Not Just Outside the Box, but Orthogonal to It

Tesseract: synonym for Hypercube. See:

Eric W. Weisstein. "Hypercube." From MathWorld--A Wolfram Web Resource.

Read, see pretty pictures, AND maneuver and rotate a simulated tesseract with the mouse. Watch the perspective shange it in fascinating way. Might give you an aesthetic/kinesthetic appreciation of hypercube/tesseract geometry!

Then click from there to other pages at Eric W. Weisstein's MathWorld... IMHO the best Math Pages on the web.

"Polytope" is the multidimensional generalization of polygon and polyhedron. It includes Tesseracts and other things.

If you google "Polytope Number" you'll find a page of mine "Table of Polytope Numbers, Sorted, Through 1,000,000."

Visualization of 4-dimensional objects is possible for some people. It is extraordinarily rare to be "born with it," if possible at all. It can be learned with varying degrees of difficulty depending on the age that you start, how good you are at 3-D to begin with, and how good you are at geometry and math in general.

One well-documented case is Alicia Boole Stott, niece of THAT Boole, who invented Boolean Logic. He gave her a set of colored blocks with instructions on what colors could go next to what others. It was a toy designed to get her visualizing 4-dimensional shapes. She proved to be VERY good at doing so, into adulthood. She could also go 5-D and 6-D to some extent.

There was a burst of approximately 1,000 publications about the 4th dimension in the late 1800s. This forever influenced all nonfiction AND science fiction.

One Swiss gentleman who could visualize 4-D was Ludwig Schlafli [umlaut over the a]

Born: 15 Jan 1814 in Grasswil, Bern, Switzerland
Died: 20 March 1895 in Bern, Switzerland

See:

Schlaflipage by St.Andrew's University Math History

"Ludwig Schlafli first studied theology, then turned to science. He worked for ten years as a school teacher in Thun. During this period he studied advanced mathematics in his spare time..."

He discovered something profoundly important. Let me summarize:

* there are an infinite number of regular polygons, like an equilateral triangle, square, pentagon, hexagon... where all edges are the same length and all angles identical.

* there are exactly 5 regular polyhedra, with all faces the same and all angles the same: Tetrahedraon (triangular pyramid), Cube, Octahedron, Dodecahedron, Icosahedron. Everyone who plays Role Playing Games knows these now as dice shapes. They are called Platonic Solids.

* Schlaflii, home, alone, over a decade, discovered that there are exactly SIX 4-D equivalents to Platonic Solids, namely the Pentatope (4-D Simplex), TESSERACT (a.k.a. hypercube, a.k.a. 4-D measure polytope), hyperoctahedron, hyperdeodecahedron, hypericosahedron, and one with no equivalent in any other dimension, the 24-cell.

* in all higher dimension, there are only 3: the equivalent of terahedron, cube, and octahedron.

His work was published, but ignored, in part, because so few could visualize. Alicia Boole Stott confirmed his work: she could "see" it was true.

When I was a child, I learned to visualize 4-D objects, in a hazy way. Later I became a mathematician, and then a part-time professor of math.

Strangely, the visualization partially returned to me at age 53. So I have written several math papers these recent years, some devoted to 4-D and higher dimension shapes. I've been emailing back and forth with real experts, including Richard Feynman's son Carl, about the hypervolume of some of these shapes, and how that is changed by truncateing them in various ways.

My son is mad at me for never being able to find out the details of the colored blocks, and wishes he could visualize 4-D. There's a famous science fiction story about a toy that teaches children to visualize 4-D, recently made into such a bad movie that I won't name it, and they use to to sort of Tesser away.

I think a visualizer of 4-D is John Forbes Nash, Jr., the subject of "A Beautiful Mind." His dissertation was only 28 pages long. It was, formally, about polytopes, but changed Economics forever.

"The King of Geometers" H.S.M. Coxeter died a couple of years ago. His book "Regular Polytopes" [a Dover paperback] helped me enormously as a child. The photos and illustrations were beautiful.
Last month I was looking in a library at later editions of that, and others of his books.

Now, much of those 2-D images of 3-D projections of 4-D objects are available on the web. I'd start at mathworld.com, if I were you. In their search box, type "pentatope" or "hypercube" and manipulate 4-D objects with your mouse.

Interesting stuff. I was going to write about this at some point, so I figured might as well now. As of now, my blog (linked at my name) has two entries up on moving into 4-D.

One key thing to remember is that being able to visualize a 4-D cube is a far cry from being able to "visualize in 4 dimensions," as some say. Doing the former is equivalent to being able to see a single square on ordinary graph paper, without being able to see the expanse of space in the different dimensions.

One of George Boole's daughters Alicia was an expert at producing three dimensional representations of four dimensional objects. She coined the concept polytope and also worked closely with Coxeter.

Sorry Jonathan having only skimmed your post I missed the fact that you had already referenced Alicia Boole who was, I repeat, one of Boole's daughters and not his niece. All four of the Boole daughters are fascinating as was his extraordinary wife Mary Everest Boole who was the niece of the man after whom the mountain is named and who grew up in the French house of Samuel Hahnemann the founder of homeopathy.

Jonathan the cubes of which you speak were invented by Charles H.Hinton another 4-D fanatic who wrote a sequel to Flatlands, coined the word tesseract and married Alicia Boole's sister Mary Ellen. If you go here to the Hinton page at Wikipedia you will find a link to a pdf that explains how the cubes work.

Dear Thony C.,

thank you for straightening me out on the history/geneology.

One 4-D paper I'm working on now is abstracted as follows.

Sequence of Hypervolumes of Convex Hulls of
4-dimensional Euclidean Points Determined by a Sum of Four Squares

Draft 3.2, 25 pages, of 23 March 2007
by Jonathan Vos Post
Computer Futures, Inc.

ABSTRACT:

What is the content (hypervolume) of the convex hull of the vertices of P(n) (i.e. of the polytope), as a function of n?
P(n), for a nonnegative integer n, is
the convex hull of all vertices which have integer coordinates in the Euclidean space Z^4, defined by all permutations of:
(+-h, +-i, +-j, +-k) such that
h^2 + i^2 + j^2 + k^2 = n.

This problem arises as a 4-dimensional analogue of a 3-dimensional problem which was posed and solved by Wouter Meeussen in 2001. It touches on known results from Coxeter, Conway, Sloane, and others.

Appendices detail specific example through
n = 32.

Partial results may be summarized in this table:

n vertices hypervolume comment
0 0 0 null polytope
1 8 8/3 hyperoctahedron {3,3,4}
2 24 8 24-cell {3,4,3}
3 32 < 144 rectified 8-cell?
4 24 32 24-cell {3,4,3}?
5 48 16 = 2a(2) elongated 24-cell
6 96 2a(3) elongated rectified 8-cell?
7 64 ~ 33.803896 truncated 8-cell
8 24 32 = 4a(2) elongated (2 axes) 24-cell
9 104 > 3a(1)
10 144 > 4a(4) runcinated 24-cell?
11 48 3a(3) elongated rectified 8-cell?
12 144

[and then more results, still incomplete]

Cervone and Banchoff's work inspired the 4D visualization interface in my software. Pics at 24 Views of the Complex Exponential Function.