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.