martes, 12 de octubre de 2010

Vedic square


From Wikipedia, the free encyclopedia
Jump to: navigation, search

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).

[edit] Algebraic Properties

If we ignore the ninth column and the ninth row, (which are all nines) we are left with the semigroup \{\mathbb{Z}/(9), \circ\} where \mathbb{Z}/(9) 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 semigroup. If {a,b} are elements of \{\mathbb{Z}/(9), \circ\} then a \circ b can be seen as, a\times b \equiv c \pmod{9}.

This does not form a group since not every non-zero element has a corresponding inverse element, for example 8\circ 3 = 6 but there is no a \in \{ 1,\dots,9 \} such that 6\circ a = 8.. That is because 9 is not a prime number: 3 and 6 which are not co-prime to 9 do not fall within the multiplicative group of integers modulo 9.

[edit] See also

[edit] References

  • Deskins, W.E. (1996). Abstract Algebra. New York: Dover. pp. 162–167. ISBN 0486688887.
  • Pritchard, Chris (2003). The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching. Great Britain: Cambridge University Press. ISBN 0521531624.

No hay comentarios:

Publicar un comentario en la entrada

yesyukan