**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“

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

**Digital Pattern of Primes**is composed from Leading and Last Digits: Diagonally Staggered “Prime Partners” are Vertical Pairs of Primes and SemiPrimes – where the SemiPrimes are Predictable Products of Prime1 x Prime2. The Pattern is always Right!

## 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”

*digital*rather than

*algebraic*terms, #PrimeNumbers become attractive decoration as #AmazingColourPatterns rather than a problem…

# 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 *complementar*y *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

*Orthogonal*Positions – Right: Additional Sums of EVEN Value in

*Diagonal*Positions

**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

*Orthogonal*Positions – Right: Additional Sums of EVEN Value in

*Diagonal*Positions

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

- EVEN + ODD = ODD

*Orthogonal*Positions – Right: Additional Sums of ODD Value in

*Diagonal*Positions

## 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.

*Orthogonal*Positions – Right: Extra Even x Even = Even in between’

*Diagonal*Products

*Orthogonal*Positions

Right: Additional Odd x Odd Products along

*Diagonals*

*Orthogonal*Positions

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: 3^{2} + 4^{2} = 5^{2}

*Arithmetic*and

*Geometric*Context:

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: