Talk:Four Color Theorem Applied to 3D Objects

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Hi Sean. I was wondering if you have any more ideas for additions or edits. I'll also be doing some editing over the next two weeks, so hopefully the page will be finished up soon. -Peter (7/9/12)

Wait, nevermind, I know about what I just asked. I looked it up and I learned that already, it was just worded differently then what I knew. Sorry for the confusion.

-Sean

Wow, that is very interesting. I like what you added, it really helped the page. The formula was quite extraordinary. I didn't know there was a formula, but I was really surprised that it's a very simple calculation! I have a question, however, how do you find the y(x) for different objects? I think maybe if that is interesting enough, maybe that could be put on the page? I will try to look it up some and see what comes up. Thanks again, it's looking great! -Sean

Hey Sean. I added a section on the connection to topology and one under the why it's interesting section about coloring other, weirder shapes where the four color theorem doesn't apply, so you should take a look. If you want to change anything or think anything should be taken out from what I wrote, go for it. Also, if you think of further stuff to research and include I would greatly encourage you to pursue it, and let me know if you want any help finding information. I might do a little more digging and editing in the next couple weeks, and will let you know if I find anything interesting.

If you have any questions about what I wrote, just let me know! -Peter (12:02, 6/12/12)


Hi, Peter. Thank you for the help on my page. It was all very useful and I put it into consideration and made the changes you suggested. It really did help the page overall.

Might I just say that I am very glad that you chose my page for your research topic, so thank you!

Well, I did like the idea of the mathematical formulas involved in this project. That would be very interesting to look in to. My experience with graph theory and this type math before doing this project was zero and I never got around to actually researching or finding a formula (and it was also somewhat out of my league for right now), so I think that would be a great asset to my page. Adding a section to the page for this would be great, and if any help is needed I would be glad in assisting you, although I maybe of little help with it, but I could always try.

Although it is summer, I will try to keep up with my page and see any changes made and help out in any way needed. I am very busy throughout the summer but I want to fit time in my schedule for the page.

I just want to say thank you again for working on this project. I was very surprised and excited knowing that someone else was interested in it as well... and that college students are working on it. And if Diana is your teacher, she is really good and helpful. She can practically help with any problems you have and is full of useful insight.

Thank you again,

-Sean M

Hi Sean! My name is Peter and I'm a student at Swarthmore College. You've got a really interesting and quite original page going here, and I've got some suggestions for you if you're interested. I've written some thoughts below. As its summer break, you may not be in the mood to work on your page, and that's totally fine too. So, would you rather (a) leave the page as is, (b) take a look at my suggestions and implement what you like, or (c) have me make some edits to the page?

Format Suggestions:
  • You should move the last two paragraphs of the basic description to the More Mathematical section, since they talk about images in that section.
  • I'd suggest breaking up the text in the More Mathematical section by putting in more smaller sections. For example, under the heading of A More Mathematical Explanation you could have a subheading for How I Conducted my Research with a list of the steps you took, a subheading for Observations with the relevant text, and a section with a heading like Connection to Graph Theory.
  • It's really cool that you related this topic to graph theory. To help your readers out even further, you should provide a link to the page about graph theory here: Graph Theory.
  • There are a couple of times when you use a term and don't define it for a little while. I know sentences get really long and confusing when you try to define the term and use it all together, but one option you have is to use mouse over definitions like this. To learn how to make these, check out the Mouse Overs section of this page: Help:Wiki Tricks.
Content and specific sentence suggestions:
  • In your first sentence you say:
    "This picture is showing a basic understanding of the four color theorem using a bumpy 3D shape."
    It might make more sense to say something like "This picture shows an extension of the four color theorem to a bumpy, 3-D surface".
  • In your basic description, you say:
    “The Four Color Theorem states that any planar map can be covered in a minimum of four colors without any regions colored the same color touching unless by vertices, which is acceptable."
    Some maps can be covered in less than three colors, so it’s not that four is the minimum. It is just that no more than four colors are ever needed. So it would sound better to say something like “The Four Color Theorem states that no more than four colors are needed to cover any planar map such that that no regions of the same color are touching unless by vertices, which is acceptable.”
  • At that that point on the page, you should mention that the four color theorem applies only to flat 2-D surfaces, and that what you were doing in your research was to see if the theorem also applies to 3-D objects.
  • In the Why It's Interesting section, you say
    “Also, the torus is the only object that can be colored in 7 colors, whereas planes, spheres, or other 3D objects need at most four colors.”
    Actually this isn’t true. The Mobius Strip is a weird, non-flat 2-D surface that can need as many as 6 colors to be colored. Here’s the link to the page about the shape: Mobius Strip. There are other 3-D shapes that require 7 colors like the torus, and unless I am mistaken, there are even 3-D shapes that could require hundreds of colors. If you would like, I could do some research to figure out more about these 3-D shapes that require more than 4 colors and add a section about it to your page. There are mathematical formulas to figure out the maximum number of colors needed to cover a shape, which I could also look into.

Overall a very impressive page. Nice work.

-Peter (12:02, 6/12/12)

You've done a really fantastic and thorough job here. I love the way you lay out your process and observations so the reader can connect with what you did. If you want to polish it up before the deadline tonight, my only (relatively minor) points are:

  • The first sentence of your second paragraph, "The Four Color Theorem statesthat any planar map can be covered in a minimum of four colors without any regions colored the same color do not touch unless by vertices, which is acceptable," looks like maybe you edited it halfway through and then didn't read the whole thing back over? Not sure what happened, but the sentence doesn't make sense.
  • When you refer to "this chart and graph below" at the end of the third paragraph, there aren't actually any charts there — they're in the more mathematical section. Instead of referring to them at all, I suggest you quickly summarize their results. You just need a sentence or two, but since those results are the bulk of the work you did, it'd be nice to see them mentioned in your basic description!
  • You mention "the tori" as though it were singular a couple of times, but the singular form is "torus;" "tori" is plural.
  • At the bottom of the page, you list some other pages on this site that you found helpful. It would actually be most helpful to the reader if you linked to them directly in your text the first or most salient time they appear. So in the one-line blurb at the top of the page, you would have, "This picture is showing a basic understanding of the Four Color Theorem using a bumpy 3D shape," and later, under "Why it's Interesting," you would have, "Also, the Torus is the only object that can be colored in 7 colors, whereas planes, spheres, or other 3D objects need at most four colors.

That's all I've got! Again, great job. You've really created a wonderful, helpful page here.

-Diana (20:58, 4/202)


Think about the example of a Rubix cube. When the cube is solved, each face is a different color. But when you scramble it, the faces get sub-divided into different sections. So, try to define how you're talking about sides, faces, and sections here. If you look at each face of a shape as being a surface with potential subsections (which we know would require at most 4 colors) how does that change the number of colors required to cover the whole shape?

Part of the reason we want you to think about this is because some of the numbers in your table don't seem to be correct for all situations. You'll need to re-think some of the math there.

-Leah and Diana (4/1/12 18:40)


I made some minor edits to fix some format issues on the page. (The main image was showing up at the bottom, because the code was off.)

I'm looking forward to seeing the images you've made so far -- the nets and the cube -- up on this page along with the chart you made. These pages support a table format you can use to display the chart; go to Wiki Tricks for info on how to do that.

Also, while there's a lot to say about the image you have up as your main image right now, I'm not sure it's the right main image for this page, since it doesn't actually display an application of the four color theorem to a 3D object. If you're sticking with the topic of coloring 3D objects, this image actually seems more like background information or an extension of your project. As you get your other work up on this page and begin to write a discussion and explanation of what you're finding, try to identify the focus of your work and choose an image that fits it.

-Diana (22:34, 3/10/12)

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