The Devil's Staircase | Infinite Series
PBS Infinite Series・2 minutes read
The Cantor set, named after Georg Cantor, is a self-similar fractal constructed by removing middle thirds at each stage, revealing points with base 3 expansions without 1's. The Cantor function, also known as the Devil's staircase, is a continuous curve with a length of 2 despite having 0 slope almost everywhere, making both mathematical objects unique with unusual properties.
Insights
- The Cantor set, constructed by removing middle thirds in stages, contains points with base 3 expansions devoid of 1's, revealing a fractal nature that allows infinite zooming despite being uncountable and having a length of 0.
- The Cantor function, or Devil's staircase, characterized by flat and diagonal lines built in stages, showcases a continuous curve with a length of 2, despite having almost zero slope throughout, highlighting its unique and intriguing mathematical properties.
Get key ideas from YouTube videos. It’s free
Recent questions
What is the Cantor set?
A collection of numbers constructed in stages.
What is the Cantor function?
A function built in stages with flat and diagonal lines.
How is the Cantor set related to base 3 expansions?
It contains points with base 3 expansions without 1's.
What are the key characteristics of the Cantor set?
It is self-similar, uncountable, and has a length of 0.
How is the Cantor function different from the Cantor set?
It is a function with flat and diagonal lines.
Related videos
TED-Ed
How big is infinity? - Dennis Wildfogel
Answers in Genesis
The Mandelbrot Set: Atheists’ WORST Nightmare
What makes it tick?
What is fractal geometry? │ The History of Mathematics with Luc de Brabandère
3Blue1Brown
Fractals are typically not self-similar
Northern Diaries
Is God A Mathematician? - Fractal Geometry of Nature
Summary
00:00
"Cantor Set and Function: Unique Mathematics"
- Cantor set, named after Georg Cantor, is a collection of numbers constructed in stages.
- In each stage, middle thirds are removed, leaving smaller intervals.
- Cantor set is not empty and contains points with base 3 expansions without 1's.
- Viewing Cantor set in base 3 reveals the points that remain after stages.
- Cantor set is a fractal, self-similar, allowing zooming in infinitely.
- Cantor set is uncountable but has a length of 0.
- Cantor function, or Devil's staircase, is built in stages with flat and diagonal lines.
- Cantor function is continuous, climbing vertically without vertical leaps.
- Cantor function's curve has a length of 2, despite having 0 slope almost everywhere.
- Cantor set and Cantor function are unique mathematical objects with unusual properties.
