InIndian mathematics, aVedic square is a variation on a typical 9 × 9multiplication table where the entry in each cell is thedigital root of the product of the column and row headings – in other words, each cell contains theremainder when the product of the row and column headings is divided by 9 (with remainder 0 represented by 9). Numerousgeometricpatterns andsymmetries can be observed in a Vedic square, some of which can be found in traditionalIslamic 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 themonoid${\displaystyle ((\mathbb{Z}/9\mathbb{Z}{)}^{\times },\{1,\circ \})}$ where${\displaystyle \mathbb{Z}/9\mathbb{Z}}$ is the set of positive integers partitioned by theresidue classesmodulo 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${\displaystyle (a\times b)\mathrm{mod}9}$, where the element 9 is representative of the residue class of 0 rather than the traditional choice of 0.

This does not form agroup because not every non-zero element has a correspondinginverse 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 acyclic group with 2 as one choice ofgenerator - this is the group of multiplicativeunits in thering${\displaystyle \mathbb{Z}/9\mathbb{Z}}$. Every column and row includes all six numbers - so this subset forms aLatin 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 eachdigital root in a three-dimensionalmultiplication table.^[2]

Vedic squares with a higherradix (or number base) can be calculated to analyse the symmetric patterns that arise. Using the calculation above,${\displaystyle (a\times b)mod(\text{base}-1)}$. 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

• Indian mathematics
• Modular arithmetic

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