**An
introduction to the triangular numbers**

**General
principles**

The *triangular
*numbers* *represent an important subset of the
infinite set of *positive integers* (alternatively known
as *natural*, *whole* or *counting*
numbers). A *triangular* number (here designated 'T') is
so called because, when expressed as a group of uniform circular
counters, it is capable of filling completely a frame having the
form of an equilateral triangle. The following diagrams explain
the principle.

Such numbers are simple derivatives of the natural numbers and of the counting process, "one, two, three, ..." where, instead, we say, "one, one plus two, one plus two plus three, ...". This is evidenced in the structure of each triangle. Each term in the series is uniquely identified by a number representing its ordinal position - this matching the number of counters forming a side of the related triangle.

Given the
order (*i*, say) of a triangular number, its value may be
calculated in one of two ways - either, (a) by adding together
the numbers named when counting to *i*, or (b) by forming
the product *i*(*i*+1) and *halving* the
result. To illustrate these alternative procedures, let us
suppose we need to calculate (1) the 7th term T(7), and (2) the
37th term T(37) of the series.

(1) Using (a): 1+2+3+4+5+6+7 = 28; using (b): 7 x 8 / 2 = 28

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

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
generator triangles (G)**

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. This subset of the triangular numbers is also
distinguished, (a) by the presence of all perfect numbers except
6 (the first) and, (b) as normally expressed in base 10 (ie *denary*
or *decimal*) notation, the digits of each term after the
first sum, penultimately, to 10. [Details of this matter may be
found here.]

Observe
that the first term of the series is *degenerate*, but is,
nevertheless, designated G(1). The first generator triangle
proper, G(2), is 10, the second 28, and the third 55 -
corresponding to T(4), T(7) and T(10), respectively. Clearly, the
order of a G-triangle is equal to the number of *rings of
counters* surrounding the central counter __plus 1__. This
principle becomes apparent when successive rings are displayed in
different colours, as in the next diagram.

Based on the foregoing diagram a number of self-evident features of G-triangles follow, viz

G(j) = G(j-1) + Outline(j); j > 1

Outline(j) = 9(j-1); j > 1

G(j) = 9j(j-1)/2+1; j > 0

G(j) = T(3j-2); j > 0

The centroid counter (represented in white) occurs as the jth element in the jth row (as counted from the base)

Additional
properties are revealed by following a different train of
thought. As the following diagrams reveal, every G-triangle 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

the order of an elemental triangle is

__1 less__than that of the corresponding G-triangleboth X and Y take their order from that of the related G- triangle

the number of elemental triangles in X is 6, and the number in Y is 12

Vernon Jenkins MSc

2003-08-08