### Arithmetic in Disguise: What is it?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny

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**.)

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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Copyright © 1996-2018 Alexander Bogomolny

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