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Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
, a
Vedic upright=1.2, The Vedas are ancient Sanskrit texts of Hinduism. Above: A page from the '' Atharvaveda''. The Vedas (, , ) are a large body of religious texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute th ...
square is a variation on a typical 9 × 9
multiplication table In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essenti ...
where the entry in each cell is the digital root of the product of the column and row headings i.e. the
remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In algeb ...
when the product of the row and column headings is divided by 9 (with remainder 0 represented by 9). Numerous
geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
patterns A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated li ...
and symmetries can be observed in a Vedic square, some of which can be found in traditional
Islamic art Islamic art is a part of Islamic culture and encompasses the visual arts produced since the 7th century CE by people who lived within territories inhabited or ruled by Muslim populations. Referring to characteristic traditions across a wide ra ...
.


Algebraic properties

The Vedic Square can be viewed as the multiplication table of the
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
((\mathbb/9\mathbb)^, \) where \mathbb/9\mathbb is the set of positive integers partitioned by the
residue class In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
es
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
nine. (the operator ''\circ'' refers to the abstract "multiplication" between the elements of this monoid). If a,b are elements of ((\mathbb/9\mathbb)^, \) then a \circ b can be defined as (a \times b) \mod, 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
because not every non-zero element has a corresponding
inverse element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
; for example 6\circ 3 = 9 but there is no a \in \ such that 9\circ a = 6..


Properties of subsets

The subset \ forms a
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
with 2 as one choice of generator - this is the group of multiplicative
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
in the ring \mathbb/9\mathbb. Every column and row includes all six numbers - so this subset forms a Latin square.


From two dimensions to three dimensions

A Vedic cube is defined as the layout of each digital root in a three-dimensional
multiplication table In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essenti ...
.


Vedic squares in a higher radix

Vedic squares with a higher
radix In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ...
(or number base) can be calculated to analyse the symmetric patterns that arise. Using the calculation above, (a \times b)\mod. 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 In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
*
Monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...


References

* * * * * {{Indian mathematics Indian mathematics Modular arithmetic