Lecture 3: Introduction to Diversity in , 2019

Tongou Yang

The University of British Columbia, Vancouver [email protected]

July 2019

Tongou Yang (UBC) 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 (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 ( 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