Vedic square

The Vedic Square can be viewed as the multiplication table of the monoid ${\displaystyle ((\mathbb {Z} /9\mathbb {Z} )^{\times },\{1,\circ \})}$ where ${\displaystyle \mathbb {Z} /9\mathbb {Z} }$ is the set of positive integers partitioned by the residue classes modulo nine. (the operator ${\displaystyle \circ }$ refers to the abstract "multiplication" between the elements of this monoid).

If ${\displaystyle a,b}$ are elements of ${\displaystyle ((\mathbb {Z} /9\mathbb {Z} )^{\times },\{1,\circ \})}$ then ${\displaystyle a\circ b}$ can be defined as ${\displaystyle (a\times b)\mod {9}}$, where the element 9 is representative of the residue class of 0 rather than the traditional choice of 0.

This does not form a group because not every non-zero element has a corresponding inverse element; for example ${\displaystyle 6\circ 3=9}$ but there is no ${\displaystyle a\in \{1,\cdots ,9\}}$ such that ${\displaystyle 9\circ a=6.}$.

Properties of subsets

The subset ${\displaystyle \{1,2,4,5,7,8\}}$ forms a cyclic group with 2 as one choice of generator - this is the group of multiplicative units in the ring ${\displaystyle \mathbb {Z} /9\mathbb {Z} }$. Every column and row includes all six numbers - so this subset forms a Latin square.

${\displaystyle \circ }$	1	2	4	5	7	8
1	1	2	4	5	7	8
2	2	4	8	1	5	7
4	4	8	7	2	1	5
5	5	1	2	7	8	4
7	7	5	1	8	4	2
8	8	7	5	4	2	1

A Vedic cube is defined as the layout of each digital root in a three-dimensional multiplication table.[1]

Vedic square in base 100 (left) and 1000 (right)

Vedic squares with a higher radix (or number base) can be calculated to analyse the symmetric patterns that arise. Using the calculation above, ${\displaystyle (a\times b)\mod {({\textrm {base}}-1)}}$. The images in this section are color-coded so that the digital root of 1 is dark and the digital root of (base-1) is light.

• Latin square
• Modular arithmetic
• Monoid

• Deskins, W.E. (1996), Abstract Algebra, New York: Dover, pp. 162–167, ISBN 0-486-68888-7
• Pritchard, Chris (2003), The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, Great Britain: Cambridge University Press, pp. 119–122, ISBN 0-521-53162-4
• Ghannam, Talal (2012), The Mystery of Numbers: Revealed Through Their Digital Root, CreateSpace Publications, pp. 68–73, ISBN 978-1-4776-7841-1
• Teknomo, Kadi (2005), Digital Root: Vedic Square
• Chia-Yu, Lin (2016), Digital Root Patterns of Three-Dimensional Space, Recreational Mathematics Magazine, pp. 9–31, ISSN 2182-1976