Topology

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Topology

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.

Topological equivalency

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 shape 1 to shape 2, even though the distance between the points, for example, between points b and d, changes significantly because shape 2 has been stretched. Thus the two shapes in Figure 1 are topologically equivalent, even if their measurements are different.

Similarly, in Figure 2, each of the closed shapes is curved, but shape 3 is more circular, and shape 4 is a flattened circle, or ellipse. However, every point on shape 3 can be mapped or transposed onto shape 4.

Shapes 1 and 2 are both topologically equivalent to each other, as are shapes 3 and 4. That is, if each were a rubber band, it could be stretched or twisted into the same shape as the other without connecting or disconnecting any of its points. However, if either of the shapes in each pair is torn or cut, or if any of the points in each pair join together, then the shapes are not topologically equivalent. In Figure 3, neither of the shapes is topologically equivalent to any of the

shapes in Figures 1 or 2, nor are shapes 5 and 6 equivalent to each other. The circles in shape 5 are fused; and the triangle in shape 6 has a broken line hanging from its apex.

Famous topologists

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 (Figure 4). Normally, cutting a strip of paper into a long, narrow rectangle and connecting the ends will result in a belt-like loop with two sides. A person cannot draw a single line with a pencil on both sides of the belt-like loop without crossing an edge. In constructing the Möbius strip, however, the strip of paper is twisted as it is looped, and the result is a one-sided object.

At first, this one-sided construction seems impossible, but if a person draws a straight, continuous line on the Möbius strip, the line will cover the entire length of both sides of the strip without ever crossing an edge, and it will return to its starting point in one long stroke.

Georg Friedrich Bernhard Riemann (1826-1866) developed some of the most important topological ideas about the stretching, bending, and twisting of surfaces, but he died prematurely at the age of 39 before he could expand significantly upon his ideas.

Felix Klein (1849-1925) is best known for the paradoxical figure which was named after him: the Klein bottle.

The Klein bottle is a one-sided object that has no edge. It is a tapered tube whose neck is bent around to enter the side of the bottle. The neck continues into the base of the bottle where it flares out and rounds off to form the outer surface of the bottle (Figure 5). Like the Möbius strip, any two points on the bottle can be joined by a continuous line without crossing an edge, which gives the impression that the inside and outside of the Klein bottle are continuous.

Classifications

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. Genus 3 shapes (most pretzels) have three holes in them. And so on.

The determining feature in classifying the topological genus of a shape is deciding if every point in one shape can be transposed or mapped onto a point in the other. Sometimes this process is easy, as in the case of a wedding ring and a donut, which are genus 1 topological shapes. But with complex genus shapes of 4 and above, the determination can be difficult.

Current research

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 strands of DNA which have become entangled. Topology, especially knot theory, helps molecular biologists solve such problems of equivalency.

Topology also has applications in synthesizing new molecules, called dendrimers, which may be used in gene therapy. Dendrimers are branching, tree like polymers (strings of smaller molecules) synthesized in laboratories by chemically attaching the polymers to a core molecule. Dendrimers are approximately the same size as many naturally occurring molecules, including DNA and proteins. In gene therapy, new genetic material needs to be transported to the nucleus of a cell to replace damaged or defective genes in human cells. Then the repaired cells can reproduce more healthy cells. However, there must be a way to transport the healthy DNA into cells, which is where dendrimers come in. Normally, DNA wraps itself around clusters of proteins called histones. Gene therapists can fool the immune system of the human body by substituting dendrimers for naturally occurring clusters of histones and thus transport healthy DNA into damaged cells. Topology is useful in this process, because its purpose is to decide mathematically which shapes are equivalent. The closer that synthetic dendrimers mimic the naturally-occurring histones, then the greater are the chances that the DNA will be transported to create healthy cells.

See also Polymer.

KEY TERMS

Dendrimer molecules —Branching, treelike polymers synthesized in laboratories by attaching the polymers to a core molecule.

DNA —Deoxyribonucleic acid, a thin, ladderlike molecular structure which contains the genetic code that defines life.

Euclidean geometry —Geometry based upon the postulates of the Greek educator, Euclid who lived roughly 300 BC Euclid’s parallel postulate that only one line may be drawn through a given point such that it is parallel to another line, was overthrown in the nineteenth century and gave way to the nonEuclidean geometries created by Gauss and Riemann.

Geometry —The area of mathematics that deals with deductions concerning the measurements and relationships between points, lines, angles, and figures in physical space.

Polymers —Strings of smaller molecules.