Self-study
P1
Fractal geometry is a mathematical theory devoted to the study of complex shapes called fractals. ████████ ██ █████ ██████████ ██ ████████ ███ ███ ████ ████████████ ████████ ████████ ███████ ███ ████████ ██ ████████████████ ███ ███████████ ██ █████████ ███████ ██ ████████ ██ █████████████ ███████ ██████ ██ ████ ████ █████ ████ ██████████ █████ █████████ ████ ███ ██████ ██ █ ██████ ███
Spotlight ·
Fractal geometry
No exact definition of fractals, but they show self-similarity: small details in shape look like the shape as a whole.
Example ·
Koch curve
The Koch curve is an example of a fractal. (Details of what the Koch curve looks like. No need to memorize this. Will come back if asked about it.)
P2
███████████████ ██ █████ ████ ███ ████████████ ███████ ██ ████████ ████████ ██ ████ █████ ███ ████ ███ ██ ███ ████████ ███████ ███ ████████ █████ ███ █████ ███ ███████ ████ ███ █████ ██ ███████ ███
More on the Koch curve ·
Can be created by computer
Interesting point about fractal geometry ·
Simple processes can create complex patterns
This is one of the major reasons people are into fractal geometry.
P3
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Enthusiasts ·
Believe fractal geometry will rival importance of calculus
Also believe it will allow mathematicians to effectively describe natural forms (such as a cloud).
Skeptics ·
Fractal geometry will have lasting ro in math only if it can support theorems and proofs
Passage Style
Critique or debate
Spotlight
21.
Which one of the following ██ ███████ ██ ███ ███████ ██ ███ ██████ █
a
illustrated by an ███████
The ability to be illustrated by an example doesn’t make sense as something that would allow a computer to generate the image.
b
uncomplicated
There’s no evidence that “fully explicit” means uncomplicated, because we have no reason to think computers aren’t able to apply complicated rules. “Fully explicit” should convey an idea that helps explain why images of the Koch curve can be generated by a computer. A set of complicated rules should still be applicable by a computer.
c
expressed unambiguously
“Expressed unambiguously” means expressed clearly. This fits as a replacement for the phrase “fully explicit,” because rules being expressed clearly make sense as part of why a computer can follow the rules. The rules are expressed clearly and are always the same; this is why computers can generate images of the Koch curve.
d
in need of ███████ ███████████
There’s no evidence that needing lengthy computation is something that helps explain why a computer would be able to generate images of the Koch curve.
e
agreed on by ███
There’s no evidence that having a consensus about the rules constitutes a reason a computer would be able to generate images of the Koch curve. A computer can apply rules that some people oppose, as long as the rules are clear.