From Math Images
I really like that you flushed out the area and volume sections of the torus, and the images you added. A few ideas on the editing you've done.
Things to edit in your work:
- //as they are simple figures with very many interesting properties.// Even though this is the basic description, you can describe roughly some of those interesting properties. My sense is that readers would prefer to be shown an interesting property, even if they end up not finding it that interesting, than just to be told that interesting properties are there. Admittedly, I don't know why the interesting properties of tori are interesting, though maybe Steve does.
- My plan is to look through a couple differential geometry books to figure out what to say and how to say it simply (perks of living with another person who does math--we've got three differential geometry books between the two of us)
- On the pictures at the beginning of the more mathematical section, can you be a bit clearer about how the cross-sections are created?
- I've tried... I think I'm being too wordy, though
- Nice job on the surface area and volume derivation. I just find myself wondering why finding the surface area and volume of a torus is something that people want to do. I don't really know what the answer is, though I think these formulas can be used as a good way of testing various formulas from differential geometry.
- //The reasons for this are a bit complicated and are omitted here.// People apparently find lines like this offensive. At least from what I'm observed, when non-math-majors (even who like math a lot) read things like "it's a bit complicated", they hear a snub. Maybe there's another way of saying the same thing?
- I sort of hand waved a symmetry argument this time around. The picture that goes with it (which was remarkably difficult to create...) will be way more helpful than the words I wrote. Let me know what you think.
Things to edit in Lizah's work (maybe you just haven't gotten around to some of these):
- The descriptions of the three types of tori need better translation (e.g. a horn torus is like a donut that actually doesn't have a hole, because the donut is so thick; this is a bad description, but I think you know what I'm getting at).
- I edited all of her doughnut references. I think I solved the problem.
- //A torus is a donut-shaped surface of revolution formed by revolving a circle about the z-axis.// The z-axis? Not necessarily. Also, surface of revolution should have a mouse-over or the term should be dropped.
- I killed the term. See if it works now, or if I need a definition of surface .
- //We can divide a torus into regions and color each region so that adjacent regions always have different colors.// The problem with this wording is that it suggests that the task of dividing and coloring a torus is an action we decide to take, rather than finding a coloring given a torus divided into regions being a problem that is "presented" to us.
- I think I fixed it .
- //Now the Cartesian coordinate of a torus symmetric about the z-axis is// "Cartesian coordinate" should be replaced with correct terminology.
//“This is a picture” should simply be “This picture”
Types of Torus: This section should be entitled Types of Tori instead of Types of Torus. Somewhere, I would write that the plural of torus is tori.//
//When you say “In general, a torus can have multiple holes.” Are there any instances in which it can’t?//
Yes. The only instant when it does not have multiple holes-which is also the most common- is when it has one hole. A torus with one hole, also called a ring torus is actually the most known type of torus.
//I get confused by the discussion of dimensions. Isn’t a circle a 2-dimensional object? Also, how does a cube wrap? Is there an image that could demonstrate this? //
A circle is a 2D object, but a line is ID. A line(1D object) bends to form a circle (a 2D object). A rectangle(a 2D object) folds to form a tube(a 3D object). The concluding sentence in that section summarizes this phenomenon where an n-dimension object can actually exist in n+1 dimensions. I've asked the Drexel folks to make images that would help illustrate this point.
//I might title this section “Multiple Holes and Multiple Dimensions” and then explain that both can be determined using the term n-torus.//
I feel like with this either way is fine, as in I could start with n-torus, then say it has two meanings or start with "Multiple Holes and Multiple Dimensions" then say they that they're both related to n-torus.
//Metric Properties of the ring torus
- I looked up the words “metric properties” on Google and wikipedia. Area and perimeter aren’t properties specific to the metric system. I’d find a different title. I’d also capitalize Ring Torus in the title. //
I changed this.
Thanks again for the helpful suggestions.
Chris Taranta 7/7
“This is a picture” should simply be “This picture”
Types of Torus: This section should be entitled Types of Tori instead of Types of Torus. Somewhere, I would write that the plural of torus is tori.
Metric Properties of the ring torus
- I looked up the words “metric properties” on Google and wikipedia. Area and perimeter aren’t properties specific to the metric system. I’d find a different title. I’d also capitalize Ring Torus in the title.
- I might title this section “Multiple Holes and Multiple Dimensions” and then explain that both can be determined using the term n-torus.
- There are two definitions of the term n-torus”, not n-tori.
- When you say “In general, a torus can have multiple holes.” Are there any instances in which it can’t? If not, simply say: “A torus can have multiple holes.”
- I get confused by the discussion of dimensions. Isn’t a circle a 2-dimensional object? Also, how does a cube wrap? Is there an image that could demonstrate this?
Are there any applications of tori?
Here are some ideas for the Drexel folks relating to this page:
1. An animation illustrating the basic description ie a circle being rotated around the z-axis to form a torus, explicitly marking the z-axis (showing that it is dofferent than the x and y axes). I could not find a suitable image online.
2.Gene suggested having an image showing the difference between R and r.
If there's anything you feel could be useful for this page, please feel free to put it in
//can you link to Four Color Theorem in your section about coloring?//
I now have, thanks
//Did you create the first image that's within the page? If not, it needs to be cited somehow. //
I did not create the image. Initially I cited it in a mouse over, but I guess the current citing is much more clear.
//Can you explain why the metric properties are like that of the cylinder, just to make it more clear to the reader?//
Thanks for the suggestions
Anna 6/29 (mostly notes for our meeting)
Did you create the first image that's within the page? If not, it needs to be cited somehow.
Can you explain why the metric properties are like that of the cylinder, just to make it more clear to the reader?
We might want to ask the drexel people to make a little interactive thing for the coloring, so people can spin it around and actually see what's going on.
I don't know if you've gone through all of Steve's comments, but I wanted to add a couple quick thoughts. The first is, can you link to Four Color Theorem in your section about coloring?
I think some of the extensions Steve is pointing to, while interesting, may be beyond the scope of this page. I'd encourage you to look them up, but work on the content you have before adding anything more.
Steve Maurer 6/23
In a More Mathematical Explanation.
A figure of a tube showing R and r would be helpful. A discussion of how these explicit and parametric equations were obtained would be helpful. Maybe I will have time to work with you on this.
The types of Tori is a specialized topic - I didn't even know it. I would put the surface and volume formulas first. Look up Pappas' Theorem. Using that theorem (which is not obvious but easy to state) it is not hard to compute the volume. That would be a good link, to have a Pappus page.
Two definitions of an n-torus. I hadn't thought about this, although I was familiar with both objects. The two definitions are not equivalent. The first sort of n-torus can be represented in 3-dimensional space - you showed how. I didn't know that the n-torus of the other sort can be represented in (n+1)-dimensional space, but now I've thought about it and see that it is true. The representation I mentioned in a meeting a week ago is in 2n-dimensional space. The discussion of this sort of torus should probably be expanded, and you have asked me to discuss with you how to do this.
Coloring of a torus. Nice addition and nice picture. I have corrected what you said about 7-coloring. You might want to look up references to the Ringel-Young Theorem (which generalizes the 7-color theorem to surfaces of all genuses - you might want to look up "genus" too, because that relates to your first definition of n-torus.