Triangular Day: Are the Figurative Families that Related?

I don't know if you noticed, but today is a triangular day. It is the first of June, and 1 is indeed a triangular number.

1 is also a special number because it is a square number as well. In fact, if you look at all of the regular polygonal figurative families, one is the first number. It is the first pentagonal number, hexagonal number, and so on. In fact, a dot can represent whatever figure you want it to, which shows the creativity of mathematics.

Because of this specialness to the number one, I thought we should analyze all of the regular polygonal figurative families. Let's look at the first fifteen numbers in each family.

Name	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Triangular	1	3	6	10	15	21	28	36	45	55	66	78	91	105	120
Square	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225
Pentagonal	1	5	12	22	35	51	70	92	117	145	176	210	247	287	330
Hexagonal	1	6	15	28	45	66	91	120	153	190	231	276	325	378	435
Heptagonal	1	7	18	34	55	81	112	148	189	235	286	342	403	469	540
Octagonal	1	8	21	40	65	96	133	176	225	280	341	408	481	560	645
Nonagonal	1	9	24	46	75	111	154	204	261	325	396	474	559	651	750
Decagonal	1	10	27	52	85	126	175	232	297	370	451	540	637	742	855

This table just looks like a regular table of numbers. However, there are tons of patterns buried inside of it.

For example, look at each row. If you take the common differences of each one, what do you find?

The very top row has a constant difference of one, as you would expect. The next row, which contains the triangular numbers, has a difference of all natural numbers. The 1 and 3 have a difference of 2, the 3 and 6 have a difference of 3, the 6 and 10 have a difference of 4, and so on. In other words, rows n-1 and n have a difference of n. This is the definition of triangular numbers, so that point is obvious.

The next row, which contains the square numbers, has a difference of all odd numbers. The 1 and 4 have a difference of 3, the 4 and 9 have a difference of 5, the 9 and 16 have a difference of 7, and so on. In other words, rows n-1 and n have a difference of 2n-1. If you want to see a quick proof of it, it isn't too hard. It just requires some algebra. Just do n squared minus n-1 squared, and you should receive 2n-1 to prove it.

n² - (n - 1)² = 2n - 1

n²  - (n² - 2n + 1) = 2n - 1
n² - n² + 2n - 1 = 2n - 1
2n - 1 = 2n - 1

If you look at the pentagonal numbers, it isn't as obvious what the pattern is. However, the differences are 4, 7, 10, 13, 16, and so on. The differences all differ by three, making the equation 3n-2. This can be proven using similar logic to above.

So, the first row has a difference of 1, which can be rewritten as 0n - (-1). The second row has difference of n, or 1n - 0. The third row has 2n - 1, the fourth row has 3n - 2. Do you see the pattern?

0n - (-1)
1n - 0
2n - 1
3n - 2

The fifth row of hexagonal numbers should have a difference of 4n - 3 if this pattern continues, right? Let's see if it worked.

1 - 0 = 1 = 4(1) - 3
6 - 1 = 5 = 4(2) - 3
15 - 6 = 9 = 4(3) - 3
28 - 15 = 13 = 4(4) - 3
45 - 28 = 17 = 4(5) - 3

The pattern indeed continued. For heptagonal numbers, the difference is 5n - 4, for octagonal numbers, the difference is 6n - 5, and so on.

Instead of looking at the rows this time, let's look at the columns and their differences. The first column has a difference of zero obviously. That is the nature of why one is in every figurative family. The second column has a difference of one. Since this number is the number defining the family itself, that isn't too surprising.

The third column has a difference of three. This is a little strange, but we can keep going. The fourth column has a difference of six. The fifth column has a difference of ten. Now do you see the pattern?

0, 1, 3, 6, 10

These are all triangular numbers! In fact, the triangular numbers have come back in the whole figurative family.

I find this table really cool to analyze. On June 15th (15 is a triangular number), we will revisit this table and look at some other neat identities within it.