# Titles of Collections

As a Table of Contents or Catalogue of Concepts – possibly best viewed in a second browser window – to accompany the Pattern Collections:

# I. “DIGITAL PYRAMIDS”

1. Pyramids of Decimal and “Digital” Numbers
2. Pyramids of “Last Digit Series”
• “Last Digits” form Vertical and Horizontal Series of 5 Digits
• Diagonally, [Top Right to Bottom Left] we see Square Numbers] ending with Last Digits 0 1 4 9 6 5 6 9 4 1 0 with 5 acting as ‘mirror’
• Symmetrically, we see “Squares – 1” with 5-Digit-Series that are mirrored by either 5 or 0 [Top Left to Bottom Right Diagonal]
3. Single “Last Digit Patterns”
• Fascinating, how every Digit reveals its own Pattern
4. Composite “Last Digit Patterns”
• Different ‘digital combinations’ form different Patterns
5. Digital Prime Diagonals

# II. “ARITHMETICAL MATRICES”

1. “Diagonal Sums” of Odd and Even Summands
• Sums can be visualised as Diagonals and on Parallel Diagonals
2. Prime Numbers as “Diagonal Sums”
• When Odd Numbers can do it, Primes can do it, too!
3. “Digital Squares” of SemiPrimes
• What do SemiPrimes show us in their Matrix of “Prime Products”?
4. The “Sequential Factorisation” of SemiPrimes
• A bit tricky to produce the Products first and then a sorted sequence as a LookUp Table – but I’ve moved from coding to designing code – so I have to do it ‘by hand’ – who will code it beautifully?

# III. “DIGITAL TABLES”

1. “Digital Tables” of Primes and SemiPrimes
• Primes are Sums; SemiPrimes are Products; Together they form “Prime Pairs” when aligned by Leading and Last Digits
2. “Digital Tables” for “SemiPrime Series”
• Turning SemiPrimes products into a LookUp Table for humans or computers is greatly helped when we SEE the underlying patterns

## IV. “ORTHOGONAL PYRAMIDS”

1. Orthogonal Pyramids for Numbers as Counts
• Vertical and Horizontal Pyramids doubled and mirrored
2. Prime Numbers in Orthogonal Pyramids
• Patterns of Non-Patterns or ‘regular irregularity’?
3. Fourfold Pyramids for Primes and Non-Primes
• Everything depends on Directions for Numbering

## V. “NUMERICAL PLANES”

1. Odd Products in “Symmetrical Number Planes”
• Prime Numbers are a subset of Odd Numbers
• When visualised in a Symmetrical Context, they show their beauty rather than their mystery…
2. “Symmetrical Number Planes” for “Orthogonal Prime Axes”
• What do we See when we highlight Prime Numbers along the Axes?
• “Vertical Pairs” of Primes and SemiPrimes – Diagonally Staggered
• revealing the Ortho-Diagonal Nature of “Digital Sums” n x 10 + m
3. The OrthoDiagonal Nature of “Digital Sums”
• A remarkable combination of “Parallel Diagonals” and “Vertical Pairs”
• and an even more remarkable Pattern of mainly Primes
4. “Digital Sums” in “Symmetrical Number Planes”
• Who comes first: the Sums or the Pattern?

# VI. SUMS and PRODUCTS

1. “Diagonal Sums” of Last Digits of Odd and Even Summands in Orthogonal and Diagonal Chessboards:
1. Odd + Odd = Even
1. Odd + Even = Odd
2. Even + Even = Even
1. Even + Odd = Odd
2. “Orthogonal Products” of Last Digits of Odd and Even Factors in Orthogonal and Diagonal Chessboards:
1. Odd x Odd = Even
1. Odd x Even = Even
2. Even x Even = Even
1. Even x Odd = Even
3. OrthoDiagonal Symmetries in Number Planes
• Numbers and their patterns as ‘digital footprints’ are distinctly beautiful!
4. OrthoDiagonal Sums, Products and Primes
• Diagonal Sums, Orthogonal Products and OrthoDiagonal Primes?
5. “Symmetrical Number Planes” for “SemiPrime Factors”
• The 4-fold Symmetry of a Number Plane is not only beautiful
• Primes as Co-Factors create SemiPrimes as Products

# VII. ROOTS and DIAGONALS: PYTHAGORAS through a DIGITAL LENS

1. Pythagoras’ Sums of Squares in “Symmetrical Number Planes”
• Patterns over Patterns… What do they tell us?