Day 25, MAT115

Fractals can be made out of just about anything, but this one is really beautiful.

¹

I got onto it because of this article Molecularly thin, two-dimensional, all-organic perovskites, which has a number of gorgeous graphics highlighting the duality between the cube and octahedron, with some tetrahedra thrown in to boot.

Knots tend to make nice features in logos, e.g. here's a Celtic heart knot for you lovers:

Is it tricolorable?

I bring it up now, because we're ready to talk about infinity -- and this one's best projections feature the infinity symbol!

Two of the three knots that are 6 or 7 crossings and tricolorable made use of a crossing where all the colors were the same (the rest of the crossings were the kind where three colors meet at the crossing).

• Bands are not knots or links, per se; they are surfaces; we sometimes say that they are "spanning surfaces" for knots or links.

  Nonetheless, we can think of a band as an unknot, say. But basically it is the edge(s) of a twisted band that are the knots or links.

• And some knots cannot be obtained by the method of cutting twisted bands, e.g. the figure-eight and three-twist knot; and some links cannot be obtained that way, e.g. the Borromean Rings.

  Those that can be are called "torus knots" or "torus links".

John Newton 1725-1807 (stanza 6 Anon); here's the Snopes fact-checked story of the author, a slave trader.

"Alice laughed. 'There's no use trying,' she said. 'One can't believe impossible things.'

I daresay you haven't had much practice,' said the Queen. 'When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast. There goes the shawl again!"

• Achilles and the Tortoise (the Tortoise beats Achilles.)
  

• The Dichotomy paradox (Motion is impossible.)
  

Mathemalchemy features "Zeno's Path", in honor of Zeno; and the Tortoise Tess.

• "I hate you."
• "I hate you twice as much as you hate me!"
• "I hate you twice what you hate me, plus one!"
• Eventually it gets to infinity, right?

Natural numbers are things which have a size, and if you add one to a natural number, you get a bigger number.

You don't get "a bigger number" -- more hate -- when you hate someone "infinity plus one".

In part because you don't have a number to begin with; but trying to add a number (1) to NaN doesn't work to give you a bigger number.

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

Intuitively: We will say that two sets have the same size if they have the same cardinality.

And there's the fact that Hilbert himself introduced the analolgy of the infinitely roomed hotel...:)

• Hilbert's Hotel: "it's always booked solid, but there's always a vacancy."
• Hilbert's Hotel and the Really Big High School Band
• Hilbert's Hotel and the Really Big Really Big High School Band Competition (RBRBHSBC)

1. The place is packed, and one person shows up;

   This result informs us that when someone on the playground hollers "I hate you infinity plus 1!", they really haven't hated any more than a simple infinity.

   Alternatively, if your lover says they love you infinitely much, you can't impress them by saying that you love them infinitely plus one. They will scoff, and perhaps leave you for a better mathematician! So take note....

2. The place is packed, and an infinite school bus shows up.

   This result informs us that when someone on the playground hollers "I hate you 2*infinity!", they really haven't hated any more than a simple infinity.

3. The place is packed, and an infinite number of infinite school buses show up! (Buses numbered 1, 2, .... with seats labelled 1, 2, ....)

   This result proves that there are just as many rational numbers (ratios of integers) as there are integers.

4. But if a bus containing the irrational numbers shows up -- now there's an infinite set! It's too big for the hotel...:( Finally the Hilbert Hotel meets its match!

   The real numbers (containing both the rational and irrational numbers) is just too big for Hilbert's Hotel. Mathematicians' guts lead them to believe (generally) that the real numbers are the next largest infinity (the first "uncountable" one).

5. The Hilbert Hotel shines a light on a strange fact about infinity: that when you get to sets that are infinitely sized, you find that their subsets can be exactly the same size as they are. This is very strange, and doesn't work with finite sets. So it's a very mysterious property for those of us accustomed to working with finite things....

1. Some mathematical facts:

   Mathematical fact zero:

   A set S is just a collection of objects; and a subset of S is just what you'd expect: some members of S. It may be all the members, in which we call it an "improper" subset, or it may be none of them (in which case we call it the empty set).

   We often denote a set by using braces, e.g. \(S=\{1,2,3\}\) is the set of the first three natural numbers.

   We say that \(a\) is an element of \(S\) if \(a\) is contained in \(S\), and we write \(a \in S\). So \(1 \in S\), \(2 \in S\), and \(3 \in S\). We deny that an object is in \(S\) this way: \(4 \notin S\).

   Mathematical fact one:

   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!

   Mathematical fact two:

   Let's call the set of all of a set S's subsets its power set, \(P(S)\). Pascal's triangle (that rascal!) shows you how many of each type of subset you have (for finite sets).

   

   

   Mathematical fact three:

   The power set of a set of $n$ elements contains $2^n$ elements.

   (We know that since each row of Pascal's triangle adds to a power of 2.)

2. Even the set containing no elements has one subset -- the set itself. So the power set has one element -- has size bigger than the set itself.

   This property holds true for all finite sets -- and it turns out to be true for infinite sets, too!

   Here's a silly video to illustrate how the power set grows with sets of increasing size. (Thanks to Dr. Towanna Roller (Asbury University) and her daughter Kristyn Roller (UK) for this one!)

3. This means that, although the natural numbers $1,2,3,\ldots$ is infinite as a set, there's a bigger set (the power set of the natural numbers).

   And the power set of that set is bigger yet, and so on forever, forever, Hallelujah, Hallelujah!

4. Upshot: infinity comes in various sizes! Infinitely many, in fact. Crazy sounding, I know -- but it's true.

   That symbol that you've been familiar with for all your lives, $\infty$: you thought it stood for a single thing; but it stands for a whole collection of monstrously big things, all too big to really think about properly. (Well, Cantor did!:)

5. That's too crazy, it seems; but, nonetheless, mathematicians know that it's true, because we can prove it! The key to determining if two sets have the same size is the one-to-one correspondence -- and that is key for infinite sets, as well.

6. So, in the end, here's what you say on the playground, when you want to love your friend more:

   "I love you more than the power set of your set of infinite love."

   

   Amen!