Curvature of Space

Circle Limit IV: Heaven and Hell by M.C. Escher (1960)

Purpose

In this lab we'll have a look at curvature and how we can think about the curvature of our own universe by looking at some examples from other "universes", first, and then how we might apply that knowledge to our own.

Procedure

Print the worksheet.

Imagine yourself in Flatland, a 2-dimensional flat space, like an infinite piece of paper. If a sphere passed down through the plane of Flatland, a Flatlander would first see a point, which would grow to a circle, reach a maximum size, shrink to a point again and disappear. What would a 4-dimensional sphere look like if it passed it through 3-dimensional space? Imagine you are a Flatlander, and a 3-dimensional cube is passed through Flatland. What possible shapes could you see as the cube passed through Flatland?
It's very hard to imagine a 3-dimensional space that's curved into a fourth dimension, so we have to refer to 2-dimensional spaces that are curved into a third dimension as an analogy. Imagine that you are a 2-dimensional creature living on the surface of a sphere. List three geometrical tests that would tell you if your universe is positively curved.
In many of his works, the Dutch graphic artist M.C. Escher explored two problems. The first was the regular division of a plane into tiles; the second was the representation of 3-dimensional objects and infinities in 2 dimensions. The two works in this assignment are projections of creatures "living" in 2-dimensional spaces, which may or may not be flat, onto the page, which is definitely flat. The creatures are all the same size in their own world - the apparent change in size of the angels and devils in Circle Limit V is an artifact of the projection onto the flat page, like the distortion of the size of Greenland on a Mercator map in an atlas. So how do you go about determining the curvature?
Recall a few facts from geometry. First of all, on any surface, of any curvature, the sum of the angles at any point is equal to 360 degrees. Secondly, the sum of the interior angles of a triangle will be exactly 180 degrees for a flat surface, less than 180 degrees for a negatively curved surface, and more than 180 degrees for a positively curved one.

In these works, the creatures are tiled, so that their bodies fit together in regular patterns. To find the curvature, find a repeating triangular shape and count how many triangles intersect at different points. From this you can determine the size of each of the angles of the triangle.

Here is an example.

Figure 2: Sun and Moon by M.C.Escher (1948).
In Figure 2, there are two types of birds, which are all roughly triangular (their vertices being the points where 6 birds are touching: the wing-tips and the beaks). The birds are all the same size in their space, so that the 6 angles at a vertex are all the same. There are always six birds coming together at a vertex, so that each of the interior angles (the white dots on the image to the right) of each bird's triangle is:

360 degrees / 6 = 60 degrees. The sum of the three angles of the triangle of each bird is therefore (60 + 60 + 60 =) 180 degrees, so that we now know that the curvature is flat.

First, guess what the curvature is in Figure 3. Then determine it using the method from the example.

Figure 3: Circle Limit 4 -- (Heaven and Hell) by M.C.Escher (1960).

Again, the angels and devils occupy identical triangles in their own universe. Note that you are not supposed to measure the angles, but that you will have to use the method described in the example above, counting triangular tiles meeting at a vertex point. Also note that the triangles are not equilateral. That is, in this picture, all three angles are not the same! Show your work (it shouldn't take much space at all).
Enough with those 2-dimensional analogies. Let's move on to our real universe and see how geometry might tell us things interesting to astronomers and you. How might you use two beams of light to figure out what the curvature of our universe is? Of course, there's a big problem here: we can't follow the paths of the two light rays throughout the universe! On top of that, the local curvature of our universe isn't necessarily the same as the global curvature at all. What dominates the curvature in the inner solar system? Is it flat, positive or negative? How do we know?