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:

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

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