Arithmetic in Disguise

Arithmetic in Disguise: What is it?
A Mathematical Droodle

The picture reminds me of a smiling koala bear that was affinely mapped onto a right-angled triangle. (Turn your head left 45° and see if you agree with me. My wife does not.)

However, it was obtained with the applet below (# Rows 64, # Colors 64, Operation x·y AND (x+y) and Colors Reversed as the low right portion of the Square.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

The applet draws a square or triangular array of dots whose color is defined through simple arithmetic and bitwise operations. x and y coordinates are counted from the upper left corner of the array. (The triangular array is just a sheared version of the lower left half of the square.) They then are combined by the selected Operation. The result is taken Modulo the number of colors. Or, as an alternative, in the Binary mode all non-zero results are made to correspond to a single quantity. The finite result becomes an index into an array of gray shades.

Note how much the low right portion of the original display (# Rows 32, # Colors 31, Operation x OR y, and Colors Reversed unchecked) resembles the fractal structure of the Sierpinski gasket or that of Pascal's triangle in modular arithmetic.

Simone Severini from the Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo, has pointed out that Sierpinski gasket comes also through with the Operation x AND y if the result of calculations is split into two classes: zero and non-zero. This is achieved by checking the Binary box.

References

C. A. Pickover, Wonders of Numbers, Oxford University Press, 2001 (p. 173)

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