It was
while I contemplated the mathematical order of our
universe and the supremacy of human reason in grasping
it that my thoughts floated toward a formula I had
recently read of on deriving the radii of circles
inscribed into right triangles, a relationship of
significant utility in architecture and engineering, as
well as simple theoretical computations. I employed the
formula to obtain the radii of the 3,4,5 triangle, which
happened to be 1, then, the 5,12,13 triangle, which
happened to be 2. “Is there a pattern?” I wondered, and
delved into any potential numerical relationships
between the numbers of a fundamental Pythagorean triple
(that which cannot be further reduced and remain
consisting of whole number values) and the radii of
circles inscribed into triangles having sides
corresponding to these triples. What I discovered
sparked sheer astonishment and exhilaration within me.
Consider the following fundamental Pythagorean triples:
a=
shorter leg b= longer leg c=
hypotenuse Inscribed circle radius
3
4 5
1
5
12 13
2
7
24 25
3
9
40 41
4
The radius of a circle inscribed into a right triangle
is well known to be defined by the formula (a+b-c)/2.
The length of the shorter leg seems to increase with
every subsequent fundamental triple by two, with 3 as
the first term. Hence, length of the shorter leg may be
defined by the arithmetic series t(R)= 3+ 2(R-1)= 2R+1.
R in this case is the number of the given term. However,
this number also corresponds to the inscribed circle
radius.
Thus, the shortest leg of a fundamental Pythagorean
triangle equals twice the radius of the inscribed circle
plus one.
Moreover, a pattern can be detected in the progression
of B. The difference between b length of the second and
first triples is 8. The difference between the third and
second is 12. The difference between the fourth and
third is sixteen. For every subsequent triple, the
increase in difference between it and the last is four
or 1*4. Therefore, any length of b in a fundamental
Pythagorean triple can be defined as 4*(1+2+….+R), with
R being the number of the fundamental triple. This can
be expressed in series notation as 4k=1RΣk
or four times the sum of all consecutive natural numbers
up to R.
In all fundamental Pythagorean triples of this sort, it
is easily observable that the length of the hypotenuse,
c, equals b+1. Henceforth, c equals one plus four times
the sum of all consecutive natural numbers up to the
radius of the inscribed circle, or 1+4k=1RΣk. Now, we can also
establish a correlation between this fact and the fact
that a=2R+1. Since (a+b-(b+1))/2= R, 2R= a-1, a=2R+1.
Moreover, every fundamental Pythagorean triple is
associated with a particular R value, which must be a
natural whole number for the following reason: a=2R+1
and is a natural whole number, also an odd number for
all fundamental Pythagorean triples. 2R= a-1, which,
consequently, must be an even number, divisible by two.
Hence, R must be a natural whole number.
From the preceding algorithms, it is possible to derive
any fundamental Pythagorean triple knowing but the
radius of its inscribed circle, or the length of a
single side. Every fundamental Pythagorean triple shall
henceforth be associated with a number, dubbed the
radial number, equaling the length of the radius of its
inscribed circle. Fundamental Pythagorean triples exist
exclusively in association with such numbers, of the
arithmetic sequence n, and their occurrence is therefore
entirely periodic. Moreover, every positive integer
radial number has a Pythagorean triple associated with
it.
Hence, Stolyarov's Theorem:
Every possible integer radial number can be
associated with a fundamental Pythagorean triple
consisting of the lengths of the sides of the right
triangle into which a circle with a radius of the radial
number is inscribed. The relationship between the radial
number R and the three numbers of the triple is as
follows:
a= 2R+1
b= 4n=4k=1RΣk
c= 1+
4k=1RΣk
Beauty is
mathematical. I have written on that subject and proved
such a contention in my
Essay on the Genuine Meaning of
Beauty. Everything
sublime and inspiring about this vast world of ours is
such due to pattern, proportion, precision, and logic.
What I had elucidated here was a new mathematical
relationship between fundamental Pythagorean triples and
inscribed circles, an infinite new treasure chest of
ideas for artists, architects, and graphic designers to
explore. Every one of these proportions between radius
and triple is distinct and unique, yet each can be
fathomed and employed by means of three simple
algorithms. Here is my challenge to the artistic
community: integrate this property into your
endeavors! Build buildings with it, furnish
intricate visual scenery in painting and mosaics, or
perhaps, using a computer, expand its beauty and
complexity beyond the power of the unarmed human hand.
Tap its potential, and, instead of arbitrary whimsical
paradigm shifts, furnish yet another magnificent leap of
progress for Man in the noblest of aesthetic and
technical realms.
G. Stolyarov II
is a science fiction novelist, independent philosophical
essayist, poet, amateur mathematician, composer,
contributor to
Enter Stage Right,
Le Quebecois Libre,
Rebirth of Reason,
and the
Ludwig von Mises Institute,
Senior Writer for
The Liberal Institute,
and Editor-in-Chief of
The Rational Argumentator,
a magazine championing the principles of reason, rights,
and progress. His newest science fiction novel is
Eden against the Colossus.
His latest non-fiction treatise is
A Rational Cosmology.
Mr. Stolyarov can be contacted at
gennadystolyarovii@yahoo.com.
This TRA feature has been edited
in accordance with TRA’s
Statement of Policy.
Click here to return to TRA's Issue
X Index.
Learn about Mr. Stolyarov's novel,
Eden against the Colossus, here.
Read
Mr. Stolyarov's new comprehensive treatise,
A Rational Cosmology,
explicating such terms as the universe, matter, space, time,
sound, light, life, consciousness, and volition, at
http://www.geocities.com/rational_argumentator/rc.html.
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