**NUMERICAL GEOMETRY - A
PRIMER**

**Preamble** - From a bag of coins of the same
denomination it is possible to construct a number of families of
simple symmetrical shapes. Each family is called a*
series*, and each member of a series is called a

Depicted below are the early terms of the 2-D, or * plane*,
series - some of which figure in

Clearly, in each case, the number of terms is without limit.
The position of any term in a series is referred to as its * order*;
this is also indicated by the number of counters forming a side.
The following table gives the order and numerical value
associated with each term of the series depicted above. For
completeness, the higher order values of X and Y have been
included.

Order | E/I | R/S | X | Y |

1 | 1 | 1 | 1 | 1 |

2 | 3 | 4 | 7 | 13 |

3 | 6 | 9 | 19 | 37 |

4 | 10 | 16 | 37 | 73 |

5 | 15 | 25 | 61 | 121 |

6 | 21 | 36 | 91 | 181 |

7 | 28 | 49 | 127 | 253 |

8 | 36 | 64 | 169 | 337 |

9 | 45 | 81 | 217 | 433 |

10 | 55 | 100 | 271 | 541 |

Before entering into the specifics of these plane
numerical structures it is important that we broaden our study to
include the 3-D, or * solid*, figurate
numbers that require for their physical realisation a supply of
uniform spheres or cubes. These include the cube, tetrahedron,
pyramid, and octahedron. However, let us begin by taking a closer
look at the 2-D figurate numbers - important in their own right,
but also the building blocks of these 3-D structures.

**The Triangle Series.**

It will be observed that two versions of this
series are depicted: the * equilateral*
(E) and the

(1) Using (a): 1+2+3+4+5+6+7+8+9 = 45; using (b): 9 x 10 / 2 = 45.

(2) Using (a): 1+2+3+4+...+35+36+37 = 703; using (b): 37 x 38 / 2 = 703.

We observe that moving from one term to the next higher in the series simply involves the addition of a further row of counters - one more than in the previous row.

Regarding perfection of form: an E-triangle has 3 axes of symmetry; an I-triangle, only 1.

An indication of the distribution of the Triangle Series in the sequence of natural numbers is given below.

Range | 1 - 10 | 1 - 100 | 1 - 1000 | 1 - 1000000 |

No of Triangles | 4 | 13 | 44 | 1413 |

**The Rhombus/Square Series.**

These series are identical in respect of the numbers involved.
The difference is visual and, as with the Triangle Series,
depends on the method used in packing the counters. Each term
after the first is observed to be the union of two adjacent
triangles. A given term may be calculated in one of two ways, (a)
by multiplying its position (ie order) by itself, or (b) by
forming the sum of an *odd number* sequence of the same
length. To illustrate, suppose it is required to calculate (1)
the 8th term, and (2) the 19th term.

(1) Using (a): 8 x 8 = 64; using (b): 1+3+5+7+9+11+13+15 = 64 (ie the total of the first 8 odd numbers).

(2) Using (a): 19 x 19 = 361; using (b): 1+3+5+7+...+33+35+37 = 361 (ie the sum of the first 19 odd numbers).

To move from a given term to the next higher in the series
requires the addition of an L-shaped * gnomon *of
counters - representing an odd number of the same order as the
resulting figure.

A rhombus has 2 axes of symmetry; a square, 4.

An indication of the distribution of these series is given below.

Range | 1 - 10 | 1 - 100 | 1 - 1000 | 1 - 1000000 |

No of Rhombus/Squares | 3 | 10 | 31 | 1000 |

**The Hexagon/Hexagram Series.**

These series are clearly related; they arise as * intersection*
(ie overlap) and

The Hexagon Series may be developed *per se* by
constructing a ring of 6 counters around the first term, a ring
of 12 around the second, a ring of 18 around the third, and so
on. Likewise, the Hexagram Series may be developed from the first
with rings of 12, 24, 36, and so on. It follows that each term in
these series may be viewed as a group of 6, or 12, triangles of
next lower order centred around a single counter. These analyses
are illustrated below..

On the left, we see the 4th hexagon of 37 counters with its equivalent of 6 triangles of order 3 (ie one less than that of the hexagon) set around the centroid counter. On the right, the corresponding hexagram of 73 is shown to be equivalent to 12 of the same triangles disposed about the centroid. Remarkably, these particular figures represent the factors of 2701 - the characteristic value of Genesis 1:1 and 73rd triangle. The symbolism depicted by these representatives of the Hexagon/Hexagram Series should not escape us: the 6/1 principle of the hexagon mirrors the Creation Week; the 12/1 principle, the Tribes of Israel and The Lord with His disciples!

The calculation of a particular term in either series is best
based upon the analyses depicted above. Simply multiply the
triangle of order *one less* by 6 (for hexagon), or 12
(for hexagram), and *add 1*. As an example, suppose we
need to know the 9th terms in the Hexagon/Hexagram Series.
Clearly, these figures are constructions based upon the 8th
triangle (= 36); hence X = 6 x 36 + 1 = 217, and Y = 12 x 36 + 1
= 433.

In general, hexagon and hexagram each possess 6 axes of symmetry.

Clearly, these numbers are less prolific than those already mentioned as the following summary reveals.

Range | 1 - 10 | 1 - 100 | 1 - 1000 | 1 - 1000000 |

Hexagon | 2 | 6 | 18 | 577 |

Hexagram | 1 | 4 | 13 | 408 |

**Symmetrical derivatives of the Square**

In the development of these figures the counters are better represented by unit squares.

**The Diamond (D) Series**

Each term in this series (after the first) is obtained by
removing 4 x I-triangles from the square having an order *twice*
that of the triangles, *plus one*. The first six terms,
together with the number of counters involved, are given below.

**The Octagon (O) Series**

This series is developed in the same manner as the foregoing
except that the order of the square is required to be *three
times* that of the triangles, *plus one. *Here are the
first six terms of this series:

Like the square, both diamond and octagon possess 4 axes of symmetry.

The following table indicates the distribution of these derivatives of the square.

Range | 1 - 10 | 1 - 100 | 1 - 1000 | 1 - 1000000 |

Diamond | 2 | 7 | 22 | 707 |

Octagon | 1 | 4 | 12 | 378 |

**Adding a Dimension**

The following table lists the first 10 terms of the solid
series: **Cube (C)**, **Gnomon (G)** -
ie the difference of adjacent cubes, **Tetrahedron (Q)**,
**Pyramid (P)**, and **Octahedron (H)**.

Order | C | G | Q | P | H |

1 | 1 | 1 | 1 | 1 | 1 |

2 | 8 | 7 | 4 | 5 | 6 |

3 | 27 | 19 | 10 | 14 | 19 |

4 | 64 | 37 | 20 | 30 | 44 |

5 | 125 | 61 | 35 | 55 | 85 |

6 | 216 | 91 | 56 | 91 | 146 |

7 | 343 | 127 | 84 | 140 | 231 |

8 | 512 | 169 | 120 | 204 | 344 |

9 | 729 | 217 | 165 | 285 | 489 |

10 | 1000 | 271 | 220 | 385 | 670 |

We observe, (a) that the G-Series is identical to the X-Series (listed earlier), (b) that, after the first, each term of the P-Series is the sum of the corresponding and preceding terms of the Q-Series, and (c) similarly, each term of the H-Series is derived as the sum of two terms from the P-Series.

The relationship between cube and corresponding gnomon is explained in the following figures:

Here, the counters take the form of unit cubes. However, for the remaining solids, uniform spheres are more appropriate. The tetrahedron may then be envisaged as a stack of consecutive E-triangles, beginning with the first; the pyramid, a similar stack of consecutive squares; and the octahedron, the union of two consecutive pyramids.

An assessment of the perfection of form of these solids is as
follows: the cube has 9 *planes of symmetry*; the gnomon,
3; the tetrahedron, 6; the pyramid, 4; and the octahedron, 9.

**The Phenomenon of Polyfiguracy.**

A consideration of the foregoing sections reveals the
interesting fact that certain numbers are * polyfigurate,
*ie they are associated with two, or even three,
symmetrical forms. Thus, 25 is both 5th rhombus/square and 4th
diamond, 36 is both 8th triangle and 6th rhombus/square, 64 is
both 8th rhombus/square and 4th cube, while 37 is 4th hexagon,
3rd hexagram, and 3rd octagon. This phenomenon is comparatively
rare as the following table makes clear. Indeed, in the whole
range of natural numbers, only two can claim the distinction of
being

Range | 2 - 10 | 2 - 100 | 2 - 1000 | 2 - 1000000 |

No of Figurates / (% of range) | 8 (89) | 40 (40) | 154 (15) | 4947 (0.5) |

No of Polyfigurates / (% of range) | 4 (44) | 14 (14) | 25 (2.5) | 61 (0.006) |

**Compound Figuracy**

The product of two or more numbers, all of which are figurate, may be represented graphically - the unit counters of the figures previously discussed now being replaced by symmetrical clusters, or clusters of clusters. As an illustration, let us consider 370, the product of 10 and 37:

Because 10 counters can be set out on a flat surface as an equilateral triangle, and the 2-D forms of 37 include hexagon and hexagram, their product, 370, can be represented as shown. Alternatively, triangular arrangements of 37's convey the same total, thus:

Clearly, many numbers will exhibit such polyfiguracy; thus, the phenomenon is not as remarkable in this context.

**Conclusion**

The earlier simple structures of numerical geometry lie at the heart of mathematics. Being universal, immutable, and completely independent of man, and of the symbols and methods he uses to represent number, they define an important class of empirical absolutes. The fact that the opening words of the Hebrew Bible and the Creator's name, as it is found in New Testament Greek, are underpinned by such structures must, therefore, be highly significant.

Vernon Jenkins

19.12.98

Modified: 28.12.98