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.

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

$\circ$	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

Algebraic properties

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

If $a,b$ are elements of $((\mathbb{\mathbb{Z}}/9\mathbb{\mathbb{Z}}{)}^{\times },\{1,\circ \})$ then $a\circ b$ can be defined as $(a\times b)mod9$, 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 $6\circ 3=9$ but there is no $a\in \{1,\cdots ,9\}$ such that $9\circ a=6.$.

Properties of subsets

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

From two dimensions to three dimensions

Slices of a Vedic cube (upper figures), and trimetric projections of the cells of given digital root d (lower figures)

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

Vedic squares in a higher radix

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,

$(a\times b)mod(\text{<mrow data-mjx-texclass="ORD">base</mrow>}-1)$

. 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.

See also

• Latin square
• Modular arithmetic
• Monoid

References

• 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
• 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

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