I – as any mathematician – know quite a few math gems that – I believe – should be at the heart of school-math… But, this one, is just exactly what school-math should be 4 ALL of us!
This is one of those mathematical pieces, you look at them once and you say “Nice!”, you look again and say “Oh!, I missed that before! Very interesting!, then you look again and say “WOW …..”
“Nice!”: You first see each natural (counting) number is shown by colourful dots!: 1 by one dot, 2 by two dots, 3 by three dots, and so on…
“Oh … Interesting!”: Then you see, the dots make different shapes (diagrams): Like pairs, triangles, squares, pentagons and circles. But then you realize! Oh!: Each diagram made of “pairs” represents a number divisible by 2 … Each diagram made of “triangles” or “triangle-like-diagram” represents a number divisible by 3 … Each diagram made of “squares” or “square-like-diagram” represents a number divisible by 4 … Each diagram made of “pentagons” or “pentagon-like-diagram” represents a number divisible by… guess!… 5… And finally, you see that each diagram made of “circles” or “circle-like-diagram” always has a prime number of dots – or of other diagrams – and represents a number divisible by that prime number.
You have found, for example, a diagram like a triangle made of squares, is a number divisible by 3 and 4 (like 12, 24, 36, 48, 60, …). Or, a circle of 7 pentagons is a number with factors 7 and 5. Or a circle of 7 circles of 7 dots each, is the number 7 x 7 = 49 (the last diagram in the picture above).
Do you see??… Almost magically, you can be factorized numbers just by “looking at them”: Each diagram actually represents a complete factorization of each natural number.
Pairs give factor 2, Triangles give factor 3, Squares give factor 4 – included by its beauty, since 4 is not a prime – , Pentagons give factor 5, and Circles give all other Prime factors.
Now… Just give yourself a minute, click and enjoy your first “Close Encounter Of The Nth Kind, for all N” with Numbers!!…
And… The mathematical magic goes on…
“WOW…..”: If you looked at the Mega-Cool animation created by datapointed.net , you will see how for bigger numbers, the different diagrams become more and more mixed up. You can see circles of squares which points are really triangles, or another circles, or… But watching more carefully – if you look for some “order” in this increasing chaos – you will find, for example: That squares, made of squares, made of squares look pretty ordered. And triangles, made of triangles, made of triangles look even amazingly. The same happens with pentagons and, more generally, with circles of p dots, which dots are actually other circles of p dots, which are also circles of p dots, … (p always a prime number!)
What you have discovered is:
The numbers that are powers of a prime number p (p x p x …) are represented by diagrams which are not only more symmetric, harmonic and beautiful, but also they are highly special, since they become fractals!!!
On the screen, the only case one can really “see” is the Sierpinski Triangle for the case 3 x 3 x 3 x … [You should also note, that the powers of 2 may also become a fractal, but we must consider another diagrams, repeating the very first two steps of the original diagrams: Dividing each dot into two smaller ones each time. Then we get a Cantor-Like Set!] (See also: El Conjunto de Cantor y Las Potencias de 2, Fractals or Sierpinski Dreieck)
From now on, every child has the chance to SEE & ENJOY numbers thanks to Brent Yorgey’s (amazing) Factorization Diagrams and the (incredible) animation by datapointed.net.
PS: But,… keep on looking and searching… There will always be something else to “see”… 😉 …