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  


Homework
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: 

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 only 

 This page is unfinished. and may be finished sometime.