![Page 1: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/1.jpg)
The bigness of things
Vaughn Climenhaga
University of Houston
Image from Wikipedia Image from Wikipedia
![Page 2: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/2.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 3: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/3.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people?
number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 4: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/4.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 5: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/5.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish?
length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 6: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/6.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 7: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/7.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city?
# of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 8: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/8.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 9: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/9.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house?
# of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 10: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/10.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 11: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/11.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment?
# of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 12: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/12.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 13: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/13.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book?
# of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 14: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/14.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 15: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/15.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook?
# of users data
the internet? # of websites data useful data
![Page 16: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/16.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 17: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/17.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet?
# of websites data useful data
![Page 18: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/18.jpg)
How big is it?
Meaning of “big” depends on what “it” is, and why we care.
How big is . . .
a crowd of people? number weight
a fish? length weight
a city? # of people diameter area
a house? # of bedrooms area volume
an assignment? # of problems time
a book? # of pages information
Facebook? # of users data
the internet? # of websites data useful data
![Page 19: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/19.jpg)
Various notions of “bigness”
Concrete, familiar meanings of “big” from the previous slide:
0. cardinality
1. length
2. area
3. volume
or “weighted” versions:
weight =
∫density d(volume)
More abstract meanings: “amount of data”?
I We are used to thinking of kB, MB, GB, TB, etc.
I But a 500 GB hard drive where every bit is set to ‘0’ doesn’thave much data on it. . .
![Page 20: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/20.jpg)
Various notions of “bigness”
Concrete, familiar meanings of “big” from the previous slide:
0. cardinality
1. length
2. area
3. volume
or “weighted” versions:
weight =
∫density d(volume)
More abstract meanings: “amount of data”?
I We are used to thinking of kB, MB, GB, TB, etc.
I But a 500 GB hard drive where every bit is set to ‘0’ doesn’thave much data on it. . .
![Page 21: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/21.jpg)
Various notions of “bigness”
Concrete, familiar meanings of “big” from the previous slide:
0. cardinality
1. length
2. area
3. volume
or “weighted” versions:
weight =
∫density d(volume)
More abstract meanings: “amount of data”?
I We are used to thinking of kB, MB, GB, TB, etc.
I But a 500 GB hard drive where every bit is set to ‘0’ doesn’thave much data on it. . .
![Page 22: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/22.jpg)
Subsets of R3
Focus on familiar meanings for now. Consider some subsets of R3.
(a) finite set (b) curve (c) surface (d) open region
0-dimensional 1-dimensional 2-dimensional 3-dimensional
![Page 23: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/23.jpg)
Subsets of R3
Focus on familiar meanings for now. Consider some subsets of R3.
(a) finite set (b) curve (c) surface (d) open region
0-dimensional 1-dimensional 2-dimensional 3-dimensional
Cardinality:
I (a): Good way to measure how big a finite set is
I (b)–(d) have infinite cardinality
![Page 24: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/24.jpg)
Subsets of R3
Focus on familiar meanings for now. Consider some subsets of R3.
(a) finite set (b) curve (c) surface (d) open region
0-dimensional 1-dimensional 2-dimensional 3-dimensional
Length:
I (a) has zero length. (Cover each point with tiny intervals)
I (b): Good way to measure how big a curve is
I (c)–(d) have infinite length: no curve of finite length can cover
![Page 25: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/25.jpg)
Subsets of R3
Focus on familiar meanings for now. Consider some subsets of R3.
(a) finite set (b) curve (c) surface (d) open region
0-dimensional 1-dimensional 2-dimensional 3-dimensional
Area:
I (a)–(b) have zero area. (Cover with tiny discs)
I (c): Good way to measure how big a surface is
I (d) has infinite area
![Page 26: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/26.jpg)
Subsets of R3
Focus on familiar meanings for now. Consider some subsets of R3.
(a) finite set (b) curve (c) surface (d) open region
0-dimensional 1-dimensional 2-dimensional 3-dimensional
Volume:
I (a)–(c) have zero volume
I (d): Good way to measure how big an open region is
![Page 27: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/27.jpg)
Subsets of R3
Focus on familiar meanings for now. Consider some subsets of R3.
(a) finite set (b) curve (c) surface (d) open region
0-dimensional 1-dimensional 2-dimensional 3-dimensional
Moral: To say how “big” a thing is, need to know its dimension.
![Page 28: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/28.jpg)
Subsets of R3
Focus on familiar meanings for now. Consider some subsets of R3.
(a) finite set (b) curve (c) surface (d) open region
0-dimensional 1-dimensional 2-dimensional 3-dimensional
Moral: To say how “big” a thing is, need to know its dimension.
I Dimension itself is a notion of bigness
I What is “dimension”? Seems to be which measure we use. . .
![Page 29: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/29.jpg)
Example 1: A Cantor set
Consider the sets
C0 = [0, 1]
C1 = [0, 13 ] ∪ [23 , 1]
C2 = [0, 19 ] ∪ [29 ,13 ] ∪ [23 ,
79 ] ∪ [89 , 1]
I Cn is disjoint union of 2n intervals of length 3−n
I Get Cn+1 from Cn by removing middle third of each interval
I The middle-third Cantor set is C =⋂
n≥0 Cn.
Fact 1: C is infinite.
I In fact, C is uncountable. (Bijection between {0, 1}N and C )
Fact 2: C has zero length. (Length of Cn is 2n3−n → 0)
What is the dimension of C? Between 0 and 1.
![Page 30: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/30.jpg)
Example 1: A Cantor set
Consider the sets
C0 = [0, 1]
C1 = [0, 13 ] ∪ [23 , 1]
C2 = [0, 19 ] ∪ [29 ,13 ] ∪ [23 ,
79 ] ∪ [89 , 1]
I Cn is disjoint union of 2n intervals of length 3−n
I Get Cn+1 from Cn by removing middle third of each interval
I The middle-third Cantor set is C =⋂
n≥0 Cn.
Fact 1: C is infinite.
I In fact, C is uncountable. (Bijection between {0, 1}N and C )
Fact 2: C has zero length. (Length of Cn is 2n3−n → 0)
What is the dimension of C? Between 0 and 1.
![Page 31: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/31.jpg)
Example 1: A Cantor set
Consider the sets
C0 = [0, 1]
C1 = [0, 13 ] ∪ [23 , 1]
C2 = [0, 19 ] ∪ [29 ,13 ] ∪ [23 ,
79 ] ∪ [89 , 1]
I Cn is disjoint union of 2n intervals of length 3−n
I Get Cn+1 from Cn by removing middle third of each interval
I The middle-third Cantor set is C =⋂
n≥0 Cn.
Fact 1: C is infinite.
I In fact, C is uncountable. (Bijection between {0, 1}N and C )
Fact 2: C has zero length. (Length of Cn is 2n3−n → 0)
What is the dimension of C? Between 0 and 1.
![Page 32: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/32.jpg)
Example 1: A Cantor set
Consider the sets
C0 = [0, 1]
C1 = [0, 13 ] ∪ [23 , 1]
C2 = [0, 19 ] ∪ [29 ,13 ] ∪ [23 ,
79 ] ∪ [89 , 1]
I Cn is disjoint union of 2n intervals of length 3−n
I Get Cn+1 from Cn by removing middle third of each interval
I The middle-third Cantor set is C =⋂
n≥0 Cn.
Fact 1: C is infinite.
I In fact, C is uncountable. (Bijection between {0, 1}N and C )
Fact 2: C has zero length. (Length of Cn is 2n3−n → 0)
What is the dimension of C? Between 0 and 1.
![Page 33: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/33.jpg)
Example 1: A Cantor set
Consider the sets
C0 = [0, 1]
C1 = [0, 13 ] ∪ [23 , 1]
C2 = [0, 19 ] ∪ [29 ,13 ] ∪ [23 ,
79 ] ∪ [89 , 1]
I Cn is disjoint union of 2n intervals of length 3−n
I Get Cn+1 from Cn by removing middle third of each interval
I The middle-third Cantor set is C =⋂
n≥0 Cn.
Fact 1: C is infinite.
I In fact, C is uncountable. (Bijection between {0, 1}N and C )
Fact 2: C has zero length. (Length of Cn is 2n3−n → 0)
What is the dimension of C? Between 0 and 1.
![Page 34: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/34.jpg)
Example 2: The Koch curve
Consider the curves
K0 =
K1 =
K2 =
. . .
I Kn has 4n line segments of length 3−n
I Get Kn+1 from Kn by replacing each line segment with ascaled-down copy of K0
I The Koch curve is K = limn→∞ Kn
Fact 1: K has infinite length. (Length of Kn is 4n3−n)
Fact 2: K has zero area. (Exercise – cover it with small rectangles)
![Page 35: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/35.jpg)
Example 2: The Koch curve
Consider the curves
K0 =
K1 =
K2 =
. . .
I Kn has 4n line segments of length 3−n
I Get Kn+1 from Kn by replacing each line segment with ascaled-down copy of K0
I The Koch curve is K = limn→∞ Kn
Fact 1: K has infinite length. (Length of Kn is 4n3−n)
Fact 2: K has zero area. (Exercise – cover it with small rectangles)
![Page 36: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/36.jpg)
Example 2: The Koch curve
Consider the curves
K0 =
K1 =
K2 =
. . .
I Kn has 4n line segments of length 3−n
I Get Kn+1 from Kn by replacing each line segment with ascaled-down copy of K0
I The Koch curve is K = limn→∞ Kn
Fact 1: K has infinite length. (Length of Kn is 4n3−n)
Fact 2: K has zero area. (Exercise – cover it with small rectangles)
![Page 37: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/37.jpg)
What is dimension?
Algebraic idea: # of parameters/coordinates. (Always an integer!)
More geometric idea: dimension is a scaling exponent.
Given λ > 0 and E ⊂ R3, let λE = {λx | x ∈ E}I volume(λE ) = λ3 · volume(E )
I area(λE ) = λ2 · area(E )
I length(λE ) = λ1 · length(E )
I cardinality(λE ) = λ0 · cardinality(E )
“Correct” thing to do now is find for each α > 0 a measure
µα : {subsets of R3} → [0,∞] such that µα(λE ) = λαµ(E )
This is α-dimensional Hausdorff measure, but requires technicalities
![Page 38: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/38.jpg)
What is dimension?
Algebraic idea: # of parameters/coordinates. (Always an integer!)
More geometric idea: dimension is a scaling exponent.
Given λ > 0 and E ⊂ R3, let λE = {λx | x ∈ E}I volume(λE ) = λ3 · volume(E )
I area(λE ) = λ2 · area(E )
I length(λE ) = λ1 · length(E )
I cardinality(λE ) = λ0 · cardinality(E )
“Correct” thing to do now is find for each α > 0 a measure
µα : {subsets of R3} → [0,∞] such that µα(λE ) = λαµ(E )
This is α-dimensional Hausdorff measure, but requires technicalities
![Page 39: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/39.jpg)
What is dimension?
Algebraic idea: # of parameters/coordinates. (Always an integer!)
More geometric idea: dimension is a scaling exponent.
Given λ > 0 and E ⊂ R3, let λE = {λx | x ∈ E}I volume(λE ) = λ3 · volume(E )
I area(λE ) = λ2 · area(E )
I length(λE ) = λ1 · length(E )
I cardinality(λE ) = λ0 · cardinality(E )
“Correct” thing to do now is find for each α > 0 a measure
µα : {subsets of R3} → [0,∞] such that µα(λE ) = λαµ(E )
This is α-dimensional Hausdorff measure, but requires technicalities
![Page 40: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/40.jpg)
Self-similarity
Previous slide highlighted self-similarity of measures.
Think about self-similarity of sets. Scale a set E by a factor of 12 .
How many copies needed to recover original shape?
E = [0, 1] 2 = 21 copies
E = [0, 1]2 4 = 22 copies
E = [0, 1]3 8 = 23 copies
Moral: If E is a union of n copies of λE , then E is self-similar, andthe dimension of E is α, where n = λ−α.
Solve this to write dimE = α = log n− log λ .
![Page 41: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/41.jpg)
Self-similarity
Previous slide highlighted self-similarity of measures.
Think about self-similarity of sets. Scale a set E by a factor of 12 .
How many copies needed to recover original shape?
E = [0, 1] 2 = 21 copies
E = [0, 1]2 4 = 22 copies
E = [0, 1]3 8 = 23 copies
Moral: If E is a union of n copies of λE , then E is self-similar, andthe dimension of E is α, where n = λ−α.
Solve this to write dimE = α = log n− log λ .
![Page 42: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/42.jpg)
Self-similarity
Previous slide highlighted self-similarity of measures.
Think about self-similarity of sets. Scale a set E by a factor of 12 .
How many copies needed to recover original shape?
E = [0, 1] 2 = 21 copies
E = [0, 1]2 4 = 22 copies
E = [0, 1]3 8 = 23 copies
Moral: If E is a union of n copies of λE , then E is self-similar, andthe dimension of E is α, where n = λ−α.
Solve this to write dimE = α = log n− log λ .
![Page 43: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/43.jpg)
Self-similarity
Previous slide highlighted self-similarity of measures.
Think about self-similarity of sets. Scale a set E by a factor of 12 .
How many copies needed to recover original shape?
E = [0, 1] 2 = 21 copies
E = [0, 1]2 4 = 22 copies
E = [0, 1]3 8 = 23 copies
Moral: If E is a union of n copies of λE , then E is self-similar, andthe dimension of E is α, where n = λ−α.
Solve this to write dimE = α = log n− log λ .
![Page 44: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/44.jpg)
Self-similarity
Previous slide highlighted self-similarity of measures.
Think about self-similarity of sets. Scale a set E by a factor of 12 .
How many copies needed to recover original shape?
E = [0, 1] 2 = 21 copies
E = [0, 1]2 4 = 22 copies
E = [0, 1]3 8 = 23 copies
Moral: If E is a union of n copies of λE , then E is self-similar, andthe dimension of E is α, where n = λ−α.
Solve this to write dimE = α = log n− log λ .
![Page 45: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/45.jpg)
ExamplesApply the formula dimE = log n
− log λ to some examples.
E λ n dimE
interval 12 2 log 2
log 2 = 1
square 12 4 log 4
log 2 = 2
cube 12 8 log 8
log 2 = 3
Cantor set 13 2 log 2
log 3 ∈ (0, 1)
Koch curve 13 4 log 4
log 3 ∈ (1, 2)
May consider other Cantor sets:
I scale by 15 , use 3 copies to build: dim = log 3
log 5
What about something like ?
![Page 46: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/46.jpg)
ExamplesApply the formula dimE = log n
− log λ to some examples.
E λ n dimE
interval 12 2 log 2
log 2 = 1
square 12 4 log 4
log 2 = 2
cube 12 8 log 8
log 2 = 3
Cantor set 13 2 log 2
log 3 ∈ (0, 1)
Koch curve 13 4 log 4
log 3 ∈ (1, 2)
May consider other Cantor sets:
I scale by 15 , use 3 copies to build: dim = log 3
log 5
What about something like ?
![Page 47: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/47.jpg)
ExamplesApply the formula dimE = log n
− log λ to some examples.
E λ n dimE
interval 12 2 log 2
log 2 = 1
square 12 4 log 4
log 2 = 2
cube 12 8 log 8
log 2 = 3
Cantor set 13 2 log 2
log 3 ∈ (0, 1)
Koch curve 13 4 log 4
log 3 ∈ (1, 2)
May consider other Cantor sets:
I scale by 15 , use 3 copies to build: dim = log 3
log 5
What about something like ?
![Page 48: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/48.jpg)
Dimension as a growth rate
Alternate way to derive dimension of our examples:
1. Given r > 0, break set into pieces of diameter ≤ r
2. N(r) = number of such pieces
Observe that N(r) ≈ r− dim
I interval: N(r) = r−1
I square: N(r) ≈ r−2
I cube: N(r) ≈ r−3
Conclusion: dim = limr→0logN(r)− log r (growth rate of N(r))
I Cantor set: N(3−n) = 2n, so logN(3−n)− log(3−n) = log 2
log 3
I Koch curve: N(3−n) = 4n, so logN(3−n)− log(3−n) = log 4
log 3
![Page 49: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/49.jpg)
Dimension as a growth rate
Alternate way to derive dimension of our examples:
1. Given r > 0, break set into pieces of diameter ≤ r
2. N(r) = number of such pieces
Observe that N(r) ≈ r− dim
I interval: N(r) = r−1
I square: N(r) ≈ r−2
I cube: N(r) ≈ r−3
Conclusion: dim = limr→0logN(r)− log r (growth rate of N(r))
I Cantor set: N(3−n) = 2n, so logN(3−n)− log(3−n) = log 2
log 3
I Koch curve: N(3−n) = 4n, so logN(3−n)− log(3−n) = log 4
log 3
![Page 50: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/50.jpg)
More general examples
Coastline of Britain
I r = size of ruler
I rN(r) = measured length
N(r) ≈ r−1.25
Measured length ≈ r−.25 →∞
Can show that N(2−(k+2)) = Fk ,the kth Fibonacci number
Use fact that Fk ≈(1+√5
2
)kto deduce that dim = log(1+
√5)
log 2 + 1
![Page 51: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/51.jpg)
More general examples
Coastline of Britain
I r = size of ruler
I rN(r) = measured length
N(r) ≈ r−1.25
Measured length ≈ r−.25 →∞
Can show that N(2−(k+2)) = Fk ,the kth Fibonacci number
Use fact that Fk ≈(1+√5
2
)kto deduce that dim = log(1+
√5)
log 2 + 1
![Page 52: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/52.jpg)
Bernoulli processes
Consider the following two stochastic processes:
1. Flip a coin repeatedly, write down outcome (H or T)
2. Roll a die repeatedly, write down the number from 1 to 6
Which one is “bigger”? The second one, but why?
I “Bigness” = amount of information to record after eachiteration of the experiment
I Information measured in number of bits
I h bits can store 2h possible sequences
For n possible outcomes, need 2h = n, so h = log2 n.
I First process: h = log2 2 = 1
I Second process: h = log2 6 ∈ (2, 3)h = entropy
![Page 53: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/53.jpg)
Bernoulli processes
Consider the following two stochastic processes:
1. Flip a coin repeatedly, write down outcome (H or T)
2. Roll a die repeatedly, write down the number from 1 to 6
Which one is “bigger”? The second one, but why?
I “Bigness” = amount of information to record after eachiteration of the experiment
I Information measured in number of bits
I h bits can store 2h possible sequences
For n possible outcomes, need 2h = n, so h = log2 n.
I First process: h = log2 2 = 1
I Second process: h = log2 6 ∈ (2, 3)h = entropy
![Page 54: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/54.jpg)
Bernoulli processes
Consider the following two stochastic processes:
1. Flip a coin repeatedly, write down outcome (H or T)
2. Roll a die repeatedly, write down the number from 1 to 6
Which one is “bigger”? The second one, but why?
I “Bigness” = amount of information to record after eachiteration of the experiment
I Information measured in number of bits
I h bits can store 2h possible sequences
For n possible outcomes, need 2h = n, so h = log2 n.
I First process: h = log2 2 = 1
I Second process: h = log2 6 ∈ (2, 3)h = entropy
![Page 55: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/55.jpg)
Unequal probabilities
What if I use a weighted coin? Say P(H) = 13 and P(T ) = 2
3 .
I More or less information? What’s the entropy?
Think of extreme case: P(H) = 11000 and P(T ) = 999
1000 .
I The event TTTTT doesn’t carry as much information now
I Most events carry less information
Definition: the information content of an event E is − log2 P(E )
I Entropy = expected information content of each experiment
Coin with weights 13 and 2
3 :
I H carries information log2(3), and T carries info log2(32)
Entropy = 13 log2(3) + 2
3 log2(32) = log2(3)− 23 < 1 (log is concave)
![Page 56: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/56.jpg)
Unequal probabilities
What if I use a weighted coin? Say P(H) = 13 and P(T ) = 2
3 .
I More or less information? What’s the entropy?
Think of extreme case: P(H) = 11000 and P(T ) = 999
1000 .
I The event TTTTT doesn’t carry as much information now
I Most events carry less information
Definition: the information content of an event E is − log2 P(E )
I Entropy = expected information content of each experiment
Coin with weights 13 and 2
3 :
I H carries information log2(3), and T carries info log2(32)
Entropy = 13 log2(3) + 2
3 log2(32) = log2(3)− 23 < 1 (log is concave)
![Page 57: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/57.jpg)
Unequal probabilities
What if I use a weighted coin? Say P(H) = 13 and P(T ) = 2
3 .
I More or less information? What’s the entropy?
Think of extreme case: P(H) = 11000 and P(T ) = 999
1000 .
I The event TTTTT doesn’t carry as much information now
I Most events carry less information
Definition: the information content of an event E is − log2 P(E )
I Entropy = expected information content of each experiment
Coin with weights 13 and 2
3 :
I H carries information log2(3), and T carries info log2(32)
Entropy = 13 log2(3) + 2
3 log2(32) = log2(3)− 23 < 1 (log is concave)
![Page 58: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/58.jpg)
Unequal probabilities
What if I use a weighted coin? Say P(H) = 13 and P(T ) = 2
3 .
I More or less information? What’s the entropy?
Think of extreme case: P(H) = 11000 and P(T ) = 999
1000 .
I The event TTTTT doesn’t carry as much information now
I Most events carry less information
Definition: the information content of an event E is − log2 P(E )
I Entropy = expected information content of each experiment
Coin with weights 13 and 2
3 :
I H carries information log2(3), and T carries info log2(32)
Entropy = 13 log2(3) + 2
3 log2(32) = log2(3)− 23 < 1 (log is concave)
![Page 59: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/59.jpg)
Maximising entropy
Suppose I use a coin with weights p and q.p, q ∈ [0, 1]p + q = 1
I Information content of event H is − log2 p
I Information content of event T is − log2 q
entropy = expected information content
= P(H)(− log2 p) + P(T )(− log2 q)
= −p log2 p − q log2 q = p log2
(1
p
)+ q log2
(1
q
)
Because log is concave down we always have entropy ≤ 1
Strictly concave ⇒ equality iff p = q = 12
![Page 60: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/60.jpg)
Maximising entropy
Suppose I use a coin with weights p and q.p, q ∈ [0, 1]p + q = 1
I Information content of event H is − log2 p
I Information content of event T is − log2 q
entropy = expected information content
= P(H)(− log2 p) + P(T )(− log2 q)
= −p log2 p − q log2 q = p log2
(1
p
)+ q log2
(1
q
)
Because log is concave down we always have entropy ≤ 1
Strictly concave ⇒ equality iff p = q = 12
![Page 61: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/61.jpg)
Maximising entropy
Suppose I use a coin with weights p and q.p, q ∈ [0, 1]p + q = 1
I Information content of event H is − log2 p
I Information content of event T is − log2 q
entropy = expected information content
= P(H)(− log2 p) + P(T )(− log2 q)
= −p log2 p − q log2 q = p log2
(1
p
)+ q log2
(1
q
)
Because log is concave down we always have entropy ≤ 1
Strictly concave ⇒ equality iff p = q = 12
![Page 62: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/62.jpg)
Relationship to dimension
Recall example of (12 ,14)-Cantor set.
After n iterations, get a set Cn with 2n intervals
I Length varies: left a times, right b times ⇒ r = (12)a(14)b
I # with this length is ( na )
I If a = pn and b = qn, then ( na ) ≈ e(−p log p−q log q)n
Instead of covering all of C , just cover the part where #left#right ≈
pq .
I r =((12)p(14)q
)n ⇒ − log r = n(p log 2 + q log 4)
I logN(r) ≥ n(−p log p − q log q)
dim ≈ logN(r)
− log r≥ −p log p − q log q
p log 2 + q log 4=
entropy
average expansion
Get actual dimension by maximising over (p, q).
![Page 63: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/63.jpg)
Relationship to dimension
Recall example of (12 ,14)-Cantor set.
After n iterations, get a set Cn with 2n intervals
I Length varies: left a times, right b times ⇒ r = (12)a(14)b
I # with this length is ( na )
I If a = pn and b = qn, then ( na ) ≈ e(−p log p−q log q)n
Instead of covering all of C , just cover the part where #left#right ≈
pq .
I r =((12)p(14)q
)n ⇒ − log r = n(p log 2 + q log 4)
I logN(r) ≥ n(−p log p − q log q)
dim ≈ logN(r)
− log r≥ −p log p − q log q
p log 2 + q log 4=
entropy
average expansion
Get actual dimension by maximising over (p, q).
![Page 64: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/64.jpg)
Relationship to dimension
Recall example of (12 ,14)-Cantor set.
After n iterations, get a set Cn with 2n intervals
I Length varies: left a times, right b times ⇒ r = (12)a(14)b
I # with this length is ( na )
I If a = pn and b = qn, then ( na ) ≈ e(−p log p−q log q)n
Instead of covering all of C , just cover the part where #left#right ≈
pq .
I r =((12)p(14)q
)n ⇒ − log r = n(p log 2 + q log 4)
I logN(r) ≥ n(−p log p − q log q)
dim ≈ logN(r)
− log r≥ −p log p − q log q
p log 2 + q log 4=
entropy
average expansion
Get actual dimension by maximising over (p, q).
![Page 65: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/65.jpg)
Relationship to dimension
Recall example of (12 ,14)-Cantor set.
After n iterations, get a set Cn with 2n intervals
I Length varies: left a times, right b times ⇒ r = (12)a(14)b
I # with this length is ( na )
I If a = pn and b = qn, then ( na ) ≈ e(−p log p−q log q)n
Instead of covering all of C , just cover the part where #left#right ≈
pq .
I r =((12)p(14)q
)n ⇒ − log r = n(p log 2 + q log 4)
I logN(r) ≥ n(−p log p − q log q)
dim ≈ logN(r)
− log r≥ −p log p − q log q
p log 2 + q log 4=
entropy
average expansion
Get actual dimension by maximising over (p, q).
![Page 66: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/66.jpg)
Relationship to dimension
Recall example of (12 ,14)-Cantor set.
After n iterations, get a set Cn with 2n intervals
I Length varies: left a times, right b times ⇒ r = (12)a(14)b
I # with this length is ( na )
I If a = pn and b = qn, then ( na ) ≈ e(−p log p−q log q)n
Instead of covering all of C , just cover the part where #left#right ≈
pq .
I r =((12)p(14)q
)n ⇒ − log r = n(p log 2 + q log 4)
I logN(r) ≥ n(−p log p − q log q)
dim ≈ logN(r)
− log r≥ −p log p − q log q
p log 2 + q log 4=
entropy
average expansion
Get actual dimension by maximising over (p, q).
![Page 67: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/67.jpg)
Information compression
Entropy measures information content
I Related: how much can data be compressed?
Shannon’s source coding theorem:
If we run n iterates of a process (IID)with entropy h, the results can be storedin nh bits of information, but no fewer.
Idea: First n results determine a subinterval of [0, 1]
I “Typical” interval has width ppnqqn = 2−nh
I Takes n bits to encode that much precision
![Page 68: University of Houstonclimenha/doc/uh-bigness.pdf · The bigness of things Vaughn Climenhaga University of Houston Image fromWikipedia Image fromWikipedia](https://reader034.vdocuments.site/reader034/viewer/2022042321/5f0aa0107e708231d42c8d35/html5/thumbnails/68.jpg)
Information compression
Entropy measures information content
I Related: how much can data be compressed?
Shannon’s source coding theorem:
If we run n iterates of a process (IID)with entropy h, the results can be storedin nh bits of information, but no fewer.
Idea: First n results determine a subinterval of [0, 1]
I “Typical” interval has width ppnqqn = 2−nh
I Takes n bits to encode that much precision
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Information contentEntropy can be used to analyse genetic data.
I Genome: string of symbols A,C ,G ,T
I Some regions more important than others
Topological entropy and topological pressure
I Quantities related to entropy discussed above
I Can be adapted to study genetic data
I High entropy/pressure ⇒ high information content⇒ more likely to be a coding region of the genome