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The Pythagorean View on Problems of Periodicity in Modern Science D. Weise International Society for the Interdisciplinary Study of Symmetry (ISIS) 121596, Russia, Moscow, Tolbukhina , Complex 7 – Building 1 – Suite 167 tel. : (095) 448 -13 -34, Е-mail: [email protected] ru

ABSTRACT The Pythagorean approach to numerical sequences in chemistry and nuclear physics has allowed offering geometrical analogies based on Figurate numbers (three-dimensional forms of the periodic Mendeleyev law and packing models of nuclei). The Pascal’s triangle is used to deduce the analytical equations. Atomic magic numbers through 12, 360 and nuclear magic numbers through 21, 400 are calculated. Key words The periodic law, the packing nuclear model, magic numbers, figurate numbers, Pascal's Triangle.

FIGURATE NUMBERS Figurate or polygonal numbers appeared in 15 th-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".

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

BLAISE PASCAL This triangle, though named after French mathematician Blaise Pascal (1623 – 1662), appears as early as the tenth century in Chinese mathematical scripts and probably is even older.

PASCAL’S TRIANGLE The entries in Pascal’s triangle are also known as binomial coefficients, and are given by: C 00 C 11 C 22 C 33 C 44 … Cnk = n! /(k! (n – k)!) C 30 C 41 C 52 C 63 C 74 … They are also known as the (n, k) binomial coefficients. C 10 C 21 C 32 C 43 C 54 … C 20 C 31 C 42 C 53 C 64 … C 40 C 51 C 62 C 73 C 84 … C 50 C 61 C 72 C 83 C 94 … … … …

GNOMON The connotation of the term gnomon is that originally given by Hero of Alexandria, namely, 'A Gnomon is that form that, when added to some form, results in a new form similar to the original. '

FIGURES & GNOMONS Some figures (white) and their gnomons (light blue)

NATURAL NUMBERS natural numbers appear in the second row and column, and are given by the formula: n = C n 1 = n

GNOMON OF THE n-th NUMBER The gnomon of the n-th number, in its familiar rows-of-balls manifestatio n, is one ball.

TRIANGULAR NUMBERS Figurate triangular numbers appear in the third row and column, and are given by the formula: n = C n 2 = (n 2 + n) /2

THE GNOMON OF THE n-th TRIANGULAR NUMBER The gnomon of the n-th triangular number, is a row of n balls.

TETRAHEDRAL NUMBERS Figurate tetrahedral numbers appear in the fourth row and column, and given by the formula: n = C n 3 = (n 3+3 n 2+2 n)/6

THE GNOMON OF THE N-TH TETRAHEDRAL NUMBER The gnomon of the n-th tetrahedral number is a ntriangular layer of balls.

FIBONACCI NUMBERS The sequence of Fibonacci numbers, 1, 1, 2, 3, 5, 8, . . . , in which each successive number is equal to the sum of the two preceding numbers. Fibonacci was the "greatest European mathematician of the middle ages“ (c. 1175 - c. 1240). His full name was Leonardo Pisano.

FIBONACCI NUMBERS Fibonacci numbers appear as the sum of the numbers ocated l along slanting lines. Fibonacci numbers appear also on plants patterns. Thus, Pascal’s Triangle is the cognitive bridge connecting wildlife with microcosm.

PERIODICITY OF ATOM PROPERTIES ATOMIC MAGIC NUMBERS In the atomic shell model, the shells are filled with electrons in order of increasing energy until they completely fill a closed shell, producing the inert core of a noble gas. The atomic magic numbers for atom 2, 10, 18, 36, 54, and 86 correspond to the total number of electrons in filled electron shells.

PASCAL'S TRIANGLE MODIFICATION “A” If the ones in the first column are replaced with twos, and if the 2 nd, and 3 rd columns are filled in according to the fundamental rule by which Pascal’s triangle is built up, there will be found: odd numbers, given by the formula: n = 2 n – 1

GNOMON OF A SQUARE Odd numbers have form of the gnomon of a square. The most familiar form of a gnomon is the L-shaped object of that name that serves as the pointer on most sundials.

SQUARE NUMBERS square numbers, given by the formula: n = n 2 Square numbers composed from Lshaped gnomons.

Magic numbers for atom in the Pascal’s Triangle modification

EXTENDED PERIODIC TABLE This periodic table is extended to include element 168. The elements 113, 115, 117, and above 118 have not yet been proven to exist, they have only generic names and symbols.

3 D MENDELEYEV's PERIODIC SYSTEM Two monochromatic buildingblocks posed one above another in every layer correspond to one electron orbital. A period corresponds to a horizontal layer. The number of building-blocks in such a twofold layer corresponds to the number of elements in the period and to the Principle Quantum Number n.

3 D MENDELEYEV's PERIODIC SYSTEM The top stratum with the thickness of 1 in each twofold layer corresponds generally to elements with uncoupled electrons. The bottom stratum corresponds to elements for which coupled electrons exist. Two monochromatic buildingblocks posed one above another in every layer correspond to one electron orbital.

3 D MENDELEYEV's PERIODIC SYSTEM 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.

RESUME 1 In terms of Pythagoreans an electron subshell is gnomon of shell. An electron shell is gnomon of all atom electronconfiguration.

ANALYTICAL REPRESENTATION OF ATOMIC MAGIC NUMBERS

ANALYTICAL REPRESENTATION OF NUMBERS OF ELEMENTS IN THE PERIODS for the numbers of the elements in n-th period: Pn = [(-1)n(2 n+3) + 2 n 2 + 6 n + 5]/4 generates numbers: 2, 8, 8, 18, 32, 50, 72, 98 … Recurrent form: Pn+5 = Pn+4 + 2 Pn+3 - 2 Pn+2 - Pn+1 + Pn

PREDICTED ATOMIC MAGIC NUMBERS The formula generates Atomic Magic Numbers: • 2, 10, 18, 36, 54, 86, • 118, 168, 218, 290, 362, 460, 558, 686, 814, 976, 1138, 1338, 1538, 1780, 2022, 2310, 2598, 2936, 3274, 3666, 4058, 4508, 4958, 5470, 5982, 6560, 7138, 7786, 8434, 9156, 9878, 10678, 11478, 12360, …

Recurrent sequence of the sixth order for inert gases: Zn+6 = 2*Zn+5 + Zn+4 - 4*Zn+3 + Zn+2 + 2*Zn+1 – Zn

PERIODICITY OF NUCLEI PROPERTIES NUCLEI MAGIC NUMBERS It is found that nuclei with even numbers of protons and neutrons are more stable than those with odd numbers. In particular, there are "magic numbers" of neutrons and protons which seem to be particularly favored in terms of nuclear stability: 2, 8, 20, 28, 50, 82, 126 Magic Numbers

PASCAL'S TRIANGLE MODIFICATION “B” The values of all numbers in the table correspond to doubled numbers of the conventional Pascal’s Triangle.

OBLONG PLANE NUMBERS Oblong plane numbers, given by the formula: n = 2 n = n 2 + n are the gnomons for pyramidal numbers

OBLONG PYRAMIDAL NUMBERS AND ITS GNOMONS Oblong pyramidal numbers, given by the formula: n = 2 n = (n 3+3 n 2+2 n)/3 and oblong plane gnomons

OBLONG PYRAMIDS IN ANCIENT ARCHITECTURE TEMPLE of the INSCRIPTIONS at Palenque Yucatan, Mexico

3 D REPRESENTATION OF NUCLEAR MAGIC NUMBERS The number of spheres changing each other is equal to magic numbers 2, 8, 20, 28, 50, 82, and 126. The segments of dark blue lines bridge the numbers, which sum is equal to the next magic number.

ANALYTICAL REPRESENTATION OF NUCLEI MAGIC NUMBERS The lowest values 2, 8, and 20 correspond to independent nucleon motion in a single particle harmonic oscillator potential. The figures for numbers 28, 50, 82, and 126 correspond to the same central potential but with strong spin-orbit coupling.

ANALYTICAL REPRESENTATION OF NUCLEAR MAGIC NUMBERS • A formula valid for all Nuclear Magic Numbers is: MNm= k * (m 2 - m) + (m 3 + 5 m)/3, IF m = 1, 2, 3 THEN ELSE k = 1 k = 0

PREDICTED NUCLEI MAGIC NUMBERS • The formula generates Nuclear Magic Numbers: • 2, 8, 20, 28, 50, 82, 126, • 184, 258, 350, 462, 596, 754, 938, 1150, 1392, 1666, 1974, 2318, 2700, 3122, 3586, 4094, 4648, 5250, 5902, 6606, 7364, 8178, 9050, 9982, 10976, 12034, 13158, 14350, 15612, 16946, 18354, 19838, 21400 …

RESUME 2 In terms of Pythagoreans a nuclei shell is gnomon of all nuclei configuration.

PERIODIC SYSTEM FOR KIDS

PERIODIC SYSTEM FOR KIDS

PERIODIC SYSTEM FOR KIDS

PERIODIC SYSTEM FOR KIDS

2 -D SYMMETRIC TRANSFORMATIONS The traditional long form of periodic system resulting from two-dimension symmetric transformations

This is the house that Jack built Shells Subshells K s This is the house that Jack built. p This is the malt that lay in s the house that Jack built. d This is the rat that ate p the malt that lay in s the house that Jack built. f This is the cat that killed d the rat that ate p the malt that lay in s the house that Jack built. g This is the dog that worried f the cat that killed d the rat that ate p the malt that lay in s the house that Jack built. L M N O This is the house that Jack built Conformity of filling of electron shells to some children's literary works by the example of a poem "This Is The House That Jack Built".

Ez az a ház, amelyet Jancsi épített Elektronhéj Alhéjak K s Ez az a ház, amelyet Jancsi épített. p Ez az a maláta, mely abban s a házban hevert, amelyet Jancsi épített. d Ez az a patkány, amelyik evett abból p a malátából, mely abban s a házban hevert, amelyet Jancsi épített. f Ez az a macska, amelyik megölte azt d a patkányt, amelyik evett abból p a malátából, mely abban s a házban hevert, amelyet Jancsi épített. g Ez az a kutya amelyik gyötörte azt f a macskát, amelyik megölte azt d a patkányt, amelyik evett abból p a malátából, mely abban s a házban hevert, amelyet Jancsi épített. L M N O Ez az a ház, amelyet Jancsi épített

Дом, который построил Джек Оболочки Подоболочки K s Вот дом, который построил Джек. p А это пшеница, которая в темном чулане хранится s в доме, который построил Джек d А это веселая птица-синица, которая ловко ворует p пшеницу, которая в темном чулане хранится s в доме, который построил Джек f Вот кот, который пугает и ловит d синицу, которая ловко ворует p пшеницу, которая в темном чулане хранится s в доме, который построил Джек. g Вот пес без хвоста, который за шиворот треплет f кота, который пугает и ловит d синицу, которая ловко ворует p пшеницу, которая в темном чулане хранится s в доме, который построил Джек. L M N O Дом, который построил Джек Соответствие заполнения электронных подоболочек атомов химических элементов композиции некоторых произведений детской литературы на примере стихотворения "Дом, который построил Джек".

Pyramidal composition Madonna and Child Painting with St Anne c. 1510, by Leonardo da Vinci Oil on wood, 168 x 130 cm Musée du Louvre, Paris

UNIFIED ELECTRON-NUCLEON MODEL