Topology - Topological Equivalency, Famous Topologists, Classifications, Current Research
shape geometry
Topology, which is often described as "rubber-sheet geometry," is a branch of geometry that focuses on distortion. Topology describes mathematically the features of a geometric shape that do not change when the shape is twisted, stretched, or squeezed. Tearing, cutting, and combining shapes do not apply to topology. Topology helps to solve problems about determining the number of colors necessary to illustrate maps, about distinguishing the characteristics of knots, and about understanding the structure and behavior of DNA molecules.
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The crucial problem in topology is deciding when two shapes are equivalent. Unlike Euclidean geometry, which focuses on the measurement of distances between points on a shape, topology focuses on the similarity and continuity of certain features of geometrical shapes. For example, in Figure 1, each of the two shapes has five points: a through e. The sequence of the points does not change from shap…
Topological ideas can be traced back to Gottfried Wilhelm Leibniz (1646-1716), but three of the most famous figures in the development of topology are Möbius, Riemann, and Klein. Augustus Ferdinand Möbius (1790-1868) is best known for his invention of the Möbius strip, which is a simple strip of paper that is twisted and connected so that it has only one side. Normally, cuttin…
Topological shapes are classified according to how many holes they have. Shapes with no holes at all-spheres, eggs, and convex or concave shapes like bowls—are regarded as genus (or type) 0 shapes. Genus 1 shapes have one hole in them: a donut (or torus), a wedding band, a pipe, or anything with a looped handle (a teacup). Genus 2 shapes have two holes in them, for example, a figure eight. …
Topology has a number of interesting applications, including molecular biology and synthesizing new chemical compounds to help in gene therapy. For example, strands of DNA (deoxyribonucleic acid, which contains the genetic code that defines life) often become knotted. Researchers need to know if the knotted mass of DNA is just one strand of DNA that has wound back upon itself, or if it is several …
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