#cantor set

[rauko]

Alexander Horned Sphere | James W. Alexander (1924)

A classic example of a pathological (counterintuitive) object in the mathematical field of Topology. It was discovered by James Waddell Alexander II upon realizing that his original 1924 proof of Schoenflies' Theorem in 3D was incorrect. He then crafted the Horned Sphere as a new topological proof for the wild embedding of a 2-Sphere in 3D.

[numb3rth30ry]

(It’s Always) Sunny set up. 

Or, if you prefer, “mathemateggs.”

[well-420]

My mind when i try to solve 1+1

I always thought that math was easy, until the alphabet gets involved

[hazelriver74]

Was anyone going to tell me the cantor set contains 1/4 or was i just supposed to be asked to prove it first

[mucholderthen]

CANTOR SETS - BASIC SELF-SIMILARITY  The work of Georg Ferdinand Ludwig Philipp Cantor [1845 – 1918] – German mathematician and, in the latter third of the 19th century, the inventor of set theory, now a fundamental theory in mathematics.

IMAGES Simple Cantor Set  [explained below] 3D Cantor Set (by Oppenheimer on deviantArt)  [Cantor dust] Cantor Set, 2004, digital c-print (by Kevin Van Aelst)

Constructing the Cantor Ternary Set The Cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments, as follows.

Start with a line segment – for example, the line [0,1] in the top row of the first image.

Cut this unbroken line into three identical parts.

Delete the middle of the three parts.   This leaves what you see in the second row. 

Repeat:   Cut each of the two lines in the second row into thirds and then delete the middle third of each of the two lines. This creates the third row.

Repeat ...  [X]

The Cantor set is the prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated.

Sources consulted included Wikipedia articles on these subjects: Cantor sets, Georg Cantor, Topology, Set theory

[mnemehoshiko]

Be honest now... what do you really think about the Cantor set? (Cor. Did you ever have Prof. Todd?)

Anytime I think about the Cantor set...I have overwhelming to start chanting in Latin.

Beyond that, meh? I MEAN IT'S A USEFUL EXAMPLE OF A NOWHERE DENSE SET? But like beyond that...*yawns*.

No, I did not have Prof. Todd. Was he any good?

[processedandsussed]

prototyping 3D navigation interfaces for data space in processing

[fathom-the-universe-blog]

If you keep dividing the line segment infinitely you get something really interesting. It is called Cantor Dust. You get an infinite number of points with a total length of zero' Fathom the Universe