“There is geometry in the humming of the strings, there is music in the spacing of the spheres.”
Often reality attempts to elude our sense of mathematical and logical intuitiveness especially in fields such as topology resulting in peculiar and interesting shapes. In mathematics, a space is generally regarded as a set of points having some specified structure. Spaces are often complicated and have a vast array classification with complex definitions (Figure 1 gives you an idea of the complexity). This article gives a basic introduction to some of these ideas such as topological spaces and manifolds.
Topology is the mathematical study of the properties that are preserved through deformations, twisting, and stretching of objects. Topology is often associated with spatial objects such as curves, surfaces, the space we call our universe, the space-time of general relativity, knots, etc.
A topological space can be understood as an ordered pair of points based on set of values. The mathematical definition of a topological space outlines certain conditions that need to be obeyed before a set of points can be called a topological space for a more rigorous system of definitions. Intuitively, we would think a topological space is a space that is used to map various geometric objects in plane dimensions however in reality the definition of a topological space has been extended for greater mathematical sense of completeness. Nevertheless, topological spaces are extremely useful when plotting out all kinds of geometric objects.
Homeomorphism is the correlation between 2 shapes that can be topologically transformed into the other. Let us consider a sphere and a cube. Geometrically if we compare these objects, they have different geometric properties such as different symmetries, edges, vertices. However, if we view these shapes topologically, they are homeomorphic as one shape could be easily deformed into the other.
Intuitively, we understand by stretching a sphere outward in a certain manner we can get a cube. However mathematically we can take every unique point on the surface of a sphere and connect it to every unique point on the surface of a cube and thus deform the cube into a sphere. To put it simply homeomorphism is a correspondence between two topologically equivalent shapes where one shape can be transformed into another shape by twisting, stretching, extending and smoothening.
When two shapes are homeomorphic, they are topologically invariant. A topological invariant is essentially just a topological property preserved when we deform and stretch a shape. An example would be the number of holes (number of closed intersecting curves) in an object. Topology allows us to stretch, twist, lengthen, smoothen objects but we are not allowed to tear or punch holes. Since there is no way we can transform a sphere into a torus (a single holed geometric object) without punching a hole in the sphere we can conclude a sphere and a torus are not topologically equivalent (not homeomorphic). This symbol ≅ is used to show 2 shapes which are homeomorphic to each other and the symbol ≇ shows 2 topologically inequivalent shape. For example:
Since the number of closed intersecting curves (number of holes) is a topological property preserved after deformation we can call the number of closed intersecting curves in a object a topological invariant. Therefore, topologists can classify geometric shapes based on the number of holes that they have. For example, they classify a shape with 1 hole (torus) to be of genus 1
This leads to the following mathematical joke: a topologist is someone who cannot distinguish between a doughnut and a coffee mug. Explanation: these objects are homeomorphic as they both are of genus 1 (have 1 hole) and one shape can be deformed into the other.
Orientability is another topological invariant. An orientable surface is one where consistent ‘orientation’ can be assigned over the entire surface. Think of orientability as moving an oriented figure on the surface of an object. If the figure maintains the same orientation throughout the surface of the object, we can then call the object orientable.
For example, consider a clock that reads 3 o clock. We see that the hour hand is 90 degrees clockwise from the minute hand. In an orientable object such as a sphere, no matter how we move this clock around the surface it will maintain the same orientation and the hour hand will always be 90 degrees clockwise form the minute hand. Since we were able to define clockwise has been defined consistently on the surface of the sphere, we can call the sphere orientable object. This may seem extremely obvious, and you may think that all shapes should be orientable, but you will discover later that this doesn’t hold true for all shapes like certain manifolds. More examples of orientable objects are cylinders, torus, etc.
In contrast with our common notion of 2-dimensional space occurring on a flat surface, 2 dimensional spaces exist on surfaces of geometric objects such as surfaces of spheres or paraboloids and may not necessarily be ‘flat’. This occurs when Euclid’s infamous 5th axiom, the parallel postulate does not hold. His infamous 5th axiom essentially states that parallel lines never intersect each other. Therefore 2 dimensional spaces can be broadly classified as elliptic, Euclidean, and hyperbolic where parallel lines intersect, remain parallel, and diverge away from each other respectively.
A manifold is a topological space that is locally homeomorphic to Euclidean space. To understand what this means it would be useful to think about the analogy of the earth. The earth is in the shape of geoid and is a topological space which is elliptic or spherical but not Euclidean. However, from the perspective of a farmer standing of a field on the surface, since the field is being looked at very closely by the observer it is enlarged by a very large scale and would appear ‘flat’ or Euclidean. The earth is a topological space and field would be local part of the earth’s surface which could be called topologically equivalent (homeomorphic) to a Euclidean space. Alternatively, we can also define a manifold as something we can completely cover with charts. Charts are simply parts of the manifold surface that we can map on to a plane with real numbers. In Figure 11 we can call green triangle to be a chart as it is part of the spheres surface and can be mapped on to a 2-dimensional plane with real numbers.
Continuing the analogy of earth as a topological space, when we try taking small patches or parts of the surface of the earth and plotting them on a plane with real numbers, we essentially just get 2 dimensional maps of places. For example, covering Europe with a chart on the globe and plotting it out in 2 dimensions would give us the map of Europe.
Since the earth is something we can completely cover with charts (maps) we can conclude that earth is a manifold. The number of dimensions required to plot the points of a manifold is the referred to the dimensions of the manifold. The earth is 3 dimensional and can be covered with 2 dimensional maps, since the points were plotted into a map of 2 dimensions, the earth is a 2-dimensional manifold.
The Möbius band is an infamous topological space and a manifold. Take a rectangular strip of paper and twist one of the ends by 180 degrees and then connect the ends.
The 2 arrows represent how the ends of the Möbius band are supposed to align, with the arrows pointing in the same direction and attached alongside each other.
Möbius bands have several interesting topological properties:
- Has only one surface: Try tracing a line along the center of the Möbius band. An interesting animation of a painting by M.C Escher has the same effect. Ants crawling along a Möbius band seem to cover the entire surface without changing direction. This implies that a Möbius band has one surface or side. However, it is truly fascinating how a strip of paper with 2 surfaces can be made into a shape with just one surface.
- Non – orientable: When we try moving a clock on the surface of the Möbius band we see that the as the clock passes the twist it changes orientation, and the hour hand is no longer clockwise from the minute hand. Since clockwise is not constant in a Möbius band we cannot assign a consistent orientation to a Möbius band and therefore it is non orientable.
- 3-dimensional topological space and a 2-dimensional manifold: To prevent the Möbius band from intersecting itself we extend the shape to the third dimension and add a half twist, thus causing the Möbius band to only exist completely in 3 dimensions. It follows with the definition of manifolds whenever we take a chart (small part of the surface) of the Möbius band we can plot it on a 2-dimensional plane and therefore it is a 2-dimensional manifold.
Klein bottles are another example of a peculiar and fascinating manifolds.
This is a common representation of a Klein bottle. By aligning the arrows of the same color in the same direction and attaching them side by side we get a Klein bottle. The following animation helps better visualize the creation of Klein bottle from this initial diagram.
The Klein bottle is topologically quite similar to the Möbius band as they share many topological properties. Klein bottles are non-orientable manifolds and are also composed entirely of one surface. Figure 21 shows that only 1 net is required to cover the entire surface of the Klein bottle proving that it has only one surface.
However, Klein bottles differ from Möbius bands in one important aspect that they are higher dimensional. The Klein bottle is a 4-dimensional structure and a 3-dimensional manifold. Therefore, due to this difference in dimensions we can conclude that the Klein bottle and the Möbius band are not homeomorphic. Analogous to how the Möbius band would intersect itself in just 2 dimensions and by extending the shape to the third dimension and adding a twist we were able to visualize it clearly, the Klein bottle is in fact 4 dimensional and the area where it intersects itself is simply where it would twist in the 4th dimension. The Klein bottle we see in images is simply a projection and not a complete depiction of an actual Klein bottle as it is not possible to draw or easily visualize 4 dimensions. A 4-dimensional Klein bottle would have charts that would be mapped on the 3-dimensional plane and therefore it would be called a 3-dimensional manifold.
Topology is a very interesting and unique branch of mathematics where we draw unseen connections between shapes that may look completely different to the human eye but in reality, are remarkably similar. By simply observing the properties that are preserved when we deform object in a certain manner, we can create correlations and conclusions that have immensely deepened our understanding of the world around us and the space in which we live in. Numerous diverse kinds of interesting shapes exist and continue to be constantly discovered, whose distinctive and peculiar properties defy our logic and capture our curiosity.