**The
Pythagorean Approach to Problems of Periodicity **

**in
Chemistry and Nuclear Physics **

**D.
Weise**

**Key
words:**

*The
periodic law, the packing nuclear model, magic numbers, figurate numbers,
Pascal's Triangle.*

__Periodicity
of atom properties__

**Figurate
numbers **

Numbers,
known as figurate or polygonal numbers, appeared in 15th-century arithmetic
books and were probably known to the ancient Chinese; but they were of
especial interest to the ancient Greek mathematicians. To the Pythagoreans
(*c.* 500 BC), numbers were of paramount significance; everything
could be explained by numbers, and numbers were invested with specific
characteristics and personalities. Among other properties of numbers, the
Pythagoreans recognized that numbers had "shapes."(Britannica)

Pascal'
Triangle is an arrangement of numbers such that each number is the sum
of two numbers immediately above it in the previous row.

**Figure
1.*** Pascal's Triangle, though named after Blaise Pascal, appears
as early as the tenth century in Chinese mathematical scripts and probably
is even older.*

**C**_{n}^{k}**
= ****n! / (k! (n - k)!)**

This
is the same as the (*n, k*) binomial coefficient.

In columns and rows of the table starting from the second row we can find:

natural
numbers (**n = ****C _{n}^{1}
= n**,),

figurate:

triangular
(**n****= C _{n}^{2}=
(n^{2} + n)/2 **) numbers, and

tetrahedral
(**n****= C _{n}^{3}=
(n^{3} + 3n^{2} + 2n)/6**) numbers.

(The
signs , , , ,
*et
cetera*, used
in this work for designation of figurate numbers are not common).

The sequence of Fibonacci numbers, 1, 1, 2, 3, 5, 8, 13, . . . , in which each successive number is equal to the sum of the two preceding numbers.

If we replace ones in the first column with twos and build the 2nd, and 3th columns according to the fundamental rule, we will see:

odd
numbers (**n****
= 2n - 1**)
- (the sign
is chosen because of its similarity to the gnomon of square),

square
numbers (**n****
= n ^{2}**).

Let's
move apart lines of the table and in the third column above each square
let's write the same square. The fourth column (**E**) will be filled
according to the of Pascal's Triangle rule (1, 4+1=5, 4+5=9,
9+9=18, 9+18=27 and so on). In the fifth column let's enter double
numbers of the fourth column (**2E**).

It
is necessary to notice, that in the fourth column the numbers of electronic
pairs of atoms of inert, or noble, gases are written. In the fifth column,
accordingly, - there are charges of these atoms, or *magic numbers*
for atom.

*Figure
2.**The modification of Pascal's Triangle*

So, we have a 3D model of Mendeleyev's periodic system, to which it is possible to give the following explanations:

**Figure
3.****
3D Mendeleyev's periodic system**

1. Periods.

A period corresponds to a horizontal layer of the submitted animation. Thickness of each period makes 2 building-blocks.

The amount of building-blocks in such a twofold layer corresponds to the amount of elements in the period and to the Principle Quantum Number N.

The top stratum withthe thickness of 1building-block in each layer corresponds generally to elements with uncoupled electrons. The bottom stratum corresponds to elements, for which coupled electrons are available.

Two monochromatic building-blocks posed one above another in every layer correspond to one electron orbital.

2. Electron subshells.

The
elements, the properties of which are determined by outside electron subshells
**s,
p, d, f, g,** are grouped together in modules of individual colour.

**s ****-
red;
**

**p****
- green;
**

**d****
- dark blue;
**

**f****
- yellow;
**

**g****
(theoretically predicted) - lilac.
**

**Outcome
of the analytical approach to mathematical singularities of the periodic
law of elements**

The figurate-numerical approach allows for the expression of inert gases atoms charges by the algebraic equation:

**Z
= [ (-1) ^{n} (3n + 6) + 2n^{3} + 12n^{2} + 25n
- 6 ] / 12**

**n
= 1, 2, 3, ****... (period
number)**

For
magic numbers 2, 8, 20, 28, 50, 82, 126 the geometrical image was offered,
and the analytical formulas were found like it was made for atom.

**Figure
4. **Another modification of the Pascal's Triangle. The values of all
numbers in the table correspond to doubled numbers of the conventional
Pascal’s Triangle.

Oblong
plane numbers **n****
= 2n
= n ^{2}
+ n **;

oblong
pyramidal numbers **n****
= 2n
= (n ^{3}+3n^{2}+2n)/3**.

The amount of spheres changing each other, is equal to magic numbers 2, 8, 20, 28, 50, 82, 126. The segments of dark blue lines bridge the numbers, which sum is equal to the next magic number. Symmetry and regularity of the marked numbers arrangement in the table attract attention.

The lowest values 2, 8, and 20 agree with independent nucleon motion into a single particle potential, like a harmonic oscillator.

**MN _{m}= (m^{3}
+ 3m^{2} + 2m)/3**

**m = 1, 2, 3.**

The figures for numbers 28, 50, 82, and 126 agree, in addition to the central potential, witha strong spin-orbit coupling ( by Maria Mayer and Jensen) have the another form.

They are featured by the formula:

**MN _{m}= (m^{3}+5m)/3**

**m = 1, 4, 5, 6, 7 ...**

The combined formula is:

**MN _{m}= 0^{(m^2 - 5m
+ 6)} *
(m^{2} - m) + (m^{3}
+ 5m)/3**

**m = 1, 2, 3 ...**

Author was surprised and pleased to have found similar
diagrams
and formulas on several pages of Linus Pauling's notebooks
(for example, the images 102, 174, 181, 185, 207, 218, 229, 230, 233).

**References:**

1. The Encyclopedia "Britannica" on CD 2000 Delux Edition & WWW pages.

2. Linus Pauling. Research
Notebooks

http://osulibrary.orst.edu/specialcollections/rnb/index.html
Research Notebook 25.

3. http://www.isis-s.unsw.edu.au/interact/gallery/image_files/wiese/weise.html