Arranging of dots, which represent numbers, into shapes or geometrical figures are not an especial new phenomenon. These figures can be found as far back as Stone Age, where they used to decorate sacred stones, rock etc. These numbers were also very important to the Pythagoreans, who imparted numbers with specific characteristics and personalities. And they believed that these characteristics and properties could explain everything in the world. However the Pythagoreans were not the only ones who added mystic and divine powers to special numbers, we find these prevalent also among the Babylonians, the ancient Maya and most other ancient cultures.

Before we’ll tell about the different types of figure numbers, we’ll tell something about the underlying patterns of these numbers. Many people keep a small store of number facts in their heads. A gross 144, is a dozen dozen 12x12=12². 16, 32,48 and 64 is the numbers of ounces in 1 to 4 pounds, but it is also known as the powers of 2.

Anyone who works with numbers naturally acquires s stock of more or less useless facts. For example the number 9 are the last single-digit number and are also a perfect square, 3x3. Is this deeply significant? No, it is more of a coincidence.

The perfect squares are among the most familiar numbers….

We can hardly avoid spotting that the differences between them get steadily larger:

Moreover, the sequence of differences is simple – very simple. It is just the sequence of odd numbers. That leads to the question: "Is the sequence of difference always simpler than the original sequence??

Let’s look at the sequence for the cubical numbers.

The cubes increases more rapidly than the squares (naturally !!). We can check just how fast, as before, by writing down the difference between them…

The last line is the difference between the differences. It is also rising steadily, but just 6 at a time. As you see, there is an underlying pattern when we look at the figurate numbers. One could almost say that the numbers are controlled by

Triangular numbers are the natural numbers which can be drawn as dots and arranged in triangular shape: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, etc.

Of all numbers, 10 were held in greatest reverence by the Pythagoreans; the sum 1 + 2 + 3 + 4 = 10 were named tetraktys, "the holy fourfouldness", representing the four elements: fire, water, air, and earth.

Gnomons are the geometric numbers forming a 90^o degree angle. The number of dots in gnomons is always an odd number.

By adding additional gnomons the Babylonians created larger squares from which they discovered many interesting connections between numbers.

A square for formed these relationships….

1+3=2²=4

1+3+5=3²=9

1+3+5+7=4²=16

1+3+5+7+9=5²=25

1+3+5+7+9+11=6²=36

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc are figured in this way:

The sum of two consecutive triangular numbers are always a square number.

3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25, etc

Tn + Tn – 1= n²

Pentagonal numbers are nothing more than numbers forming pentagons ( not very exiting, agree….)

We may add the scope of figurate numbers to three-dimensional space. We will then get the tetrahedral numbers, pyramidal numbers, cubic numbers etc.

To "build" cubic numbers, 1, 8, 27, 64, 125, 216 etc., we successively stack n*n squares n high.

There are several different tetrahedral figures. They can be found with trinagular, square og polygonial "bottom". To "build" a triangular tetrahedral number, we can just stack n successive triangular numbers starting with 1. Then each new "level" will be next triangular number in the line. The square tetrahedral is the sameas the triangular one execpt that it is a stack of squares instead. However we are very sorry we don´t have any pictures of them.

We can easily extend the scope of figurate numbers, just by adding another dimensions. However these figures impossible to visualize (so we are not going to show any pictures of them). Here is table of some of the numbers we have looked at :

1-D the n-th counting number

2-D triangular and square numbers

3-D tetrahedral and cubic numbers

and we can go on to

4-D the super tetrahedral numbers

and on

5-D were the figurate numbers haven’t been named

and on

and on... because there will always be another dimension.