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You are not logged in. #1 20140125 02:59:03
four color cubea four dimensional hypercube might be realized in 3D with a cube with the corners colored with four colors, maybe. igloo myrtilles fourmis #2 20140130 00:38:03
Re: four color cubeThe rumor is that there are 402 ways to color a hypercubes corners with two colors. igloo myrtilles fourmis #3 20140130 05:49:47
Re: four color cubeI have started out with a depiction of the 22 shapes you igloo myrtilles fourmis #4 20140130 06:37:48
Re: four color cubeI was just pondering what I am doing and it seems to me, just casually that there are going to be more than Last edited by John E. Franklin (20140130 07:06:29) igloo myrtilles fourmis #5 20140130 07:17:20
Re: four color cubeSo let's see. I am trying to reduce a 4d, length [0,1] on each axis, to a 3d cube with some rules associated with it, essentially.
That always bugged me, because isn't that the fun of math, running down the wrong route!! Last edited by John E. Franklin (20140130 07:20:01) igloo myrtilles fourmis #6 20140130 09:13:52
Re: four color cubeCode:say you have a 2booleanvariable truth table expressed with a square. Here is the square: tt tf so three outputs are true and one is false. the inputs are the x and y axes along the edge of the square. x and y inputs vary from 0 to 1. Now wikipedia does a pretty bad job explaining Karnaugh maps. I also do a bad job because they are not easy to teach without some interaction and a good lecture hall and an occasional question. Why should I ramble about Karnaugh maps? The reason why is because maybe the electrical engineers could donate valuable information to the math guys and the mathematicians could develop their boolean algebra a little more if they are lucky. So here is another Karnaugh map in 2variables: AB CD A,B,C,andD are just positions to talk about. Note you can start at any location and build a tree. A /\ B C The tree then closes up to D. Basically in my mind, we are taking a 2d object, a square, and making it into an object that is either 2d, a tree, or 1d, the same tree, depending how you view it. If you view the tree one vertex at a time and move through the tree like a flowchart, following paths, then it might be a 1d object. The reason I mentioned that is because I'm trying to take the 16 vertex graph called a hypercube, and create 3d or 2d trees that I like so I can make comparisons either on paper or otherwise. (again although this is "help me", I hope I'm not breaking rules here) (basically I'm just thinking through my project and thought someone might get interested and post a comment to spawn ideas for me, thanks in advance, but I don't expect that to happen so please just let me ramble since graph theory and boolean algebra are important things to work on. Never mind knot theory! Here is how a 4variable (hypercube) Karnaugh map is set up in 2d for EE majors in the 1980s: 0 1 3 2  0  Z A C B  1  D E G F  3  L M O N  2  H I K J Note that E is at (1,1) input location. Output locations A, D, G, and M are the four locations where the input coordinates vary by only one variable. The reason I believe Maurice 1953 Karnaugh did this is because when you combine terms or locations with a loop drawn around them, the boolean expression naturally get smaller. For example, let's label two boolean variables x and y and make them go horizontally 0 1 3 2 with y as the 1's place and x as the 2's place. Now make z and w go downward with 0 1 3 2 too(, so 00 01 11 10) for z and w making those numbers. Now the column CGOK is when x and y make 3 and x=1 and y=1 or their input values in the corresponding boolean truth table. The row LMON is when z and w are both true or 1. Location E is at (1,1), rightdown or 0101/xyzw so x=0 y=1 z=0 w=1 So for GO, xy both are true (1) and on the vertical z/w are 1/3. zwzw 0111 So for GO the variable z is both 0 and 1, so just drop it out of the equation. And GO has w=1 in both places so use w. So equation for GO is xyw. A boolean truth table has an output value and likewise, the location GO could be filled with two 1's or two 0's if you circle it together. If two 1's, then output value is 1=xyw or OutputValue(x,y,z,w)=xyw. The output value will be one when those three xyw variables multiply to 1, they are all 1. They are actually AND'd together, but multiplying does the same thing. A little more explanation here is needed ofcourse, actually a lot, but I'm just trying to help so you can piece it together from various sources if you are trying to learn it. Below I am trying to show that z is true for half of karnaugh map grid at LMON and HIKJ see parenthesis to the right and binary inputs z,w. Note that w is true for the middle two horizontal rows. From this info, wz=LMON. w OR z = DEFG LMON HIKJ. 0 1 3 2  z,w 0  Z A C B 0 0  1  D E G F \ 0 1  w 3  L M O N \ / 1 1  z 2  H I K J / 1 0 w XOR z = DEFG OR HIKJ. If you don't know this stuff you kind should piece it together from many angles since few people have time to post a comprehensive explanation, but I'll do piecemeal at times to aid a little. Last edited by John E. Franklin (20140130 10:27:53) igloo myrtilles fourmis #7 20140130 10:45:03
Re: four color cubeCode:0 1 3 2  z,w 0  Z A C B 0 0  1  D E G F \ 0 1  w 3  L M O N \ / 1 1  z 2  H I K J / 1 0 LO, EI, MN, MG, OJ, KN, DH all have equations that are identical looking except the variables are switched, but the form is the same. Note LO and MN are disjoint on a row but OJ and KN are diagonal. This is the only flaw I know of with the 2d representatiion over using a 4d hypercube. But some comprimise was needed to make it a 2d array. Another neat thing I learned is that if you rotate the four quadrants of the 4d Kmap, it is still equivalent, but the variables change. So 0 1 3 2  z,w 0  Z A C B 0 0 rotate stuff D Z B F  E A C G 1  D E G F \ 0 1  w M I K O 3  L M O N \ / 1 1 L H J N this rotation creates a similar equation with different variables.  z it is like a "gear"rotation among the four quadrants and helps to realize 2  H I K J / 1 0 shapes that are not easily realized in this 2d format. Notice the gear stuff on the right, I made that up to compensate for the 4d in 2d losses. Last edited by John E. Franklin (20140130 10:47:46) igloo myrtilles fourmis #8 20140201 01:57:29
Re: four color cubeIn graph theory, igloo myrtilles fourmis #9 20140201 03:03:46
Re: four color cubePerhaps this 3d igloo myrtilles fourmis #10 20140201 04:26:26
Re: four color cubeThe Karnuagh map intended originally to Last edited by John E. Franklin (20140223 11:36:20) igloo myrtilles fourmis #11 20140201 05:01:11
Re: four color cubeNow a hypercube of any igloo myrtilles fourmis #12 20140201 12:27:00
Re: four color cubeCan the faces of the hypercube be represented as a planar graph? 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #13 20140203 05:44:36
Re: four color cubeThe following is just info on the cuff with little proven: igloo myrtilles fourmis #14 20140211 07:43:41
Re: four color cubeFollowing from post#3, I have changed the order of the 22 shapes igloo myrtilles fourmis #15 20140216 12:52:19
Re: four color cubeI spent a couple days drawing this out on a pad of paper and then I entered it into the computer. igloo myrtilles fourmis 