# Pattern Collections

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...

# II. “ARITHMETICAL MATRICES”: 6 – 9

## 8. “Digital Squares” of SemiPrimes

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

# III. “DIGITAL TABLES”: 10 – 11

## 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:

# VI. SUMS and PRODUCTS: 17 – 21

## 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.

# 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.

## 25. Last Digits of Pythagoras’ Sums of Squares

And now just the Last Digits in their individual uniqueness: