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Joint Session: Measuring FractalsDiversity in Mathematics, 2019
Tongou Yang
The University of British Columbia, Vancouver
toyang@math.ubc.ca
July 2019
Tongou Yang (UBC) Fractal Lectures July 2019 1 / 12
Let’s Talk Geography
Which country has the longest coastline?
What is the longest river on earth?
Tongou Yang (UBC) Fractal Lectures July 2019 2 / 12
Which country has the longest coastline?
List of countries by length of coastline
(a) Map of Canada (b) Map of Nunavut
Tongou Yang (UBC) Fractal Lectures July 2019 3 / 12
Which country has the longest coastline?
List of countries by length of coastline
(a) Map of Canada
(b) Map of Nunavut
Tongou Yang (UBC) Fractal Lectures July 2019 3 / 12
Which country has the longest coastline?
List of countries by length of coastline
(a) Map of Canada (b) Map of Nunavut
Tongou Yang (UBC) Fractal Lectures July 2019 3 / 12
What’s the Longest River on Earth?
A Youtube Video
Tongou Yang (UBC) Fractal Lectures July 2019 4 / 12
Measuring a Smooth Curve
Example: use a rope
Figure: Boundary between CA and US on the Great Lakes
Tongou Yang (UBC) Fractal Lectures July 2019 5 / 12
Measuring a Rugged Coastline
Figure: Coast of Nova Scotia
Tongou Yang (UBC) Fractal Lectures July 2019 6 / 12
Covering by Grids
0 1 2 3 40
1
2
3
4
Tongou Yang (UBC) Fractal Lectures July 2019 7 / 12
Counting Grids
Number of Grids=15
Side Length ≈ 125km
Coastline ≈ 15× 125 = 1845km.
Tongou Yang (UBC) Fractal Lectures July 2019 8 / 12
Covering by Finer Grids
0 1 2 3 4 5 6 7 8 90
1
2
3
4
5
6
7
8
9
Tongou Yang (UBC) Fractal Lectures July 2019 9 / 12
Counting Grids
Number of Grids=34
Side Length ≈ 62km
Coastline ≈ 34 × 62 = 2108km.
Tongou Yang (UBC) Fractal Lectures July 2019 10 / 12
The Coastline Paradox
In general, finer grids lead to ”longer” coastlines, and in theory, manyfractal has infinite length!
Tongou Yang (UBC) Fractal Lectures July 2019 11 / 12
Practice Time!
Tongou Yang (UBC) Fractal Lectures July 2019 12 / 12
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