Factorization Diagram

Natural numbers greater than 1 are either composite numbers or they are prime numbers. We can decompose a composite number into prime factors. For example, the number 75 can be decomposed into 3 * 5 * 5, where 3 and 5 are prime numbers.

Here is a function that decomposes a number into a list of its prime factors.

Some numbers, like 2, or 97, are not composite numbers, they cannot be further decomposed. They are prime numbers.

For prime numbers, the "decomposition" is a list with just one element, the number itself.

Visualizing a Factorization

In 2012 computer scientist Brent Yorgey published a blog post describing a nice way to visualize the prime decomposition of a number. He called his visualization a factorization diagram.

The Number 75

For example, the number 75 = 3 * 5 * 5 is represented as follows:

The first prime factor, 3, is represented by the green triangle. At each of the three corners we find a cyan pentagon (the second prime factor, 5), and at each of the corners of those pentagons, we again find a pentagon (the third prime factor, also a 5). At each corner of those pentagons we then find a black dot.

The total number of black dots is 3 times 5 times 5 = 75. The diagram represents the number 75 by recursively decomposing it into substructures.

The Number 1155

Here is another diagram, for the number 1155 = 3 * 5 * 7 * 11:

At the top level we have a triangle, breaking the number into three equal components (each with a total of 1155 / 3 = 385 dots). At the core of each component is a pentagon, breaking that component down into 5 equal subcomponents (each with a total of 385 / 5 = 77 dots). Each of those subcomponents is a heptagon (a seven-sided polygon), further breaking it down into 7 equal subcomponents (each with a total of 77 / 7 = 11 dots). At the bottom of the decomposition we have hendecagons (eleven-sided polygons), breaking it down into 11 dots.

Prime Numbers

Prime numbers cannot be further decomposed (that's the whole point of prime numbers). Thus, their factorization diagrams consist of a single polygon.

Here are the first few prime numbers: