Cosmology & Topology : Problem 3

Imagine we lived on a flat two-dimensional surface, a Flatland. We would not have any knowledge of the third-dimension. We could not point to it, move into it, or even describe it. If visited by someone in the third dimension, their powers would seem magical. If they dropped three-dimensional objects through the plane, we'd see some strange things. To a Flatlander confined to living in the plane, what would a sphere look like as it passed through the plane? (Hint: what is the intersection of sphere and a plane?). What would a cube look like as it fell through the plane face down, or at an angle? Try making playdough models of various shapes and slice them through to investigate what the intersections would look like.

As far as we know, we don't live in four dimensions. If a fourth dimension existed what would a four-dimensional hypersphere look like to us as it dropped through our flat three dimensions? (Hint: extrapolate from the two-dimensional analogy).

Imagine now that we are still two-dimensional creatures but that we don't live on Flatland. We live on the surface of a sphere, Sphereland. Again, we have no knowledge of the third dimension but are only aware of the surface of our sphere. As we move along this surface, what is the shortest distance between two points? In Flatland it is a straight line. What is the shape of this path in Sphereland? (Hint: imagine drawing a straight line on a flat sheet (eg. an OHP transparency) and stretching it over the sphere).

The sum of the interior angles of a triangle on a flat surface is always 180° or Π radians. In general however this is not true. On a curved surface

where Κ is the curvature of the surface and A is the area of the triangle. A sphere has positive curvature everywhere of Κ = 1/R² (this means that the smaller the radius, the greater the degree of curvature of the sphere). Now consider a sphere with unit radius R = 1. The great circles define the straightest lines possible on the curved space. The triangle is drawn with edges along these natural arcs of the sphere.

What is the total surface area of the sphere?
What can you say about the smallest and largest triangles that can be drawn on the sphere? What would be bounds for their areas be?
What are the bounds for the sum of the interior angles of the smallest and largest triangles on the surface of a sphere?

Try to prove (1). It might help you to look at the area of lunes (areas bounded by great circles on a sphere).

2 great circles intersecting at N and S divide the spheres into 4 lunes. How might you find the area of one such lune?

Find the shaded area. Can you work out how to find the angle sum of a triangle from here?

Teacher's notes

Flatland and Sphereland

This starts with imagining we are 2-dimensional creatures living on a 2- dimensional surface, Flatland. What would our world be like? The book Flatland by Edwin A. Abbott (ISBN: 1570624380) could be a useful resource for this. The students then go on to discuss what interactions between 3-dimensional objects and Flatland would be like. Then they consider living on the flat 2-dimensional surface of a sphere, Sphereland, and repeat the exercise. They can then extrapolate this to think about how we, in our 3-dimensional world, might experience 4- dimensional objects if they exist.

The rest of the project is about looking at spherical triangles on the surface of a sphere. The formula (1) given in the project is

Angle sum = Π + Κ × A,

where ( is the curvature of the surface (Κ = 1/R² for a sphere), and A is the area of the triangle.

We can establish limits for the area of a spherical triangle, if the sphere has radius 1, of 0 < A < 4Π, and hence limits for the angle sum of Π < Σ(angles) < 5Π. It would probably help the students to establish these practically first, by measuring angles, and estimating areas.

The project goes on to ask the students to prove this formula. This is done by considering the area of a lune (an area bounded by two great circles on a sphere), as in the diagram in the project. First, the students should establish that the area of the lune is given by

A = (θ/2Π) × 4ΠR² (for a sphere of radius R)

They should then look at the diagram in the project which shows two great circles AA' and CC' intersecting at B and B'. It might be helpful to draw this out on a polystyrene sphere, or stick paper great circles around a beachball. This should enable them to establish that Area (Δ ABC) = Area (Δ A'B'C'), and Area (Δ A'BC') = Area (Δ AB'C).

Then focus on just the hemisphere visible if you look down on the sphere from above: that is, the two great circles AA' and CC' and the equator bounding the hemisphere. The area between AA' and the equator is a lune, similarly for that between CC' and the equator, and between AA' and CC', as in the diagrams below.

Write down the area for each of these lunes. Then add them together to obtain the area of the hemisphere plus 2 × Area (Δ ABC). They should be able to go from here to equation (1) of the project.

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