# Figuring Numbers

Introduction

Gematria is founded upon correlations between the meanings carried by words and their numerical values, calculated under various systems of alphabetic numeration. The numerical relationship between words may be a simple equality or the sharing of important factors. However, in some cases the link is based upon the numerical properties of geometric figures. 

Regular arrangements of objects such as coins on a surface can give rise to representations of plane figures such as the triangle, square, pentagon and hexagon. Similarly, spheres or cubes can be regularly packed to build solid objects such as the cube, tetrahedron and pyramid. Ever larger versions of these figures can be created, therefore each geometric figure is associated with a series of numbers. All such numbers are described as figurate.

Units can be packed in different formations, each lending itself to the creation of characteristic shapes. We begin with the most natural formation: triangular packing.

Plane Figures: Triangular

Units on a surface can be packed in triangular formation to give a series of regular figures. The first seven members of the series of triangular numbers are illustrated below. All series of figurate numbers begin with a single unit, the second member of each series always being the first to show its characteristic shape. Each triangle in the series is given a positional value. Triangle 10 is the fourth in the series and can alternatively be referred to as T(4).

The series of triangular numbers is created by simply adding the natural numbers in order, something that can be intuited by observing that the positional value of any triangle in the series is the same as the number of units along its base. The formula for calculating the nth triangular number is as follows: T(n) = 1/2n(n + 1).

Three more regular geometric figures that can be created from triangular packing are the hexagon, hexagram and trapezium. Each are associated with their own series of numbers. Hexagon 19 and hexagram 37 are closely related to triangle 28, because they are produced by its intersection with an inverted copy of itself. Every third triangle can undergo this process, which can be repeated to create approximations of the Koch snowflake. Trapezium 22 is also obviously related to triangle 28, but less obviously to triangle 66, which can be trisected into three such trapezia .

Plane Figures: Square

Units can pack in square formation to give more plane figures: principally the square, the diamond and the octagon (note that this is not a regular octagon, which has its own numerical series). Diamond 25 and octagon 37 can both be derived from square 49. Every odd square will yield a diamond in this way, and every third square will give an octagon.

Plane Figures: Pentagonal and Hexagonal

Units can be arranged in pentagonal formation to give the series of pentagons, in hexagonal formation to create hexagons, and so on. Here are pentagon 22 and hexagon 28. Centred Plane Figures

Note that pentagon 22 and hexagon 28 extend outwards from an edge (in this case, from the top). Most of the figures I have shown so far do likewise. But centred hexagon 19 and diamond 25, shown earlier, grow out from a central unit and are called centred figures. Some more centred figures are shown next. Observe that the centred square is just the diamond rotated through 45 degrees. A figure of particular interest in the New Bible Code is the centred pentagram. The five points on the pentagram are slightly foreshortened, but it is nevertheless a regular plane figure.

Centred Pentagram 151 Solid Figures

Regular solids, such as the cube, tetrahedron and square pyramid can similarly be represented by tightly packed arrangements of spheres or cubes. As with plane figures, all solids are associated with their own numerical series. Polyfiguracy

Some numbers can be represented by two or more regular shapes. These are known as polyfigurate numbers. If two shapes can be created, the number is bifigurate. A very few numbers, such as 37 and 91, are trifigurate. Structural Properties

All plane and solid shapes of any complexity have structural properties. Some basic structural properties of the triangle and the cube are shown here.  Series of Figurate Numbers

The first 16 terms of some regular plane figures and solids are shown in the table. The table below shows the approximate percentage of numbers that are figurate in two or three dimensions for three ranges. The proportion of figurate numbers decreases as the range increases, so there will be less likelihood of large numbers being figurate, and therefore more significance to the phenomenon. Bill Downie 17/5/04

Latest update 25/9/12

Notes

1.By 'geometric figures' I mean regular plane and solid figures (polygons and polyhedrons), including fractal snowflakes and star figures, that can be created by tightly packed arrangements of regular units. I now include figures packed in triangular, square, pentagonal and hexagonal formation and all centred figures, although I have excluded higher dimensional figures. I may include more figures in future pages.

2. Trapezia of different heights can be constructed, but the one shown has equal numbers of units on the top and side and could therefore be said to be regular. It is numerically equal to the pentagonal numbers.