Day 36, MAT115

Reading for next time: The Loneliest Numbers (p. 201)

Today's topic is based on the Hilbert Hotel (p. 249). For your final you'll be expected to show familiarity with these topics (in particular, I will assume that you have carefully read each section).

The Topologist's favorite riddle: What's the difference between a donut and a coffee cup?

Answer: There isn't any! (to a topologist....)

• Let's make some!
• A mobius band is made from a regular band using only a single twist.
• By the way, did you know that there are two different Mobius bands? They're mirror images of each other! (Symmetry, again....)
  • Check yours against someone else's, and convince yourself that they're really different.
  • Make one of each!
• Properties of a mobius band:
  • A mobius band has only one side.
  • A mobius band has only one edge.

• What happens when you cut a mobius band "in half"? Let's check out a bug on a band....
• What if you cut it into "thirds"?

• Conveyor belts
• Kazakhstan's National Library in Astana
• Recycling Topology
  • The official recycling symbol is mobius, but over time an alternative has crept in:
    

  • My dad loved mobius bands so much....
  • a history of the recycling symbol
  • AARP article on the history of the creation of the recycling symbol
  • Recycling -- Mobius, or only Seemingly Mobius?
• A joke: "Why did the chicken cross the Mobius band?"
• Mobius music:
  • First of all, Vi Hart's most beautiful performance of Music Box music: Pachelbel's Music Box Canon in D
  • But what really got me excited was when Vi introduced me to "mobius music"
  • I got so excited that I bought a box of my own, and I've been making my own "mobius music". But I use math to make the music we play as a mobius band (since I haven't any talent as a composer...).

• What kind of a knot is the edge of a Mobius band? (Cut one in half!)
• Do you get a knot with a twice-twisted Mobius band? (Cut one in half!)
• What objects (knots, links) did we get by cutting the Mobius band in "thirds"?
• What kind of a knot is the alternative recycling symbol shown above?

Eternity is tied up with the concept of infinity, frequently.

About another Newton (other than Sir Isaac): John Newton 1725-1807; ironically Newton was a slave trader (who ultimately saw the error of his ways).

• The modern understanding of infinity dates to work done by Georg Cantor, in the late 1800s.

• Our author quotes David Hilbert (himself a very famous and brilliant mathematician), who said "No one shall expel us from the Paradise that Cantor has Created."

  Strogatz says that his objective is to "give you a glimpse of this paradise." (p. 253)

• Welcome to the Hilbert Hotel!
  • there is a room for each natural number.
  • Everyone can be asked to move (but only once per night).

• Notions of infinity rely on the idea of one-to-one correspondence, which you may recall from our early study of numbers.
• Can the Hilbert Hotel really be full?
• Hilbert Hotel and the infinite schoolbus
• Hilbert Hotel and the limitless stable of schoolbuses

• infinity is NAN: not a number (and is therefore neither even nor odd, as Strogatz explains to Kim Forbes and her son Ben).

• infinity is NAS: not a size, either! the reason is that there is more than one size of infinity! this is the most important thing I want you to remember (this fact will allow you to win on the playground, when someone tells you that they hate you an infinite amount -- you can still hate them more!;).

• The natural numbers are infinite in number ().

• cardinality

  two sets have the same cardinality if there is a one-to-one correspondence between them.

  intuitively: two sets are the same size if they have the same cardinality.

• Here's a sensible rule you can be sure of: a subset is never bigger in size than the set itself.

  and if the sets are finite, the proper subset is always smaller....
  but if the set is infinite, we may actually be able to throw away elements of a set and not change the size of the set!

• We'll assume that they're not bigger -- that there is a one-to-one correspondence -- then show that that is impossible.
• We'll need this fact: every real number can be expressed by an infinite decimal.
• We need to be aware, however, that some real numbers are expressed by more than one decimal (for example, 1=.99999.... -- did you know that?)
• We'll show that just the real numbers between 0 and 1 are too numerous to be put in one-to-one correspondence with the natural numbers.
• We show this using a procedure that we might call "dodge ball".

There are infinitely many sizes of infinity. It turns out that the power set of a set is always of larger cardinality than the set itself. Thus every infinite set is smaller than its power set, which is an infinite set, which is smaller than its power set, etc....