Day 28, MAT115

This session will be captured on Zoom, if I remember to turn it on, and record it.

Our page of zooms and the play-by-plays.

Plus an article from the New York Times, which describes why geometry is our special human superpower....

For those interested in a little extra practice with Egyptian division, here is another example, using the Unit Fraction Table.

• Where have we used symmetry already?
  • Yanghui's triangle had an error in it....
  • There's a certain symmetry between Egyptian multiplication and division.
  • There's a type of symmetry in the golden rectangle: within a golden rectangle is found a perfect scale copy of itself (see Connor, above).
  • Here's a video clip of Fred Astaire and Ginger Rogers. They say Fred Astaire was pretty good (well, "the best dancer ever"); but Ginger Rogers did everything Fred did, only backwards (and in heels...).

    This is often referred to as a "mirror reflection": a type of symmetry.

  Where else does symmetry appear, and how do we use it?

• 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. I hear that it's 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! Here are some other creatures -- which ones also have bilateral symmetry? What other kinds of symmetry do you see? Symmetry of lifeforms on Earth

• "Our sensory organs and central nervous system are, as the result of evolutionary development, genetically programmed to recognize regularity, and hence order."
  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 strong resembled the average face of the population in which they lived, but which was characterized by more child-like features."

• Symmetry can be used 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. Let's start with the latter:
  • Contradancing
  • Haeckel and the radiolaria: 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."

• 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 at the definitions.

• In the plane of 2-Dimensions,
  • What is the most symmetric rectangle?
  • What are 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?