August 12, 2006
I recently came across this illustration of a tesseract (the 4-dimensional cube). Note the letters- they indicate which faces of the component cubes adjoin in a real tesseract.
This model, though a common representation, can be confusing becaue it’s actually a net of a tesseract. Think of it as the tesseract spread out to be viewable in three dimensions. (For comparison, this is a net of a cube). By indicating the adjoining faces with letters, you can get a clearer understanding of how a tesseract is truly structured.
This page, from the Union College Department of Mathmatics site, has several great blurbs and animations on understanding hyper-dimensional space. One idea they mention helps with some difficulty I have conceiving time as a dimension:
“Flatlanders [two-dimensional beings] can understand a sphere as a sequence of circles changing over time. The flatlanders see time as a third dimension, but we see the third dimension as a physical one. Similarly, we can understand a hypersphere from the fourth dimension as a sequence of spheres changing over time. We use time as a means of representing a fourth physical dimension.”
Another cool thing at that site is a pair of animations that compare the relationship between the two-dimensional shadow of a cube and the three-dimensional “shadow” of a tesseract.