Visualizing the Fourth Dimension Using Color

But what about dimension 8½?

Color can be used for more than exploring special relativity. Assigning color to every point on a surface is a way to envision higher dimensions. Let's start with a simple example: a disc with concentric colors. If we map hue to height linearly, the result is a cone.

[disc with concentric gradient] [cone with concentric gradient] [matching disc and cone]

We get a cone because we chose a linear mapping. If we'd picked a different mapping, we would have gotten another shape, such as a hemisphere. More exotic mappings are possible.

[hemisphere with concentric gradient]

A more interesting example is a shape that intersects itself, such as the figure eight on the left below. Again, identify each hue with a different height. A shape will intersect itself only if it contains two points which have both the same three-dimensional position and the same color. In this case, the bluer the region, the further it is below the plane containing the figure; the redder it is, the higher it is above the same plane. It's as if we pulled the red parts of the figure up while pulling the blue parts down. The resulting figure shows that the shape can be embedded in a three-dimensional space without self-intersection. (In the illustrations, I've given the figures some thickness to make self-intersections more obvious.)

[self-intersecting planar figure 8] [modified figure 8 which avoids self-intersections by extending into the third dimension]

This technique is particularly handy when dealing with knots. For example, the planar trefoil knot on the left can be embedded in 3D space without self-intersection, as shown in the second illustration.

[planar trefoil knot] [3D trefoil knot which avoids self-intersection]

Unknotting a Knot Using Four Dimensions

This mental tool really starts to shine when we kick everything up a dimension. Rather than measuring depth in three-dimensional space, color will now measure depth in four-dimensional space. With this, we can perform a neat trick: if we embed a knot in a space with four physical dimensions, we can undo it without cutting. The trick is to move a loop of it "around" another loop in a direction perpendicular to our 3D directions. (The 4D directions are called ana and kata, and they're analogous to the familiar left/right, up/down, and forward/back directions of 3D space.) To us, moving in the ana or kata direction will appear as changing color.

Taking again the trefoil knot, let white indicate a point in our 3D space. The deeper green the point, the further it is in 4D space (in the ana direction) from our 3D space. The first step in undoing the knot is to lift a loop in the ana direction. To us, the loop turns green.

[completely white knot] [a loop of the knot becomes pale green] [loop is now a rich green]

Next, pull the now-4D (green) loop through another part of the knot still in (white) 3D space. Remember, the green and white parts of the knot occupy different locations in four-dimensional space, so they don't intersect each other.

[green loop begins to move] [green loop apparently intersects a white loop] [green loop has almost passed 'through' white loop] [green loop now on the other side of white loop]

Finally, move the loop in the kata direction to return it to our 3D space. (The green loop becomes white again.)

[green loop begins to fade] [green loop is now a pale green] [entire figure now white and unknotted]

Voilà: what was a knot is now an unknotted closed loop.

What would we see if you watched this happen in real life? Since we can't see anything outside our 3D slice of 4D space, from our perspective the moving (green) loop would disappear, to reappear later in the unknotted position.

By the way, this technique works for all knotted closed loops. Any knotted closed loop can be unknotted by embedding it in 4D space.

Color and Complex Surfaces

Employing color to indicate 4D depth may be most useful when envisioning surfaces that can't be embedded in 3D space without self-intersection. The simplest example is the Klein bottle, a closed surface with only one side (unlike, say, a sphere's surface, which has an inside and an outside). Using color to distinguish four-dimensional depth makes clear how a Klein bottle can exist in four dimensions without self-intersection.

[Klein bottle with differing colors where it self-intersects]

Last updated 30 August 2004
All contents ©2004 Mark L. Irons