Hypercubed?
Published Friday, April 1st, 2005 at 7:20 pm
Some of you (is anyone actually there?) may be wondering what it is to be Hypercubed. Well Hypercubed is the inflected verb meaning to form into a Hypercube. According to Merriam-Webster a Hypercube is “a geometric figure… in Euclidean space of n dimensions that is analogous to a cube in three dimensions” (see reference). Notice it is a geometric figure in space of n dimensions. Therefore Hypercubed means to be formed into a n-dimensional equivalent of a cube. However, when people mention a hypercube they most commonly are referring to the tesseract, the four-dimensional version of a cube. Doesn’t Hypercubed sounds better then tesseracted?What is a hypercube (tesseract) look like? A line is a one (1) dimensional object. Now imagine if you were to make a duplicate of this line and then connect these two lines by more lines of the same length. This would be a two (2) dimensional square. Now take this square and connect it to a duplicate square using more duplicate squares. You now have two squares connected by four squares, six faces in all. This is a three (3) dimensional cube (look here). Now the hard part; imagine two cubes where each face of the first cube is connected to one face of the other cube by a duplicate cube. This is difficult to imagine because we live and think in three dimensions. But mathematically there is no difference between space with three dimensions and space with four dimensions. Well obviously, due to limitations of 3-d space, we cannot see a four dimensional object. However, using the same techniques that one uses when drawing a 3 dimensional cube on a two dimensional piece of paper we can project an image of a four dimensional hypercube (or the wire frame of one) into three dimensions. In 2000 I posted an article on my website explaining how to use POV-RAY to display four dimensional objects here. The image below is a projection of a hypercube made with POV-RAY.
There are other ways to view a hypercube. We all know how to unfold a 3D cube; it is also possible to unfold a hypercube. Check out Salvador Dali’s painting “Crucifixion” (aka “Corpus Hypercubus”).
Related Links
- Hypercube - Visualization
- Hypercube’s Hompage - What are Hypercubes?
- Hypercubed - Wolfram Research
- Four-Space Visualization of 4D Objects - Masters Degree Thesis by Steven Richard Hollasch
March 20th, 2006 at 12:09 am
[...] In early Dec. 1999 I purchased the hypercubed.net domain name (followed shortly by hypercubed.com in April). While working on the look an feel of the hypercubed website I wanted to create some graphics of hypercube 3D projections to decorate the front page. But I didn’t want a simple picture of a hypercube I wanted it to rotate four dimensionally. I could have simply created a few images of a hypercube by hand and combine them into an animated gif but first my graphics skills were (are) very poor. Plus it wouldn’t be very cool. So I decided to create a POV-RAY script to create hypercubes for me. The script I made didn’t simply draw a hypercube but was actually a set of macros that allowed me to define and view projections of any four dimensional object. Using the macros I was able to create the hypercube animated gifs that still adorns the hypercubed.com website today. In March 2000 (6 years now) I wrote a how-to article documenting my POV-RAY macros for visualizing four dimension objects. This page has been sitting on the hypercubed.com, relativity unnoticed, ever since. So here for your retro viewing pleasure is the original text of that article.[...]