This applet is based on the Mathematical Recreations column of
Ian Stewart, featured in the August 2000 issue of
Scientific American .
The original paper was Fractal Images of Formal Systems ( Journal of Philosophical Logic , Vol.26, No.2, p.181-222,1997) by Patrick Grim and Paul St. Denis (both at Stony Brook University, NY).
Basically, the applet functions as follows: the decimal numbers 0 to 15
are noted in their binary equivalent (0000 to 1111) and are each
given a color code (I used Java's standard colors and three self defined
colors which I hope will show up OK on all systems...).
These colors are indicated in the colorbars on the left and on the top. The binary values are matched to the colors on the left.
Next, we enter propositional calculus by performing any of the indicated
logical operations on each of the binary number pairs.
I.E.: in row 3 and column 5 we find 0010 and 0100 (start counting from 0, remember?), so if the "NAND" (Not AND) radiobutton is checked (which is default) we find the result of '0010 NAND 0100' which we can divide bitwise into:
0 NAND 0 --> 1 0 NAND 1 --> 1 1 NAND 0 --> 1 0 NAND 0 --> 1So the result is 1111 binary (15 in decimal notation) which corresponds to the color black. The square with coordinates (5,3) shows up black.
Surprisingly (?) we see the appearance of a Sierpinski's gasket-like figure (in black for NAND). This is not a real fractal, though. It's only a pseudo-fractal since it has limited level of detail (a true fractal would go on indefinitely).
To view the results with the other logical operators, first select the appropriate operator, then click 'Redraw'.
The 'Stop!' button is present because the applet runs in a separate
thread which will wait eternally for you to do something. Since you probably
won't spend eternity toying around with it, it's a good idea to stop the
thread when you're leaving.
Your browser should stop the thread anyway, but hey: better safe than sorry ;-).
For source code, comments, suggestions etc. you can always E-mail me at: