Day 32, 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.

I will drop a couple of your lowest homework scores.

• The exam
• High score: 97 (before the curve!)
• key
• curve: add 6 to your score.
• comments?

You want to start thinking about your Mathematical logo. We have a couple of "logo-popular" topics left (e.g. fractals, knots, infinity); but you may have already seen something that inspires you (some people like spirals -- like CINSAM!).

At any rate, you want to start thinking about this. Two of our final days will be dedicated to presenting our logos in class (or on-line for the on-line section).

In-person presentations include a one-page, type-written description of your logo -- what it is (specifically, mathematically -- what are its mathematical elements?), why you chose to use the components you did.

But, in contrast to the regular polygons (of which there are infinitely many), there are only five!

The Five Convex Regular Polyhedra (Platonic solids) -- thanks Wikipedia!
Tetrahedron	Hexahedron or Cube	Octahedron	Icosahedron	Dodecahedron
(Animation)	(Animation)	(Animation)	(Animation)	(Animation)
fire	earth	air	water	universe

I'll turn you loose in a bit, to build your own magnetically beautifully solids, but, before we go, I want to remind you of a couple of ideas -- one in particular which points to our future (fractals):

"In the Hindu tradition, the icosahedron represents purusha, the male, spiritual principle, and generates the dodecahedron, representing prakriti, the female, material principle." (It's interesting that they chose duals, don't you think?)

I hope that this has you imagining "Russian Dolls", which are those which stack in such a way that there is a perfect copy of a doll inside each doll, only smaller (self-similarity at a smaller scale):

• The tetrahedron can be drawn to be a "distorted" Mercedes symbol, as we've seen.
• The cube you've undoubtedly done before.
• The others are more complicated, but here are all the solids:
  

  Practice drawing them! Color them and make them pretty....

  Can you see the duality?

What conclusions can we draw from this data? Is there a pattern? (Of course there is!:) The pattern leads to the concept of "Duality".

We can also verify that the Platonic solids satisfy Euler's formula, even though they're three dimensional objects! Their graphs are planar, however, as you can see!