Transcript Irrelevant Topics in Physics (part deux)

Travis Hoppe

Plus-size numbers
 They’re just big boned, more to love really

Venn-diagrams
 Forget everything you’ve forgotten about them from
third-grade

Football betting
 Did I mention I’m from Vegas?

First a contest!
 Write the largest number you can think of on the note
card.
 Use standard mathematical functions or define your
own.
 The number must be verifiably finite and computable.
 The number must be completely defined on the card.

First-grader:
1000000000

Third-grader:
9999999999

Sixth-grader:
999999999

Twelfth-grader:
99
9
99
9
9
99

Addition, multiplication and exponentiation are
simply higher orders of the same function:
ab  a  a  a
b copies of a
a b  a  a  a
b copies of a
The idea is not to generate the
largest number, per se, but
rather the largest growing
function...
Many different styles: Conway’s
chained arrow, hypergeometric, and of course
Knuth’s up-arrow

Each arrow starting from exponentiation forms
the higher operators:
a  b  a  a  a  a
b
b copies of a
a  b  a 
b
a
a
.
..
a
b copies of a
 a  a  a
b copies of a
3  2  3  27
3
3  3  3  327  7625597484987
33
3  4  3
3
33
3
7625597484987
Note that the operator is right-associative:
3  3  3  327
33
3  3  (3 )  3
3 3
9

Clearly we can grow larger numbers by simply
adding more arrows onto the expression:
a  b  a  a    a
b copies of a
a n b  a n1 a n1 n1 a
b copies of a

We have primitive brains –
 For small numbers we can only think spatially, four
cows, three hens etc …
 Abstract numerical systems allow us understand larger
quantities

If you build it ….
 Large numbers systems were invented because of their
necessity. For example …

Graham’s number is so big that even Knuth’s up
arrow notation is insufficient to contain it. It is the
best known upper-bound to the problem:
Consider an n-dimensional hypercube, and connect each pair of vertices to
obtain a complete graph on 2^n vertices. Then color each of the edges of this
graph using only the colors red and black. What is the smallest value of n for
which every possible such coloring must necessarily contain a single-colored
complete sub-graph with 4 vertices which lie in a plane?
g1  3  3
g n  3  gn1 3
G  g64

This is an upper bound to the problem. It has
been proven that the lower bound solution is at
least 11. The authors state that there is, “some
room for improvement”.
You, at the conclusion of this talk
Irrelevant,
yet
nonetheless
interesting
things Venn
diagrams
Things you
know about
Venn
Diagrams

Let C = { C1, C2, ..., Cn } be a collection of simple closed curves
drawn in the plane. The collection C is said to be an independent
family if the region formed by the intersection of X1, X2, ..., Xn is
nonempty, where each Xi is either int(Ci ) (the interior of Ci ) or is
ext(Ci ) (the exterior of Ci ).

If, in addition, each such region is connected and there are only
finitely many points of intersection between curves, then C is a
Venn diagram, or an n-Venn diagram if we wish to emphasize the
number of curves in the diagram.

In other words – every subset built from a collection of n
objects has to be represented only once ….
A
B
Not Venn as A U B is not represented, this
is still known as an Euler diagram
A
B
C
Constructing graphs allows different Venn
diagrams of the same order to be compared.
Diagrams are isomorphic if their graphs are
Isomorphic.

Number of vertices in the graph is no more than:
 2n  2 
v

 n 1 


Must display “n-fold” symmetry.
Can be shown that these only exist when n is
prime
Football
betting

Which squares are better
and by how much?

Should some squares cost
more?

Are you more likely to be
a win or lose?

Is the post-season
different from regular
season?
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9

Data collection – wrote script to dump all scores from 19942007 season (no box scores were found pre-1994).

Partitioned data into regular season and post (wildcard,
playoffs and Super Bowl) games.

Took into account home-field advantage; 07 is different from
70, with the first score the home team (irrelevant in games
where neither team has home field advantage).

Each score (x,y) was given an expected value:
EX ( x, y )  10 PQ1 ( x, y )  20 PQ 2 ( x, y )  30PQ 3 ( x, y )  40PQ 4 ( x, y )