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Fractal geometry is a mathematical theory devoted to the study of

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New Project RC Butler 2019 - Practice 2 RC Passages Everyday
Passage # 349, Date : 24-Sep-2019
This post is a part of New Project RC Butler 2019. Click here for Details


Fractal geometry is a mathematical theory devoted to the study of complex shapes called fractals. Although an exact definition of fractals has not been established, fractals commonly exhibit the property of self-similarity: the reiteration of irregular details or patterns at progressively smaller scales so that each part, when magnified, looks basically like the object as a whole. The Koch curve is a significant fractal in mathematics and examining it provides some insight into fractal geometry. To generate the Koch curve, one begins with a straight line. The middle third of the line is removed and replaced with two line segments, each as long as the removed piece, which are positioned so as to meet and form the top of a triangle. At this stage, the curve consists of four connected segments of equal length that form a pointed protrusion in the middle. This process is repeated on the four segments so that all the protrusions are on the same side of the curve, and then the process is repeated indefinitely on the segments at each stage of the construction.

Self-similarity is built into the construction process by treating segments at each stage the same way as the original segment was treated. Since the rules for getting from one stage to another are fully explicit and always the same, images of successive stages of the process can be generated by computer. Theoretically, the Koch curve is the result of infinitely many steps in the construction process, but the finest image approximating the Koch curve will be limited by the fact that eventually the segments will get too short to be drawn or displayed. However, using computer graphics to produce images of successive stages of the construction process dramatically illustrates a major attraction of fractal geometry: simple processes can be responsible for incredibly complex patterns.

A worldwide public has become captivated by fractal geometry after viewing astonishing computer generated images of fractals; enthusiastic practitioners in the field of fractal geometry consider it a new language for describing complex natural and mathematical forms. They anticipate that fractal geometry’s significance will rival that of calculus and expect that proficiency in fractal geometry will allow mathematicians to describe the form of a cloud as easily and precisely as an architect can describe a house using the language of traditional geometry. Other mathematicians have reservations about the fractal geometers’ preoccupation with computer-generated graphic images and their lack of interest in theory. These mathematicians point out that traditional mathematics consists of proving theorems, and while many theorems about fractals have already been proven using the notions of pre-fractal mathematics, fractal geometers have proven only a handful of theorems that could not have been proven with pre-fractal mathematics. According to these mathematicians, fractal geometry can attain a lasting role in mathematics only if it becomes a precise language supporting a system of theorems and proofs.

1. Which one of the following most accurately expresses the main point of the passage?

(A) Because of its unique forms, fractal geometry is especially adaptable to computer technology and is therefore likely to grow in importance and render pre-fractal mathematics obsolete.
(B) Though its use in the generation of extremely complex forms makes fractal geometry an intriguing new mathematical theory, it is not yet universally regarded as having attained the theoretical rigor of traditional mathematics.
(C) Fractal geometry is significant because of its use of self-similarity, a concept that has enabled geometers to generate extremely detailed computer images of natural forms.
(D) Using the Koch curve as a model, fractal geometers have developed a new mathematical language that is especially useful in technological contexts because it does not rely on theorems.
(E) Though fractal geometry has thus far been of great value for its capacity to define abstract mathematical shapes, it is not expected to be useful for the description of ordinary natural shapes.


2. Which one of the following is closest to the meaning of the phrase “fully explicit” (Highlighted)

(A) illustrated by an example
(B) uncomplicated
(C) expressed unambiguously
(D) in need of lengthy computation
(E) agreed on by all


3. According to the description in the passage, each one of the following illustrates the concept of self-similarity EXCEPT:

(A) Any branch broken off a tree looks like the tree itself.
(B) Each portion of the intricately patterned frost on a window looks like the pattern as a whole.
(C) The pattern of blood vessels in each part of the human body is similar to the pattern of blood vessels in the entire body.
(D) The seeds of several subspecies of maple tree resemble one another in shape despite differences in size.
(E) The florets composing a cauliflower head resemble the entire cauliflower head.


4. The explanation of how a Koch curve is generated (lines 08–17) (Text in red) serves primarily to

(A) show how fractal geometry can be reduced to traditional geometry
(B) give an example of a natural form that can be described by fractal geometry
(C) anticipate the objection that fractal geometry is not a precise language
(D) illustrate the concept of self-similarity
(E) provide an exact definition of fractals


5. Which one of the following does the author present as a characteristic of fractal geometry?

(A) It is potentially much more important than calculus.
(B) Its role in traditional mathematics will expand as computers become faster.
(C) It is the fastest-growing field of mathematics.
(D) It encourages the use of computer programs to prove mathematical theorems.
(E) It enables geometers to generate complex forms using simple processes.


6. Each of the following statements about the Koch curve can be properly deduced from the information given in the passage EXCEPT:

(A) The total number of protrusions in the Koch curve at any stage of the construction depends on the length of the initial line chosen for the construction.
(B) The line segments at each successive stage of the construction of the Koch curve are shorter than the segments at the previous stage.
(C) Theoretically, as the Koch curve is constructed its line segments become infinitely small.
(D) At every stage of constructing the Koch curve, all the line segments composing it are of equal length.
(E) The length of the line segments in the Koch curve at any stage of its construction depends on the length of the initial line chosen for the construction.


7. The enthusiastic practitioners of fractal geometry mentioned in lines 39–40 (Lines Bolded) would be most likely to agree with which one of the following statements?

(A) The Koch curve is the most easily generated, and therefore the most important, of the forms studied by fractal geometers.
(B) Fractal geometry will eventually be able to be used in the same applications for which traditional geometry is now used.
(C) The greatest importance of computer images of fractals is their ability to bring fractal geometry to the attention of a wider public.
(D) Studying self-similarity was impossible before the development of sophisticated computer technologies.
(E) Certain complex natural forms exhibit a type of self-similarity like that exhibited by fractals.


8. The information in the passage best supports which one of the following assertions?

(A) The appeal of a mathematical theory is limited to those individuals who can grasp the theorems and proofs produced in that theory.
(B) Most of the important recent breakthroughs in mathematical theory would not have been possible without the ability of computers to graphically represent complex shapes.
(C) Fractal geometry holds the potential to replace traditional geometry in most of its engineering applications.
(D) A mathematical theory can be developed and find applications even before it establishes a precise definition of its subject matter.
(E) Only a mathematical theory that supports a system of theorems and proofs will gain enthusiastic support among a significant number of mathematicians.



  • Source: LSAT Official PrepTest 57 (June 2009)
  • Difficulty Level: 650

Originally posted by hero_with_1000_faces on 23 Sep 2019, 09:24.
Last edited by SajjadAhmad on 16 Oct 2019, 00:21, edited 10 times in total.
Updated - Complete topic (1032).
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Re: Fractal geometry is a mathematical theory devoted to the study of  [#permalink]

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New post 23 Sep 2019, 10:19
Hi hero_with_1000_faces

I really appreciate your efforts you have learned how to post an RC passage, i would like to suggest you few more things which make the passage easy to understand and presentable.

  • Bold the question statement (as i did in this passage for you).
  • Add the OA right below the timer.
  • Highlight the text where required.

    As required in question #2 in this passage.

    2. Which one of the following is closest to the meaning of the phrase “fully explicit” as used in lines 24–25?

    You can highlight the text in the passage and change the question statement as

    2. Which one of the following is closest to the meaning of the phrase “fully explicit” as (Highlighted)?

    Or mentioned the correct line # otherwise the confusion will remain.

    Q #2, 4 and 7 requires edits please edit accordingly.

  • Paste the starting passage text as topic name i.e in this passage

    "Fractal geometry is a mathematical theory devoted to the study of" is the topic name. I have just edited.
  • Add the source tag, if the source in not from the available tags add it manually at the very end of the passage.

Thanks for your cooperation.
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Re: Fractal geometry is a mathematical theory devoted to the study of  [#permalink]

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New post 23 Sep 2019, 21:31
hero_with_1000_faces wrote:
SajjadAhmad

Thank You sajjad, Im learning :)



hero_with_1000_faces - I want to understand the reasoning behind Question#1 - solution
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New post 25 Sep 2019, 00:10
dushyanta wrote:
hero_with_1000_faces wrote:
SajjadAhmad

Thank You sajjad, Im learning :)



hero_with_1000_faces - I want to understand the reasoning behind Question#1 - solution



1. Which one of the following most accurately expresses the main point of the passage?

(A) Because of its unique forms, fractal geometry is especially adaptable to computer technology and is therefore likely to grow in importance and render pre-fractal mathematics obsolete.


This cannot be the Main Idea as "Fractal Geometry is therefore likely to grow in importance and render pre-fractal mathematics obsolete." as the it is in fact opposed in last para by citing mathematicians who think Fractal Geometry has proved only handful theorem.




(C) Fractal geometry is significant because of its use of self-similarity, a concept that has enabled geometers to generate extremely detailed computer images of natural forms.

They anticipate that fractal geometry’s significance will rival that of calculus and expect that proficiency in fractal geometry will allow mathematicians to describe the form of a cloud as easily

Incorrect, as it is only the anticipation of enthusiastic practitioners not of the passage or author.


(D) Using the Koch curve as a model, fractal geometers have developed a new mathematical language that is especially useful in technological contexts because it does not rely on theorems.

This cannot be inferred in any way. nothing mentioned about technological contexts and not relying on theorems is used as a criticism for Fractal geo.

(E) Though fractal geometry has thus far been of great value for its capacity to define abstract mathematical shapes, it is not expected to be useful for the description of ordinary natural shapes.

last para

A worldwide public has become captivated by fractal geometry

No where in the passage it has been mentioned that the Fractal geometry has been of great value, there is only an anticipation by enthusiastic practitioners that it might be of value. Also it cannot be inferred that FG wont be useful for description of ordinary natural shapes, the criticism is only for 'fractal geometers' on the grounds of their lack of interest in theory i.e proving theorems etc.

(B) Though its use in the generation of extremely complex forms makes fractal geometry an intriguing new mathematical theory, it is not yet universally regarded as having attained the theoretical rigor of traditional mathematics.

From the authors tone in the first para it can be inferred that he finds FG interesting, and also as some Mathematicians do not have high regards as per last sentence of passage, we can say that it is not universally accepted.

Hence B is right
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Fractal geometry is a mathematical theory devoted to the study of  [#permalink]

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New post Updated on: 27 Nov 2019, 02:24
Hi everyone,
Solved this one in 16 minutes.

--------------------------------------------------------------------------------


P1

Paragraph one talks about fractals and what they are. A fractal is a figure that if decomposed will resemble to the figure itself. The further we analyze a fractal the smaller it will be. The Khoche curve is an example and we know that each component of each section of it is of the same size.

Brief summary: What are fractals and an example: the Khoche curve.

P2

The concept of self similarity is detailed. We are given that thanks to the repetitiveness we can represent fractals on computer but because fractals stages tend to get smaller and smaller at some point it will be difficult to illustrate such stages well. Finally the author says that fractals can be used to detail a complex figure in a simple way.

Brief summary: Self similarity and scope of fractals

P3

Paragraph 3 tells us that many people support fractal theory and that these people think of fractal geometry as a rival of calculus and as a very precise way of describing objects. Mathematicians on the other hand have doubts because fractal geometry alone has proved only some theorems without mathematic concepts.

Brief summary:Fractal enthusiasts and skeptics


Main point

The main point is to discuss the extent of fractals


--------------------------------------------------------------------------------




1. Which one of the following most accurately expresses the main point of the passage?

Pre-thinking

Main point question

Refer to main point above


(A) Because of its unique forms, fractal geometry is especially adaptable to computer technology and is therefore likely to grow in importance and render pre-fractal mathematics obsolete.
Not in line with pre-thinking

(B) Though its use in the generation of extremely complex forms makes fractal geometry an intriguing new mathematical theory, it is not yet universally regarded as having attained the theoretical rigor of traditional mathematics.
in line with pre-thinking

(C) Fractal geometry is significant because of its use of self-similarity, a concept that has enabled geometers to generate extremely detailed computer images of natural forms.
Partial scope

(D) Using the Koch curve as a model, fractal geometers have developed a new mathematical language that is especially useful in technological contexts because it does not rely on theorems.
Out of scope

(E) Though fractal geometry has thus far been of great value for its capacity to define abstract mathematical shapes, it is not expected to be useful for the description of ordinary natural shapes.
[b]Not in line with pre-thinking[/b]

--------------------------------------------------------------------------------




2. Which one of the following is closest to the meaning of the phrase “fully explicit” (Highlighted)

Pre-thinking

Inference question

[b]Refer to the highlighted part in the text. The meaning of this phrase is that the process is very detailed and it can be repeated infinite times.

[/b]

(A) illustrated by an example
Not in line with pre-thinking

(B) uncomplicated
Not in line with pre-thinking

(C) expressed unambiguously
in line with pre-thinking

(D) in need of lengthy computation
Not in line with pre-thinking

(E) agreed on by all
Not in line with pre-thinking

--------------------------------------------------------------------------------



3. According to the description in the passage, each one of the following illustrates the concept of self-similarity EXCEPT:

Pre-thinking

Detail question

"Self-similarity is built into the construction process by treating segments at each stage the same way as the original segment was treated. "
Let's read the answer choice


(A) Any branch broken off a tree looks like the tree itself.
Similar

(B) Each portion of the intricately patterned frost on a window looks like the pattern as a whole.
Similar

(C) The pattern of blood vessels in each part of the human body is similar to the pattern of blood vessels in the entire body.
Similar

(D) The seeds of several subspecies of maple tree resemble one another in shape despite differences in size.
Not similar because each seed should resemble the maple tree.

(E) The florets composing a cauliflower head resemble the entire cauliflower head.
Similar


--------------------------------------------------------------------------------


4. The explanation of how a Koch curve is generated (lines 08–17) (Text in red) serves primarily to

Pre-thinking

Purpose question

"The Koch curve is a significant fractal in mathematics and examining it provides some insight into fractal geometry."
The purpose is to provide insights in to FG


(A) show how fractal geometry can be reduced to traditional geometry
Not in line with pre-thinking

(B) give an example of a natural form that can be described by fractal geometry
The Koch curve is not a natural form

(C) anticipate the objection that fractal geometry is not a precise language
Not in line with pre-thinking

(D) illustrate the concept of self-similarity
Not in line with pre-thinking but illustrating self similarity is also a purpose. "This process is repeated on the four segments so that all the protrusions are on the same side of the curve, and then the process is repeated indefinitely on the segments at each stage of the construction."

(E) provide an exact definition of fractals
Not in line with pre-thinking


--------------------------------------------------------------------------------



5. Which one of the following does the author present as a characteristic of fractal geometry?

Pre-thinking

Detail question

Let's analyze the answer choices


(A) It is potentially much more important than calculus.
Per the passage it may rival calculus

(B) Its role in traditional mathematics will expand as computers become faster.
not given

(C) It is the fastest-growing field of mathematics.
not given

(D) It encourages the use of computer programs to prove mathematical theorems.
not given

(E) It enables geometers to generate complex forms using simple processes.
"simple processes can be responsible for incredibly complex patterns."

--------------------------------------------------------------------------------



6. Each of the following statements about the Koch curve can be properly deduced from the information given in the passage EXCEPT:

Pre-thinking

Inference question

Let's analyze the answer choices


(A) The total number of protrusions in the Koch curve at any stage of the construction depends on the length of the initial line chosen for the construction.
Per the passage at each stage we should have just one point of protrusion.

" At this stage, the curve consists of four connected segments of equal length that form a pointed protrusion in the middle."


(B) The line segments at each successive stage of the construction of the Koch curve are shorter than the segments at the previous stage.
Can be inferred from here: "the reiteration of irregular details or patterns at progressively smaller scales so that each part,"

(C) Theoretically, as the Koch curve is constructed its line segments become infinitely small.
same as B

(D) At every stage of constructing the Koch curve, all the line segments composing it are of equal length.
Can be inferred from here: "At this stage, the curve consists of four connected segments of equal length that form a pointed protrusion in the middle."

(E) The length of the line segments in the Koch curve at any stage of its construction depends on the length of the initial line chosen for the construction.
Can be inferred from here: "To generate the Koch curve, one begins with a straight line. The middle third of the line is removed and replaced with two line segments, each as long as the removed piece, which are positioned so as to meet and form the top of a triangle."


--------------------------------------------------------------------------------


7. The enthusiastic practitioners of fractal geometry mentioned in lines 39–40 (Lines Bolded) would be most likely to agree with which one of the following statements?

Pre-thinking

Inference question

Let's refer to the initial part of the last paragraph to answer this question


(A) The Koch curve is the most easily generated, and therefore the most important, of the forms studied by fractal geometers.
The most makes this answer too extreme

(B) Fractal geometry will eventually be able to be used in the same applications for which traditional geometry is now used.
Can be inferred from here: "They anticipate that fractal geometry’s significance will rival that of calculus and expect that proficiency in fractal geometry will allow mathematicians to describe the form of a cloud as easily and precisely as an architect can describe a house using the language of traditional geometry."

(C) The greatest importance of computer images of fractals is their ability to bring fractal geometry to the attention of a wider public.
Cannot be inferred

(D) Studying self-similarity was impossible before the development of sophisticated computer technologies.
Too extreme

(E) Certain complex natural forms exhibit a type of self-similarity like that exhibited by fractals.
out of scope


--------------------------------------------------------------------------------


8. The information in the passage best supports which one of the following assertions?

Pre-thinking

Inference question

Let's analyze the answer choices


(A) The appeal of a mathematical theory is limited to those individuals who can grasp the theorems and proofs produced in that theory.
out of scope

(B) Most of the important recent breakthroughs in mathematical theory would not have been possible without the ability of computers to graphically represent complex shapes.
out of scope

(C) Fractal geometry holds the potential to replace traditional geometry in most of its engineering applications.
engineering applications are not even discussed

(D) A mathematical theory can be developed and find applications even before it establishes a precise definition of its subject matter.
Chosen because of POE. [b]hero_with_1000_faces Thanks for the explanation:) Can you also point which are the applications of fractals per the passage?[/b]

(E) Only a mathematical theory that supports a system of theorems and proofs will gain enthusiastic support among a significant number of mathematicians.
enthusiastic support among mathematicians is not discussed

--------------------------------------------------------------------------------


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Originally posted by auradediligodo on 27 Nov 2019, 02:01.
Last edited by auradediligodo on 27 Nov 2019, 02:24, edited 1 time in total.
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Re: Fractal geometry is a mathematical theory devoted to the study of  [#permalink]

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New post 27 Nov 2019, 02:20
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auradediligodo
the passage tells us there is no exact definition of "fractals" but then proceeds to show us how this theoretical area has developed and found applications even so.
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Re: Fractal geometry is a mathematical theory devoted to the study of   [#permalink] 27 Nov 2019, 02:20
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