The Pyramid Table. The Four Matrices; Top Level, Matrix; x = 1. Matrix; x = 2
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1 The Janet Periodic Table (first printed 1928) is also known as the Left Step Table. The Janet table orders the elements according to the filling sequence of the atom. This table may be re-organized as four square matrices. Each matrix is a different size. If the cells of each matrix are represented as a cube, then the matrices may be stacked to form a stepped Pyramid with four levels. Each level is a matrix. If each cell represents a chemical element, then the stepped pyramid becomes a 3D Table of Elements. Each level may be viewed as a floor plan (square matrix). It is also possible to view vertical sections cut through the pyramid to reveal vertical relationships of the elements. Each element is associated with a cube which has a location within the pyramid. Six location numbers are required to locate any cube. The six location numbers may also be associated with physical properties of the natural elements. The pyramid table is an interesting organization of the elements. The Four Matrices; The Janet table may be re-arranged as a set of four matrices. Each matrix is a different size. If the cells are represented as cubes, then the matrices may stack vertically with the four core cells in alignment. The result appears as a stepped pyramid. Each cell represents a chemical element identified by the atomic number (Z a ) shown below as the upper number in each cell. Each matrix is identified by a matrix number (x) which is also a level number in the pyramid structure. Top Level, Matrix; x = Matrix; x =
2 Matrix; x = Bottom Level, Matrix; x = The Location Matrix; The pyramid table is a rearrangement of the Janet Periodic Table into four square matrices. The location numbers (x,y,z,m x,m y,mz) define the location of any element cube within the pyramid. The location numbers may be associated with physical properties of the element. Location numbers may be grouped as a location matrix ; x y z m x m y m z February 1, 2018 Page 2
3 Where; x gives the level (matrix number) y gives the ring number (concentric square rings) z gives the displacement within a ring from the nearest corner cube m x gives the upper or lower half of any matrix m y gives the left or right half of any matrix m z gives the displacement (clockwise, ccw) within any ring Cell Location; Any cell (or cube within the pyramidal structure) may be identified by the six location numbers. Each matrix is identified by a matrix number or level number (x) ; x = 1,2,3,4 x = 1 represents the top level (smallest matrix) x = 4 represents the bottom level (largest matrix) A matrix is composed of a 2x2 core surrounded by concentric square rings. The core and each ring are identified by a ring number (y); y = 0,1,2,3, y min = 0, y max = x-1 The core is; y = 0 Sequential cells within a ring are identified as a displacement from the nearest main diagonal (corner cube). Each cell is identified by a displacement number (z); z = 0,1,2,3, z min = 0, z max = y If the cube lies on a major diagonal then; z = 0 A matrix is divided into two vertical portions. A half matrix is identified by a vertical portion number (m x ). The upper half is; m x = -½ The lower half is; m x = +½ February 1, 2018 Page 3
4 A matrix is also divided into two horizontal portions. A half matrix is identified by a horizontal portion number (m y ). The left half is; m y = -½ The right half is; m y = +½ A cube which is not located on a major diagonal is displaced from the corner cube within a ring. The direction of displacement (m z ) takes values; Clockwise displacement is; m z = +½ Counter clockwise displacement is; m z = -½ Location Functions; The location functions (F n ) are functions of the location numbers. They are defined as; F x = ⅔x(x+1)(2x+1) + 2x 2 (m x - ½) F y = (m y - ½) - 2y(y m y ) F z = 2zm z Each element may be represented by atomic number. The element is located within the pyramid by the location numbers. The atomic number must be related to the location numbers. The sum of location functions gives the atomic number (Z a ); Z a = F x + F y + F z Location Calculations; The following examples are calculations for different elements. Cerium; Z a = 58 Location matrix; ½ -½ -½ Location functions; F x = ⅔x(x+1)(2x+1) + 2x 2 (m x - ½) = ⅔(4)(4+1)(8+1) + 2(4 2 )(-1) = = 88 F y = (m y - ½) - 2y(y m y ) = (-1) - 6(4 + ½) = -28 F z = 2zm z = 2(-1) = -2 February 1, 2018 Page 4
5 Atomic number; Z a = F x + F y + F z = = 58 Krypton; Z a = 36 Location matrix; ½ +½ +½ Location functions; F x = ⅔x(x+1)(2x+1) + 2x 2 (m x - ½) = ⅔(3)(3+1)(6+1) + 2(3 2 )(-1) = = 38 F y = (m y - ½) - 2y(y m y ) = (0) - 2(2 - ½) = -3 F z = 2zm z = 2(½) = 1 Atomic number; Z a = F x + F y + F z = = 36 Alumimium; Z a = 13 Location matrix; ½ -½ -½ Location functions; F x = ⅔x(x+1)(2x+1) + 2x 2 (m x - ½) = ⅔(2)(2+1)(4+1) + 2(2 2 )(0) = = 20 F y = (m y - ½) - 2y(y m y ) = (-1) - 2(2 + ½) = -6 F z = 2zm z = 2(-½) = -1 Atomic number; Z a = F x + F y + F z = = 13 February 1, 2018 Page 5
6 Slicing; A 3D periodic table may be created if each cell of a matrix is represented as a cubic block, and the matrices are stacked vertically. Matrix01 (x=1) is the top level and Matrix04 (x=4) is the base. The resulting 3D table resembles a stepped pyramid. Vertical slices through the structure reveal vertical relationships between the elements. Major slicing reveals all four levels of the structure. Minor slicing does not account for all levels and also reveals vertical relationships. Two major slices are shown below; Major Slice ( North View); La Sc Lu B Ga Tl H Na Rb Fr He Mg Sr Ra Ne Kr Rn Zn Hg Yb Major Diagonal Slice ( North-West View); U Nb Db Si Sn Fl Li K Cs 119 He Mg Sr Ra F Br At Ni Pt Ho Janet (Left Step) Periodic Table; The Janet PT is displayed below in two parts (A,B). Each cell represents a chemical element represented by the atomic number (Z), shown as the lower number. A cell also contains the orbital (nl) of the most significant electron, shown as the upper tag number. Each row has a common sum (n+l) of quantum numbers. Where; L = 0,1,2,3 = s,p,d,f and n = 1.8 February 1, 2018 Page 6
7 Janet PT (Part A); Janet PT (Part B); Conclusion; The Periodic Table may be represented in 3D as a stepped pyramid having four levels. Relationships of the elements may also be revealed by vertical slicing. February 1, 2018 Page 7
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