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Cubes for adults are not toys


The peculiarity of the periodic system of elements offered below is, that the form is born as a result of certain operations with numbers and figures based on operations of symmetry. So the child playing with cubes (lego), creates architectural and other designs.

An impulse for the present work was the find discovered at manipulations with the famous Pascal's Triangle*. (As I learnt later, such manipulations relate to the method of finite differences). 

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

1    1
1    2    1
1    3    3    1
1    4    6    4    1
1    5    10   10   5    1
1    6    15   20   15   6    1
1    7    21   35   35   21   7    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.

The numbers can be arranged a little differently. Let's turn (rotate) the array so that units of one side of the triangle are posed in the first column, and units of the second side of the triangle are posed in the first line of the table.

The table in the "rectangular" form was published in " Trattato di numeri et misure", 1556-60; ("Treatise on Numbers and Measures"), issued partially only after death of the author, which was outstanding Venetian mathematician Niccolo Tartaglia. The Pascal's Triangle appeared in the Blaise Pascal's work "Traitise du triangle arithmetique avec quelques autres petits traites sur la meme matiere" (written 1654, printed 1665 also after death of the author)[3].

In our work such principle of arrangement of numbers in the tables is used  which wasused by Tartaglia and Pascal.

Figure 7. Pascal's Triangle (Tartaglia's Table). 

Many patterns are found in the Triangle making it an intricate and complicated mathematical device. Some  of them  will be necessary  for us. 

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

natural numbers (n), 


triangular (n)  numbers, and

tetrahedral (n) numbers.

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

** Figurate numbers:

Triangular numbers 1, 3, 6, 10, 15, 21, etc., were visualized as points or dots arranged in the shape of a triangle.

Square numbers are the squares of natural numbers, such as 1, 4, 9, 16, 25, etc., and can be represented by square arrays of dots. Inspection reveals that the sum of any two adjacent triangular numbers is always a square number.

Oblong numbers are the numbers of dots that can be placed in rows and columns in a rectangular array, each row containing one more dot than each column. An oblong number is formed by doubling any triangular number.

The gnomons include all of the odd numbers; these can be represented by a right angle, or a carpenter's square. Gnomons were extremely useful to the Pythagoreans...

Polygonal number series can also be added to form threedimensional figurate numbers; these sequences are called pyramidal numbers.


'... But the gentleman dressed in white paper leaned 
forwards and whispered in her ear, 'Never mind what 
they all say, my dear, but take a return-ticket every 
time the train stops.'
Lewis Carrol,  "Through the Looking Glass"

If Alice follows the advice, then with the equal price for travelling between stations the full amount paid for the tickets at each train stop will make the triangular numbers series.

On the other hand the Pascal triangle contains famous Fibonacci numbers (1, 1, 2, 3, 5, 8...), which have direct relation to the phyllotaxis mentioned above.

Figure 8. Pascal's Triangle, Fibonacci numbers and combinatorics. Fibonacci rows in 
the table are shown only in one of the two directions symmetric relatively to the bisectors 
of the angles of Pascal triangle. The phyllotaxis regularity is associated with 
Fibonacci numbers (the quantity of spirals of forming elements on plants).

For combinations, k objects are selected from a set of n objects to produce subsets without ordering. The number of such subsets is denoted by nCk, read "n choose k."

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


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