As one begins to study more sophisticated concepts in science, one will encounter the concept of a manifold. This weird-sounding word has a strong root in mathematics, in particular geometry. Here we will intuitively explore what a manifold is with various examples.
Ricky Simanjuntak
Introduction
Here we have a world map.
(Earthcenteredat0°Elongitude)
This map tries to project the surface of the Earth onto a rectangular, flat surface. Let us compare it to the three-dimensional representation of the Earth.
Notice the North Pole is shown as a big white area that spans from America to Asia on the flat map. On the contrary, in the 3D graphic of the Earth, the North Pole is shown as a small, white circular area not even the size of North America.
Take a look at the actual data.
NorthPole
CITY
area
,Total@
NorthAmerica
COUNTRIES
totalarea
4.1670812916
2
mi
,
9.46094×
6
10
2
mi
The 3D sphere representation is more accurate. This difference demonstrates how the flat map is not an accurate representation of the Earth. We use the flat map because we experience the Earth as a flat surface in our daily experience, but globally it is not correct since it gives some misinformation, as we have shown in the case of the North Pole.
This leads us to the intuitive definition of a manifold: something that locally looks like a plane, but globally can be something else. The Earth is a manifold.
Deformation
The first step in understanding a manifold is to think of it as a flat plane that gets bent. Consider the following example.
The main reason why we prohibit this is because to define deformation mathematically, we need to associate each point on flat surface to a point in the deformed surface in a one-to-one manner. This can’t happen if we join two points together.
2. No tearing the surface into two separate parts.
This is because we want deformation to happen in both directions. If A can be deformed to B, then B can be deformed to A. If we can tear apart A, then to turn it back we need to glue together some points, which is not allowed.
3. No creating holes.
In order to deform a holed surface, we need to join the boundary together, which is not allowed.
Manifold
More Dimension
What Is Not a Manifold?
Anything that has a point so that in order to make it from flat object we have to violate the 3 rules we mentioned before is not a manifold.
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Two cones stacked together, as shown above, are not a 2-manifold since the point where they are connected is two points from both cones joined together, which is not allowed.
ParametricPlot[{Cos[t],Cos[t]*Sin[t]},{t,0,2π}]
The above curve is not a manifold either, since it has a self-intersecting point.