Factorization Diagrams

Factorization diagrams are depictions of numbers using dots but grouped by factors. They were invented by PhD student Brent Yorgey “ … in an idle moment a while ago …” (October 2012). Here are the first forty nine factorization diagrams, compiled by Michael Naylor from Brent's original post.

Starting at the top left you can see one dot, two dots, three dots, a pair of two dots (4), five dots, a pair of three dots (6), seven dots, & so on. Prime numbers are just a circle of dots, and any above 42 or so are pretty much indistinguishable.

The posters include color coding for each factor and a polygon background. The colors were generated so that close numerical neighbors are very distinct from each other and the colors don't repeat (although each cycle of eight or so is quite close). The color scheme is displayed by the top row of plain circles on the poster and consists of the first dozen prime factors [ 1 2 3 5 7 11 13 17 19 23 29 31 ] and is continued on the bottom with the next dozen [ 37 41 43 47 53 59 61 67 71 73 79 83 89 91 97 ].

As an example, factorization diagram three is three dots over a triangle and is the same color as dot three in the color scheme. Now looking at the big picture, underlying triangles are the same color as are dots in a triangular formation, indicating a factor of three. Similarly the next color in the scheme paints a prime factor of five and this color can be seen in underlying pentagons and pentagonal formations. (A different color scheme was used for the illustrations in these notes.) Since primes have no factors they end up as a circle of dots, which circumscribe a disc (the underlying polygon). Numbers with two factors can be depicted in two different ways – so fifteen is three by five or five by three.  Numbers with three factors have six different depictions.

Four factors have twenty four depictions. It's factorial !

The diagram actually used in the poster is arbitrary, but posters have been produced which always use the smallest factor (more intriguing formations), and which always use the largest factor (more colorful).

Four arrangements of factor diagrams have been produced. Since factor diagrams are graphical depictions of numbers, and since number arrangements sometimes highlight mathematical relationships between numbers, these combinations facilitate further explorations. Linear The standard linear arrangement of rows of 12 factorization diagrams results in the default size of one square inch each.

 Ulam Spiral The Ulam spiral starts in the middle (look for the big shiny dot) and spirals out in a square spiral. The Polish Mathematician Stanislaw Ulam (1909-1984) noticed (whilst doodling) that primes in a square spiral tend to fall on the diagonals.  Hexagonal Hexagonal grids have a natural elegance and provide a nice base for concentric rings of factorization diagrams.

 Phyllotactic Spiral Phyllotaxis (leaf arrangement) is the well known natural spiral as seen in daisies and sunflower seed heads. It seems like each factor generates its own wave of spiral arms simultaneously with all the others. 