Sunday, March 2, 2008

Hypercubes and Plato's Cave

Edit: We've explored the visual ideas in this entry from a number of perspectives since it was published, here are some more recent entries:
Playing Games in Extra Dimensions
O is for Omniverse G to J
What's Around the Corner?
When's a Knot Not a Knot?
The Holographic Universe
Slices of Reality
Why Do We Need More Than 3 Dimensions?
Time in 3 Dimensions

One of the most common questions about this way of visualizing dimensions is whether the four dimensions of spacetime really are four spatial dimensions, or just three spatial plus one of time. I argue that for us "time" really is in the fourth spatial dimension which we, as creatures built from chemical reactions obeying the laws of entropy, are experiencing in a unique way. This relates closely to a phrase that is being uttered by many physicists nowadays: "time is an illusion". Saying that time is an illusion doesn't mean that we don't experience time from moment to moment, but rather it means that what we are experiencing as time is only a tiny window into a much greater underlying fabric, which ultimately encompasses the multiverse of all possible universes and quantum indeterminacy.

Hypercubes
The above youtube video shows what is commonly known as a four-dimensional cube: a hypercube, or tesseract. Before you click on the "play" button, we are not really "seeing" the hypercube, because a 4D object needs to be rotated for us to appreciate its higher dimensionality. In other words, without adding a time element to our appreciation of the shape shown in this animation, a significant part of what makes this a unique shape remains unseen.

In the above animation, we are looking on our computer screen at a flat 2D representation of a 3D shadow of a 4D object. Confused? If you go to the bottom of this blog entry there is another youtube movie, again showing a rotating hypercube. In this one, if you move your eyes so close to the monitor that your left eye sees the left half of the image and your other eye sees the right, your brain (with a bit of practice) can then merge those two halves into a stereoscopic visualization, from which you can get a hint of what we're really talking about here: "shadows" of a 4D shape being seen from the 3rd dimension.

Plato's Cave
Are you familiar with the allegory of Plato's Cave? It tells of some hypothetical people who spend their lives trapped in a cave, unable to see out into the real world, and all that they can surmise about reality is based upon the shadows cast upon the cave's walls as objects or people pass by the entrance to the cave. Trying to visualize higher dimensions is a similar exercise: our 3D reality is created by higher dimensional patterns, and what we witness from moment to moment, from day to day, from big bang to entropy and beyond those two extremes, is really just shadows of those higher-dimensional shapes and patterns. Ultimately, all of those shapes and patterns exist as potential within the underlying fabric of quantum indeterminacy.

Enjoy the journey,

Rob