28sep19GF## Review

Wednesday: (1IH No projector and screen)

- 1. Five minute reports by students.
- 2. Greenboard presentation of dim(Sierpinski Gasket)=irrational (Barnsley)
Thursday: (IGL 121AH with TV screen )

- 1. Sinewheel
- 2. Tavares's cubes
- 3. Thurston's sphere eversion by corrugations
- 4. 3D Koch aka Fahauer fractals, aka Yulia's recursive kochblue.py
## Please consider using David Eck's superb and free-on-line computer graphics The University of Minnesota has a nice webpage for both on-line and hard-back copies here:

Homework

https://open.umn.edu/opentextbooks/textbooks/introduction-to-computer-graphics

Hobar&Smith Prof. David Eck (http://math.hws.edu/eck/) is both mathematician and computer scientist, who did his PhD at Brandeis with Richard Palais.The important thing about his approach is that it is "hands-on" even when he is discussing obsolete or OpenGL languages. He does that by simulating the older languages and libraries with HTML5/CANVAS/Javascript/WebGL software. And (!) he is offering his glsim.js Python/OpenGL excerpt for translating such RTICAs into WebGL painlessly.

But the final two chapter has eye-candy quality examples in full-blown WebGL applications. Everything is inspectable. Nothing is hidden. And it's free! (except the hardback, which is cheap!).

## Mini Projects Continued

## From Sinewheel to Allerton

Background:sinewheel.html is a HTML5/CANVAS implementation by Zach Reizner of a 40 year of Apple BASIC rtica to explain trigonometric functions by how their graph is generated, or (geometrically) how to pass from the parametrization of a circle from polar coordinates ($r= const$) to Cartesian ($ x=r \sin t, y= r \cos t$) coordinates.Question: Does this rtica (animated yes, but barely interactive) illustrate this well enough? Did you catch it's meaning when you played with it?

As such, it became the beginning of a lesson on how to graph functions in analytic geometry that are better (contain more information) than the one you learned in high school. Note that the sine function graph is generated as an illusion by the corners of a sequence of polygons:

origin:radius:circle:horizontal:corner:vertical:pos number axis.

If you buy it that for the first(!) polygon the distance along the circle is the same as the distance moved on the positive reals, and then count steps, then the endpoint falls a distance equal to the circumferential distance travelled then the corner sits on the graph of the sine of the arclength of the angle in radians.The second part was a similar generation of the parabola $y=x(1-x)$ in the unit square. This lead to using a cobweb graph to illustrate logistic chaos. See Lisa Li and Manting Huang's IGL project

Miniproject:Improve the way sinewheel tells its story. Make it interactive. E.g. have the student step through the formation of the illusion. Try a 2D animation explaining both sine and cosine. Invent!

Resources:

- http://new.math.uiuc.edu/quasiweb17/allertonJSweb.html
Tools:2D line drawing using HTML5/CANVAS onlyThis page is unfinished.and may be finished sometime.