# 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...

# I. “DECIMAL PYRAMIDS”: 1 – 5

## 1. Pyramids of Decimal and “Digital” Numbers This Collection illustrates how the Decimal System depends on the Last Digits 0 and 5. 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” The LAST Digits of Decimal Numbers are Indicators for Odd or Even [1 3 5 7 9] and Prime or Non-Prime [1 3 7 9] as well as Multiples of 5 or 2 x 5 .

## 4. “Composite Last Digit” Patterns This Collection develops Combinations of Last Digits thus illustrating their unique Individual Role in the Overall Scheme of Odd and Even Last Digits

## 5. “Digital Prime Diagonals” One Last Digit leads to another, and one Pattern to another, so that Four Diagonals of Prime Numbers emerge with equal ‘Steps’ between the Primes: 2 Right, 1 Up to go BACK in Value or 1 Down, 2 Left to go UP in Value. What does this tell us about the ‘mechanism’ of Multiplication vs Addition? The Cells in Pyramids are simply Counts or Quantifiers!

# 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: The Intricate TAPESTRY of PRODUCTS created from PRIME NUMBERS as FACTORSMost 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 This collection shows the ‘awkward overlap’ between ‘visible geometry’ and ‘invisible arithmetic’: the triangle (no 61) as half a matrix shows a diagonal of “PRIME SQUARES”. 62 and 63 show the “Prime Factors” next to the SemiPrime result, 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 This 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: Primes and SemiPrimes altogether. The Pattern is always right: Primes are given. SemiPrimes ‘happen’ to fit and are derived from a LookUp Table compiled from sorting Factorised SemiPrimes.

# 12. Pyramid Shapes as Building Blocks of “Numerical Planes” for Numbers as Counts – (rather than Results of Arithmetical Operations or Quantifiers of Measuring Units) The Top Down Pyramid becomes One HALF of Vertical and Horizontal Pyramids and A QUARTER of Orthogonal Pyramids in a “Symmetrical Number Plane”

# V. “NUMERICAL PLANES”: 14 – 16

## 14. Odd Products in “Symmetrical Number Planes” Number PLANES expand the Number LINE and offer 4-fold Symmetry. Looking at the fundamental NUMERICAL characteristics of quarks gives us the ‘match’ between Number as a model and Physics as ‘events’ or interactions between the tiniest of particles.

# 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. Left: Odd x Odd = Odd Products in Orthogonal Positions – Right: Extra Even x Even = Even in between’ Diagonal Products

# 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 The Difference of Numbers in an Arithmetic and Geometric Context:a Value as a Result of an Operation and the Visualisation of that Operation