- Well, we've been from 1 to infinity. That's a long way.
Actually, we went back to before 1: to zero (my hero), too; back to the Mayans; back to the Babylonians and Egyptians.
From there we went forward, and ended up studying some of the most modern of mathematics (fractals). More on that in a moment....
On the practical side, I've taught you to read minds, and to win money: what more could you want!:) Speaking of reading minds, I've read yours and I'm worried that you might have forgotten something:
- Your logos are due today, Thursday, by midnight. Here's what was
said in our syllabus:
LOGO: Companies have logos; you should have one, too! Your math logo will be something (like a family crest) that represents you. It will be created using elements from this course (or other mathematical elements of your own choosing). You will type up a one-page sheet, illustrating and explaining your choice. These will be exhibited on-line.
So I'll have you submit to me your logo and a one-page paper as an assignment, and then I'll have you put your logo and a description (you could just copy the text from your paper) onto a discussion board, so that everyone may enjoy them.
It will be a digital poster session, essentially.
This is your final assignment prior to the final.
- Your comprehensive final will be available Tuesday, May 4th, at 8:00 am through
Thursday midnight.
As in previous exams, you will have an IMath component, and a Canvas component (which you should do immediately following your IMath portion).
- Materials for today:
- A video over today's review material: review
- Next week, finals week: I'll keep ordinary office hours. You can
reach out during those hours, and you can certainly contact me
outside of those if necessary. I'm often available.
- Good luck to you!
- Let's start with a brief review of what we learned about last time:
- We discovered something almost beyond belief (but which
mathematicians know to be true): there are infinitely many
sizes of infinity.
If you thought infinity was big, you didn't know about power sets! Infinity gets bigger, and bigger, and....
- If you've done your reading, then you knew that the real
numbers -- numbers including all the natural numbers, plus all
the fractions, plus all the irrational numbers like $\sqrt{2}$
and $\pi$ -- is a set bigger than the set of natural numbers.
- It was Georg Cantor who created this "paradise" (David Hilbert's word) of infinities.
And Strogatz showed us exactly how Cantor demonstrated that the real numbers are bigger than the natural numbers in our final reading).
- We discovered something almost beyond belief (but which
mathematicians know to be true): there are infinitely many
sizes of infinity.
- So let's do a little review of our course. Let's see where we've
been this semester. I took a screen-shot of the agendas for the
semester:
Quite a wide variety of topics there!
First of all, your exam will come in two parts: new and old. There is plenty of material since our last exam, and that will make up half of your exam.
- There was a review page for the second
exam and other material on our Day 20
agenda, and that would be good to review.
- I've created a similar review page for
the first exam material.
- So let's have a gander at the things we've discussed since that
last exam. There are basically three topics:
- Bands, Links, Knots
- Fractals
- Infinity
- Bands, Links, Knots
- We studied the bizarre band that keeps on playing: the
mobius band (one-sided, one-edged band).
- When we snipped it in "two", it did something funny: it stayed in one.
- When we snipped a thrice-twisted band in "two", it did something funny: it snipped into one, but it was a trefoil knot!
- The link that the band turned into is called a
Hopf link. There's also the unlink (two
"wedding bands"), Solomon's "knot", and the
Borromean rings. Those are the four links we studied.
- The unknot and the trefoil are the simplest. Then
there's the figure eight knot, and two
five-crossing knots: the cinquefoil (a torus
knot), and the five-twist knot. Those are the
five knots that I expect you to know, and be
able to draw at the drop of knot.
- How do we distinguish knots, or links? Two ways:
- Reidemeister moves (RI, RII, RIII)
- Tri-colorability
- We studied the bizarre band that keeps on playing: the
mobius band (one-sided, one-edged band).
- Fractals: a world within a world....
Do it again, do it again, ....
Nature loves patterns, especially the Fibonacci numbers, which are created by a fractal process: add the two preceding numbers; now do it again! Nature probably loves them because they're fractal, because nature loves fractals.
Fractals are so important because, from a simple rule, and simple repetition, beautiful and complex-seeming things appear.
The bunny tree is fractal:
A world within a world.
The key is that there is something (e.g. a stick), to which a process is applied and it results in more of the something (e.g. more sticks); and then do it again, do it again, etc. Forever. Ad infinitum. When we do this, we can end up with infinitely long objects within finites spaces, which is a very strange notion....
- Infinity
- Once again the one-to-one correspondence proved
to be the key.
- Power sets -- the set of all subsets of a set --
linked Pascal's triangle back in, and we discovered
that power sets are always bigger than the set from
which they are formed.
This turned out to be true for infinite sets, as well, so that there are an infinite number of bigger and bigger infinities. Mind-blowing!
The natural numbers, even natural numbers, integers, and even the rational numbers (ratios of integers) are all small infinities.
A bigger infinity is the real numbers (because it includes all the irrational numbers, like $\pi$, which are as big a set as all the real numbers themselves).
- Once again the one-to-one correspondence proved
to be the key.