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