To be watched as Carousels by clicking on the first picture of each collection. However, with a view to appreciating / grasping and understanding what you see, it may be best to open another browser and look at the “Titles of Collections” in parallel. As a “Catalogue of Concepts”, they act like ‘power points’ and feed your mind rather than your eyes...

I. “DECIMAL PYRAMIDS”: 1 – 5

1. Pyramids of Decimal and “Digital” Numbers

2. Pyramids of “Last Digit Series”

3. “Single Last Digit” Patterns

4. “Composite Last Digit” Patterns

5. “Digital Prime Diagonals”

II. “ARITHMETICAL MATRICES”: 6 – 9

6. “Diagonal Sums” of Odd and Even Summands

7. Prime Numbers as “Diagonal Sums“

8. “Digital Squares” of SemiPrimes

The Triangle (half a square) of Vertical x Horizontal Primes results in more attractive and “diagonally symmetrical” patterns:

9. The “Sequential Factorisation” of SemiPrimes

III. “DIGITAL TABLES”: 10 – 11

10. Primes and SemiPrimes

11. “Series of SemiPrimes”

SemiPrimes are the Product of Vertical Primes x Horizontal Primes or, in diagonal symmetry, Horizontal Primes x Vertical Primes.

This collection shows the SemiPrime Series resulting from Factors 7, 11 and 13 and where they fit into the Pattern of Primes:

IV. “ORTHOGONAL PYRAMIDS”: 12 – 13

12. Pyramid Shapes as Building Blocks of “Numerical Planes” for Numbers as Counts – (rather than Results of Arithmetical Operations or Quantifiers of Measuring Units)

13. Prime Numbers in “Numerical Planes” of “Orthogonal Pyramids”

V. “NUMERICAL PLANES”: 14 – 16

14. Odd Products in “Symmetrical Number Planes”

15. “Symmetrical Number Planes” for “Orthogonal Prime Axes”

16. The Ortho-Diagonal Nature of “Digital Sums”: Leading Digits x 10 + Last Digit

VI. SUMS and PRODUCTS: 17 – 21

17. “Digital Sums” in “Symmetrical Number Planes”

18. “Diagonal Sums” of Last Digits – in Orthogonal and Diagonal Chessboards

Last Digit SUMS are the ‘arithmetical nugget’ of ‘geometrically diagonal‘ Sums.
But there are a number of complementary concepts that need to be bridged
for ‘grand unification’:

Arithmetic: Odd versus Even, ‘Player‘ versus Result;
Geometry: Vertical versus Horizontal, Orthogonal versus Diagonal,
Visualisation: Label versus Content, Single Quadrant versus 4-Fold Symmetrical Plane, Orthogonal versus Diagonal Chessboard.

With focus on odd and even SUMS of odd and even DIGITS and NUMBERS, there are four cases of combination:

First: ODD Last Digits as Summands in Vertical Column and in Horizontal Rows:

• ODD + ODD = EVEN

Second: ODD Last Digits in Vertical Column and EVEN Last Digits in Horizontal Rows:

• ODD + EVEN = ODD

Third: EVEN Last Digits in Vertical Column plus EVEN Digits in Horizontal Rows:

• EVEN + EVEN = EVEN

Fourth: EVEN Last Digits in Vertical Column, ODD Last Digits in Horizontal Rows:

• EVEN + ODD = ODD

19. “Orthogonal Products” of Last Digits in “Orthogonal and Diagonal Chessboards”

Geometrically, sums are diagonal to express the multitude of combinations of their summands, i.e. the ‘plurality’ of their ‘value composition’ – in a diagonal line.

Products are orthogonal because every unique combination of 2 factors produces a rectangle that is unique in its shape, e.g. 24 = 3 x 8 = 4 x 6 = 2 x 12.

20. “Ortho-Diagonal Symmetries” in “Numerical Planes”

21. Diagonal Sums + Orthogonal Products => Ortho-Diagonal Primes

VII. PYTHAGORAS’ Triangles & Squares, Sums, Roots and Diagonals: 22 – 25

22. Pythagoras’ Sums of Squares as Triangles

Here are larger matrices to ‘intensify’ the impression the pattern is making.

23. Pythagoras’ Famous Equation: 3² + 4² = 5²

24. Pythagoras’ Sums of Squares in “Symmetrical Number Planes”

25. Last Digits of Pythagoras’ Sums of Squares

And now just the Last Digits in their individual uniqueness:

NEXT: Gallery – One Pattern at a Time