This puppy is thick with cool patterns and other fun facts.
Blaise was really onto something. The
triangle itself was actually first created by the ancient
Chinese, but Pascal is the one who found its
patterns. Here's a few of 'em:
The sum of the numbers in any row is equal to 2 raised to the power of the row number.
20 = 1 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
If the second number of any row is prime, all other numbers in the row (excluding the ones) are divisible by the prime.
row 7: 01 07 21 35 35 21 07 01
| | | | | |
V V V V V V
divisible by 7
There is an odd "hockey stick" pattern. Start at a 1 on the side of the triangle. Draw a diagonal line down from the 1, and end it somewhere in the middle of the triangle. The sum of all the numbers in the diagonal will equal the number below the end of your "selection" which is not on the same diagonal itself.
01
01 01
01 02 01
01 03 03 01
01 04 06 04 01
01 05 10 10 05 01
01 06 15 20 15 06 01
The sums of diagonal rows (shown in alternately bold and plain text) produce the Fibonnacci Sequence.
01 = 1
= 1
01 01 = 2
= 3
01 02 01 = 5
= 8
01 03 03 01 = 13
= 21
01 04 06 04 01 = 34
---
01 05 10 10 ---
---
01 06 15 ---
---
01 07 ---
---
01 ---
Pascal's triangle is often drawn with each number in a small hexagon. If you fill in the odd numbers on the triangle and leave the even numbers empty, you'll produce the recursive Sierpinski Triangle fractal. Here's a rather poor approximation of that fractal:
~<
~B8/
(@V<@%
<@%@%@$V
<$$ VB0
<$/3@ C@X%B
X@@g@@@%@@%@@0
X$( `$0`
%$G@X ^@%G@`
0$@C@@% <@@0$@@`
G$ `@X ^@< X@^
00g0` `$G$G <@G@X %BG$^
$0gV@$(@@(Bg8V@@<@$XG00^@@^
@8^^^^^^^^^^^^`^^^^^^^^^^^X@^
`$g@X ~$G@<
`B$^Gg% <B$ $B<
`@G%VCC0% /@VVCXV@/
`@g$ /@B0 /@g8 G$@(
~@8 g@ C@C~@B /@/ $$ 0@`<@V
^@%8%8CVGC8%G8@ %@%8GV8%G%CGV$V
<@@% $$@` C@@< <@$G
X@C<@8 `$g`0$` C@<C@/ (@C^$G
XB/G%%8g ^BC8%GCB` 0GVCG%$/ C$C8%%GB
/@@. 8@8 B@X ($@ BB8 `$$C (@@^ g0B
G@X%@^ 0@/@$ <@%X@X /@/%@^ $B(0$ `@%/@X X@XV@^ 8$(G$
XBg8BC$88B8$g%$C@0%@%@88@C$8803@g8@%$8g@3@g8@3B88$3@G%
I'll put more patterns up here as soon as I can figure out how to describe them with HTML and ASCII. ;)