September 3, 2007
The universe is incomprehensibly big. It takes millions of years for anything significant to happen on a cosmic scale.
Might beings of hyperdimensionality sufficient to perceive time as a lower dimension be able to construct entire universes?
To what end?
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.
July 14, 2006
Seifert Surface has built a tesseract in Second Life.
He built his model of the 4-dimensional cube as a Victorian home. Here’s what you’ll see if you visit the home in SL:
Not exactly astonishing. You appear to just loop around and wind up back where you started. But stay with me here.
A tesseract is the extension of a cube into 4-dimensional space, and it’s very difficult (if it’s even possible) for humans to conceptualize it. Try to think of it geometrically, not realistically. As the square is to a line, and a cube to a square, so is the tesseract to the cube.
Another way to understand it is to imagine the vertices. In a square, each vertex extends in two directions. In a cube, each vertex extends in three directions. What happens when you add another dimension and extend the cube’s vertices in four directions? That’s a tesseract.
Hard to conceptualize, right? Take another look at someone walking through Surface’s tesseract house, this time with the camera zoomed far out:
Is it making more sense? Of course, this isn’t really what a tesseract looks like, but it’s an effective illustration. As in a cube, where each side of its component squares adjoin another side, each face of the tesseract’s component cubes adjoin another face. To represent a 4-dimensional object in 3-dimensional space,* the cubes move so their faces can adjoin.
Meg and Charles Wallace Murray never had it so good.
If you really want your mind blown, here’s a functional tesseract version of Rubik’s Cube.
SL users can portal to Surface’s “Crooked House” here.
More info here.
Still too pedestrian for you? A 5-dimensional Rubik’s Cube here.
*Technically, this is representing a 4D object in a 2D representation of 3D. But lets keep it simple.