From Math Images
Hey ya'll, this is my page. On this page, I want to discuss the 4 Color Theorem using 3D objects. I will try to see if I can find counterexamples of the theorem when using 3D objects, even though it was mainly focused on flat, 2D surfaces.
I want discuss any patterns between the number of sides and the minimum number of colors used to cover the area. I will use nets of the objects I will be exploring to show similarities and differences between the numbers of colors and sides used to cover the object as a 2D shape and a 3D shape. I also want to explore the difference between 3D objects with defined faces and 3D objects with only one face when applying the 4 Color Theorem.
22 February, 2012- I took the nets of different shapes (cube, pyramid, triangular prism, rectangular prism, another form of a triangular prism, a cylinder, and a pentagon sort thing) and looked at the differences between the minimum number of colors needed to color each shape as a 2D and 3D object.
6 March 2012- I made a 3D model of a square and I made a chart comparing the number of sides to the minimum number of colors used to cover the shape as a 2D and 3D object. This is a link to my image page http://mathforum.org/mathimages/index.php/Four_Color_Theorem_Applied_to_3D_Objects
One thing to think about is how you can be sure that you're getting the minimum number of colors required for any map over a 3-D surface. For example, you know you can color the faces of a cube with three colors, so that opposite sides are the same color and adjacent sides are different colors, but what happens if you divide all those sides into sub-sections like those in the 2-D version of the theorem? Try to really challenge your findings surrounding the minimum number of colors needed for any shape.
-Diana (18:14 3/4/12)
Sorry, my work graphed is not coming up at all... but there is a table with the work in it. So if you want to see the graph, I guess ask me for it.
27 March 2012- Sorry, i have forgotten really to write on this... but anyways, i've been trying to put my table and graph of my work in, but its not really working. i do, however, have a slight pattern that i've found... in 3D objects, the minimum number of colors used is higher than in the 2D objects EXCEPT for the cylinder. i think this is so because it has a curved side, not flat sides like all the other shapes. i also think this happened because the curved side can touch all the other faces, but in the other objects, most sides couldn't touch all the other ones, so there would be a repeat colored side. idk if that makes sense really, but i get it hahaa
http://mathforum.org/mathimages/index.php/Torus this is the link to the tori, which actually needs more than 4 colors to cover its surface when set up a certain way (: juss for extra stuff to look at
12 April 2012- I think all of ya'll know what happened over spring break, so I didn't really have any time to go on my page and work on it. I have been writing stuff on paper to put on and that's what i'm doing right now. I have some more stuff on graph theory to explain as well as a few patterns. Hopefully ya'll can look past this large jump from the last time, this was sincerely the first time I could get on since then.
Ok. So I have put up a lot of work. I cannot get any pictures or charts and graphs into my page and i greatly need that so if anyone can help me please do. I have looked it up on WikiTricks but the format doesn't want to work on my page :/ its a mess. The pictures don't even show up.
13 April 2012- So I am almost finished. I will go through all my work and see if I can put anything else in, and where. I bet there will be more stuff coming in.... but I am basically done!!! I figured out how to put the pictures in. Thanks to Anea (: I wrote in the math area about bipartite graphs and graph theory as well as what I noticed in my project. Please look at my page if you can and write some comments for me if you have any. Thanks!!
16 April 2012- I added stuff into the why its interesting section about the torus because that is the only known object in that certain layout of regions that needs more than 4 colors to cover the surface... that is, it needs 7 colors
17 April 2012- I think I am pretty much done. I don't have much to do, so I am just gonna proof read today I guess.
DONE!!!!!!!!!! THANK GOODNESS. THE PAGE IS COMPLETE..... I HOPE (: