### Numerology

March 2, 2015

The

Greek philosopher,

Pythagoras, is known to everyone for the

Pythagorean theorem, but his greatest impact in his own time was as founder of the

religious movement,

Pythagoreanism. Little is known about this

pre-Socratic philosopher, but it's thought that disciples of Pythagoreanism believed that

numbers were the ultimate

reality. This idea that reality is different from what we observe is a feature, also, of the philosophy of

Plato, and it's conjectured that Plato was influenced by Pythagoras.

| *Pythagoras (Πυθαγορας)* c. 570 - c. 495.
(From Mathematicians and astronomers: twenty portraits, engraving by J.W. Cook, 1825, via Wikimedia Commons.) |

Pythagoras had the idea that 10 was the most perfect number, possibly through examination of his

fingers and

toes.

Mathematicians have their own

perfect numbers, which are numbers whose value equals that of all their

divisors excluding themselves. Thus, ten is not a perfect number in today's sense, since 1 + 2 + 5 ≠ 10, but six is (1 + 2 + 3 = 6). Whether or not there are an

infinity of perfect numbers is an

unsolved problem in number theory.

The Pythagoreans represented the perfection of ten by a graphic, called the

tetractys (τετρακτυς), as shown below.

There was a lot of

symbolism in the tetractys aside from the number of points adding to ten. The last row of four relates to the

four classical elements,

Earth,

Air,

Fire and

Water.

As you can see from the figure, the tetractys acts as a

template for the

coefficients of the

polynomial expansion of (x + 1)

^{n}; for example, the last row represents (x + 1)

^{3} = 1x

^{3} + 3x

^{2} + 3x

^{1} + 1x

^{0}. This

triangular array of coefficients is known as

Pascal's triangle, named after the

French mathematician,

Blaise Pascal. The coefficients are series

A007318 in the

On-Line Encyclopedia of Integer Sequences. I wrote about Pascal in a

previous article (Blaise Pascal, June 19, 2012).

Physics is typically the realm of numbers with

dimensions that are mixtures of

powers of

length,

mass,

time, and sometimes

charge. That's why the predominant system of measurement before the

International System of Units was "

MKS," for meter-kilogram-second. However, it's possible to arrange

fundamental physical constants to yield

dimensionless numbers. One example of this is the fine structure constant, denoted by the

Greek letter,

**α**.

The fine structure constant is related to some rather fundamental things; namely, the

elementary charge **e**, Planck's constant **h**, the speed of light **c** and the mathematical constant **π**,
α = (2 π e^{2})/(h c)

This constant, which is quite close to the reciprocal of 137 (~1/137.036), expresses the strength of the

interaction of charged particles. It's fundamental to

electromagnetism, and it combines those three fundamental physical constants, **e**, **h** and **c**. At one time, its calculated value was very close to 1/136. Now, it's very close to 1/137.
Eminent physicist,

Arthur Eddington, thought that there was something "magical" about this number, and he proposed some

*a priori* reasons why it should have that value. Eddington continued to investigate dimensionless numbers, deriving an

*a priori* value for the number of

protons in the

universe. This number, now called the

Eddington number, is calculated to be 136 x 2

^{256}, or 137 x 2

^{256}, or about 10

^{80}.

Paul Dirac made similar observations.

To most physicist of the time, all this seemed to be a retreat, from the way modern physics is practiced, to the philosophy of

Aristotle. Some compared it to the ancient practice of

numerology. In fact,

Guido Beck,

Hans Bethe, and

Wolfgang Riezler published a

parody of this approach in the respected physics

journal,

Die Naturwissenschaften,[1-2] in which they related the dimensioned value of

absolute zero temperature, T

_{0} = -273 °C, to the fine structure constant; viz.,

T_{0} = -(2/α - 1)

Since many people still believe in "

lucky numbers," (not to be confused with a mathematician's

lucky numbers) it's interesting to see how words are transformed into single

digits. You can associate the digits, 0-9, with

letters of the alphabet, as shown in the table. All this is rather arbitrary, but we're not operating in the realm of

science when we do numerology.

We can then add together the numbers associated with every letter in a word. Taking

*Tikalon* as an example, 2 + 9 + 2 + 1 + 3 + 6 + 5 = 28. If we're interested in just a single digit, we do repeated partitions and sums to yield 2 + 8 = 10, and 1 + 0 = 1. Thus, Tikalon's "lucky number" is one. As a scientist, I'm not inclined to play 111 in the

New Jersey State Lottery; or, for that matter, play the

lottery at all.

In a recent paper on

arXiv, mathematicians

Steve Butler,

Ron Graham, and

Richard Stong examine this "partition and sum" process. They treat a more general case in which just a single plus sign is inserted into the number, or as many as one plus sign between every digit, and they consider numbers written in

binary (base-two) format. They are able to show that very few

iterations of such operations are needed to get down to the single digit, one.[3]

In examining the

bound on how many steps it takes to get a binary number down to 1, they observe that inserting as many plus signs as possible (that is, summing all the digits) for a number n results in a new number of the order

log(n). The problem then reduces to how long it takes for log(log(· · · (log(n)) · · · )) to get to a value less than one. As you can see, you might need an

infinite number of iterations, but the number of iterations grows very much more slowly than n.[3]

### References:

- G. Beck, H. Bethe, and W. Riezler, "Remarks on the quantum theory of the absolute zero of temperature," Die Naturwissenschaften, vol. 2 (1931), p. 39.
- Ben Weiner, A parody paper in solid state physics, published in 1931," Tohoku University Web Site, April 25, 1997. This is an English translation of ref. 1.
- Steve Butler, Ron Graham, and Richard Stong, "Partition and sum is fast," arXiv, January 14, 2015.