X-Sender: Thomas_Banchoff@postoffice.brown.edu Mime-Version: 1.0
Date: Fri, 23 Feb 1996 00:46:07 -0400
To: Jennifer_Clare@Brown.edu (Jennifer Clare) From: tfb@cs.brown.edu (Thomas Banchoff) Subject: Re: ch 2
>Dear Jennie,
Soon, I hope, all the html difficulties will be behind us. Until then, let me try some responses.
Firstly, to respond to your question regarding my artist friend who
>is studying Flatland and the fourth dimension. She is a student at ISOMATA, which is a fine arts high school where they actually get fine arts high school diplomas. She is a photography/mixed media major, planning to go on in film or sculpture at Sarah Lawrence or Art Institute of Chicago. All of their classes are related to art and perception in some way, and her math class chose to focus on space and relationships in space. This obviously relates to photography form/space/composition. This led them right into dimensionality and the study of the hypercube. I have been sending her some of the class dialogues. That is okay, right? They are reading another book besides Flatland. I will try to ask her what that book is, if you would be interested.
By all means, feel free to share our dialogues with your friend, especially if she is willing to share some responses with us. I wonder if anyone in her class would be interested in B3D. I certainly would like to know about the other book or books they use. Is there a syllabus for their course?
>The other topic brought up by this chapter is the building of my hypercube, which is a major challenge. After bonding my fingers together with Superglue, and piercing my leg with a tack, I decided that this is harder than it looked when I took it on. I am trying to build a hypercube out of four unit hypercubes. I first did this in the second dimension, by cutting out four unit squares, and putting them together to form another square. Then I made four cubes, fit them together and made a cube. Can I extend this pattern into the fourth dimension? In the book you use numbers other than four. You portray the side length as m, and conclude that a square can be split up into m^2 unit squares. A cube with side length m can be split up into m^3 unit cubes. So a hypercube would be split up into m^4 unit hypercubes? But the book says we are not able to build a model. This just means that some of the cubes will be compromised. So if I build a hypercube with side length 16, it should be split up into 16 unit hypercubes. This is a lot of hypercubes to make. Can I compromise and make four? Or am I missing out on something? I am trying to save myself work without creating something that does not make sense.
I think a hypercube with sidelength 2 is already interesting. That would lead to 16 hypercubes, the way we get 8 for a cube in three-space or 4 for a square. Froebel had one gift which consisted of eight cublets. It isn't exacly clear how to handle the 16 hypercubelets. Perhaps you are moving toward filling a portion of three-space with a certain number of rhombic dodecahedra, each of which is a projection of a hypercube? That might be difficult to do, unless you photocopy a net of one of those twelve-sided figures and ask fifteen of your classmates to help you out.
>A question. On page 27, the book says that "Each time a cube is formed in a new dimension, it means adding edges in a direction perpendicular to all of the directions so far." That makes sense. Then it goes on to say that the final edge is perpendicular in particular to the longest diagonal of the previous cube. How is this? I would think that the final edge is perpendicular to the face of the original cube. Why the diagonal? I can't visualize this model.
Try the analogy--if we draw a diagonal on a rectangle that is one face of a rectangular block, then the other edge is perpendicular to the whole rectangle, edges, diagonal, and all.
>Another question: I was sitting at Ben and Jerry's, discussing the fourth dimension with two friends, and we were talking about shadows. Our shadows are two dimensional representations of our three dimensional figures. What kind of light source could create a shadow in the fourth dimension? Would the shadows be three dimensional? Or would they be "flat" as well? Would the projection of light be different in different dimensions?
The shadow of something from the fourth dimenison would probably be cloud-like in our world. Do you remember the travel sequences in "A Wrinkle in Time"? Meg definitely got the impression of a shadow presence. We can try to deal with those shadows even without knowing the light source that causes them, the way A Square might try to cope with elliptical shadows of the Sphere, cast from some oblique source that he could not begin to comprehend.
There is a lot more to be said about shadows, and we will.
Prof. B.
I am still having problems with the html, so this will be
>another email.
>jennie clare
Ask one of the people who has finally succeeded--they'll probably be happy to help. When you do drop this in the Week 5 slot, include my comments perhaps? As a link, if you want--try Jeremy Kahn on that idea.
Prof. B.