- We finished up with Pascal's triangle, and some strange new connections:
- Pascal's triangle sits "on a sea of zeros" -- we added those in.
- Then we realized that we can push Pascal's triangle
"backwards," creating rows above what we thought of as the
first row -- starting with just a 1.
- We can do it in infinitely many ways, and, although I am
fascinated with symmetry, we decided to push it up to the left
-- creating another sea of zeros to the right.
- So above the top 1 of Pascal's triangle, we added two
entries: "1 0". Add them together and you get the 1 below.
- Once we made that choice, and decided that the line of
ones starting from bottom right would be preserved, all else
was determined. The pattern of adding two values and
getting the value below would dictate the rest of the
new "triangle".
- This preserved some patterns, such as the line of ones from
bottom right to top; but ended other patterns, such as the line
of ones from bottom left to top!
But some patterns were preserved.
- One particularly interesting pattern that was preserved
was the pattern of Fibonacci numbers: this extension of the
Fibonacci numbers (also backwards!) fell out:
\(\ldots,-55,34,-21,13,-8,5,-3,2,-1,1,0,1,1,2,3,5,8,13,21,34,55,\ldots\) - But the most important pattern preserved was that the sum
of the rows is a power of two. But this had some strange
consequences:
\[
S_1 = 1-1+1-1+1-1+1-1+\ldots = \frac{1}{2} = 2^{-1}
\]
\[
S_2 = 1-2+3-4+5-6+7-8+\ldots = \frac{1}{4} = 2^{-2}
\]
\[
S_3 = 1-3+6-10+15-21+28-\ldots = \frac{1}{8} = 2^{-3}
\]
and so on....
- Using those very strange infinite sums, I was able to
derive a final and even stranger infinite sum:
\[
S = 1+2+3+4+5+6+7+8+\ldots = \frac{-1}{12}
\]
However odd this sum may look, it turns out that even the great
Euler came to the same very strange conclusion!
Baffling, I know, but infinity is very strange!:)
- Pascal's triangle sits "on a sea of zeros" -- we added those in.
- You'll each have about three minutes to present your logo. You should
describe both
- the mathematical elements, and why they're important to you; and
- any other personal connections in your logo.
- Here's the batting order:
