Star Numbers

“Star Numbers” are numbers which can be represented in the shape of "Centered Hexagrams" (Six Pointed Stars). These appear as concentric layers of dots, each layer being arranged in the shape of a hexagram, around a central dot.

Below are representations of the first few:

The different layers are represented here in different colors, so that each star is seen as containing the previous (for instance in the star of 121, is seen that of 73, itself seen as containing the star of 37, which contains the star of 13: the primordial star of 12 around 1).

The center is represented as amber; the first layer, as blue (with 2 shades, to distinguish the inner hexagon from the outer points), the next one as pink, then orange, then green…

The formula to determine a Star Number is:

6n(n-1)+1 for n=position from 1.

For instance, the first Star Number is: 6x1 (1-1) + 1 = 6 (0) + 1 = 0 +1 = 1

The second Star Number is: 6x2 (2-1) + 1 = 12 (1) + 1 = 12 + 1= 13

And so on…

The sequence of the Star Numbers can be seen as starting with 1 (which appears as the center of all the different kinds of “centered figurate numbers”), or with 13, which is the first number actually representable in the shape of a hexagram.

The formula for n=position from 13 is :

6(n²)+6(n)+1

So, the first “literal” Star Number is:

6 (1²) + 6 (1) + 1 = 6 (1) + 6 (1) + 1 = 6 + 6 + 1 = 13.

The second is:

6 (2²) + 6 (2) + 1 = 6 (4) + 6 (2) + 1 = 24 + 12 + 1 = 37.

And so on…

There is another formula for determining a Star Number for n= position from 13:

12 (T n) + 1

The symbol T stands for “Triangle”; the triangle of a number being the sum of all the numbers from 1 to that number.

For example, the 1st Star Number from 13 is:

12 (T 1) + 1= 12 (1) + 1 = 12 + 1 = 13

The second is:

12 (T 2) + 1 = 12 (1+2) + 1 = 12 (3) + 1 = 36 + 1 = 37

And so on…

This latter formula reflects another dimension of the Star Numbers: the Hexagram being composed of 12 Triangles (6 in the inner hexagonal portion, and 6 in the outer portion) around the center, as appears in this representation of the Star of 37:

In each hexagram are 12 triangles, the base of each being equivalent to the number of layers around the center.

So in the Star of 37, as we see in the above representation, each of the triangles is of base 2 (and 1 + 2 = 3, so 3 dots), the same as its number of layers, as we see again in the representation below.

Here is the list of the Star Numbers from 1 to 1633, the number of Letters within the representation of the Holy Verses as a six pointed star, with indications of their positions both from 1 and from 13:

9	433	8
10	541	9
11	661	10
12	793	11
13	937	12
14	1093	13
15	1261	14
16	1441	15
17	1633	16

Position from 1	Star Number	Position from 13
1	1
2	13	1
3	37	2
4	73	3
5	121	4
6	181	5
7	253	6
8	337	7