Day 13, MAT115

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Announcements:
- This week's quiz will be over symmetry. You'll want to get right on your homework....
Last time:
- We had an exam! Let's talk about it:
  - The test was actually out of 104 points, by accident; but I'll just treat your score as a percentage (as though it were out of 100).
  - Median score: 83 (top scores: 104! And there were two...)
  - Key
  - A couple of comments:
    - Make sure that I added up your points correctly: the amount that you earned on a page should be in the bottom left corner.
    - Feel free to argue if you feel that I didn't give you points you deserved. Sometimes I miss your point, and if you can make a case for yourself, do so!
    - In graphs there is some confusion about the meaning of the term "edges": an edge is just a connection between two vertices. It doesn't have to come at the "edge" of a diagram:
      
      Here there are 0, 1, 3, and 6 edges, and these are the triangular numbers.
    - Best moment....
Today's Question of the Day:

Are you two-faced?
- We're starting a new topic, one which I think is beautiful (and I hope that you'll agree). Let's talk about symmetry:
  
  Symmetry, as wide or as narrow as you define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection. Hermann Weyl (German Mathematician; 1885 - 1955)
  - Where have we seen (or used) symmetry already?
    - Yanghui's triangle had an error in it, that we found by assuming symmetry.
    - We used symmetry when drawing complete graphs: "Start with a regular polygon..." I might have said.
      
      I'm terribly fond of this graphic I made to illustrate all simple graphs with five vertices: I tried to use symmetry to advantage:
    - There's a beautiful symmetry in Pascal's triangle, and in its mathematical usefulness:
      - the problem of choosing 3 friends from 5 to go to the show
        is symmetric to
      - the problem choosing 2 (= 5-3) friends from 5 to not go to the show.
    - There's a type of symmetry in the Fibonacci (or golden) rectangle: within a golden rectangle is found a perfect scale copy of itself.
      We might call this "symmetry of scale" -- two identical copies of a thing, but of different sizes:
    Where else does symmetry appear, and how do we use it?
  - Here's a video clip of Fred Astaire and Ginger Rogers.
    - You'll notice many times when Fred and Ginger are "mirroring" each other; this is sometimes referred to as a "mirror reflection" (or "reflection symmetry"): one type of symmetry.
    - But in this clip you'll also see symmetry of scale, and what we might call "translational symmetry": a copy displaced from itself.
    - And notice that it is funny, or jolting, when symmetry is broken:
      
      Just like it was "jolting" when Yanghui's triangle failed symmetry in the 35 position....
    By the way: they say Fred Astaire was pretty good (well, "the best dancer ever"); but Ginger Rogers did everything Fred did, only backwards (and in heels...).
  - Along the same lines is this clip of Lucille Ball and Groucho Marx:
    I Love Lucy!
  - Are humans symmetric? Turns out that we're purported to have a "good side" and an "evil side"!
    Years ago a couple of my students tried the facial symmetry trick. It came back, on TikTok.
    Humans possess bilateral symmetry: we have mirrored sides. At least to external appearances. Inside, of course, some of our organs are on one side or the other. That's an interesting twist!
    "Our sensory organs and central nervous system are, as the result of evolutionary development, genetically programmed to recognize regularity, and hence order."
    "Patterns" are, in some sense, a symmetry: that the same thing is repeated. It leads to predictability.
    1. Which "face" did babies focus on?
    2. Composite versus designed faces
      - "In experiments, test subjects found computer-generated composite (averaged) images of faces more attractive than the many individual faces, from which the composite images were generated." (p. 23)
      - "[test subjects] developed a face which strongly resembled the average face of the population in which they lived, but which was characterized by more child-like features."
  - Here are some other of Earth's creatures -- which ones also have bilateral symmetry? What other kinds of symmetry do you see? Symmetry of life forms on Earth
    A phenomenal scientist and artist, Ernst Haeckel, discovered and documented many of the radiolaria. Let's check out a little of a video featuring many of his fantastic images: On the Discovery Docket: Proteus
    
    "They are like an alphabet of possibilities, as if the ancient sea were dreaming in its depths all the future permutations of organic and invented form. From backbones to bridges, and from the earth to the stars."
  - Symmetry can be used by us humans in two different ways:
    - to simplify, and
    - to complexify.
    We can use symmetry to solve some problems, and we can use symmetry to make things more interesting.
    - Making things simpler (too simple?): graphic designers "fake" the true recycling symbol with one arrow, which they copy and rotate to make their "faux recycling symbol" (the true one is on the right, and you'll see that the arrows are not all the same).
    - Complexifying: Making things more interesting!
      Contradancing
      These dances (and others, like square dances) effectively illustrate rotational, mirror, and translational symmetries.
  - Now to help us break down some of the most important elements of these topics, I want to distribute this worthy handout, or four pages of it at any rate.
    It covers two kinds of symmetry that are very important: rotational and reflective.
    So let's take a look! And get a start. Your homework is to complete it.
  - In the plane of 2-Dimensions,
    - What is the most symmetric rectangle?
    - How would you define the regular polygons:
    - How many regular polygons are there?
    - What kinds of symmetry do they possess?
    - Which regular polygons can be used exclusively to "tile" the plane, like a bathroom floor?
    - Using symmetry in a process, to build complete graphs. Let's try \(K_6\)!
Links:
- Symmetry, by Hermann Weyl (Princeton University Press, 1952)
- Symmetry of lifeforms on Earth
- A fun reading on symmetry
- Hexagonal Paper

Website maintained by Andy Long. Comments appreciated.

Fibonacci's solution in Liber Abaci,, chapter 12:
Because the above written pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second montth, and thus there are in the second month 3 pairs; of these in one month two are pregnant and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; ...
there will be 144 pairs in this [the tenth] month; to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.
To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year.
You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.