Talk:Dual Polyhedron
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[edit] Abram 8/5
Nice job. In the inversion about a point section, where you describe the polar of a line, you may still want a sentence somewhere that explicitly states the polar of point B is the vertical line going through point C. You can decide whether or not you want to make this change, though, and, at any rate, there's no need to check with me no matter what you decide. I declare the page ready.
[edit] MaeBeth 8/5
Think I've fixed everything. How does it look?
[edit] Abram 8/5
This page is really great. Three extremely small things that would clean it up.
There's a typo in the basic description, where "each vertex has 3 faces" appears both with a mouse over and as regular text.
At the beginning of the more mathematical section, the mouse over for polars ("the faces of the dual polyhedron"), and actually, that whole sentence, is a little bit confusing, but I'm not entirely sure why. Maybe instead of a mouse-over, you should have an actual in-page sentence that states that the polar plane of the vertices will become the faces of the dual and vice-versa.
In the reciprocation about a circle section, your explanation is really clear. After " The point is labeled B; the inversion point is C; and the center of the circle is A," maybe point out in text that the vertical line going through C is the polar (just b/c the text "polar" in the figure is so small). Similarly, after, "The pole of a line with respect to a circle is the inverse of the point on the line that is closest to the center of the circle," could you point out that the pole of the vertical line going through C is the point B?
[edit] Anna 8/3
It looks good to me! Why don't you put ready? by it on the S09 page, and I'll encourage Abram to take a look at it soon.
[edit] MaeBeth 7/30
So i think that I have most everything fixed. Just let me know if there's anything i haven't done right or if there's something else that needs changing.
[edit] Anna 7/29
I'm going to highly two of Abram's comments that I really do think need to be addressed. The first is that question "What is the dual to an n-gon?" It feels awkward and is confusing, so I'd encourage you to either just scrap it or have some type of mouse over response.
Also, I'd like to see you point much more to your image in the Dorman-Luke construction, because I think that will be really helpful to the reader.
[edit] Abram 7/22
Nice job with the edits. The page is really clear, well-laid out, and interesting.
A few small fixes:
//As you can see in the image, the dual polyhedra of the Platonic solids are all Platonic solids themselves//
Make it clear which image you are referring to.
// (If you stick to 2 dimensions, you get dual polygons. What is the dual to an n-gon?)//
OK, I think we can leave that text intact if we drop the question, and rewrite the first sentence as "Reciprocating about a circle in two dimensions allows you to transform a polygon into a corresponding dual polygon".
//This new dual statement is of course: An octahedron has 8 faces and 6 vertices.//
I still think one more example would really help people see the pattern. How about "Each face of a cube has 4 vertices and each vertex has 3 faces [with a mouse over defining what you mean by a vertex "having a face"]," which turns into "Each vertex of an octahedron has 4 faces and each face has 3 vertices".
//This sphere will be analogous to the circle we used in the previous section.//
Great job helping the reader connect the different sections to each other
//Dorman-Luke construction//
Put the first use of this term in boldface.
//Then construct the circumcircle around the vertex figure at each of its vertices. //
Refer the reader to an image. The mathworld picture you have below this works great, especially if you add a caption to it.
The one big thing I would say this page can use is an explanation of what these different construction methods buy you, as a mathematician, or what makes them interesting. Statements like, "The Dorman-Luke construction can be used to do blah-blah-blah in geometry" or "Reciprocation about a sphere allows one to exploit principles of duality to prove statements about blah-blah-blah" would be great. Though maybe you don't actually know these applications, which is fine, if that's the case.
[edit] MaeBeth 7/15
So i think that ive finished with up with the edits that you all have suggested. Except for one. Abram- the suggestion that i should get rid of //(If you stick to 2 dimensions, you get dual polygons. What is the dual to an n-gon?) // im not sure about. I agree with you that it sounds confusing and that taking it out wont distract from the page but prof maurer wrote it not me.
so if y'all could just take a look and see if there's anything i missed that would be great!
[edit] MaeBeth 7/15
Thanks for all your comments! Ill get to them right away. First of all, I dont know why but i don't seem to get emails about this page being updated so i didnt see your comments until now. But now that I know about it I'll just keep an eye out for them. Anna- I already fixed the extra apostrophes. They were there because the balloon thingy doesnt like special text so I had Maria help me with that. Drexel folks are working on applets for this page which im really excited about. They should be in 3D!!
[edit] Anna 7/14
I've got a couple of things to point out.
- In your first mouse over, you have some stray apostrophe's
- "There exist five of these polyhedra." is mathspeak! Saying "There are five of these polyhedra" flows better. Highly mathematical language like that should be avoided in the basic description, if only because it will seem a bit off to a not mathematically sophisticated reader.
- Can this "More specifically, the polar of a point, B , in a circle is the line that goes through the inversion point of B, let's call it C, and is also perpendicular to the line containing B and the center of the circle, A. " be two sentences instead of one?
- Are the drexel folks working on applets to help with the reciprocation sections? I feel like that would make a big difference.
I really like your introduction of duality, as well.
[edit] Abram 7/14
Hi Mae Beth,
Nice job with the edits. I like your explanation of duality a lot. I have a few specific suggestions, most of which are about changing word order.
// It is interesting to observe that polyhedra obey the rules of duality. This means that if you find the dual of a polyhedron and then find the dual of the dual you end up back with your original polyhedron.//
Drop the first sentence and move the second sentence until after you have explained how duality works with polyhedra (i.e. move it a few paragraphs later).
//As you can see in the main image and learn more about later, the dual of the cube is the octahedron and vice versa.//
Put this sentence before you give the statement "An octahedron has...".
//An octahedron has 8 faces and 6 vertices//
I think one more example would be nice. How about "Each face of a cube has 4 vertices and each vertex has 3 faces [with a mouse over defining what you mean by a vertex "having a face"]," which turns into "Each vertex of an octahedron has 4 faces and each face has 3 vertices".
//The simplest way to create the dual polyhedron for a Platonic solid is by finding the midpoints of each of the faces, and then connecting these midpoints so that they become the vertices of the new dual polyhedon.// Maybe refer the reader back to the illustration with the octahedron and the cube?
//(If you stick to 2 dimensions, you get dual polygons. What is the dual to an n-gon?) //
It's not really fair to ask this question, because you haven't yet described how to create dual shapes. I think you could drop all of this with no real loss.
//The vertices of the dual polyhedron can also be found by the process of reciprocation because they are the poles that are reciprocal to the face planes of the original polyhedron.//
Point out the specific dual statements that arise, much like you did with the cubes and octahedrons. Something like: "The faces of a polyhedron and the vertices of the dual polyhedron are reciprocals of each other. Swapping "face" with "vertex" gives another true statement: The vertices of a polyhedron and the faces of the dual polyhedron are reciprocals."
//After finding the vertex figure, construct the circumcircle around the vertex figure at each vertex.//
Drop the phrase "After finding the vertex figure". Also, clarify whether "each vertex" refers to vertices of the polyhedron or vertices of the vertex figure.
[edit] Abram 7/8
Really nice basic description. The overall approach and the way you explain ideas is really clear. Two big suggestions.
First, your picture of the octahedron embedded in the cube is great, and any text that would be aided by looking at that image should specifically direct the reader to the image. Adding another related picture would also be a good idea.
Second, if there's any way to flesh out a bit more about cool properties of duality, that would be great. I mean, you have the statement about a cube has 8 vertices and 6 faces... Is there another statement you can make that exploits duality.
Here are a bunch of small suggestions that I think will really help tighten this up.
- It is interesting to observe that polyhedra obey the rules of duality.
This sentence structure doesn't make it clear that duality is this "unique counterpart" you refer to in the preceding sentence. What about something like, "The way in which the counterpart is similar to the original uses a mathematical concept called duality.."
- involution operation
Replace this term with the text that is in its mouseover. There's no need to use a word that big here. Also, I would move this statement a paragraph or two later, after you have explained what duality is.
- A cube has 8 vertices and 6 faces. Because of duality, we are able to make another true statement about the dual of a cube just by switching the terms 'vertex' and 'face'. This statement is of course: An octahedron has 8 faces and 6 vertices.
You haven't told anybody yet that the octahedron is the dual of a cube. Maybe you can add something like: "As you can see in the main image of this page, and you will learn why later, the octrahedron is the dual of a cube."
- A special case of dual polyhedra occurs with the Platonic solids.
Put a transition sentence before this, like: "In general, if you start with a polyhedron, it is a complicated process to create the dual polyhedron that has this unique relationship with the original."
[edit] Abram 7/1
So I hear you would like some assistance figuring out how to explain duality? Do you have time to meet this afternoon (Wednesday) b/w 1:30 and 4? Sorry for not sending this message via email, I just don't have your email address.
[edit] Abram 6/24
Hey Mae Beth (MaeBeth?), so I think you've edited this page since Steve made his comments, but if not, and you are feeling overwhelmed by comments, just ignore mine for the time being.
I think the main thing that throws me off about this page is that I'm not clear whether you are basically writing about duals of polyhedra in general or duals specifically about Platonic solids. The title of the page is "Dual Polyhedron", while your main image is titled "Duals of Platonic Solids" but it has pictures of other polyhedra, and you basic description is mostly about Platonic solids and their duals, but a little about compounds.
I agree with Steve that one thing that might really give this page some zip is to decide/learn what's important about duality and talk about that. I'm also thinking that one thing that will help you make this decision is to decide what types of polyhedra you are writing about.
Also, check out the wikipedia page on Platonic solids. It's incredibly well-written. I actually have the beginnings of a Platonic Solid page which I have largely abandoned because everything I was going to say was presented so well in the wikipedia page. You won't run into that issue, because you're writing about different subtopics than I was, but the page still has a lot of stuff you may want to borrow, steal, or link to.
[edit] Steve Maurer 6/23
I've put a bunch of edits on the page itself; see the differences with the history option. I agree with Abram that what you wrote is generally quite clear. The real issue is what more ought to be here.
For instance, somewhere there needs to be a general description of what we mean by duality. I guess that's another page, but here you can say what duality means for polyhedra and why it's a useful idea. One thing it means is an operation that when you do it twice, you get back to the original. Thus the dual of the cube in the octahedron, and the dual of the octahedron is the cube. More than that, it means that true statements come in dual pairs. For instance, the dual of the true statement "Every vertex of a cube meets 3 faces" is "Every face of an octahedron has 3 vertices".
As you recognize, it would be great to have more figures that explain reciprocation in 3-D. More than that, it would be great to have visuals (maybe dynamic) that shows how reciprocation gives the dual, say, or a cube. Unlike your basic description, reciprocation does not generally give the dual so that the vertices are in the middle of the faces of the original. The astute reader may have already figured this out from your description and wondered if this is a problem. It isn't because you just get a bigger or smaller dual figure, but one needs to see the construction to believe this.
There is a great linear algebra explanation of duality; you are actually taking orthogonal complements or otherwise put, dual linear maps. I am not expecting you to know about this or explain it. Some more advanced reader might develop this once the site is public.
In the Dorman Luke construction you might rotate the left figure so that the rectangle lines up with the corresponding rectangle in the right figure. This would help one to follow the paragraph above the figures.
Please write me if you don't see any reason for some edit I made. Often I did them to make clear when you were introducing a term readers should not be expected to know.
[edit] MaeBeth 6/17
From what I read it seems that the method of finding the midpoints of the faces and connecting them only works for Platonic solids and other regular polyhedra. I could be wrong, but this was the impression i got.
[edit] Abram 6/16
Overall, the basic description is a great summary of what a dual Platonic solid is. As long as the reader knows what a tetrahedron, octahedron, etc, are, your description both defines a dual of a Platonic solid clearly, and states the most important fact about it.
Two things you might want to change in this section:
- Change the first sentence to say that every Platonic solid has a corresponding dual polyhedron, or strike the first sentence entirely. In either case, you don't talk at all about dual polyhedra for non-Platonic solids, so your one caveat at the end of the basic description should be the only reference you make to non-Platonic solids.
- Include an image somewhere that shows, maybe in table format, the five Platonic solids, all named, and their duals, also all named. Your main image has all of this except the names.
For the more mathematical section, the stuff you currently have could use a bit of editing. You express ideas very precisely, which is great, but a little bit at the cost of awkward language (this isn't surprising -- it's really hard to be both precise and clear). I can give you suggestions, or you just generally look for ways to use fewer words and embedded clauses to express the same idea.
Before that, however, you might want to include other facts about dual Platonic solids that are slightly more relevant (and actually interesting). These facts include:
- If a Platonic solid has E edges, F faces, and V vertices, its dual has E edges, V faces, and F vertices.
- The dual of the dual of a Platonic solid is itself (you allude to this fact, but don't state it outright), and a proof thereof.
- Maybe other things (I don't know enough about this topic to know everything that's important).
This section about ways of constructing the dual are really interesting -- it might just be a bit less significant than other stuff. Also, I'm curious: it seems like the easiest way to construct a dual is precisely what you described in the introduction -- put dots in the centers of the faces, then connect adjacent dots, and voila, you've got the dual. Why go through these more complicated methods?
