Packing rectangles: Does rotation ever help?Box stacking problemPlacing Axis-parallel rectangles on 2-D...
Packing rectangles: Does rotation ever help?
Box stacking problemPlacing Axis-parallel rectangles on 2-D planeUpper bound for tetrahedron packing?A circle packing conjectureDoes list of distances define points uniquely?rational rotation vector and closed curvesSpace packing fraction of tetrahedron and octahedronTranslative packing constant strictly larger than lattice packing constantTiling a square with rectanglesCompose/decompose rotation matrix from/to plane of rotation and angleSome inequalities on chain of circle packing
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Dominic van der Zypen posed an interesting Box stacking problem.
This is a spin-off question.
Let a collection of rectangles $r_1,ldots,r_n$ be given by their side lengths in $mathbb{R}$.
Let $R$ be a rectangle of minimum area enclosing the rectangles arranged
in the plane without overlap (i.e., with disjoint interiors).
Q. Is there an example where not all the rectangles have sides aligned with
the sides of $R$?
In other words, where at least one rectangle's sides are not parallel to the
sides of $R$? Is it ever advantageous to "tilt" one or more rectangles to
achieve a minimal area?
mg.metric-geometry plane-geometry
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
86.7k16240714
$endgroup$
add a comment |
$begingroup$
Dominic van der Zypen posed an interesting Box stacking problem.
This is a spin-off question.
Let a collection of rectangles $r_1,ldots,r_n$ be given by their side lengths in $mathbb{R}$.
Let $R$ be a rectangle of minimum area enclosing the rectangles arranged
in the plane without overlap (i.e., with disjoint interiors).
Q. Is there an example where not all the rectangles have sides aligned with
the sides of $R$?
In other words, where at least one rectangle's sides are not parallel to the
sides of $R$? Is it ever advantageous to "tilt" one or more rectangles to
achieve a minimal area?
mg.metric-geometry plane-geometry
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
86.7k16240714
$endgroup$
6
$begingroup$
I think this should work. Consider 2 rectangles with r1 having dimensions 1x11 and r2 having area 10x100. Then the packings that have aligned sides have enclosing areas at least 1100, but if you "lean" r1 against the end of r2 then you can get an enclosing rectangle with area around 1025.
$endgroup$
– Yosemite Stan
1 hour ago
$begingroup$
There was mention of some research showing how one could pack (1+ delta)n^2 unit squares in a square of side (1 + epsilon)n. Consider searching "Packing squares in a square. There is also "shipping a pool cue" in the diagonal of a packing box. Gerhard "Can't Name Any Names Yet" Paseman, 2019.04.27.
$endgroup$
– Gerhard Paseman
1 hour ago
add a comment |
$begingroup$
Dominic van der Zypen posed an interesting Box stacking problem.
This is a spin-off question.
Let a collection of rectangles $r_1,ldots,r_n$ be given by their side lengths in $mathbb{R}$.
Let $R$ be a rectangle of minimum area enclosing the rectangles arranged
in the plane without overlap (i.e., with disjoint interiors).
Q. Is there an example where not all the rectangles have sides aligned with
the sides of $R$?
In other words, where at least one rectangle's sides are not parallel to the
sides of $R$? Is it ever advantageous to "tilt" one or more rectangles to
achieve a minimal area?
mg.metric-geometry plane-geometry
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
86.7k16240714
$endgroup$
Dominic van der Zypen posed an interesting Box stacking problem.
This is a spin-off question.
Let a collection of rectangles $r_1,ldots,r_n$ be given by their side lengths in $mathbb{R}$.
Let $R$ be a rectangle of minimum area enclosing the rectangles arranged
in the plane without overlap (i.e., with disjoint interiors).
Q. Is there an example where not all the rectangles have sides aligned with
the sides of $R$?
In other words, where at least one rectangle's sides are not parallel to the
sides of $R$? Is it ever advantageous to "tilt" one or more rectangles to
achieve a minimal area?
mg.metric-geometry plane-geometry
mg.metric-geometry plane-geometry
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
86.7k16240714
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
86.7k16240714
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
86.7k16240714
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
86.7k16240714
asked 1 hour ago
Joseph O'RourkeJoseph O'Rourke
86.7k16240714
86.7k16240714
6
$begingroup$
I think this should work. Consider 2 rectangles with r1 having dimensions 1x11 and r2 having area 10x100. Then the packings that have aligned sides have enclosing areas at least 1100, but if you "lean" r1 against the end of r2 then you can get an enclosing rectangle with area around 1025.
$endgroup$
– Yosemite Stan
1 hour ago
$begingroup$
There was mention of some research showing how one could pack (1+ delta)n^2 unit squares in a square of side (1 + epsilon)n. Consider searching "Packing squares in a square. There is also "shipping a pool cue" in the diagonal of a packing box. Gerhard "Can't Name Any Names Yet" Paseman, 2019.04.27.
$endgroup$
– Gerhard Paseman
1 hour ago
add a comment |
6
$begingroup$
I think this should work. Consider 2 rectangles with r1 having dimensions 1x11 and r2 having area 10x100. Then the packings that have aligned sides have enclosing areas at least 1100, but if you "lean" r1 against the end of r2 then you can get an enclosing rectangle with area around 1025.
$endgroup$
– Yosemite Stan
1 hour ago
$begingroup$
There was mention of some research showing how one could pack (1+ delta)n^2 unit squares in a square of side (1 + epsilon)n. Consider searching "Packing squares in a square. There is also "shipping a pool cue" in the diagonal of a packing box. Gerhard "Can't Name Any Names Yet" Paseman, 2019.04.27.
$endgroup$
– Gerhard Paseman
1 hour ago
6
6
$begingroup$
I think this should work. Consider 2 rectangles with r1 having dimensions 1x11 and r2 having area 10x100. Then the packings that have aligned sides have enclosing areas at least 1100, but if you "lean" r1 against the end of r2 then you can get an enclosing rectangle with area around 1025.
$endgroup$
– Yosemite Stan
1 hour ago
$begingroup$
I think this should work. Consider 2 rectangles with r1 having dimensions 1x11 and r2 having area 10x100. Then the packings that have aligned sides have enclosing areas at least 1100, but if you "lean" r1 against the end of r2 then you can get an enclosing rectangle with area around 1025.
$endgroup$
– Yosemite Stan
1 hour ago
$begingroup$
There was mention of some research showing how one could pack (1+ delta)n^2 unit squares in a square of side (1 + epsilon)n. Consider searching "Packing squares in a square. There is also "shipping a pool cue" in the diagonal of a packing box. Gerhard "Can't Name Any Names Yet" Paseman, 2019.04.27.
$endgroup$
– Gerhard Paseman
1 hour ago
$begingroup$
There was mention of some research showing how one could pack (1+ delta)n^2 unit squares in a square of side (1 + epsilon)n. Consider searching "Packing squares in a square. There is also "shipping a pool cue" in the diagonal of a packing box. Gerhard "Can't Name Any Names Yet" Paseman, 2019.04.27.
$endgroup$
– Gerhard Paseman
1 hour ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
@YosemiteStan's example.
$endgroup$
1
$begingroup$
Actually, in the first packing the smaller rectangle should lie on the top of the larger one, giving the intended area of 11*100=1100.
$endgroup$
– Victor Protsak
35 mins ago
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@VictorProtsak: Thanks; corrected.
$endgroup$
– Joseph O'Rourke
31 mins ago
add a comment |
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1 Answer
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$begingroup$
@YosemiteStan's example.
$endgroup$
1
$begingroup$
Actually, in the first packing the smaller rectangle should lie on the top of the larger one, giving the intended area of 11*100=1100.
$endgroup$
– Victor Protsak
35 mins ago
$begingroup$
@VictorProtsak: Thanks; corrected.
$endgroup$
– Joseph O'Rourke
31 mins ago
add a comment |
$begingroup$
@YosemiteStan's example.
$endgroup$
1
$begingroup$
Actually, in the first packing the smaller rectangle should lie on the top of the larger one, giving the intended area of 11*100=1100.
$endgroup$
– Victor Protsak
35 mins ago
$begingroup$
@VictorProtsak: Thanks; corrected.
$endgroup$
– Joseph O'Rourke
31 mins ago
add a comment |
$begingroup$
@YosemiteStan's example.
$endgroup$
@YosemiteStan's example.
edited 31 mins ago
community wiki
2 revs
Joseph O'Rourke
1
$begingroup$
Actually, in the first packing the smaller rectangle should lie on the top of the larger one, giving the intended area of 11*100=1100.
$endgroup$
– Victor Protsak
35 mins ago
$begingroup$
@VictorProtsak: Thanks; corrected.
$endgroup$
– Joseph O'Rourke
31 mins ago
add a comment |
1
$begingroup$
Actually, in the first packing the smaller rectangle should lie on the top of the larger one, giving the intended area of 11*100=1100.
$endgroup$
– Victor Protsak
35 mins ago
$begingroup$
@VictorProtsak: Thanks; corrected.
$endgroup$
– Joseph O'Rourke
31 mins ago
1
1
$begingroup$
Actually, in the first packing the smaller rectangle should lie on the top of the larger one, giving the intended area of 11*100=1100.
$endgroup$
– Victor Protsak
35 mins ago
$begingroup$
Actually, in the first packing the smaller rectangle should lie on the top of the larger one, giving the intended area of 11*100=1100.
$endgroup$
– Victor Protsak
35 mins ago
$begingroup$
@VictorProtsak: Thanks; corrected.
$endgroup$
– Joseph O'Rourke
31 mins ago
$begingroup$
@VictorProtsak: Thanks; corrected.
$endgroup$
– Joseph O'Rourke
31 mins ago
add a comment |
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$begingroup$
I think this should work. Consider 2 rectangles with r1 having dimensions 1x11 and r2 having area 10x100. Then the packings that have aligned sides have enclosing areas at least 1100, but if you "lean" r1 against the end of r2 then you can get an enclosing rectangle with area around 1025.
$endgroup$
– Yosemite Stan
1 hour ago
$begingroup$
There was mention of some research showing how one could pack (1+ delta)n^2 unit squares in a square of side (1 + epsilon)n. Consider searching "Packing squares in a square. There is also "shipping a pool cue" in the diagonal of a packing box. Gerhard "Can't Name Any Names Yet" Paseman, 2019.04.27.
$endgroup$
– Gerhard Paseman
1 hour ago