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