## 17. “Diagonal Sums” of Last Digits – in Orthogonal and Diagonal Chessboards

**Last Digit SUMS** are the ‘arithmetical nugget’ of ‘geometrically *diagonal*‘ Sums.

But there are a number of *complementar*y *concepts *that need to be bridged for ‘grand unification’:

**Number: ***Odd* versus *Even*, ‘*Player*‘ versus *Result*;**Geometry: ***Vertical *versus *Horizontal*, *Orthogonal *versus *Diagonal*,**Visualisation: ***Label *versus *Content*, *Single Quadrant *versus *Symmetrical Plane*, *Orthogonal* versus *Diagonal *Chessboard.

With focus on ODD and EVEN sums of ODD and EVEN digits and numbers, there are four cases of combination:

**First: **ODD Last Digits as Summands in Vertical Column and in Horizontal Rows:

- ODD + ODD = EVEN

**Second: **ODD Last Digits in Vertical Column and EVEN Last Digits in Horizontal Rows:

- ODD + EVEN = ODD

**Third**: EVEN Last Digits in Vertical Column plus EVEN Digits in Horizontal Rows:

- EVEN + EVEN = EVEN

**Fourth:** EVEN Last Digits in Vertical Column, ODD Last Digits in Horizontal Rows:

- EVEN + ODD = ODD

## 18. “Orthogonal Products” of Last Digits in Orthogonal and Diagonal Chessboards

**Geometrically, **sums are *diagonal *to express the multitude of combinations of their summands, i.e. the ‘plurality’ of their ‘value composition’ – in a diagonal *line.*

Products are *orthogonal* because every unique combination of 2 factors produces a *rectangle *that is unique in its shape, e.g. 24 = 3 x 8 = 4 x 6 = 2 x 12.

Right: Additional Odd x Odd Products along Diagonals

Right: Additional Odd x Odd = Even Products in Diagonal Positions

## 19. “OrthoDiagonal Symmetries” in “Number Planes”

Horizontal Top ↕ Bottom

Orthogonal + Diagonal = OrthoDiagonal

for ‘grand unification’ between physics and maths – via software.