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Topic: Manhole: Shape matters (Read 4309 times) |
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ERIC H. ZENGULUS
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Manhole: Shape matters
« on: Sep 24th, 2003, 2:33pm » |
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There is a famous microsoft manhole problem.
Why manhole is round shape?
One answer is the cover will not fall into the hole.
This leads me into asking this:
Are there any other shape that will not fall into (itself)?
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| « Last Edit: Sep 24th, 2003, 7:11pm by Icarus » |
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Lightboxes
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How about...
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*****An equalateral triangle? My reasoning is that the shortest side can't fit anywhere WITHIN the equal. triangle area. Any other triangle (one with a short side that will fit slightly within the longest side) is not a solution depending on the lip that is used to actually HOLD the lid in place.
*****A regular pentagon. The longest width is from one vertex to another vertex proven by drawing a line (Y) from one vertex to another, then drawing the perpendicular bisector (I think is what it is called) to the opposite side and the triangle that is created shows that the Y (the hypontenus) is always bigger then one of the sides of a right triangle. Hence, if there is a lip to hold the lid in place, then it cannot pass through.
*****I'm starting to think any regular polygon (not = 4 sides) will work. Maybe even a rhombus depending on the angles?
*****Oh, I almost forgot, and any shape in 2 dimensional space.  Or am I wrong about that?
*****And of course in the pic. An odd number of fins where a line drawn from two fins, end to end, are longer than Z. Or basically, what I'm saying is an non-symetrical shape.
added:
I KNOW I KNOW, an infinite number of shapes from an infinite number of regular polygons with an odd number of vertices (like the pic), as long as the vertices are left alone, AND the distance of one fin point to another fin point is greater than Z!
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| « Last Edit: Sep 24th, 2003, 3:50pm by Lightboxes » |
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ERIC H. ZENGULUS
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Re: New Puzzle: Manhole: Shape matters
« Reply #2 on: Sep 24th, 2003, 4:07pm » |
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forget to say:
the shape must contained in a 2-D plane
and must be convex.
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Icarus
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Re: Manhole: Shape matters
« Reply #4 on: Sep 24th, 2003, 8:20pm » |
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SWF - Eric is asking a different question that has not been directly addressed. Though I find some of it easy to answer (curiously enough I was reading about a class of curves that satisfy this condition easily just last night), others may not, so I can't dispute his placement in this forum yet. (As for the other issue: It's been fixed!)
Lightboxes: None of your specific examples work (at least how I interpret the problem). In particular, Eric has no need to add his "convex" condition, as any non-convex solution is just a variation of a convex one.
Perhaps you are thinking of the hole as being a mathematical cylinder: The same size and shape all the way down. I interpret the hole as being a penetration through thin surface. Thus, there is only one choke point, which if you can get past, things open up again.
Start with that equilateral triangle: Pick up the cover, turn it sideways, swing it around so that its surface is almost up against the side of the hole, and drop. Since the lid's height is [sqrt]3/2 [approx] 0.86, there is plenty of room along the 1 wide edge of the hole for it to slip through. (This is even true of the cylinder-type hole.)
The same is true for any polygon: They all have diagonals that are longer than the "width" of the polygon.
Some terminology: Let L be a finite shape in the plane. The diagonal width of L is the length of the largest line segment entirely contained within L. For each point P of L, define the width of L at P to be the max distance from P to any other element of L. The short diameter of L is the minimum of the widths of L at P for all points P on L.
Generally, a manhole cover whose shape L has short diameter = diagonal width will not fall through its hole. This follows because every point of L has a least one far enough away that it hits the edge going in. If the condition is not satisfied, then there is USUALLY a way to fit it through (I do not know of any exceptions, but I also do not know of any proof).
In the case of your pentagonal shape: Z is the short diameter, not Y. Turning the line for Z a little gives a longer line to provide the diagonal width. This would indicate that it should be possible to put that cover through a slightly smaller hole - and that is indeed the case. Put one spikes through and slide the lid horizontally until you can get the next spike in. Now slide the lid back so that the edge of the hole comes up to the middle of the depression between two spikes. This leaves room at the other end to slip the third spike through. The remaining two spikes follow easily after.
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SWF
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Re: Manhole: Shape matters
« Reply #5 on: Sep 24th, 2003, 10:34pm » |
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on Sep 24th, 2003, 8:20pm, Icarus wrote:| SWF - Eric is asking a different question that has not been directly addressed. |
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Since the question still has not been directly answered in this thread, I will quote from the other thread linked to above:
Question from kenny's post:
Quote:| Question: Are there any other convex shapes, aside from circles, that have the same property, that the cover couldn't be rotated to fit past even an infinitesimal lip? |
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Answer from Yossarian:
Quote:| Actually there is an infinite number of so-called equal width shapes. An example is Releaux shapes. |
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Although, I believe the correct spelling is Reuleaux.
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BNC
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Re: Manhole: Shape matters
« Reply #6 on: Sep 24th, 2003, 11:26pm » |
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A thread with a specific example (2nd last post) is here
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How about supercalifragilisticexpialidociouspuzzler [Towr, 2007]
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James Fingas
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Re: Manhole: Shape matters
« Reply #7 on: Sep 25th, 2003, 5:35am » |
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I'm thinking that a polygon with a hole cut in the middle also works (since it decreases the diagonal width without decreasing the width). But it would be hard to construct a hole with a non-hole in the middle.
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rmsgrey
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Re: Manhole: Shape matters
« Reply #8 on: Sep 26th, 2003, 7:12am » |
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on Sep 25th, 2003, 5:35am, James Fingas wrote:| I'm thinking that a polygon with a hole cut in the middle also works (since it decreases the diagonal width without decreasing the width). But it would be hard to construct a hole with a non-hole in the middle. |
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How about this shape (non-convex): take a regular haxagon, draw a second concentric one slightly smaller with the same orientation and then connect the inside to the outside with a thin strip centered on one of the edges.
Which doesn't work because you can slip enough in to get the gap to the level of the manhole, and then move the gap around at that level to fit the rest in (unless the hole is cylindrical, in which case it should be a solution - I think)
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xpitxbullx
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Re: Manhole: Shape matters
« Reply #9 on: Aug 10th, 2004, 12:29am » |
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Shape matters. If the manhole is round, the center of gravity will always be higher than the hole therefore causing the manhole cover to teeter back to a parallel state with the hole making it easier to adjust to close.
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JocK
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Re: Manhole: Shape matters
« Reply #10 on: Aug 10th, 2004, 2:56pm » |
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Yes, odd-sided Reuleux polygons -- constructed from circular arcs with centres at the vertices of an odd-sided regular polygon, each drawn between the two opposing vertices -- would be perfect manholes.
Including the circle as a limit case of such odd-sided curvilinear polygons, it seems that such polygonic curves are the only convex shapes that satisfy the criteria. But don't know how to prove that.
J CK
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Leo Broukhis
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Re: Manhole: Shape matters
« Reply #11 on: Aug 10th, 2004, 3:46pm » |
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on Sep 25th, 2003, 5:35am, James Fingas wrote:| I'm thinking that a polygon with a hole cut in the middle also works (since it decreases the diagonal width without decreasing the width). But it would be hard to construct a hole with a non-hole in the middle. |
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A non-hole in the middle of a hole is called a pole.
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Icarus
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Re: Manhole: Shape matters
« Reply #12 on: Aug 10th, 2004, 5:51pm » |
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on Aug 10th, 2004, 2:56pm, JocK wrote:| it seems that such polygonic curves are the only convex shapes that satisfy the criteria. |
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Actually there are uncountably many shapes "of constant width" - i.e. from any point on the boundary, the max distance to other points in the shape is constant - which are not "polygonic". I have even seen a procedure for generating such a curve from a large class of starting curves.
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JocK
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Re: Manhole: Shape matters
« Reply #13 on: Aug 11th, 2004, 2:19pm » |
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on Aug 10th, 2004, 5:51pm, Icarus wrote:
Actually there are uncountably many shapes "of constant width" <..> which are not "polygonic". I have even seen a procedure for generating such a curve from a large class of starting curves. |
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Interesting. Do you have a reference/URL?
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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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Icarus
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Re: Manhole: Shape matters
« Reply #14 on: Aug 11th, 2004, 7:43pm » |
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Martin Gardner, of course. He discusses them in "The Colossal Book of Mathematics.":
"A curve of constant width need not consist of circular arcs. In fact, you can draw a highly arbitrary convex curve from the top to the bottom of a square and touching its left side ..., and this curve will be the left side of a uniquely determined curve of constant width. To find the missing part, rule a large number of lines, each parallel to a tangent of [the arc] and separated from the tangent by a distance equal to the side of the square."
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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Nigel_Parsons
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Re: Manhole: Shape matters
« Reply #15 on: Aug 13th, 2004, 1:22pm » |
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I think we may be discussing British coinage here!
The 20p And 50p coins are described as "equilateral Curve Heptagons" such that any diameter is of equal length.
For those not au fait with British coinage, see:
http://www.24carat.co.uk/fiftypencestory.html
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