One might pose similar questions concerning unit line segments, or unit cubes. When in possession of a supply of Lego bricks, I have a habit of constructing a minimal (?) knotted cubical "canal surface" (as it were). The infant owners remain invariably unimpressed by my virtuosity, setting to work immediately to dismantle the artwork with gusto. Serve 'em right if they turn into unappreciated mathematicians too ... WFL On 3/17/18, Dan Asimov <dasimov@earthlink.net> wrote:
Let a "strip of squares in space" (SSS) satisfy these conditions:
1. An SSS X is a union of a collection of unit squares in R^3, with all vertices having integer coordinates.
2. Each square Q in X intersects exactly 2 other squares in the strip along *entire edges*, and these are *adjacent edges* in Q.
3. There may exist additional intersections between pairs of squares of X, as long as these are only along common vertices.
Let the "size" of X be how many squares are in X.
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?
(Actually I shouldn't call it a puzzle, since I don't know the answers.)
—Dan
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