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Final Review

Messages to the Future

  • No comments seemed necessary.

References and footnotes

  • All images are properly attributed in the page you see when you click on the image. Attributions include original source and remarks if you've modified.
  • No direct quotes are used.
  • References for text are at the end of the page, with option links to the footnotes within the text.

SteveC The Wikipedia reference is an outdated page. The current one is practically identical, but should be referenced.

Good writing

The following items are just meant to be reminders. If one of these items needs clarification, or seems like a great idea that you don't know how to implement, see What Makes a Good Math Images Page?.

Context (aka Generating interest aka Who cares?)

  • I explain the one-sidedness and non-orientability. The Why It's Interesting section mostly points out that the Cross-cap is interesting as it is a model of the Real Projective Plane.

  • Rebecca 02:08, 21 July 2011 (UTC) If you have time, I'd like to see more about how it's used to model human vision. You could even just add it to future directions for the page. That's discussed in the Why It's Interesting section on the Real Projective Plane page. It does not lend itself well to discussion with the cross-capped disk.

Quality of prose and page structuring

  • The beginning paragraph(s) of the page introduce the cross-capped disk. I realize that they are a bit technical, however I have done my best to explain the terminology I use and provide links.
  • Although it is a bit conceptual, I included a discussion of dimensions in the Basic Description as I believe it is crucial to understanding the self intersection of the Cross-cap that often frustrates people.
  • I placed the equations that parametrize the Cross-capped disk toward the end of the More Mathematical Explanation, and had the explanation and construction of the Cross-cap at the beginning. The

Integration of Images and Text

  • Wherever an image or animation is used to help with an explanation, the reader is explicitly instructed to refer to the image.

  • Rebecca 02:07, 21 July 2011 (UTC) The applets are so cool and it's nice that they're hidden because they slow the page down a bit. You might want to add a disclaimer "This applet may take up to a minute to load" or something along those lines so people don't give up on it. It just took a while (20 seconds?) on my computer. CHECK

Connections to other mathematical topics

Examples, Calculations, Applications, Proofs

  • I introduce new ideas with explanations and analogies.
  • Most of the proofs on this page are visual of conceptual, and are explained step by step.

Mathematical Accuracy and precision of language

  • In terms of the mathematical and technical accuracy of the page: I sent it to Don Shimmomoto for review, and he approved it with a few comments, which I have since addressed.
  • I try to make my statements as precise as they can be without overwhelming the reader with too many words or dense symbols. I realize that some parts of the page are slightly denser than others, and did my best to thin technical heavy parts into slightly longer, less dense sections.
  • I have many explanations, analogies, mouse-overs, and links to aid readers with new terms and concepts .

  • Rebecca 02:05, 21 July 2011 (UTC) "Surface that is non-orientable, and has only one side." - no commaCHECK
  • Consider bolding "4 embeddable"CHECK


  • Text is in short paragraphs, and broken up by relevant images throughout.
  • Hide and mouse-over features are used as appropriate to reduce clutter and scariness.
    • I did hide a second method of constructing of the Cross-cap. The interactive applets are also hidden, as they take up a lot of page space.
    • I did not hide the parametrization section. I figured that it was unnecessary as the ME section is already hidden at the start.
  • To whatever extent possible, pages do not have large, awkward chunks of white space.

  • Rebecca 02:06, 21 July 2011 (UTC) The layout of your pictures is pretty spaced out in my window. Did you consider making image 8 and 10 smaller? I believe that they need to be that large in order for people to be able to see the details in the diagrams.

Harrison's Notes to Self

  • Items in Bullets are From the Wikipedia Discussion Page Coppied by

Htasoff 14:15, 8 June 2011 (UTC)

    • Accuracy of the article
      • There is virtually nothing in the article (other than the lovely graphics) that is not false. Starting with the misguided attempt to roughly define a cross-cap in the first sentence.
        • First of all, a cross-cap is never topologically equivalent to a Moebius strip. It is a continuous image by a certain type of map of the (closed) Moebius strip into 3-space, a map that has an open interval's worth of double-points.
        • As a topological subspace of 3-space, it is the space obtained by starting with a closed disk D2, choosing an interval in the disk's interior -- say the image of [-1,1] via an embedding h: [-1,1] → int(D2) -- and then identifying the points h(x) and h(-x) with each other for all x in (0,1].
        • A cross-cap may have a boundary that is a round (perfect) circle, but is required only to have a boundary that is an unknotted simple closed curve.
        • Further, there are continuous deformations of the usual picture of a Moebius band to a picture that is still topologically a Moebius band embedded in 3-space (i.e., with no self-intersections), such that its boundary is a perfect circle. So this perfect-circle-boundary property in no way characterizes a cross-cap.
        • The word cross-cap has been erroneously used to mean a cross-cap with a disk glued on to its boundary (making a continuous image of a projective plane). But this is an error, and it should not be perpetuated in this article. Rather, the article should warn people to avoid this misuse of the word.Daqu 00:53, 4 December 2006 (UTC)Daqu 01:18, 4 December 2006 (UTC)

Text in MME is getting truncated when viewed through edit with form, but still appears on the page.

Note to self: f: X --> Y means X is a subspace of the original Y.

Htasoff 20:09, 2 June 2011 (UTC) Need the mathematical accuracy of this simplification to be checked: closed surface Useful?

General Comments

Jenn Chan (chanj)
Hey, I noticed that you have a request for an applet for the cross-cap page, and I found this web site that has what you're looking for.
Cross-cap is on the list.

Thank you. I have seen that one and it is really nice. Reza at Drexel made some applets for me that display the exact figures defined by the parametrization on the page, which I made in Mathematica. So all is well. Your capital letters seem funky to me. Should "Cross-cap" be capitalized? What about "Real Projective Plane"? -Richard 6/9

Kate 18:27, 9 June 2011 (UTC): Agreed.

Section-specific comments


I don't exactly know our policy on this, but do we want links to other pages in our opening sentence/paragraph? I almost think it's better to wait to link til you introduce the word in the basic description. -Richard 6/9

Basic Description

  • The closed-surface mouseover seems incomplete to me. It's just adjectives, not a definition?CHECK
  • Second paragraph: would it be better to talk about the main image first? I'm getting confused here with all of "the upper surface of the outside of the Cross-cap" It's hard to keep track.
  • what if you start out by saying I'll refer to this part as ceiling, this part as floor, this part as roof/top and this part as the bottom. Maybe with a diagram?CHECK, added Labels to image.
  • Repeated "the". "So the the top of the Cross-cap..."


  • You sound very informal.Rectified
  • "Here is why."
  • Now to explain..."
  • When you talk about seeing the cross-cap in the fourth dimension, I really like your description. Might it also be helpful if you used an example with time at the fourth dimension?
  • At time t = 1, you could be at (8,7,6) for one parametrization and at the same point when t=5 for another parametrization. These intersect in 3 dimensions, but not 4.I had been thinking about that, and have yet to figure out the best way to convey this. The proper animation would be great, but is beyond my knowledge of how to put together. If someone could hunt down how to do this, it would be great.
  • What's "4 embeddable"?????? Maybe a mouseover here if the definition is already on another page.Switched the order of the sentence to, hopefully, improve clarity.

-Richard 6/9

Kate 14:22, 9 June 2011 (UTC):

  • Most of this section does a good job of being as basic as possible, but your first paragraph is really intimidating, mostly because it sort of starts off with all this terminology that a reader might never have seen before.

  • Unlike another, more well-known, one-sided object, the Mobius strip, the Cross-cap is a closed surface, and, as a result, 3 dimensional models of it intersect, or pass through, themselves.CHECK
  • This sentence has way too many commas in it. Most of them aren't incorrect, but it took me several read-throughs to figure that out. I'd try and rephrase it so it's simpler, even if that takes two sentences.CHECK

  • The discussion of why it self-intersects made sense, but I think it'd be even clearer if you had a picture illustrating the ball example. CHECK
Yup, picture helps a lot.

  • This section has a lot of text at once. I think it'd be good if you broke it up with sub-sections, and maybe added more images. (Perhaps an image showing what you mean by floor and ceiling would be helpful.)CHECK

A More Mathematical Explanation

Maybe make "u = [0, 2π) and v = [0, \tfrac{\pi}{2}]" and "u = [0, 2π) and v = [0, \tfrac{\pi}{2}]" be in math writing.CHECK -Richard 6/9

Kate 14:22, 9 June 2011 (UTC):

  • Putting the domain for your parametrizations as bullet points under the equations doesn't make sense to me.

I did away with the bullet point. Hopefully this will avoid confusion.

Yes, I find it much less confusing now.

  • Both generate the same shape, they merely construct it in a different orientation. The following parametrization, generates a the Cross-cap in a different way from the first two, producing a model slightly more similar to the one at the top of the page.
  • I would suggest changing this to:
Both sets of equations generate the same shape, they merely construct it in a different orientation. The following parametrization generates a Cross-cap in a different way from the first two, producing a model slightly more similar to the one at the top of the page.CHECK

  • Can you put in images for the different parametrizations, so I can see how they differ from each other?

I left a note to the Rensselaer people asking if they could help me upload 3D graphics.

  • Typo: Sitching the cos(u)… - should be "Switching"CHECK

  • I think it would be much more helpful to have a bubble on homeomorphic than a red link.CHECK

It's a big idea that I don't think can be summarized in a bubble.

Even just saying "Two forms/shapes/things are homeomorphic if they are considered topologically the same" would be more helpful than nothing. I think this would be an ideal place to use a combined bubble/link to helper page, if Abram can find the code for that.

  • Should there be something in this section other than just all of these equations? I think I'd really like it if there were more text… I think so, too. Any Suggestions?
All I know about cross-caps is what you taught me on this page, so no, sorry. -Kate 18:32, 9 June 2011 (UTC)I'm on it.
I have written a surface visualizer using graphics shader techniques, and I plugged in one of the equation sets (and its partial derivatives, to get normals) into the visualizer and got a good result. The equation sets are a good resource. SteveC