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.

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.

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.

• 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