Lecture 3: Introduction to 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 / 11 Different bases? Higher dimensions? Notation: Middle-third Cantor set Generalisation of Cantor Set Question: How can you generalise the definition of the Cantor set? Tongou Yang (UBC) Fractal Lectures July 2019 2 / 11 Higher dimensions? Notation: Middle-third Cantor set Generalisation of Cantor Set Question: How can you generalise the definition of the Cantor set? Different bases? Tongou Yang (UBC) Fractal Lectures July 2019 2 / 11 Notation: Middle-third Cantor set Generalisation of Cantor Set Question: How can you generalise the definition of the Cantor set? Different bases? Higher dimensions? Tongou Yang (UBC) Fractal Lectures July 2019 2 / 11 Generalisation of Cantor Set Question: How can you generalise the definition of the Cantor set? Different bases? Higher dimensions? Notation: Middle-third Cantor set Tongou Yang (UBC) Fractal Lectures July 2019 2 / 11 Number of intervals removed. The positions of intervals being removed. Question: How does the dimension of the resulting Cantor set change? Theorem If the base is A and we choose B intervals at each time (regardless of their positions), then the resulting Cantor set has dimension log B : log A Using a Different Base Examples: Different bases. Tongou Yang (UBC) Fractal Lectures July 2019 3 / 11 The positions of intervals being removed. Question: How does the dimension of the resulting Cantor set change? Theorem If the base is A and we choose B intervals at each time (regardless of their positions), then the resulting Cantor set has dimension log B : log A Using a Different Base Examples: Different bases. Number of intervals removed. Tongou Yang (UBC) Fractal Lectures July 2019 3 / 11 Question: How does the dimension of the resulting Cantor set change? Theorem If the base is A and we choose B intervals at each time (regardless of their positions), then the resulting Cantor set has dimension log B : log A Using a Different Base Examples: Different bases. Number of intervals removed. The positions of intervals being removed. Tongou Yang (UBC) Fractal Lectures July 2019 3 / 11 Theorem If the base is A and we choose B intervals at each time (regardless of their positions), then the resulting Cantor set has dimension log B : log A Using a Different Base Examples: Different bases. Number of intervals removed. The positions of intervals being removed. Question: How does the dimension of the resulting Cantor set change? Tongou Yang (UBC) Fractal Lectures July 2019 3 / 11 Using a Different Base Examples: Different bases. Number of intervals removed. The positions of intervals being removed. Question: How does the dimension of the resulting Cantor set change? Theorem If the base is A and we choose B intervals at each time (regardless of their positions), then the resulting Cantor set has dimension log B : log A Tongou Yang (UBC) Fractal Lectures July 2019 3 / 11 Pick another integer B < A. Roll a \dice": Only once Roll a dice at each step. Create Your Own Cantor Set Pick an integer A. Tongou Yang (UBC) Fractal Lectures July 2019 4 / 11 Roll a \dice": Only once Roll a dice at each step. Create Your Own Cantor Set Pick an integer A. Pick another integer B < A. Tongou Yang (UBC) Fractal Lectures July 2019 4 / 11 Only once Roll a dice at each step. Create Your Own Cantor Set Pick an integer A. Pick another integer B < A. Roll a \dice": Tongou Yang (UBC) Fractal Lectures July 2019 4 / 11 Roll a dice at each step. Create Your Own Cantor Set Pick an integer A. Pick another integer B < A. Roll a \dice": Only once Tongou Yang (UBC) Fractal Lectures July 2019 4 / 11 Create Your Own Cantor Set Pick an integer A. Pick another integer B < A. Roll a \dice": Only once Roll a dice at each step. Tongou Yang (UBC) Fractal Lectures July 2019 4 / 11 Question: Find the dimension. Higher Dimensions Figure: 2D middle-third Cantor Dust Tongou Yang (UBC) Fractal Lectures July 2019 5 / 11 Higher Dimensions Figure: 2D middle-third Cantor Dust Question:Tongou Yang Find (UBC) the dimension. Fractal Lectures July 2019 5 / 11 Question: Find the dimension. Higher Dimensions Figure: 3D Cantor Dust Tongou Yang (UBC) Fractal Lectures July 2019 6 / 11 Higher Dimensions Figure: 3D Cantor Dust Question: Find the dimension. Tongou Yang (UBC) Fractal Lectures July 2019 6 / 11 Higher Dimensions (b) Sierpinski Pyramid (a) Sierpinski Triangle What is the dimension of Sierpi´nskitriangle? Tongou Yang (UBC) Fractal Lectures July 2019 7 / 11 Additional Topics The Smith-Volterra-Cantor set (\fat" Cantor set). Collatz conjecture (3n + 1 conjecture) and Collatz fractal. Tongou Yang (UBC) Fractal Lectures July 2019 8 / 11 Fractal with Positive Length At step n, remove the middle intervals with length 4−n from each interval remaining. What is the length remaining after infinitely many steps? Is there any \solid" interval left? Tongou Yang (UBC) Fractal Lectures July 2019 9 / 11 The Collatz Fractal Define a function ( 3n + 1; if n is odd f (n) = n 2 ; if n is even Conjecture: Figure: The Collatz Fractal Tongou Yang (UBC) Fractal Lectures July 2019 10 / 11 The End Tongou Yang (UBC) Fractal Lectures July 2019 11 / 11.