There are many ways to create basic design patterns from
modular arithmetic operations tables. The traditional way is to assign
a patterned square to each numbered cell in the interior of a table, say
a mod 5 addition table. Notice that the interior of the table is a 5 x
5 array of cells.
We begin by assigning patterned squares to the numbers appearing in the interior of the table. In the patterns below, additive inverses have been represented by complementary forms of a selected pattern. (This is not required, but adds an additional mathematical component to the activity.) Then we fill each square in the interior of the table with the pattern
according to the number marked in it.
We can now use this combination of 25 squares as a basic design pattern, which we can translate (slide), reflect (flip), or rotate (turn) to form larger, more artistic designs. First the basic design pattern is copied in one of the four corners or quadrants of a 2 x 2 grid. In the diagram below, the basic pattern has been copied in the upper left quadrant. Then the basic design pattern is reflected from the upper left quadrant onto the other three quadrants of the grid  in a vertical line of reflection for the upper right quadrant, and each of these in a horizontal line of reflection for the lower two quadrants. In the next basic design pattern, the original grid of patterned squares
is distorted so that each patterned square (or distorted rectangle) is
a fixed percent (70% illustrated) of the width of the patterned square
(or distorted rectangle) to its immediate left and the same percent (70%)
the height of the square (or distorted rectangle) directly above. Mathematics
refer to this as a logarithmic grid. For aesthetics, it had been
enlarged to the same size as the original basic design pattern.
The basic design pattern is then copied in the upper left quadrant of a quadrantal system and the design completed by reflection as before. A modular multiplication table, say mod 5, can also be used to generate the basic design pattern. To circumvent the undistinguished row and column of zeros in the interior of the table, we will use only the 4 x 4 array of nonzero numbers therein. Patterned squares are assigned to each of the numbers in the selected array as before. Once again additive inverses have been represented by complementary forms of a selected pattern. For variety, we have arranged the lines of the basic pattern grid in
an irregular way in order to produce a kaleidoscopic effect. As usual,
the patterns are entered into the regions of the grid according to the
number marked in it. Here this will require the deformation or stretching
of each pattern to fit the region to which it is assigned.
Then the basic design pattern is copied in the upper left quadrant of a quadrantal system and the design completed by reflection as before. Of course, there is a variety of distorted grids that you can employ
to create the final design. The only limit is your imagination.

Jill Britton Home Page artwork based on Math/Art Posters by Troutman & Forseth 
08May2005
Copyright Jill Britton 