[math-fun] Strip of squares in space puzzle
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
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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Great question. I recall Gardner wrote about minimal knotted loops with unit integer segments. The minimal cubical torus has 6 cubes (but an infinitesimal hole), 8 will produce a simple square torus with a real hole. I worked on the minimum cubical mobius (twisted) loop, which has 10 cubes. Of course 4 squares makes a cylinder. The Mobius strip I can do in 9 squares with a genuine hole. Can anyone do less? On Sat, Mar 17, 2018 at 4:19 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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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participants (3)
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Dan Asimov -
Fred Lunnon -
Scott Kim