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The Mathematics of the Tetraviews

The three "Tetraviews" that appear in the gallery are an attempt to better understand graphs of complex functions. The graph of a function of a single complex variable lies in complex two-space, which can be associate in a natural way with real four-space. That is, if z = x + iy and w = f(z) = u + iv, then the point (z,f(z)) on the graph of f can be though of as (x,y,u,v) in four-space, and so the graph is then a surface in four-space. How can we investigate this surface?

Before answering that question, let's first consider how we can understand an object in three-space, say a cube with corners at (± 1, ±1, ± 1). There are three mathematically natural views of this cube in three-space: one looking at it from a point on the positive x-axis, one from the positive y-axis and one from the positive z-axis. In each case, we see a square face of the cube (although a different face in each case). Of course, there are many other views of the cube as well, since there are many other directions from which to view it. One way to think of these directions is by imagining a large sphere enclosing the cube; every point on the sphere represents a different viewpoint, and hence a different view of the cube. The three views we described above are from where the positive coordinate axes intersect the sphere.

These three points form the vertices of a spherical triangle, and we can ask: What does the view look like from different points on this triangle? If we move along an edge of the triangle from one vertex to another, we go from looking at one face of the cube to looking at another. Our intermediate views show one face shrinking down until we see it edge-on and it become just a line, while another face that we had been seeing edge-on expands to become a full square. This corresponds to a rotation of the cube about the axis whose vertex is opposite the edge we are traversing; in this way, each edge yields a rotation about one of the axes. Half-way between two vertices on the spherical triangle our view is directly at one of the edges of the cube and both adjacent faces are seen as the same size, though neither looks square at this point. If we view the cube from the point at the center of the spherical triangle, we will be looking directly at a corner of the cube, and again we have a symmetric view, but this time including all three faces, again with some distortion.

[3 views of a cube]
Three views of a cube: looking directly at a face (left), directly at an edge (center), or directly at a corner (right). These correspond to viewpoints located at various spots on a spherical triangle: at a vertex, the center of an edge, or the center of the triangle.

Now suppose the cube is transparent and we place some object inside the cube. Then the three views from the vertices of the spherical triangle give us the three views of the object through the cube's three sides (these are like an architect's three views: the floor-plan from above, the side elevation and the front elevation). As we move along the edges of the triangle, we rotate between these views of the object inside the cube (e.g., moving from the front to the side elevation). Looking from the center of the triangle we can see into all three sides of the cube at once, giving a combination view that is, in a sense, the average of the other three (it corresponds to the architect's perspective drawing of a house).

The tetraviews carry out this same process in four-space. The cube is now a hypercube in four-space (it is transparent so it doesn't appear in the images itself), and the object inside is the graph of a complex function. The different viewpoints lie on a large three-sphere in four-space that contains the hypercube, and since there are four axes, these intersect the sphere at four points. These points form the vertices of a spherical tetrahedron on the three-sphere (thus the name "tetraview"). The four views from the corners of this tetrahedron represent projections of the function graph along each of the coordinate axes. These are shown in the picture at the four corners of the image and are arranged so as to suggest a tetrahedron: two are farther back (lower left and upper right) while two are farther to the front (upper left and lower right). The back corners are the projections into xyu-space and xyv-space, and so represent the graphs of the real and imaginary parts of the function, while the other two corners are projections into xuv- and yuv-space, which represent the real and imaginary parts of the inverse relation for the function. The image at the center of the picture is the view from the center of the spherical tetrahedron, which represents a combination of the other four, the most general view of the surface in four-space.

As with the spherical triangle in three-space, the paths along the edges of the spherical tetrahedron in four-space represent rotations of the surface inside the hypercube. This idea is explored more fully in the article "Understanding Complex Function Graphs", which includes interactive methods of navigating the views from the spherical tetrahedron.

The surfaces shown in the three tetraviews are the complex squaring function w = z2, the complex cubing function w = z3 and the complex exponential function w = ez. To determine the surfaces in terms of the four real coordinates, we use the fact that z = x + iy, w = u + iv and i2 = - 1. Then for w = z2 we have

w = (x + iy)2
= x2 + 2ixy + i2y2
= x2 - y2 + 2ixy,

so u = x2 - y2 and v = 2xy. This gives the graph parametrically as (x, y, x2- y2, 2xy). The other surfaces are treated similarly.