Cantor Set

History

    The Cantor Set is a set of points that are lying on one line that has unintuitive properties. Surprisingly Cantor was not the first to discover the Cantor set. The reason why it was named after Cantor was because he started working on it and helped make progress. The first person to discover and publish any information related to the Cantor set is H. J. S. Smith. He had found the same set except Smith’s set is closed and the Cantor set is infinite. H. J. S. Smith observed “Let m be any given integer number greater than 2. Divide the interval from 0 to 1 into m equal parts; and exempt the last segment from any subsequent division. Divide each of the remaining m-1 segments into m equal parts; and exempt the last segments from any subsequent subdivision. If this operation be continued ad infinitum, we shall obtain an infinite number of points of division P upon the line from 0 to 1. These points lie in loose order…” (Fleron, 1994, pg.137).

    After Cantor wrote his first paper he then started to study and discover information in point set topology in 1867. Topology is the study of geometric properties and spatial relations that are unaffected by the continuous shape or size of figures. Cantor started to work with Eduard Heine, who was a German mathematician, on trying to answer the question of the uniqueness of trigonometric series. A trigonometric series is a mathematical series that has terms proceeded by sines and cosines of integral multiples of a variable angle. Cantor didn’t just stop there. He continued his research further by deriving sets that would play an important role in him discovering the Cantor set. From 1879 to 1884 Cantor wrote a series of papers on set point topology. In these papers he talked about the first systematic treatment for point set topology of a real line. As Cantor continued to improve his paper, he talked about how sets need to have two components. The first component is being reducible, being able to turn into a lower degree or number, and the second is being perfect, being equal to the sum of its divisors. He talks about how a perfect set does not need to be everywhere dense, the closure is the entire space, and in the footnotes includes a statement that later became the Cantor set. What Cantor said was “the set of real numbers in the form \[x=c_1/3+⋯+c_v/3^v +⋯\] where c_v is 0 or 2 for each integer v,” (Fleron, 1994, pg.138). Cantor notes “this set is an infinite, perfect set with the property that it is not everywhere dense in any interval, regardless of how small the interval is taken to be” (Fleron, 1994, pg.138). While it is not known how Cantor came upon this set, it may have come upon purely arithmetically instead of geometrically, which is how it is usually described.

Exploration of Mathematics

    There are two different ways to think about the math of the Cantor set. It can be thought of just using English and also purely abstract thought. The English version of what the set means is: middle thirds is a line segment that is split into thirds while removing the center third and then the same thing is done for the next two lines and so on. It can also be described purely abstractly which is “\[C=⋂_{n=1}^∞I_n\] , where I_(n+1) is constructed by trisecting I_n and removing the middle third. I_0 is the closed real interval from [0,1]” (Nelson, 2019, pg.1). The Cantor fractal set is an uncountable set. This means that it is an infinite set that contains too many elements to count. This definition can also be defined to any middle segment that has a length of 0 to 1/2. Now to generalize the Cantor set it will be \[C=[0,1]-⋃_{i=0}^∞I_i\] . Another interesting thing about the Cantor set is that by using the formulas that are here they can be extended to be used on 2 dimensions and also 3 dimensions using the unit box or cube. Sometime when dealing with 3 dimensions it can be called the Cantor Dust.