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Topic: Tiling a circular hall with circular tiles (Read 2017 times) |
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JocK
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Tiling a circular hall with circular tiles
« on: Jul 24th, 2004, 12:48pm » |
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You need to tile a circular hall. You have contacted a company that can construct circular tiles of any specified diameter up to giant-sized tiles with diameter equal to the radius of your hall. The sales manager of this company wants to sell a lot of tiles and makes you a special offer that is only valid in case you ensure that you cover at least 90.7% (the hexagonal packing fraction = sqrt(3)[pi]/6) of your hall with their tiles. Expecting that this requirement will cause you to order many identical tiny tiles, he offers to deliver the tiles for a fixed price of 1$ per tile regardless of the size, even if each tile has a unique specified size.
You are very tight on budget, and you want to minimize your cost. Based on the above offer, what minimum amount do you need to pay the tile company to ensure you can tile your hall? What are the sizes of the tiles you are going to order? Are you sure one cannot tile the hall with one less tile?
J CK
PS. You can't break the tiles, and also the tiles are not allowed to overlap!
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| « Last Edit: Jul 24th, 2004, 1:31pm by JocK » |
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Jack Huizenga
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Re: Tiling a circular hall with circular tiles
« Reply #1 on: Jul 24th, 2004, 2:55pm » |
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Sorry, but this ridde doesn't make any sense. You haven't attributed a cost to the tiles besides the delivery cost, and you haven't given a reason why the "one big tile" solution isn't optimal...
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JocK
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Re: Tiling a circular hall with circular tiles
« Reply #2 on: Jul 24th, 2004, 4:48pm » |
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on Jul 24th, 2004, 2:55pm, Jack Huizenga wrote:| Sorry, but this ridde doesn't make any sense. You haven't attributed a cost to the tiles besides the delivery cost, |
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??
on Jul 24th, 2004, 2:55pm, Jack Huizenga wrote:| and you haven't given a reason why the "one big tile" solution isn't optimal... |
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It says: "any specified diameter up to giant-sized tiles with diameter equal to the radius of your hall". So one tile would cover only 25% of the hall. Not even close to the required ~ 90.7%.
If the story is difficult to follow; forget about it:
What is the minimum number of non-overlapping circular disks that you need to fit inside a circle so as to fill it to at least hexagonal close-packing density (sqrt(3)[pi]/6), under the constraint that none of the disks can have a diameter exceeding the circle radius.
JCK
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| « Last Edit: Jul 24th, 2004, 4:58pm by JocK » |
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Jack Huizenga
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Re: Tiling a circular hall with circular tiles
« Reply #3 on: Jul 24th, 2004, 4:57pm » |
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Oh ok sorry about that, I misread the question.
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SWF
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Re: Tiling a circular hall with circular tiles
« Reply #4 on: Jul 26th, 2004, 9:41pm » |
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After trying a couple of possibilities, best I could find so far is 20:
Start with three mutually tangent circles of same diameter that all are inside and tangent to the bigger circle. Then keep adding the largest circle that fits in the remaining space until 20 circles.
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JocK
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Re: Tiling a circular hall with circular tiles
« Reply #5 on: Jul 27th, 2004, 1:04pm » |
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If I'm not mistaken, the 20 biggest cicrles of this Apollonian packing together exceed 90 % packing density, but they don't reach the 90.69 % hexagonal packing density. Even for 22 circles only 90.67 % is reached (still short by 0.02%...!).
This all is relevant as the best result I found so far is 22 circles.
JCK
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Virat Agarwal
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Re: Tiling a circular hall with circular tiles
« Reply #6 on: Oct 7th, 2004, 5:22am » |
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Yes there is also available the circular tiles in natural stones which can be interlocked and can be laid down to any area on the floor.these tiles are developed in the natural stones like sandstones,limestones and granites
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John_Gaughan
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Re: Tiling a circular hall with circular tiles
« Reply #7 on: Oct 7th, 2004, 5:39am » |
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on Oct 7th, 2004, 5:22am, Virat Agarwal wrote:| Yes there is also available the circular tiles in natural stones which can be interlocked and can be laid down to any area on the floor.these tiles are developed in the natural stones like sandstones,limestones and granites |
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Nonoverlapping circles can only touch at a tangent point, leaving space around that point. Interlocking is out of the question. Polygons, especially regular polygons such as squares and hexagons, can interlock quite well.
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