to do 5, 7, 11, ...? How to mix 3 with 5, which one takes precedence?

This reminds me of "Triangle of Power" - by 3blue1brown Providing a visual method of calculating exponents, logarithms, and roots which uses triangles.

How he imagines number 2 ?

i have always given all numbers and letters genders

Nice vihart reference

When Simon said Triangle Triangle Triangle, I know one RU-vidr who should love it.

Very cool video. I liked the kind of fractal structure that it sort of creates as you go through to higher powers of 3 (and I assume any other prime, based on how Simon explains it). I wonder if there's deeper mathematical structure to these geometric interpretations of the numbers beyond just the intuition behind them (and that they're visually satisfying).

This really shouldn't be a new way of thinking of numbers, but it's still a bit fascinating

There is one minor flaw, If I try to do numbers like 544 it almost looks like a circle. 17*2^5

You could consider 4 to be a "pseudo-prime" in this visualization system and use squares. So 8 would be two squares, with one rotated 45 degrees. The only positive numbers you couldn't visualize like this are 1, and 2. Any number divisible by 4 would involve squares, so 12 would be a square with triangles on the points and 24 would be two of that shape rotated 45 degrees.

This kid has also inspired me to visualize numbers by means of polygons some time ago. What I did looks a bit different, as the polygons and clusters of polygons are not covering up each other, but lying edge-to-edge. Highlighting the different prime numbers with different colors further improves the recognizability. However the simplicity of the approach seems to get lost rather quickly when the numbers get a bit larger. Firstly with growing number of prime factors there are more and more possibilities that the same number can take. Representing 2*2 as a square 60 = 2*2*3*5 can take 6 different shapes. Secondly you can hardly distinguish a 11-gon from a 13-gon and when depicting numbers with those prime factors, the outer polygons soon become very small in order not to cover up each other. One possibility to avoid this is to decompose larger prime factors into a sum of smaller ones. That gives the numbers again a more distinct appearance. However there is some arbitrariness in this and some of the elegance and simplicity is lost. This way of imagining numbers makes it very simple to multiply, divide and draw roots (provided the result is an integer number). You also can deal with rational numbers more easily, but it makes it extremely difficult add and subtract (apart from some easy special cases, at least I haven't found a way to imagine it) and I can't think of a way for a generalization to real numbers. Would be nice to hear some further thoughts about these issues.

So how does he see the number 2?

The really cool thing about this representation is that it does not depend on any symbol or any base.... like if you wanted to communicate a number with some alien who understood numbers in base 19, this representation is great!

I am not sure what the flashing means, but I guess it is 390

Is there a technical criteria for who qualifies as a "savant"? I've only seen savants that are mentally challenged but through that are able to intuit amazing abilities. That kid seems normal, just really really good at math.

On the one hand it's great to see this kid playing around with numbers, but I find it just as sad, how lost people watching him get at such basic stuff like prime factorization... The most amazing part for most people seems to be not how he imagines the numbers, but just how he knows that 54 is 2*3*3*3.

Don't we already imagine numbers as shapes... 1, 2, 3,

How is 2, 4, 8, 16, etc. represented? Also, this system quickly becomes difficult to handle the moment addition gets involved.

Hello , this is a really interesnting way of displaying numbers, but i have a question. You mentioned that this is a decomposition in prime numbers. For 2 you overlap 2 triangles, for 3 you draw a triangle on the vertexes. What about bigger numbers, 5, 7 , ... , 97 and so on ? How can you generate a general pattern for such numbers that can't be exprimed with a general formula?

I got poor grades in math, 6th grade, because I wouldn't show my work although my answers were correct. I was re-inventing maths and that wasn't in the curriculum. Public schools yeah!

That's great kid! Now show me how you imagine this RSA public key number!

Where is Jacob Barnett now? How are the rest of the numbers visualized and arithmetic methods please!!!! Thanks

Jake Barnett is not a divine creature that is unreachable, you guys should try to contact him. I bet he would like your channel and probably will help. Just a heads-up.

390 - so 0390? Thank's Simon ;)

666th! I think this number would be a picture of satan.

Towards the end of Carl Sagan's book, "Contact" (not the film version), it talks about finding the representation of a circle found in the Base-11 notation of Pi. Of course, this is fiction, but Carl's observation of how humans are fixated with Base-10 numbering is absolutely correct.

Pleeeaaase get in contact with him, i want to see more!

I don't see how this is very useful. How can it be used for more complex math?

"Taking what we got taught at school and having fun with it" - anyone who can do that is a class A genius. ;)

Eh, how does he visualize 2^n*x, where n>1? Like 4, or 56?

So in this pattern, how would you view the number 5?

Go and interview him

Can Simon send Jacob just a email or a letter and ask how Jacob would solve a few creative math problems please? If Jacob replies, Simon can figure out allot more then any television interview ever recorded about Jacob. I don't believe in savant stuff as in a excuse that we can't do it, but I do believe some people like Jacob are remarkable creative.

54 can be nicely decomposed, but the next number 53 is a prime so it is going to be drawn in a polygon with 53 sides. Striking that there seems to be no continuity in adding 1 !

Amazing! Thanks. :) Prime decomposition...now I know what it's called. Really genius to be able to "see" numeric figures with two-dimensional shapes.

There's a gorgeous music video that treats prime numbers as floating creatures that merge by addition. "Wonderful World" by Lost Lander. Look it up. Easily one of my favorite music videos. Great song too!

I wonder what 7 looks like for this kid

This is also beautiful art. Also, Simon you are a beautiful man.

this is way too 'illuminate confirmed' moment for me, i cannot handle it all. it is some alien alphabet.

I think the coolest thing about this is that just by looking at the representations of the numbers, you can immediately see all of it's factors!

*Vi Hart screeching in the background*

Why doesn't 2 represent a line?

So, er, what's the point? We can draw a picture corresponding to any number of the form 3^n or 2x3^n. Now what?

54 has a very Jewishy vibe

How does 2 look like? And how does 36 look like?

The number at 4:30 is 13 * 5 * 3 * 2 = 390

you can't get 0, 1 or 2 with triangles :^) or you have to somehow generate a line using triangles? or a point in space? and 0 would be, no triangles?

I remember seeing a report of an autistic savant that pictured numbers as shapes and could multiply them based on their shapes somehow. It basically went something like (don't remember the exact numbers, but I'm pretty sure they were three-digits): "Well, 234 looks like this", *draws amorphous blob*, "317 looks like this" *draws another amorphous blob*, "and if you multiply the two shapes you get this shape" *draws third amorphous blob* "which is 74,178." And I'm just like...wat?

Is the PIN number 390? (2*3*5*13)

I personally do quite well on base 2, since i've memorized for fun powers of 2 up to 2^19 Here is the list of them: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288.

Another math savant, Daniel Tammet, who sees numbers as shapes and colours went on David Letterman's show and told Letterman that he looks like the number 117. If I were to guess what number Simon looks like I would guess ... 83.

So he think a number as a multiplication of 3s? HOW THE F... IS THAT GENIUS? Oh yeah, 3 it's a triangle, and 9 it's 3 triangles, and 18 it's 2 times 3 triangles, or 6 triangles. I'm geniusderp.

Do imaginary numbers as shapes! Or at least explain how they would be in our head. I wanted to know, others want to know. SUPPLY THE DEMAD GOSH DAMMIT

Funny, the host calls this representation intuitive and IT'S NOT, it helps visualize the factors of a number BUT NOT THE NUMBER ITSELF! If I show you one of these patterns you're not going to see easily what number it actually represents.

2 x (3x3x3=27) Imagine just one 3x3x3, then just two of them and over lap them. So, two(2) big triangles multiple by three(3) points of the big triangle, then by three(3) triangle at the point of the big triangle and then multiplying by the three(3) small triangle at the point of the triangle at the point of the big triangle. Maybe 6x9 = 6x3x3 too. 6 Green triangle times 9 corners/points of 3 yellow triangle per green triangle. 3:50 Maybe something.

You don't need to be a savant to have fun with math. Most kids do and they simply move on. Everybody's a gangsta with prime factorization until numbers like 199 or 10007 show up. Even just using 37 or 61 hurts the brain in many ways. If he needed the points but not the sides he could've just left the dots on the plane without the sides. It would've been a lot more interesting if the intersection of the sides would've counted as additions to the number.

Keemstar' brother is a genius

Very bad of the reporter. Not even simplifying the method can make the kid look like he's just choosing arbitrary shapes, wich isn't at all fascinating but rather feels like a scam

Fun is underrated. And overrated. It's not that you should make things fun (which is overrated) but you should have fun while you're learning (which is underrated). The point is who's responcible for the fun. I didn't expect to say that. Also, I really hope that wasn't anyone's PIN.

"You don't know what the number is just by looking at it." You could say the same thing about decimal notation. Or tallies (above about 5). I don't think there exists a notation that readily shows you all the things that a number is.

I actually don't like the putting a triangle on top of the previous triangle for multiplying by two. It's a special rule while the other numbers just put stuff at the vertices. Why can't everything be done at the vertices? Multiplying by two should just put a single piece of line on the vertices (turning each vertex into two vertices). So a 2 would look like a line (let's say horizontal line but that doesn't matter). A 4 would look like a tie fighter. An 8 would look like a capital H with serifs. And so on. This system would work with 3*2 as well.

It's cool but any large primes would just be almost cirlces. even 13 or 17 wouldn't be easily used.

Pythagoras what is it with you? Always with the triangles, your solution to everything is triangles! There are problems in that can't be solved by triangles. Will you shut up already with the triangles! Everything is triangles, you're driving me crazy!

Hey I'm curious about 0 over 0, shouldn't it be one? I've never heard a satisfying answer as to why not, always just 'because that's the way we've always done it'. I understand why x over 0 (x cannot = 0) is undefined but this seems to pit two math rules against each other ( x over x = 1 vs. Anything over 0 is undefined)

How would you do a number that's a multiple of 4? Say 12 for a simple example. The best I can come up with is just 4 non-overlapping triangles.

I do something similar, but without the geometry. I just like the concept of a number system where everything is just prime multiplication and adding one :)

I’d like to see an animation showing the different patterns in consecutive numbers. I think it would be more useful to use stars rather than regular polygons for the larger primes (a 37-pointed star is way easier to parse than a 37-gon which is already very close to a circle)

There are lots of things o would like to know about this. How does he do big prime, prioritise the order, do negatives, 1, fractions, surds, the advantages of using this over the normal system, etc

So, what's 250? A pentagon with pentagons on each vertex and mirrored onto itself? It looks like this gets really hard to see quite quickly unless it's dominantly 3s.

That's simple. You can do it with any shape, and is quite easy to visualize, but so are numbers... 4×4: A square has 4 verticies. Put a square on each verticie of the first square. 16 verticies. Same as having 4 fours. One 4 for each corner of the square.