Comments on: What shape petit fours can I cut from a sheet cake s.t....
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Comments on Ask MetaFilter post What shape petit fours can I cut from a sheet cake s.t....Fri, 28 Sep 2018 19:39:26 -0800Fri, 28 Sep 2018 20:02:02 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Question: What shape petit fours can I cut from a sheet cake s.t....
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st
(1) Each cut is a straight line across an entire piece of cake, but not necessarily across the entire pan. (2) Cake is wasted only at the edges of the pan. (3) All the resulting petit fours come out the same shape and size (except for the wasted bits at the edges). Equilateral triangles work, as do isosceles triangles. Rectangles work, either in a grid or staggered like brickwork. Hexagons do not work. I think I'm looking for tilings of the plane by one convex polygon, but not necessarily a regular, edge-to-edge, or even periodic tiling. Help?post:ask.metafilter.com,2018:site.326902Fri, 28 Sep 2018 19:39:26 -0800meaty shoe puppetmathbakingBy: Eyebrows McGee
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4707972
Diamonds, parallelograms, trapezoids all work.comment:ask.metafilter.com,2018:site.326902-4707972Fri, 28 Sep 2018 20:02:02 -0800Eyebrows McGeeBy: aws17576
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4707976
All triangles work. That's the only solution I can think of that hasn't been posted yet.<br>
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Can we use condition (1) to show that the average internal angle of each piece is 90° or less? If that's true, it would limit the possible polygons to triangles and quadrilaterals.comment:ask.metafilter.com,2018:site.326902-4707976Fri, 28 Sep 2018 20:24:10 -0800aws17576By: ktkt
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4707978
I am interpreting your condition 1 to mean you only want convex polygons.<br>
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Can you say what your issue with regular hexagons is? They seem to fit your conditions, though they would be very annoying to actually cut. <br>
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There are also lots of pentagon tilings that might suit you.comment:ask.metafilter.com,2018:site.326902-4707978Fri, 28 Sep 2018 20:44:26 -0800ktktBy: zompist
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4707979
You can get hexagons if you relax condition (3): <br>
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<a href="https://morphingtiling.files.wordpress.com/2010/12/reg08.gif">https://morphingtiling.files.wordpress.com/2010/12/reg08.gif</a><br>
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By relaxing (1) you could get L shapes:<br>
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<a href="https://uva.onlinejudge.org/external/10/p3275.jpg">https://uva.onlinejudge.org/external/10/p3275.jpg</a>comment:ask.metafilter.com,2018:site.326902-4707979Fri, 28 Sep 2018 20:55:30 -0800zompistBy: oceano
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4707995
Parallelogram?comment:ask.metafilter.com,2018:site.326902-4707995Fri, 28 Sep 2018 22:14:55 -0800oceanoBy: aws17576
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4708004
<em>Can you say what your issue with regular hexagons is?</em><br>
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I'm not the original poster, but I think these violate rule 1. If you're going for a honeycomb, you can't start with a straight cut all the way across the cake. My interpretation of rule 1 is that each cut must begin and end either at an edge of the pan or at an edge created by a previous cut.<br>
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Returning to the problem, I believe I can flesh out my previous comment about angles, though rule 2 poses a complication. I'll start by making the argument under the assumption that no cake can be discarded, then I'll comment on how to accommodate rule 2.<br>
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So: Assuming no cake gets discarded, I claim that the average of all interior angles of all pieces of cake must always be 90° or less. This is true at the beginning, when you have a single rectangle. Each subsequent cut creates new angles at its endpoints, which may land either at the corner of an existing piece or in the middle of an edge. In the former case, the total number of interior angles of all pieces increases by one, but the total of the angles does not, so the average angle decreases. In the latter case, two new angles are created which add up to 180°. The average of the new angles is 90°, and the average of the existing angles is 90° or less, so the overall average remains at or below 90°. This proves my claim.<br>
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As a result, if the pieces end up all being the same shape, then that shape must be a triangle or quadrilateral, since these are the only polygons whose average angle is 90° or less. (The sum of angles of an n-sided polygon is (n–2)*180°.)<br>
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Now, as for rule 2: I believe some clarification may be required, since I don't see anything in your question that technically rules out cutting away whatever you want at the edges to make a single petit four of any convex shape. But it's pretty clear that the intention is not to allow that. The mention of tilings of the plane suggests to me that you are looking for solutions with a fixed polygon that could be carried out on an arbitrarily large cake, producing an arbitrarily large number of petits fours. In that case, the boundary becomes relatively insignificant for large cakes. So if you have a large enough cake, you shouldn't be able to discard enough triangles from the edges to raise the average angle of the remaining pieces to 108° (which is what it would need to be to allow a solution with pentagons or higher).<br>
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If this reasoning holds up, then the only possible solutions are triangles and quadrilaterals, but it remains to determine which quadrilaterals work. <a href="https://latex.artofproblemsolving.com/t/u/e/tuevdwtl.png?time=1538199680621">Here's a solution with a quadrilateral that is not a parallelogram or trapezoid.</a>comment:ask.metafilter.com,2018:site.326902-4708004Fri, 28 Sep 2018 22:44:17 -0800aws17576By: suedehead
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4708010
If you use this, you could get hexagons: Ateco Stainless Steel Hexagon Cutter https://www.amazon.com/dp/B001BQXKQS/ref=cm_sw_r_cp_api_G1XRBbD5HM99C<br>
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Otherwise I think there's more patterns that are createable if you cut, then remove pieces, then cut, remove pieces, etc.comment:ask.metafilter.com,2018:site.326902-4708010Fri, 28 Sep 2018 23:39:01 -0800suedeheadBy: Gilgamesh's Chauffeur
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4708252
I assumed the actual point was "shapes that are practical to cut" and not a Putnam exam problem. If that's the case, the most aesthetically pleasing solution is parallelograms (as mentioned previously). Evidence: traditional shape for cutting baklava.<br>
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I would suggest that cutting into various shapes is better, though, because there are always some deviant individuals that want "just a little piece" vs. the normal folks that want a huge one...comment:ask.metafilter.com,2018:site.326902-4708252Sat, 29 Sep 2018 18:34:42 -0800Gilgamesh's ChauffeurBy: meaty shoe puppet
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4708296
<em>I assumed the actual point was "shapes that are practical to cut" and not a Putnam exam problem.</em><br>
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That's an entirely reasonable assumption to make. And while none of us have sat the Putnam, I got as far as the AIME, my wife did the AMC, and I would love to serve a selection of petit fours with proof that I had exhausted the space of petit four shapes satisfying some criteria.comment:ask.metafilter.com,2018:site.326902-4708296Sat, 29 Sep 2018 22:04:52 -0800meaty shoe puppetBy: cortex
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4708403
If you're not gonna hew to strict tiling but want to stick to long, consistent cuts across the board, zompist's hexagon slices with waste triangles suggestion would be my first thought (and you wouldn't have to do regular hexagons, if you wanted something a little more unconventional with the skew). Another take on that, still not tiling but with slightly less waste and a little more unusual zazzle: remove half of the cuts on one of the three axes of that hexagon cut and <a href="https://twitter.com/joshmillard/status/1046430747051999232">do irregular pentagons</a>.comment:ask.metafilter.com,2018:site.326902-4708403Sun, 30 Sep 2018 09:31:53 -0800cortexBy: aws17576
http://ask.metafilter.com/326902/What-shape-petit-fours-can-I-cut-from-a-sheet-cake-st#4708526
In the spirit of Gilgamesh's Chauffeur's comment about people preferring different sizes, <a href="https://latex.artofproblemsolving.com/f/p/w/fpwgcjtw.png?time=1538350644176">here's a pattern that obeys rules 1 and 2 and produces pieces of the same shape but three different sizes</a>. Or, <a href="https://latex.artofproblemsolving.com/b/x/w/bxwzhjuo.png?time=1538350585708">infinitely many sizes</a>.comment:ask.metafilter.com,2018:site.326902-4708526Sun, 30 Sep 2018 16:40:17 -0800aws17576