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Tagged with tiling packing-and-covering
8 questions
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Which pentagon gives least packing density?
We extend Which convex pentagon gives least packing density? by going from convex pentagons to general ones.
Question: Which pentagon gives the least packing density on the Euclidean plane?
Note: All ...
- 5,979
1
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How to fill a rectangle with smaller rectangles of given sizes?
I have a problem. I try to find an algorithm to fill up a given rectangle with smaller ones. Something like in this picture:
I know the size of the big rectangle, the size of all the little ...
- 11
2
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1
answer
164
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Packing densities of non-centrally symmetric planar convex regions
Reference: https://en.wikipedia.org/wiki/Smoothed_octagon
Background: The smoothed octagon is conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex ...
- 5,979
1
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Which polygons tessellate the hyperbolic plane?
The packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing.
It is well known that in Euclidean geometry, all triangles and all ...
- 5,979
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On Covering a Planar Region with Copies of a Tile of Different Shape
Background: Consider trying to cover the largest possible scaled copy of a planar region $C$ with specified shape with n instances of a tile $T$ of specified shape and size. Several families of this ...
- 5,979
4
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146
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Tiling squares with oblongs
An oblong is a rectangle whose width and length are consecutive integers: 1x2, 2x3, 3x4, etc. Does N exist such that it is possible to split the first N oblongs into 2 or more non-intersecting sets so ...
- 3,707
9
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282
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Thinnest covering of the plane by regular pentagons
Q. Is it known what is the thinnest covering of the infinite plane by regular pentagons?
By covering I mean every point of the plane is covered.
By thinnest I mean the proportion of the plane covered ...
- 151k
8
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1
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Translative packing constant strictly larger than lattice packing constant
Simply put, my question is this: what is the smallest dimension, if any,
where we can know for sure that a convex body exists whose translative
packing constant is strictly larger than its lattice ...
- 5,971