Vedic square

In Indian mathematics, a Vedic square is a variation on a typical 9 × 9 multiplication table where the entry in each cell is the digital root of the product of the column and row headings – in other words, each cell contains the remainder when the product of the row and column headings is divided by 9 (with remainder 0 represented by 9). Numerous geometric patterns and symmetries can be observed in a Vedic square, some of which can be found in traditional Islamic art.

${\displaystyle \circ }$	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	1	3	5	7	9
3	3	6	9	3	6	9	3	6	9
4	4	8	3	7	2	6	1	5	9
5	5	1	6	2	7	3	8	4	9
6	6	3	9	6	3	9	6	3	9
7	7	5	3	1	8	6	4	2	9
8	8	7	6	5	4	3	2	1	9
9	9	9	9	9	9	9	9	9	9

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 ${\displaystyle 9\circ a=6.}$.

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.²

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.