5. Pythagoras’ “DIAGONAL” Roots of Sums of Squares

Pythagoras’ famous theorem covers three operations:

  1. squaring two numbers a and b into a2 and b2
  2. calculating the sum of two squares: c2 = a2 + b2
  3. calculating or looking up the square root of that sum: c = SQRT (a2 + b2).

The most beautiful example is 52 = 32 + 42 or 25 = 9 + 16.

Here I use this colour combination to illustrate the fundamentally different approach to Number by arithmetical operations and by geometric visualisations:

Thanks to spreadsheets, we can progress Pythagoras’ theorem from Triangle to Matrix [Square or Rectangle] and Octagons, before we enter “Diagonal Rootland”:

± Roots of these sums are

  • Primes, but only some: 13, 17, 29, 37, 41, 53, 61, 89, 97
  • Non-Primes, but only some: 39 (3×13), 51 (3×17), 87 (3×19)
  • and Multiples of 5: 15, 25, 35, 45, 55, 65, 75, 85, 95, 100
    • some of them multiple times: 25 (2x), 65 (4x), 75 (2x), 85 (3x).

Visually, ± Roots form Diagonals across the a2 + b2 Diagonal:

± Roots of ‘Octagonal’ Sums of Squares – Visualised as Diagonals

Geometrically, ± Roots can thus be visualised as Corners of Squares around Diagonals and we find that the squares come in different sizes:

Diagonals are now seen Inside Squares – Multiple Times for 25, 65, 75, 85

The sizes of the squares created by Multiples of 5 reveal fascinating numerical patterns:

  • 5: 2 x 2
  • 15: 4 x 4
  • 25: 6 x 6
    • 18 x 18 [3×6 x 3×6]
  • 35: 8 x 8
  • 45: 10 x 10
  • 55: 12 x 12
  • 65: 14 x 14
    • 24 x 24 [4×6 x 4×6]
    • 36 x 36 [6×6 x 6×6]
    • 48 x 48 [8×6 x 8×6]]
  • 75: 16 x 16
    • 52 x 52 [4×13 x 4×13]
  • 85: 18 x 18
    • 36 x 36 [2×18 x 2×18]
    • 42 x 42 [3×18 x 3×18]]
  • 95: 20 x 20
  • 100: 12 x 12 [3×4 x 3×4]

NEXT: 6. Multi-Dimensional Sums

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