T O P O L O G Y
T O P O L O G Y
Length, width, height, distance become irrelevant.
Instead, other features of figures that remain the same (invariants) when figures are distorted are studied. For example, a circle and a square are usually considered quite distinct objects; in topology, however, they are both "simple, closed curves" - essentially, this means that they are both curves that do not intersect themselves anywhere and are connected, one end to the other (as opposed to a shoelace-type thing that has two ends that are not connected). So a square and a circle are said to be topologically equivalent! In fact, any polygon - of any size (remember that lengths don't matter) - would be topologically equivalent to a circle. A "figure-eight" shape, however, is different because it contains a crossing (one would have to assume that there is an actual connection at that crossing and that the figure is not just a twisted loop in three dimensions that can be untwisted).
Topology is often referred to as "The Mathematics of Distortion" or "Rubber Sheet Geometry", because if one figure can be transformed into another by just stretching (but without cutting or creating new connections), then they are said to be topologically equivalent.
In this class we look at 2-dimensional topology and a basic introduction to 3-D topology.
Networks, trees, knot theory, and variations on the Moebius Strip, are some of the topics we touch on.
A network (or "graph" in the language of graph theory) is simply an idealized version of connectedness between points in a plane. A graph is made up of points (nodes, vertices), lines (edges), and faces (enclosed areas). Topology and network theory began with Leonard Euler's analysis of the Bridges of Konigsberg problem.
This image shows the connectedness of the bridges:
The leap that Euler made was not only to consider only the connectedness of the landmasses, ignoring the size and shape of the masses and the bridges (see the image below), but also to do an analysis of the class of problems associated with visiting points connected by lines, without travelling on any line more than once. The image below shows an example of how the Bridges problem can be distilled into a graph theory/network theory problem:
Graph Theory has since found applications in Astronomy, Biology, Computer Science, Economics, etc. and has spawned entirely new fields such as "Data Mining".
As often happens with mathematics, network theory began as a theoretical field based on some concrete problems, became more and more abstract, and then eventualy became essential to solving real-world problems in many fields (especially computer science, cosmology, and data mining).
"Today, graph theory is a highly developed field of mathematics, and is both a fertile ground for the creation of new mathematics and an area with many, many applications. Many research problems in graph theory are easily stated and easily understood (although perhaps not easily solved). A few of the applications of graph theory include transportation and warehousing applications, planning and scheduling, analysis of electrical networks, and even understanding the Internet! " - John Carroll University Mathematical Vignettes
Knot Theory is another subfield of Topology that investigates various properties and invariants of knots.
The field was begun accidentally when Lord Kelvin (famous for the Kelvin temperature scale), had the idea that the elements (H, N, Fe, etc.) were made up of "knotted vortices" and that the orientation of the knots and their crossings would differentiate between the elements. He was wrong, of course (if he were correct, today our periodic table of elements would be a big chart showing the various knot configurations). However, his idea got scientists and some mathematicians to begin to try to classify different knots and understand their properties (to try to understand their related elements).
Although Knot Theory began as an "incorrect" theory, Knot Theory today (in the last couple of decades) is finding applications in many different fields.
Brief History of Knot Theory Origins
Some Topology Links:
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