Key words: The periodic law, the shell nuclear model, magic numbers, figurate numbers, Pascal's Triangle, the interdisciplinary approach.

D. I. Mendeleyev formulated the Periodic law in 1869. The shell nuclear model was first proposed by physicists J. Hans D. Jensen and Maria Goeppert Mayer working independently in 1949. Both these scientific approaches are united by the concept of Magic numbers^*.

Is there connection between magic and figurate numbers? If this connection exists, in what figures is it possible to combine magic numbers?

The present work written specially for VM magazine, represents an attempt to select graphic images, to deduce the analytical formulas for the mentioned scientific abstraction and to establish some interdisciplinary connections.
 

* Magic numbers in physics, in the shell models of both atomic and nuclear structure, any of a series of numbers that connote stable structure. They designate the sum of electrons in atoms or the sum of either protons or neutrons in nuclei that occupy completely filled, or closed, shells.

The magic numbers for atoms are 2, 10, 18, 36, 54, and 86, corresponding to the total number of electrons in filled electron shells.

The magic numbers for nuclei are 2, 8, 20, 28, 50, 82, and 126, corresponding to the total number of protons or neutrons in filled nuclear shells.

(Britannica)

**Among the many relationships of numbers that have fascinated man are those that suggest (or were derived from) the arrangement of points representing numbers into series of geometrical figures. Such 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)