The mathematical structure of the Periodic System Table 

Dipl.ing. Aco Z. Muradjan

Email: maks06@gmail.com



The Periodic Table with a new double numerical structure, presented here is an attempt to find a table form which will in some new way represent the periodicity and symmetry of the Elements, with the Periodic System its base. Also, this tetrahedral laminar table structure may someday become a base for developing a new shell structure of the atomic nucleus.

 This new rearrangement of the chemical element is based on a mathematical formula whose result is, simply, the length of the periods.

 Keywords: Periodic System, Table, Law, mirror, new, form, electron configuration, shell, period


The form or the shape of the Periodic Table, whose basis is The Periodic System of the Elements, has been a subject of many kinds of research all over the world. The fact that there are a hundred and more different forms or shapes of the Periodic Table signifies that the final shape of the Table isn’t reached yet.

As it is well known, the length of the periods in the present standard Periodic Table doesn’t have a mathematical basis. The quantum mechanical explanation of this seven periods’ table is based on the shell model with four quantum numbers n, l, ms and s and Aufbau principle (Madelung rule) based on n and l quantum numbers. 

The maximum number of electrons in the periods is N = 2n2 (N = 2, 8, 18, 32,), where “n” is a principal quantum number. But the seven periods’ table has a different order, i.e. 2, 8, 8, 18, 18, 32, which doesn’t correspond to this formula, and which also indicates that something is wrong with this period order.


New model

With some rearrangements of the order of the periods and elements in the periods, whose order corresponds to the reality or to the electronic configuration, a different order will be obtained (N = 2, 2, 8, 8, 18, 18, 32, 32). Here each period has own mirrored structure.

A strict model or table which explains the physical and chemical properties of the elements based on their atomic number, electron configuration  and above order  is presented by a French scientist Charles Janet in 1929,


and also a similar table, but with a new modified notation scheme, different from the currently accepted numeration of periods and groups, is presented in Muradjan’s Periodic Table from 2001. Here, new marks are added for the shells with capital letters from K to R, a new quantum number for the shells ns and “spdf” notation scheme for the blocks.

The length of the shells (periods) in this table is also given with the formula N = Ns = 2n2 (2, 8, 18, 32,), where Ns is the length of the shells. The length of the same doubled mirrored shells (periods), or dyads according to Janet is expressed with the formula Nd = 4*n2 (4, 16, 36, 64,).

Here, this simple formula Nd = 4 n2 is taken as a basis for the new rearrangement of the elements. 

First number group          I - 1+4+9+16

Second number group    II - 1+4+9+16

Third number group       III - 1+4+9+16

Fourth number group     IV - 1+4+9+16


The Table which expresses this order only with numbers, same as the atomic numbers in the periodic table, is given below.


And a same table but with chemical element’s notation.


With this new rearrangement of the groups, the following tetrahedral laminar structure or form of the Periodic Table will be obtained.

There are four groups of the elements with 30 elements each.

I - Hydrogen group

II - Helium group

III  - Lithium group  and

IV – Berilium group


The aufbau principle for this table shape is a lemninscate build up principle.


 If the notation scheme from Muradjan’s Periodic Table is applied on the above Table the next form will be obtained.

Every element of each group from this table can be symmetrically pictured in all other three groups, up or down, left or right, or diagonally. For the chemical or physical properties, the left-right symmetry has a preferred role. But for the atomic nucleus structure, all symmetries are possible.

The color and enumeration of the above table are presented as doubled “spdf” scheme. This scheme is shown in the picture below.

Left-right step “spdf” scheme


The scheme presenter with the chemical elements in groups will show that the mirrored structure of the Periodic Table (the table below) is based on the left-right symmetry. 



Nature is full of different forms and shapes. The laws that describe these facts are based on several basic principles. One of them is symmetry. Up and down, left and right, back and forward, pair and odd, particle and antiparticle, parity or mirror symmetry and so on, are all samples of symmetry.

The Periodic System with its graphical presentation, i.e. the Periodic Table, is an example where the mathematical aspects and the symmetry are fully included. The present seven periods Periodic Table form doesn’t have such symmetry. The natural mathematical aspects of the Periodic Table’s structure presented here must be taken as a basis for a revision of the currently used seven periods’ Periodic Table.


20.09.2013  maks                                             © Dipl. ing. Aco Z. Muradjan

                           All rights reserved ® Copyright © MA 2013  


Katz, G. An Eight-Period Table for the 21st Century, Chem. Educator 2001, 6, pp324-332.

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During  One  Hundred  Years; University of Alabama Press: Tuscaloosa, 1974.

Eric R. Scerri, “The Periodic Table: Its Story and its Significance”, Oxford University Press, N.Y., 2007






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