Reading Comprehension

PT129 · S4 · P4 · Q21 Fractal Geometry

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Fractal geometry is a mathematical theory devoted to the study of complex shapes called fractals.

Topic

Fractal geometry: its core concepts, appeal, and the debate over its mathematical legitimacy.

Framework

Present Debate

Main Point

Fractal geometry is a fascinating and visually striking branch of mathematics centered on self-similar patterns, but while many view it as revolutionary, some mathematicians remain skeptical about its value unless it develops a solid theoretical foundation of theorems and proofs. ( — last sentence of paragraph 3.)

P1: Introduction to Fractals and the Koch Curve

Fractal geometry studies complex, self-similar shapes called fractals. The passage introduces the Koch curve—a classic example—explaining its construction: starting from a line, segments are replaced to form a spiky pattern, and the process is repeated infinitely.

P2: Self-Similarity and Computer Visualization

This paragraph explains how self-similarity in fractals works using the Koch curve, and highlights how computers can generate images of increasingly detailed steps in the construction. The author points out that computer graphics make it easy to appreciate how simple rules can create intricate patterns—a key reason fractals are so intriguing.

P3: The Popularity and Controversy of Fractal Geometry

Here, the author describes two contrasting views: enthusiasts hail fractal geometry as a revolutionary tool for describing complex forms (even predicting it could rival calculus in importance), while more traditional mathematicians criticize the focus on flashy computer images over rigorous mathematical proofs. The skeptics argue that for fractal geometry to earn lasting respect, it needs intellectual rigor—precise language, theorems, and proofs.

21.

Which one of the following is closest to the meaning of the phrase "fully explicit" as used in the second sentence of the second paragraph?

  1. illustrated by an example

    Unrelated to Goal

    We can try substituting this in and it doesn't make any sense: Since the rules for getting from one stage to another are illustrated by an example, images of successive stages can be generated by computer. We were looking for something like "clear / precise".

    5% picked this

  2. uncomplicated

    Weak Match

    We can try substituting this in and it might seem to make some sense. Since the rules for getting from one stage to another are uncomplicated, images of successive stages can be generated by computer. The more uncomplicated the rules, the easier it would be to program a computer to do something, so that makes some sense. And the rules of the Koch curve didn't seem too complicated. For it being a description of how a fractal is built, we might have been impressed that we were able to follow the instructions in paragraph 1! But maybe other people thought the rules did sound complicated. How do we know whether the rules of the Koch curve for complicated or simple? And since the passage is saying that computers can generate images of other fractals, too, should we really be anchoring this sentence to what we know about the rules for the Koch curve? This sentence seems like a more general statement about how computers can generate images of a fractals because it can keep applying the same rules at any stage of the design (whether the rules are simple or complicated). This answer is worth keeping on a first pass, but when we compare it to (C), there is stronger logic to thinking, "In order for a computer to generate successive stages, the rules for getting from one stage to another must be .... clear vs. must be ... simple." A computer doesn't need simple rules, but it does need clear, unambiguous rules.

    9% picked this

  3. Correct

    expressed unambiguously

    Why this is right

    The author is stressing how a computer could understand the logic of how to build a fractal. To go from one stage to the next, you always follow the same rules, and the rules are very clear ... they are fully explicit ... they contain no ambiguity. If you want to feel this usage in a more everyday sense, pretend your friend has a crush on Melanie and wants you to go with him to her party. You're wondering if he was really invited to her party. "are you sure she wants you to come? I've seen you interpret really ambiguous things Melanie says as though they're expressing interest in your company." "Yes, yes, I'm totally sure. There was no ambiguity. She was fully explicit and said 'I want you to come to my party'. And you can bring your dorky friend." In order for a computer to be able to model each successive stage of design, it needs clear, explicit rules.

    Skill tested: Meaning in Context · how this choice captures the passage's function is the move to repeat next time.

    81% picked this

  4. in need of lengthy computation

    Unrelated to Goal

    We can try substituting this in and it doesn't make any sense: Since the rules for getting from one stage to another are in need of lengthy computation, images of successive stages can be generated by computer. There's no causal common sense that says, "Rule that are in need of lengthy computation mean that a computer can generate successive stages." Yes, a computer might be the tool we need to handle something that involves lengthy computation, but we'd say "Since this stuff takes a while to compute, we need a computer to generate images", not "Since this stuff takes a while to computer, images can be generated by computer". We want a trait that is conducive to being modeled on a computer.

    0% picked this

  5. agreed on by all

    Unrelated to Goal

    We can try substituting this in and it doesn't make any sense: Since the rules for getting from one stage to another are agreed on by all, images of successive stages can be generated by computer. We were looking for something like "clear / precise". There's no context to suggest that the reason a computer can generate successive stages is because "everyone voted and came to a consensus about it".

    4% picked this

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