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






It also shows Horizontal ROWS for SERIES of Decimal Numbers, Vertical COLUMNS for ODD & EVEN Numbers as well as a DIAGONAL of Square Numbers with their ODD and EVEN Roots – just by Counting Cells in a Pyramid Shape
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“




Arithmetically, every possible Sum is formed from all possible Summands.
8. “Digital Squares” of SemiPrimes
The Triangle (half a square) of Vertical x Horizontal Primes results in more attractive and “diagonally symmetrical” patterns:






Most intriguingly, only possibly comparable to the Strings of DNA, the Prime Factors with Last Digits 1 3 7 9 form this “Pattern of Regular Irregularity”: Products of Primes that act as “Vertical Twin Partners” in the ORTHOGONAL Pattern of Primes
9. The “Sequential Factorisation” of SemiPrimes







just as 64 – but ‘tightened up’: no empty lines for Non-Primes.
65 shows SemiPrimes as “Prime Pair Partners” in the format of a “DIGITAL Table”.
66 shows the UNSORTED List and 67 the List of SORTED SemiPrimes,
using Prime 1 as secondary Sorting Criterion.
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)








and A QUARTER of Orthogonal Pyramids in a “Symmetrical Number Plane”
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.






Right: Additional Odd x Odd Products along Diagonals


Right: Additional Odd x Odd = Even Products in Ortho-Diagonal Positions
20. “Ortho-Diagonal Symmetries” in “Numerical Planes”



Horizontal Top ↕ Bottom
Orthogonal + Diagonal = Ortho-Diagonal


in Symmetrical Number 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








And what does Factor 5 reveal?
Here are larger matrices to ‘intensify’ the impression the pattern is making.


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




a Value as a Result of an Operation and the Visualisation of that Operation
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:






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