
Squeezing the Phase Space 
What would a 2dimensional space be like? What would it be like to live in a 2dimensional space? Think about it. This question was brilliantly and amusingly answered by Edwin A. Abbott, a British schoolmaster who lived from 1838 to 1926. He wrote a book called `` Flatland, A Romance of Many Dimensions.'' The book is about life in a 2dimensional world. As the title suggests, a 2dimensional space is flat, like the surface of the blackboard, or like the floor in this room. We can use our earlier understanding of dimension here too: it takes two numbers to locate a point on the blackboard; likewise on the Earth's surface, which is flat in this sense, any point can be located by its longitude and latitude.
How about a 1dimensional space? The simplest is a line, or part of a line. There a point can be located by exactly one coordinate. An example of a 0dimensional space is a single point. In that space there is nowhere to go, and it takes no numbers to tell where you are.
Geometric shapes in dimensions 1, 2, 3.Geometric shapes are the simplest objects that can exist in space. In our 0dimensional space the only object is the point itself. The simplest 1dimensional geometric shape is a line segment: let us make it of length one to be specific. In dimension 2 we have various choices. What are they? Let us pick one example, a onebyone square. Similarly for a simple 3dimensional object we pick a onebyonebyone cube. Look at the lineup:
point Can you see any relations between them? After you think a while, maybe you will have noticed that
Here is another relation you may have seen:
Can you find any other patterns like these?
Perspective.In art, perspective means the technique of drawing threedimensional things on a twodimensional sheet of paper. Suppose we want to draw our 3dimensional cube on a 2dimensional sheet, or on the 2dimensional blackboard. What does the picture look like? Try it. Do you all get the same picture?The picture most people get looks like this:
But suppose your eye was just outside one face of the cube. Then you would see this:
(For more mature students) How far would your eye have to be from the face for the back face to seem half as wide as the front?
On to 4 dimensions: the hypercube.The list of simple geometric shapes does not have to stop with the cube! We can talk about a 4dimensional shape that would be a onebyonebyonebyone hypercube. Suppose we tried to extend the two patterns of relations. What would we get? Try to extrapolate the patterns yourselves. You will probably guess:
Exactly! But how can we move a cube ``perpendicularly to itself?'' We can't, of course, not in threedimensional space. This maneuver can only be carried out in a 4dimensional space. Does that mean it is impossible? Not mathematically! After all, when we talk about ``moving the square a distance of one unit perpendicularly to itself'' we don't actually have to physically move it, we can just imagine moving it. In the same way we can imagine moving the cube a distance of one unit through a fourth dimension. In ``Flatland,'' Abbott makes us think about how hard it is for the twodimensional Flatlanders to imagine a third dimension. But just as they can imagine three dimensions without being able to ``see'' them, we can imagine four. Just as we can draw a 2dimensional perspective picture of a 3dimensional cube, we can build a 3dimensional perspective model of a 4dimensional hypercube. We will use the perspective in which our eye is one unit away from a ``face'' of the hypercube, so the back ``face'' seems half as wide as the front. Then the perspective model will look like this:
A 3dimensional perspective model of the hypercube. Building a model of the hypercube.The model is made of drinking straws. They are joined by connectors, each of which is an `X'shape made of two 2inch lengths of straw stapled together at their centers.
The outside threedimensional cube is made of 12 fulllength straws joined with connectors.
The inside threedimensional cube is made of 12 halflength straws joined with connectors.
The two cubes are joined corner to corner by eight linking straws. (More mature students) How long should the linking straws be? For practical assembly they should be made somewhat shorter than the ideal length.
A partly assembled hypercube model. Note the two unattached connectors Topics for Further Thought.1. Can you push the patterns further and describe a 5dimensional cube, a 6dimensional cube, ... ?2. How many vertices do the point, segment, square, cube, hypercube have? Can you figure out how many vertices an ndimensional cube would have? Hint: one of the patterns of relations will help you. 3. The Euler characteristic (pronounced ``oiler'') of a geometric shape is (number of vertices)  (number of edges) + (number of 2dimensional faces)  (number of 3dimensional faces) + ... alternating plus and minus up to the dimension of the shape, where there is always one topdimensional ``face,'' the shape itself. Check that for the point, segment, square, cube and hypercube the Euler characteristic is exactly 1. Hard problem: prove this for the ndimensional cube. 4. Another simple 2dimensional shape is the equilateral triangle of sidelength one. The corresponding 3dimensional shape is the tetrahedron of sidelength one. This gives a new family: point, segment, triangle, tetrahedron,... . Notice that the triangle can be constructed from the segment by choosing a point not on the line of the segment, and joining it to each point of the segment. Then the point can be positioned to make all the new edges of unit length. Check that the segment can be constructed from the point in this way, and the tetrahedron from the triangle. The hypertetrahedron is the next member of the family. Let us call the point the 0simplex, the segment the 1simplex, the triangle the 2simplex and the tetrahedron the 3simplex, so the hypertetrahedron is the 4simplex. Find the analogue, for the family of simplexes, of the first pattern of relations in the family of cubes. 5. Build a 3dimensional perspective model of a hypertetrahedron. Work by analogy with what you did with cube and hypercube. What is the analogous 2dimensional perspective drawing of a tetrahedron? Calculate the lengths of the new elements in the perspective model (hard); otherwise fit them by trial and error. 6. Check, for n = 0, 1, 2, 3 and p < = n that the number of pdimensional faces of the nsimplex is equal to the binomial coefficient C(n+1,p+1). Hard problem: prove this for a general n and p < = n.
7. Prove that the Euler characteristic of the nsimplex
is 1.
