Meeting 2
Thursday September 1st
Chris McMurdie presents his weekly Putnam-type problem
This Weeks problem is considered a "Median problem", not too difficult but not
too easy, and not representative of the average difficulty level though.
QUESTION:
"Shaneel O'Quil"
Player Shaneel O'Quil has a certain freethrow %, Y% < 80%
At some point during the season she improves the r score greater than 80%
Is there a time during the year she is at exactly 80%?
ANSWER:
Yes
REASON:
A player's freethrow % (ft%) can be expressed in terms of total shots taken (t) and missed (m): ft% = (t-m)/t.
Using this formula and basic algebra, we can show that if the ft% is less than 4/5, then it follows that: t<5*m.
Since 5*m is an integer, and eventually ft% is greater than 4/5 (so t>5*m), t must equal 5*m at some point.
If t=5*m, then ft%=4/5 because ft% = (t-m)/t = (5m-m)/5m = 4m/5m = 4/5.
Volume of the n-dimensional sphere
Chris McMurdie generalized what we know about dimension to find the volume of
the n-dimensional sphere. It should be accessible to everyone with calculus.
[append data from chris's slides to fill this out for the web version]
Example: 2 Dimensional Sphere is a cricle, 3 dimensional sphere is a globe as we
know it.
How to generalize for larger dimensions
First we need to change our coordinate system. A rectangular coordinate system
is a little difficult to integrate and it would be better to find an
alternative.
If we use spherical coordinates then we have to calculate the jacobian of the
change-of-coordinate matrix from rectangular to spherical.
We will use the Vn(r) the formula for the volume of the n-sphere of radius r.
Then Vn+1(r) can be expressed as an integral involving Vn(r), resulting in a
generalized formula. For example think of the upper hemisphere of a 3
dimensional being sliced into circles, we can take the sum of the calculation
of all of these slices, which will be an integral. With this basic idea we can
try to generalize this information into an n dimensional sphere.
When calculating the jacobians for each dimension for the spherical coordinates
you can find a pattern in the calculations and derive a simpler formula for
finding the determinant of the jacobian. This greatly assists in calculating
the generalized form of an n dimensional sphere.
From here we have n nested integrals that we have to evaluate, which with a
little bit of math and pattern identifying can be reduced into a final form.
So by using Spherical coordinates and nested integrals it is easy to generalize
the n dimensional sphere.
Interesting Facts about high dimensional spheres:
The volume of the n dimensional unit sphere peaks somehwere between the 5th and
6th dimension. The unit sphere as n gets arbitrarily large has a volume that
approaches zero. The unit cube however remains the same volume.
Caveats:
The 0th dimension and negative dimensions are not well defined by this method.
[Footnotes: The topic of this talk may be a little too difficult for students
who have not taken all the core math courses.]
Winding down
The Rubik's cubes will be brought out afterwards, and as many of you expressed
a desire to learn a solution, we will be informally teaching it during the
unstructured part of the meeting.
[Footnotes: We should probably have some pictures and maybe a video of us doing
a rubix cube on the web page]