How to See in Higher Dimensions

Did that get your attention? One cannot of course really see in higher dimensions in the physical sense. One can, however, devise ways to obtain information about surfaces in multiple directions simultaneously and present the results visually. This presentation will make use of multiple slices of surfaces in several dimensions and display them as interactive graphics for exploration of higher-dimensional surfaces.

Begin with a one-dimensional surface, otherwise known as a line. For the simple parabola

$f = x^{2}$

one has the simple and expected graph over some region around an arbitrary initial point:

Numbering of points along the axes is suppressed as nonessential to the point of the presentation: only the overall shape is what one wants to see in this context.

If the parabola is rotated about the vertical axis one gets a two-dimensional paraboloid surface

$f = x^{2} + y^{2}$

While this can be visualized easily enough in three dimensions, instead look at side-by-side slices in the directions of the two independent variables:

These two graphs show the overall shape of the surface in either direction from the specified initial point. The behavior is the same for this rotationally symmetric surface, so this example is meant merely for orientation in seeing this way.

Things get a bit more interesting when there is different behavior in each direction, as for example the mixed quadratic and cubic

$f = x^{2} + y^{3}$

Slices of the surface in each direction are now

While there is something different in each output window, the additive structure of the function implies that half of the function remains essentially the same when only on variable is altered, just merely displaced vertically. What would be more interesting is to allow a rotation of directions along which the surface is sliced, e.g. look along the directions

$\begin{array}{l} x^{'} = x cos α - y sin α \\ y^{'} = x sin α + y cos α \end{array}$

Keeping in mind that the rotation needs to be applied at the initial point and not the origin, the corresponding surface slices are

Now beginning from some point one can pan around in the sliced space to get a feel for how the surface is changing in perpendicular directions. A full rotation of $\frac{π}{2}$ interchanges the axes, so that the cubic becomes a parabola and vice versa.

But why stop with two independent dimensions? How about a surface where each additional variable has a different power, such as

$f = x^{2} + y^{3} + z^{4} + w^{5}$

which produces the graphic slices

These are slices of a four-dimensional surface, which would require a space of at least five dimensions to contain it. This is a surface that truly cannot be viewed physically but requires an alternate mode of visualization.

The slices here again barely change as the point of reference is moved due to the additive structure of the function. Including rotated variables for this case is trickier than the last, because there are now six pairs of coordinate directions that can define a higher-dimensional rotation, and the order in which rotations are applied makes a difference in the final result. As an example, let the first rotation be in the xw-plane,

$\begin{array}{l} x^{'} = x cos β - w sin β \\ w^{'} = x sin β + w cos β \end{array}$

followed by a rotation in the zw-plane:

$\begin{array}{l} z^{'} = z cos γ - w^{'} sin γ \\ w^{''} = z sin γ + w^{'} cos γ \end{array}$

Again applying the successive rotations at the initial point, the corresponding surface slices are

Now that is something you just cannot see every day!

While the value of the function forms a natural direction orthogonal to each coordinate axis, one can also visualize parts of the same surface using pairs of variables:

The top row contains the pairs xy, xz and xw and the bottom row yz, yw and zw. The rotation about the initial point has been removed because the graphics themselves can be rotated as one likes.

A constrained surface has one degree of freedom removed and so represents a surface of one dimension less than the number of independent variables. As an example consider the three-dimensional surface

$x^{2} + y^{3} + z^{4} + w^{5} = 1$

Instead of solving for each individual variable in turn and patching together different branches for a complete graphic, it is more expedient to use a search method to find contours of the surface. For one-dimensional contours on a two-dimensional plot, one can employ marching squares over pairs of independent variables:

The plot may be empty if the initial point is farther away from the surface than the default region around that point.

Alternately one can look for two-dimensional contours in a three-dimensional space using marching cubes over triples of variables:

The top row here has the triples xyz and xyw while the bottom has xzw and yzw. The graphics may again be empty is the initial point is too far from the surface.

Another application of multiple interactive graphics is to visualize simultaneously the real and imaginary parts of complex functions, for instance the Jacobi elliptic sine function:

$f = sn (z | m) = sn (x + i y | m)$

For an arbitrary elliptic parameter and initial point, one can simply show the real and imaginary parts side by side:

One can also look at two-dimensional slices along real and imaginary axes

or along axes rotated in the complex plane just as the physical variables above in the first rotation example:

Lots of possibilities here. Cheers!