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”

- Pyramids of Decimal and “Digital” Numbers
- The ‘big ideas’ of “Digital Numbers” and “Digital Visualisations”

- 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]

- “Last Digits” form
- Single “Last Digit Patterns”
- Fascinating, how every Digit reveals its own Pattern

- Composite “Last Digit Patterns”
- Different ‘digital combinations’ form different Patterns

- “Digital Prime Diagonals“
- First on Parallel Diagonals ‘outside’ the Pyramid
- Then inside the “Pyramid of Counts”

# II. “ARITHMETICAL MATRICES”

- “Diagonal Sums” of Odd and Even Summands
- Sums can be visualised as Diagonals and on Parallel Diagonals

- Prime Numbers as “Diagonal Sums”
- When Odd Numbers can do it, Primes can do it, too!

- “Digital Squares” of SemiPrimes
- What do SemiPrimes show us in their Matrix of “Prime Products”?

- 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”

- “Digital Tables” of Primes and SemiPrimes
- Primes are Sums; SemiPrimes are Products; Together they form “Prime Pairs” when aligned by
*Leading*and*Last*Digits

- Primes are Sums; SemiPrimes are Products; Together they form “Prime Pairs” when aligned by
- “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”

- Orthogonal Pyramids for Numbers as Counts
- Vertical and Horizontal Pyramids doubled and mirrored

- Prime Numbers in Orthogonal Pyramids
- Patterns of Non-Patterns or ‘regular irregularity’?

- Fourfold Pyramids for Primes and Non-Primes
- Everything depends on Directions for Numbering

## V. “NUMERICAL PLANES”

- 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…

- “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

- The OrthoDiagonal Nature of “Digital Sums”
- A remarkable combination of “Parallel Diagonals” and “Vertical Pairs”
- and an even more remarkable Pattern of mainly Primes

- “Digital Sums” in “Symmetrical Number Planes”
- Who comes first: the Sums or the Pattern?

# VI. SUMS and PRODUCTS

- “Diagonal
**Sums**” of Last Digits of*Odd*and*Even***Summands**in Orthogonal and Diagonal Chessboards:- Odd
**+**Odd = Even- Odd
**+**Even = Odd

- Odd
- Even
**+**Even = Even- Even
**+**Odd = Odd

- Even

- Odd
- “Orthogonal
**Products**” of Last Digits of*Odd*and*Even***Factors**in Orthogonal and Diagonal Chessboards:- Odd
**x**Odd = Even- Odd
**x**Even = Even

- Odd
- Even
**x**Even = Even- Even
**x**Odd = Even

- Even

- Odd
- OrthoDiagonal Symmetries in Number Planes
- Numbers and their patterns as ‘digital footprints’ are distinctly beautiful!

- OrthoDiagonal Sums, Products and Primes
- Diagonal Sums, Orthogonal Products and OrthoDiagonal Primes?

- “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

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

NEXT: “Digital Visualisations”