Emily Merrill
March 3, 2011

I constructed an odd cycle (colored yellow) in which each brick was an equal distance from one specific other brick in the complex, like what Joe uploaded as a hand-drawn picture. It's pretty complicated, and I haven't annotated things well, but I figured I'd get it out to you faster this way. I may re-upload a version with more stuff (or anything) annotated later on. The first version has two color schemes - first red/blue close to the odd cycle, and then pink/purple further away (pink=red, purple=blue, and the last red/first pink is colored yellow) - this was purely for me to help keep track of them as I was bringing the loops together and counting. I left it in because it might be helpful to other people in confirming that the distance to each brick in the odd cycle is equal. The next block of code colors it red/blue in a standard way.

MakeBox[{x_,y_,z_},{dx_,dy_,dz_}]:=Cuboid[{x,y,z},{x,y,z}+{dx,dy,dz}];

topcycleA={Yellow,
MakeBox[{0,0,0},{1,1,1}],(* 1 *) 
MakeBox[{1,0,0},{1,1,1}],(* 2 *)
MakeBox[{2,0,0},{1,1,1}],(* 3 *)
MakeBox[{2,0,1},{1,1,1}],(* 4 *)
MakeBox[{2,0,2},{1,1,1}],(* 5 *)
MakeBox[{2,0,3},{1,1,1}],(* 6 *)
MakeBox[{1,0,3},{1,1,1}],(* 7 *)
MakeBox[{0,0,3},{1,1,1}],(* 8 *)
MakeBox[{0,0,1},{1,1,2}]    (* 9 *)
};
cycle1A={
Blue,MakeBox[{0,1,0},{1,1,1}],Red,MakeBox[{0,2,0},{1,1,1}]
};
cycle2A={
Blue,MakeBox[{1,0,-1},{1,1,1}],Red,MakeBox[{1,0,-2},{1,1,1}],Blue,MakeBox[{1,1,-2},{1,2,1}],Red,MakeBox[{1,3,-2},{1,1,1}]
};
cycle3A={
Blue,MakeBox[{2,1,0},{1,1,1}],Red,MakeBox[{2,2,0},{1,1,1}]
};
cycle4A={
Blue,MakeBox[{3,0,1},{1,1,1}],Red,MakeBox[{4,0,1},{1,1,1}]
};
cycle5A={
Blue,MakeBox[{2,1,2},{1,.5,1}],Red,MakeBox[{2,1.5,2},{1,.5,1}],Blue,MakeBox[{2,2,2},{1,.5,1}],Red,MakeBox[{2,2.5,2},{1,.5,1}],Blue,MakeBox[{2,3,2},{1,1,1}]
};
cycle6A={
Blue,MakeBox[{3,0,3},{1,1,1}],Red,MakeBox[{4,0,3},{1,1,1}]
};
cycle7A={
Blue,MakeBox[{1,1,3},{1,1,1}],Red,MakeBox[{1,2,3},{1,1,1}],Blue,MakeBox[{1,2,4},{1,1,1}],Red,MakeBox[{0,2,4},{1,1,1}],Blue,MakeBox[{-1,2,4},{1,1,1}]
};
cycle8A={
Blue,MakeBox[{0,0,4},{1,1,1}],Red,MakeBox[{-.5,0,4},{.5,1,1}],Blue,MakeBox[{-1,0,4},{.5,1,1}]
};
cycle9A={
Blue,MakeBox[{-1,0,1},{1,1,2}],Red,MakeBox[{-2,0,1},{1,1,2}],Blue,MakeBox[{-2,0,3},{1,1,1}]
};
cycle13A={Blue,MakeBox[{1,2,0},{1,1,1}],Red,MakeBox[{1,3,0},{1,1,1}]
};
cycle132A={Blue,MakeBox[{1,3,-1},{1,1,1}],Yellow, (*Pink,*)MakeBox[{1,4,-1},{1,1,1}],Purple,MakeBox[{1,5,-1},{1,1,1}],Pink,MakeBox[{1,5,0},{1,1,.5}],Purple,MakeBox[{1,5,.5},{1,1,.5}],Pink,MakeBox[{1,5,1},{1,1,.5}],Purple,MakeBox[{1,5,1.5},{1,1,.5}]
};  (*FINAL 3*)
cycle46A={Blue,MakeBox[{4,0,2},{1,1,1}],Red,MakeBox[{4,1,2},{1,2,1}],Blue,MakeBox[{4,3,2},{1,1,1}]
};
cycle465A={Yellow,(* Pink,*)MakeBox[{3,3,2},{1,1,1}],Purple,MakeBox[{3,4,2},{1,1,1}],Pink,MakeBox[{3,5,2},{1,1,1}],Purple,MakeBox[{2.7,5,2},{.3,1,1}],Pink,MakeBox[{2.3,5,2},{.4,1,1}],Purple,MakeBox[{2,5,2},{.3,1,1}]
};   (*FINAL 3*)
cycle89A={Red,MakeBox[{-2,0,4},{1,1,1}],Blue,MakeBox[{-2,1,4},{1,1,1}]
};
cycle897A={Yellow, (*Pink,*)MakeBox[{-2,2,4},{1,1,1}],Purple,MakeBox[{-2,3,4},{1,2,1}],Pink,MakeBox[{-2,5,4},{1,1,1}],Purple,MakeBox[{-1,5,4},{2,1,1}],Pink,MakeBox[{1,5,4},{1,1,1}],Purple,MakeBox[{1,5,3},{1,1,1}]
};   (*FINAL 3*)
allcycleA={Orange,MakeBox[{1,5,2},{1,1,1}]};

Graphics3D[{topcycleA,cycle1A,cycle2A,cycle3A,cycle4A,cycle5A,cycle6A,cycle7A,cycle8A,cycle9A,cycle13A,cycle132A,cycle46A,cycle465A,cycle89A,cycle897A,allcycleA}]

Blue, Red, Blue, Red, Blue, YELLOW(red/pink), purple, pink, purple, pink, purple, ORANGE
  The orange is d=0, and then purple/blues are odds and pinks/reds are evens. Each block in the (odd) 7-cycle at the bottom is touching a block with d=11, so therefore each block in that cycle has d=12.

Now, following, I have a copy with all of the colors consistent (and moderately transparent):

topcycle={Yellow,
MakeBox[{0,0,0},{1,1,1}],(* 1 *) 
MakeBox[{1,0,0},{1,1,1}],(* 2 *)
MakeBox[{2,0,0},{1,1,1}],(* 3 *)
MakeBox[{2,0,1},{1,1,1}],(* 4 *)
MakeBox[{2,0,2},{1,1,1}],(* 5 *)
MakeBox[{2,0,3},{1,1,1}],(* 6 *)
MakeBox[{1,0,3},{1,1,1}],(* 7 *)
MakeBox[{0,0,3},{1,1,1}],(* 8 *)
MakeBox[{0,0,1},{1,1,2}]    (* 9 *)
};
cycle1={
Blue,MakeBox[{0,1,0},{1,1,1}],Red,MakeBox[{0,2,0},{1,1,1}]
};
cycle2={
Blue,MakeBox[{1,0,-1},{1,1,1}],Red,MakeBox[{1,0,-2},{1,1,1}],Blue,MakeBox[{1,1,-2},{1,2,1}],Red,MakeBox[{1,3,-2},{1,1,1}]
};
cycle3={
Blue,MakeBox[{2,1,0},{1,1,1}],Red,MakeBox[{2,2,0},{1,1,1}]
};
cycle4={
Blue,MakeBox[{3,0,1},{1,1,1}],Red,MakeBox[{4,0,1},{1,1,1}]
};
cycle5={
Blue,MakeBox[{2,1,2},{1,.5,1}],Red,MakeBox[{2,1.5,2},{1,.5,1}],Blue,MakeBox[{2,2,2},{1,.5,1}],Red,MakeBox[{2,2.5,2},{1,.5,1}],Blue,MakeBox[{2,3,2},{1,1,1}]
};
cycle6={
Blue,MakeBox[{3,0,3},{1,1,1}],Red,MakeBox[{4,0,3},{1,1,1}]
};
cycle7={
Blue,MakeBox[{1,1,3},{1,1,1}],Red,MakeBox[{1,2,3},{1,1,1}],Blue,MakeBox[{1,2,4},{1,1,1}],Red,MakeBox[{0,2,4},{1,1,1}],Blue,MakeBox[{-1,2,4},{1,1,1}]
};
cycle8={
Blue,MakeBox[{0,0,4},{1,1,1}],Red,MakeBox[{-.5,0,4},{.5,1,1}],Blue,MakeBox[{-1,0,4},{.5,1,1}]
};
cycle9={
Blue,MakeBox[{-1,0,1},{1,1,2}],Red,MakeBox[{-2,0,1},{1,1,2}],Blue,MakeBox[{-2,0,3},{1,1,1}]
};
cycle13={Blue,MakeBox[{1,2,0},{1,1,1}],Red,MakeBox[{1,3,0},{1,1,1}]
};
cycle132={Blue,MakeBox[{1,3,-1},{1,1,1}],(*Yellow,*)Red,MakeBox[{1,4,-1},{1,1,1}],Blue,MakeBox[{1,5,-1},{1,1,1}],Red,MakeBox[{1,5,0},{1,1,.5}],Blue,MakeBox[{1,5,.5},{1,1,.5}],Red,MakeBox[{1,5,1},{1,1,.5}],Blue,MakeBox[{1,5,1.5},{1,1,.5}]
};  (*FINAL 3*)
cycle46={Blue,MakeBox[{4,0,2},{1,1,1}],Red,MakeBox[{4,1,2},{1,2,1}],Blue,MakeBox[{4,3,2},{1,1,1}]
};
cycle465={(*Yellow,*)Red,MakeBox[{3,3,2},{1,1,1}],Blue,MakeBox[{3,4,2},{1,1,1}],Red,MakeBox[{3,5,2},{1,1,1}],Blue,MakeBox[{2.7,5,2},{.3,1,1}],Red,MakeBox[{2.3,5,2},{.4,1,1}],Blue,MakeBox[{2,5,2},{.3,1,1}]
};   (*FINAL 3*)
cycle89={Red,MakeBox[{-2,0,4},{1,1,1}],Blue,MakeBox[{-2,1,4},{1,1,1}]
};
cycle897={(*Yellow,*)Red,MakeBox[{-2,2,4},{1,1,1}],Blue,MakeBox[{-2,3,4},{1,2,1}],Red,MakeBox[{-2,5,4},{1,1,1}],Blue,MakeBox[{-1,5,4},{2,1,1}],Red,MakeBox[{1,5,4},{1,1,1}],Blue,MakeBox[{1,5,3},{1,1,1}]
};   (*FINAL 3*)
allcycle={Yellow,MakeBox[{1,5,2},{1,1,1}]};

Graphics3D[{Opacity[0.6],topcycle,cycle1,cycle2,cycle3,cycle4,cycle5,cycle6,cycle7,cycle8,cycle9,cycle13,cycle132,cycle46,cycle465,cycle89,cycle897,allcycle}(*, Axes->True, Boxed->False*)]