#geometryblr

I don't know if mathblr is a thing (it probably is) but I figured the following should go out of my head so that it exists even if I forget it.

The following is a proof by contradiction that shows that the 5 color map theory has no equivalent with 3d reigons of space - that is to say, infinitely many solids can all touch eachother on at least one surface.

I use real world objects in this proof because they're easier to communicate than abstract terms, but I'm really just using those to refer to divisions of space. This is why I can start with:

So, say you have an infinitely long plank of wood. It's two inches wide, half an inch thing, and infinitely long.

Now say you take a drill, and drill a hole 1 inch wide down through the center of the width of the board, and insert an infinitely long 1 inch dowel into it, so it's perpendicular to the board, and extends infinitely above and below the board.

Now say you got a second infinitely long board an a second infinitely long dowel, and drill an extra hole next to the dowel in each board, so you can make it so the boards can be stacked on top of eachother without removing the dowels.

Now say you added a third board and dowel combo, and drilled the extra holes.

Each board has a hole for each dowel, so each board can touch each dowel. If we fuse each board with one dowel, now each board and dowel combo is touching each other board and dowel combo.

Now say you add dowel and board pairs infinitely many times.