This is a Java applet which shows the entries of the first
240 rows of Pascal's Triangle as small dots of a color based on the
remainder upon division by n, a number between 2 and 22. To
change n, use the slider at the top of the screen; n is
initialized as 2, and the only color turned on is 1; the pattern made
by 1 (mod 2) is known as Sierpinski's Gasket or Sierpinski's Triangle;
choosing different n and turning on different colors gives you
different patterns; when a number k is chosen, the phrase k (mod
n) is written on the screen in the color that will be used for the
dots corresponding to that number.
Changing colors: In math, k (mod n) means
the set of all numbers that can be written as k + rn,
where r is any integer. For example, 8 and 15 are both elements
of 1 (mod 7). If you want a different color to be used in a pattern, turn
off the lower values in that set and turn on a higher value; when you do
this, an asterisk * will appear after the phrase. In the example of mod
7, you can get the mod 1 color (red) to change to the mod 8 color (light
gray) by turning off 1 and turning on 8; likewise, you can see the color
for 15 (cyan) by turning on 15 and off 8 and 1.
Educational value: While there are some identities in Category
5 that can be proven by inspecting the Generalized Sierpinski's Gasket (mod
p) where p is a prime, the main use of this applet is to look
at the pretty colors.
The applet looks best when run in full screen mode. If the
box below is just a grey rectangle, you can put Java on your machine for
free through this link.
Some pretty patterns:
RGB specials: Choose mod 7, turn on 1, 2, 3, 11, 12, 13. You
can also try turning on 0 (grey background) or turn off 0 and turn on 7 (yellow
background). These colors are also interesting at mod 17 and mod 19.
Patterns in grey: Colors 8, 10, 18, 20 and 21 are all shades of grey;
so is color 0, but it tends to add too much; some of the patterns using these
colors are very nice. I especially like mod 11 and mod 13, and I like
mod 8 and mod 16 when color 8 is turned off.
Hot colors: Colors 1, 4, 7, 9, 11, 14, 17 and 19 give you patterns
in red, yellow, orange and pink. Take a look at these colors turned
on in mod 5, mod 6, mod 10, mod11, mod 13, mod 21 and 22.
