**Appendix 4 - An
Introduction to Numerical Geometry**

Mathematics is
universally respected as a primary source of objective truth -
hence its distinguished appellation *Queen of the Sciences*.
In essence it is concerned with *number* and *form*
- concepts that are found uniquely combined in a particular class
of objects: the *figurate numbers. *The simplest of these
arises directly from the operation of *counting *by
reciting the sequence of natural numbers thus: "One, one
plus two, one plus two plus three, ...", so generating the
series (or sequence) of *triangular numbers* - 1, 3, 6,
..., without limit. A typical instance is seen in the game of
snooker where the 15 'reds' are arranged in close equilateral
triangle formation at the beginning of each frame. Here is a
depiction of this triangle:

Clearly, the threefold
symmetry of form exemplified here is a fundamental property of 15
- one that is completely independent of any other symbolism that
might be invented to represent this number, and one that is *direct*,
*universal* and *immutable*.

Two classes of the
figurate numbers may be recognised, viz * plane*
and

* Numerical
geometry* is that branch of mathematics concerned
with the study of such entities.

**Some
examples:**

(1) Here
are the opening terms of the plane series of *triangular
numbers*:

Observe
that the first term - represented by a single counter - is common
to all series, whether plane or solid (it is said to be *degenerate*
because it offers no clue as to the subsequent developments).
Here, it is followed by the second (2 more than the first), the
third (3 more than the second), the fourth (4 more than the
third), and so on - these appearing as successive rows in the
growing structure. Each term is uniquely designated by the number
of counters forming a side. Thus the 'snooker' triangle
(occupying the 5th position in the series) may be conveniently
designated T(5). Interestingly, T(4) is the *tetraktys* of
the ancient Greeks.

Such figures have 3 axes of symmetry. In general, T(n) = the sum of the numbers named when counting to n. Thus, for example, T(9) - the 9th triangular number = 1+2+3+4+5+6+7+8+9 = 45. Alternatively, T(n) = n(n+1)/2 = 9x10/2 = 45, as before.

The triangle series is fundamental to the study of numerical geometry because, ultimately, all other figurate series are derived from it.

(2) The
sum of any two adjacent terms of the triangle series yields a *rhombu*s.
This is demonstrated in the rhombus series depicted below.

As may be observed, a rhombus possesses 2 axes of symmetry.

(3) The
more familiar *square* series is derived from the latter
by a simple adjustment of the counter packing arrangement -
accompanied, maybe, by a more appropriate choice of counter
shape, thus:

Squares posses 4 axes of symmetry with S(n) = R(n) = n^2.

[Observe here the use of the uparrow (^) to signify 'to the power of'.]

(4)
Another form capable of completely filling a square frame is the *diamond*.

The number of counters forming each of these structures is obtained by doubling the product, n(n-1), and adding 1. Thus, D(n) = 2n(n-1) + 1.

(5) A
given numerical triangle may be united with an inverted copy of
itself to create a structure that has the form of an *hour-glass*.
Here are the opening terms of this series:

Such figures possess 2 axes of symmetry with H(n) = n(n+1) -1

(6) The
symmetrical truncation of every third term of the square series
results in a sequence of *octagons*, each of which has 4
axes of symmetry, thus:

The characteristic formula here is O(n) = 7n^2 - 10n + 4

(7) Before
introducing the *hexagon* and *hexagram* series
(each possessing 6 axes of symmetry) it is necessary that we
first develop a broader understanding of the triangle series.

Based upon
the nature of its geometrical structure a triangular number may
be classified under one of three headings , viz * 1-centre*,

Observe
that T(4) - *and every 3rd term thereafter* - has a single
central counter; T(5) - *and every 3rd term thereafter* -
has a central group of 3; and T(6) - *and every 3rd term
thereafter* - has a central group of 6. Clearly, proceeding
from one term to the next higher in the *general* series
simply involves the addition of a further *row* of
counters, whereas proceeding to the next higher in any of the*
3 groups *requires the addition of a complete *ring*
of counters.

The *1-centre*
triangles - as defined in the previous section - may be aptly
termed *generator* *triangles* for the simple
reason that each is capable of generating a symmetrical *hexagon/hexagram*
pair by self-intersection/union about the central counter. Here
are the early terms of the generator triangle series:

Again,
observe that the first term of the series is *degenerate. *The
first 3 generator triangles proper are therefore G(2), G(3) and
G(4) - corresponding to T(4), T(7) and T(10), respectively.
Clearly, the order of a G-triangle - as thus defined - is equal
to the number of *rings of counters* surrounding the
central counter *plus 1*.

As the
following diagrams reveal, every G-triangle proper may be
envisaged as comprising 9 congruent* elemental *triangles
(drawn from the general series) set symmetrically about the
central counter. Two colours have been used to make the principle
clear.

It is this feature that allows any G-triangle to function as the generator of a symmetrical hexagon/hexagram pair (X/Y) by self-intersection/union, as the following example shows.

Observe here that

as with G, the order number of the X/Y pair is

*one more*than that of the elemental triangle involved in their structures; thus, in the above example, the 4th X/Y pair involve the 3rd triangle of the general seriesthe number of elemental triangles in X is 6, and the number in Y is 12

It therefore follows that, if n be the order of the elemental triangle,

- G(n+1) = 9n(n+1)/2 + 1
X(n+1) = 6n(n+1)/2 + 1

Y(n+1) = 12n(n+1)/2 + 1

(8) Some simple
examples of figurate solids are *cube*, *tetrahedron*
and *pyramid*.

These comprise
stacks of unit cubes or spheres - each form being associated with
its own infinite series. Thus the tetrahedron may be envisaged as
a stack of consecutive triangular numbers; and the pyramid as a
stack of consecutive squares. The *perfection of shape* is
here expressed in terms of *planes* of symmetry - the cube
having 9, the tetrahedron 4, and the pyramid 6. The
characteristic formulae associated with these are as follows:

Cube: C(n) = n^3

Tetrahedron: Q(n) = n(n+1)(n+2) / 6

Pyramid: P(n) = n(n+1)(2n+1) / 6

Associated with the
study of these entities is the concept of *polyfiguracy*.
In other words, certain whole number may be expressed visually as
a symmetrical arrangement of counters in more ways than one.
Thus, D(4) = S(5) = 25; S(8) = C(4) = 64; X(3) = H(4) = 19; and
so on. Such numbers are said to be *bifigurate*. Just two
are observed to be *trifigurate*; they are 37 and 91.

Thus, Y(3) = X(4) = O(3) = 37:

and P(6) = X(6) = T(13) = 91:

Within the parameters defined, the author is not aware of instances involving a higher order of figuracy.

In general, it can be said of
the figurate numbers that such self-evident associations of
number and form represent precious and irrefutable outcrops of *absolute
truth* in a confused world.

Vernon Jenkins MSc

2006-01-30