# Vedic square

In ancient Indian mathematics, a Vedic square is a variation on a typical 9 × 9 multiplication table. The entry in each cell is the digital root of the product of the column and row headings i.e. the remainder when the product of the row and column headings is divided by 9 (with remainder 0 represented by 9).

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
Highlighting specific numbers within the Vedic square reveals distinct shapes each with some form of reflection symmetry.

Numerous geometric patterns and symmetries can be observed in a Vedic square some of which can be found in traditional Islamic art.(Pritchard 2003, pp. 119–122)

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

## Algebraic Properties

This table can be viewed as the multiplication table of the monoid $((\mathbb{Z}/9\mathbb{Z})^{\times}, \{1, \circ\})$ where $\mathbb{Z}/9\mathbb{Z}$ is the set of positive integers partitioned by the residue classes modulo nine. Also, the operator $\circ$ means the abstract "multiplication" between the elements of this monoid. If $a,b$ are elements of $((\mathbb{Z}/9\mathbb{Z})^{\times}, \{1, \circ\})$ then $a \circ b$ can be defined as $(a \times b) \mod{9}$ by using the modulus operator mod, where we take the element 9 as the representative of the residue class of 0 rather than the traditional choice of 0.

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

If we consider the subset $\{1,2,4,5,7,8\}$, however, this does form a group. It forms a cyclic group with 2 as one choice of generator. In fact, this is just the group of multiplicative units in the ring $\mathbb{Z}/9\mathbb{Z}$.

$\circ$ 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

We can see the every columns and rows has all six cells. It shows that $\{1, 2, 4, 5, 7, 8\}$ forms a Latin square.

$\circ$ 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