Re: [math-fun] Strip of squares in space puzzle (new & improved) (final version?)
(Absolutely final version?): As always (at least that part was right) I need to correct what I wrote. This time I want to delete (what was) rule 5.: ----- There may exist additional intersections between pairs of squares of X, as long as these are only along common vertices ----- because, my role isn't to tell anyone what may or may not happen, and *besides* with the new rule 5. (essentially forbidding triple intersections) I don't even know if they are possible globally, though I strongly suspect so. The point is, for any 3 squares P, Q, R in order, they are always congruent to what I shall call the squiggle: 3 unit grid squares of which A \int B is one edge of B, and B \int C is an *adjacent* edge of B, such that A, B, and C together *do not* form a "cube corner configuration": the SSS has no CCCs. —Dan ----- As always, I need to adjust the wording of a previous post, the SSS puzzle (corrected version below). I neglected to add one essential condition. So I will restate the whole thing from scratch with better numbering. The only actual difference is the addition of 4., which I had accidentally left out — sorry! — and question C. Strip of squares in space puzzles (new & improved): --------------------------------------------------- Let a "strip of squares in space" (SSS) satisfy these conditions: 1. An SSS X is a union, of some collection of unit squares in R^3, with all vertices having integer coordinates. 2. Each square Q in X intersects exactly 2 other squares (Q-, Q+) in the strip along *entire edges*. 3. The edges in 2. are *adjacent edges* in Q. 4. The 3 squares Q, Q-, Q+ mentioned in 2. lie in mutually perpendicular planes. 5. 6. No three squares share a common vertex. PUZZLES: ------- A. What is the smallest number of squares in an SSS that is topologically a cylinder, if possible? B. What is the smallest number of squares in an SSS that is topologically a Möbius band, if possible? C. What is the smallest number of squares in an SSS that is knotted? (I still don't know the answers.) —Dan ----- -----
These rules seem contradictory. Let's suppose that the first square, A, is in the xy plane, at (0,0,0),(0,1,0), (1,0,0), (1,1,0), and one of the squares that intersects it, B is at (0,0,0), (0,1,0), (0,1,1), (0,0,1).. These intersect on the edge from (0,0,0) to (0,1,0). By rule 2, there is another square, C, adjacent to B, and it must intersect B on an edge adjacent to the one from (0,0,0) to (0,1,0), that is, C contains either the edge from (0,0,0) to (0,0,1), or the edge from (0,1,0) to (0,1,1). But in the former case, A, B, and C intersect at (0,0,0), while in the later case, A, B, and C intersect at (0,1,0). So rule 6 is violated in either case. Maybe rule 6 should say that no *four* squares share a common vertex? Andy On Sat, Mar 17, 2018 at 11:38 PM, Dan Asimov <dasimov@earthlink.net> wrote:
(Absolutely final version?):
As always (at least that part was right) I need to correct what I wrote.
This time I want to delete (what was) rule 5.: ----- There may exist additional intersections between pairs of squares of X, as long as these are only along common vertices ----- because, my role isn't to tell anyone what may or may not happen, and *besides* with the new rule 5. (essentially forbidding triple intersections) I don't even know if they are possible globally, though I strongly suspect so.
The point is, for any 3 squares P, Q, R in order, they are always congruent to what I shall call the squiggle: 3 unit grid squares of which A \int B is one edge of B, and B \int C is an *adjacent* edge of B, such that A, B, and C together *do not* form a "cube corner configuration": the SSS has no CCCs.
—Dan
----- As always, I need to adjust the wording of a previous post, the SSS puzzle (corrected version below).
I neglected to add one essential condition. So I will restate the whole thing from scratch with better numbering. The only actual difference is the addition of 4., which I had accidentally left out — sorry! — and question C.
Strip of squares in space puzzles (new & improved): --------------------------------------------------- Let a "strip of squares in space" (SSS) satisfy these conditions:
1. An SSS X is a union, of some collection of unit squares in R^3, with all vertices having integer coordinates.
2. Each square Q in X intersects exactly 2 other squares (Q-, Q+) in the strip along *entire edges*.
3. The edges in 2. are *adjacent edges* in Q.
4. The 3 squares Q, Q-, Q+ mentioned in 2. lie in mutually perpendicular planes.
5. 6. No three squares share a common vertex.
PUZZLES: ------- A. What is the smallest number of squares in an SSS that is topologically a cylinder, if possible?
B. What is the smallest number of squares in an SSS that is topologically a Möbius band, if possible?
C. What is the smallest number of squares in an SSS that is knotted?
(I still don't know the answers.)
—Dan ----- -----
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Andy Latto -
Dan Asimov