Joint Session: Measuring Fractals Diversity in Mathematics, 2019

Tongou Yang

The University of British Columbia, Vancouver [email protected]

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 (a) Map of Canada (b) Map of Nunavut

Which country has the longest coastline?

List of countries by length of coastline

Tongou Yang (UBC) Fractal Lectures July 2019 3 / 12 (b) Map of Nunavut

Which country has the longest coastline?

List of countries by length of coastline

(a) Map of Canada

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

4

3

2

1

0 0 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

9 8 7 6 5 4 3 2 1 0 0 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, many fractal has infinite length!

Tongou Yang (UBC) Fractal Lectures July 2019 11 / 12 Practice Time!

Tongou Yang (UBC) Fractal Lectures July 2019 12 / 12