## Pentagon And Hexagon

##### 2021-02-06   |   by CusiGO

As we’ve seen in the past few weeks, the order of triangles is 1, 3, 6, 10, 15, 21… The order of squares is 1, 4, 9, 16, 25, 36… What’s the order of pentagons?

First of all, it should be pointed out that, unlike triangles and squares, Pentagon numbers are formed according to the outline pattern.

For example, to find the fifth pentagon number, we first form a regular pentagon with five edge points, and then from one of the vertices, we use this common vertex to form a continuous Pentagon point, as shown in the figure. In this way, we get the first five Pentagon numbers, each of which is The sum of his own girth plus the previous girth. Like a triangle, the number of the first Pentagon is considered to be 1 (a Pentagon with a common vertex as 0 side), so the sequence is as follows:

1，5，12，22，35。。。

What’s the sixth Pentagon? What about the seventh? What about the tenth? Is there a general formula that allows us to find the nth pentagons?

And vice versa. Given an arbitrary number, we can easily find out whether it is a Pentagon?

The structure of hexagon numbers is the same as that of Pentagon, but they are not Pentagon. Instead, they use a common vertex to form a continuous hexagon.

As shown in the figure, the sequence is as follows:

1，6，15，28。。。

What is the number of the fifth hexagon? What about the sixth one? What about the tenth? Is there a formula to find the nth hexagon number?

And vice versa. Given an arbitrary number, how do we know if it is a hexagon?

The next polygon number is obviously heptagonal. The sequence is as follows:

1，7，18，34，55，81，112，148，189，235。。。

The formula for finding the nth seventh degree is: (5N 2 – 3n) / 2

The more simple formula is the octagonal number formula: 3N 2 – 2n, and its sequence is as follows:

1，8，21，40，65，96，133，176，225，280。。。

The formula of decimal number is 4N 2 – 3N, which is the same as that of octal number. Both coefficients are increased by one unit. What does this sequence of formulas tell us?

In addition to being continuous polygons, pentagons and hexagons can also be combined to cover (conveniently bend) spheres, just like the well-known football. A regular hexagon, along with equilateral triangles and squares, is one of three regular polygons that can “fill” a plane. The Pentagon can’t, but it’s working with its neighbors to launch other interesting coatings.

How many pentagons and hexagons are needed on the surface of a sphere? Or the same thing, how many pieces does a football skin consist of? Is this the only solution?

Carlo frebetty is a writer and mathematician at the New York Academy of Sciences. He has published more than 50 popular science books for adults, children and teenagers, including damned physics, damned mathematics or great games. He’s a screenwriter for crystal ball.