**Pythagoras’ famous theorem** covers three operations:

two numbers a and b into a**squaring**^{2}and b^{2}- calculating the
of two squares: c**sum**^{2}= a^{2}+ b^{2} - calculating or looking up the square
of that sum: c = SQRT (a**root**^{2}+ b^{2}).

The most beautiful example is 5^{2} = 3^{2} + 4^{2} 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).

- some of them multiple times:

*Visually, ***± Roots** form Diagonals across the a^{2} + b^{2} 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:

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]