Now i want (to build) a drag and drop toy where i can multiply or summarize numbers that are represented in this way. To see how factors move (like boids).
Kind of makes me wish that there were recognizable shapes for primes bigger 2 (pair), 3 (triangle), 4 (square) and 5 (pentagon) that didn't just look like circles. Because my favorite part about this is how you can see at a glance what the factors are. Except for primes 7 or greater I find myself cheating and looking at the top left for which prime it is.
Is there some non-regular polygon that would be more distinctly recognizable to use for 7, 11, etc?
drdeca · 3h ago
4 isn’t prime.
You could probably use the binary expansion to group the dots?
So, 1 is •
2 is ••
3 is
_•
•_•
5 is
_•
•_•
•_•
7 is
____•
_•_____•
•_•___•_•
11 is
____•
_•_____•
•_•___•_•
•_•___•_•
And so on.
(So, 2n is represented as n next to n, unless n is 2 in which case it is n above n, and 2n+1 is 1 above 2n )
Alternatively, using stars instead of n-gons could also be clearer?
worldsayshi · 5h ago
Couldn't you draw it in a recognizable way using summation?
7 = 2*3+1
11 = (2*2+1)*2+1
etc...
CGMthrowaway · 4h ago
Interesting idea
Liftyee · 3h ago
Agree. I watched for a while to see some larger primes and was a little disappointed.
Filled polygons would offer some more shapes. Filled hexagon = 7, etc etc...
GaggiX · 5h ago
Aren't 2 (pair), 3 (triangle), 4 (square) and 5 (pentagon) also "circles" with less resolution? The visualization is just consistent.
CGMthrowaway · 4h ago
Yes I dont disagree and it is elegant as is, but the way our eyes/ brain works it's much harder to ID septagon, nonagon, triacontahenagon etc at a glance. A non-regular shape would be better fit for purpose
ashwinsundar · 5h ago
I believe it's called prime factorization. Each number is placed in a group of numbers (or group of groups, etc...)
E.g. 24 -> 2 * 3 * 4 = Two groups of (three groups of (four items))
The diagrams for powers of three form the Sierpinski triangle. Makes total sense once you see it, but I hadn't seen it until today!
robot_jesus · 6h ago
Same. I loved this unique insight that the visualization provided. It definitely unlocked something in my brain for how I should think about that shape.
If anyone is curious, 6561 (3^8) is the highest pure Sierpinski available in the animation since it caps at 10K.
pvg · 7h ago
Threads (with some explainy links) from a million and a million and a bit years ago
Does it let you put your own number and see what diagram it makes?
kccqzy · 4h ago
I wish the animation could play at a slower pace so you have time to count the number of groups and the circles within each group. I also wish each time a new circle would animate from the edge of the screen and then arranged to show the addition of the new circle clearly. Otherwise, excellent visualization!
gavmor · 4h ago
The jumps between neighbors are sometimes so drastic—are we sure our numbers are in the right order?
jerf · 2h ago
That's the difference between the additive view of the world and the multiplicative one. A lot of number theory involves trying to bridge that gap, and even this simplest view of numbers can rapidly fling you into the mathematical unknown. The "simplest hard problem", the Collatz conjecture, can be viewed as coming from this space. You either take a step in multiplicative space, or a step in multiplicative space and then additive space, and ask a very simple question about where those steps can take you.
And that's all it takes to end up at an unsolved problem in math.
You can spend a lifetime on this simple observation that "the jumps between neighbors are so dramatic", just trying to reconcile the complex relationships between the additive view of the world and the multiplicative one.
And we haven't even done anything like bring in complex numbers, or rationals, or introduce exponentiation!
gavmor · 1h ago
How can a layperson approach and develop correct intuitions for "the multiplicative view" of numbers? Is it practical?
jhanschoo · 3h ago
I don't know what you mean by that, but for an example, 16=2^4 and is therefore arranged as a grid, whereas 17 is prime, and must therefore be arranged as 17 dots on a circle.
gavmor · 1h ago
The primes are some of the worst offenders, eg the transition from 647 (prime) to 648 (3×3×3×3×2×2×2), but as we approach infinity, the visualizations increasingly appear circular, and it's the least-primey (?) that break from the trend.
eg 854-856, & 857 (prime) are all virtually or perfectly circular. Or maybe I'm observing not mathematical but neuro-visual principles.
jderick · 4h ago
Can you put them all on one page and zoom in/out? Might be interesting to see some patterns in the sequence. Maybe allow filters for different factors or number ranges or different groupings.
glaucon · 3h ago
Really good. I would appreciate it if it could be slowed down, or allow the numbers to be stepped through.
dtjohnnymonkey · 2h ago
After some time I find myself waiting for highly composite numbers rather than primes.
ape4 · 3h ago
I think the sum of the area of the circles should remain constant. ie be the number that's being factored.
tocs3 · 6h ago
It makes me wonder what the algorithm for arranging the dots looks like.
GrantMoyer · 6h ago
1. Set var factors to the prime factors of the integer
2. Sort factors in ascending order
3. Set var diagram to a single dot
4. Foreach factor in factors
4.1. Set var diagram to factor copies of diagram aranged in a circle
tocs3 · 5h ago
I was thinking about triangles and squares and the answer was circles.
No comments yet
carterschonwald · 5h ago
I think this was originally invented by Brent yorgey
liendolucas · 3h ago
This is cool! Lets use it to display the digits of a clock :-)
ajjenkins · 6h ago
This would make a cool progress bar replacement. Replace percentage with the number of dots (0-100).
chrsw · 7h ago
This looks cool. Could also be a screensaver (do people still use those)?
apples_oranges · 7h ago
Macs now have them again. OLED screens made them create animated login screens. (At least I think that's what happened.)
andrewrn · 6h ago
This is very cool. It looks like you used the canvas api for this, but I had expected that you'd use D3.js since its so common for data visualizations. I am curious what your thinking was there?
pona-a · 5h ago
But it's not CRUD data visualization, it's a custom animation. Why use a heavy library if the browser draws circles just fine?
vivzkestrel · 4h ago
sliders should be added on this page that control everything. colors and speed are starters
simojo · 7h ago
it took me a few seconds before I realized that it wasn't the page loading
gus_massa · 5h ago
Slightly related: If you have small kids, I recommend https://www.youtube.com/@Numberblocks that is a cartoon that has characters that are numbers made by blocks, and they split to show sum and rearrange to show the factorization. It's fun for kids and the technical part is correct.
No comments yet
ttoinou · 4h ago
This is pure genius, congrats, and I’m disappointed at myself I didn’t think about that earlier (:
aaroninsf · 6h ago
I wish that all the factors were shown,
e.g. when on 12, I'd like to see both 3x4 and 2x6, with a visual indicator of when the animation is showing the different factors... maybe the whole thing shrinks and additional factorizations fill in tiles subdividing the space
Knowing the number of different factorizations is an interesting and visually presentable quality that interacts in an interesting way with the factors themselves
blueflow · 7h ago
I thought this was a waiting animation and the website is broken.
kccqzy · 2h ago
It would function pretty well as a waiting animation too.
nurumaik · 1h ago
Waiting animation that ends after the last prime number
Now i want (to build) a drag and drop toy where i can multiply or summarize numbers that are represented in this way. To see how factors move (like boids).
Is this visualization algorithm called something? The explanation link from a previous HN post seems to be broken: http://mathlesstraveled.com/2012/10/05/factorization-diagram...
Is there some non-regular polygon that would be more distinctly recognizable to use for 7, 11, etc?
You could probably use the binary expansion to group the dots? So, 1 is • 2 is •• 3 is _• •_•
5 is
_• •_• •_•
7 is ____• _•_____• •_•___•_•
11 is ____• _•_____• •_•___•_• •_•___•_•
And so on.
(So, 2n is represented as n next to n, unless n is 2 in which case it is n above n, and 2n+1 is 1 above 2n )
Alternatively, using stars instead of n-gons could also be clearer?
7 = 2*3+1
11 = (2*2+1)*2+1
etc...
Filled polygons would offer some more shapes. Filled hexagon = 7, etc etc...
E.g. 24 -> 2 * 3 * 4 = Two groups of (three groups of (four items))
Also try this for the archived version of that explanation -> https://web.archive.org/web/20130206023100/http://mathlesstr...
No comments yet
If anyone is curious, 6561 (3^8) is the highest pure Sierpinski available in the animation since it caps at 10K.
https://news.ycombinator.com/item?id=10776019
https://news.ycombinator.com/item?id=4788224
Factorizer - https://news.ycombinator.com/item?id=10776019 - Dec 2015 (30 comments)
Animated Factorisation Diagrams - https://news.ycombinator.com/item?id=4788224 - Nov 2012 (72 comments)
Animated Factorization Diagrams - https://news.ycombinator.com/item?id=4713048 - Oct 2012 (2 comments)
And that's all it takes to end up at an unsolved problem in math.
You can spend a lifetime on this simple observation that "the jumps between neighbors are so dramatic", just trying to reconcile the complex relationships between the additive view of the world and the multiplicative one.
And we haven't even done anything like bring in complex numbers, or rationals, or introduce exponentiation!
eg 854-856, & 857 (prime) are all virtually or perfectly circular. Or maybe I'm observing not mathematical but neuro-visual principles.
2. Sort factors in ascending order
3. Set var diagram to a single dot
4. Foreach factor in factors
4.1. Set var diagram to factor copies of diagram aranged in a circle
No comments yet
No comments yet
e.g. when on 12, I'd like to see both 3x4 and 2x6, with a visual indicator of when the animation is showing the different factors... maybe the whole thing shrinks and additional factorizations fill in tiles subdividing the space
Knowing the number of different factorizations is an interesting and visually presentable quality that interacts in an interesting way with the factors themselves