####Pentagonal Numbers

Pentagonal numbers refer to the sequence of numbers that describe the growth pattern of a series of pentagons, starting with a single dot (abstract pentagonal ). Regular pentagons are usually used a a geometric reference. Around any pentagon, the dots are equally spaced along its perimeter. At the n-th step , the pentagonal number is the total numbers of dots at that step. At step 1, it is 1; at step 2, it is 5, and, subsequently, 12, 22, 35, 51, ... Using a formula, it is n(3n-1)/2, which is not as playful as the physical models.

In fact, the formula can be derived from the geometric structure of the pentagons. Let's assume that we know how to calculate the n-th triangular number, which is n(n+1)/2, the sum of {1, 2, 3, 4, ..., n}. At the n-th step, the pentagonal number consists of three triangular numbers, with two overlapping sides in the middle. Therefore, the n-th pentagonal number is * 3n(n+1)/2-2n,* which is * n(3n-1)/2*.

In this design, we start with step 2, a pentagon with 5 dots. There are two versions provided. They can be mixed up for various patterns. By virtue of the design, the pieces can be assembled in geometrically diverse ways. They do not snap into each other; but they fit together nicely.

A tolerance of 0.3mm is left between the steps. So they can be printed together. They can also be printed one by one, using various colors, for pretty patterns. The space between two dots is 15 mm.

####References

1. https://en.wikipedia.org/wiki/Pentagonal_number
2. https://www.qc.edu.hk/math/Junior%20Secondary/Polygon%20number.htm