# Talk:Projection of a Torus

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## Contents

#### Abram 7/10

In Cartesian Product page, "n-tuples" still should say "ordered pairs".

The torus page still refers to both a "four-dimensional torus" and a "4-D torus", which is still ambiguous terminology. I know what you're getting at, but what if you replace the first use with something like, "A torus is often thought of as a donut shape, but there is another definition of the torus, according to which a torus lives in four-dimensional space." The "4-D" in "4-D torus" can just be dropped.

#### Anna 7/9

I'm seconding all of what Abram said, and FYI, I decided to go in and edit Cartesian Product really quickly.

#### Abram 7/9

Well done. A number of small comments:

• Where you write "there are ways to capture parts of the four-dimensional object in three-dimensional space," maybe replace "parts" with "features" or "properties".
• In your Cartesian Product helper page, replace "n-tuple" with "ordered pairs".
• Also in your Cartesian Product helper page, in the right-hand-side of the equation
$\{1,4,6\} \times \{2,9\} = \{\{1,2\}, \{1,9\}, \{4,2\}, \{4, 9\}, \{6,2\}, \{6,9\}\}$
all the braces except for the very first opening brace and the very last closing brace should be replaced with parentheses (your notation as is actually implies that order doesn't matter).
• Replace "A stereographic projection is used to map this 4-D object into 3-D" with "Stereographic projection is used to map this object embedded in R^4 into R^3" or "Stereographic projection is used to map this object, which lives in four-dimensional space, into three-dimensional space". The point is that somehow the wording has to be changed to indicate that it is the space in which the torus lives, not the torus itself, which is 4-dimensional.
• Similarly, replace other uses of 4-D and 3-D with a substitution of your choice.
• Finally, can you say something about where the projection point is for the funky projection on the right hand side of the main image, either the coordinates of the point or a general location?

#### Anna 7/9

Ok, I like what you've done... only the definition of the cartesian product is too abstract. Maybe make a *very* short helper page with a definition and two or three short examples?

#### Anna 7/8

Yes, 'cartesian product' is a much better term. Can you either do a mouse over or link to a definition?

Also, I'd like to see at least one reference given at the bottom of the page, even if it is just the main page where you found the image.

#### Brendan 7/7

Anna, I remember I wrestled for some time to find the correct way to write that sentence. Cross product here really means 'Cartesian Product', which I suppose is a bit clearer, so I'll go ahead and replace it.

#### Anna 7/6

Are you sure you're using the term cross product correctly here: "The 4-D torus is thus the cross product of two circles."?

#### Anna 6/29

You should link to the Torus page.

#### GK 6/8

Brendan,

Should the image be called "Projection of a Four-Dimensional Torus"? since that gives a big kick to the article?

Basic Description is too big a paragraph and should be broken up.

Your "parametrically" formula needs spaces after the commas on both sides.

Wild stuff. Even with projections with two different points I got no sense.

4-D is so hard to capture well. Wish Banchoff had at least used different colors or something--maybe a sequence of pictures. Wikipedia's 4D animation of a hypercube ain't too helpful even. Well, it's fun to start folks thinking.