Topic: of topic-i do not have a off topic post to send today (Read 4286 times)

zevele1 · « **on:** July 23, 2002, 12:41:41 pm »

But this does not mind that this forum will run in a more coherent way

Gatobrit · « **Reply #1 on:** July 23, 2002, 01:16:02 pm »

This is an old one...

What is the best shape for a manhole cover and why?

(Manhole cover = the (usually) cast iron / metal sheet placed over a hole in a pavement / sidewalk to allow access to utilities etc.)

Scronch · « **Reply #2 on:** July 23, 2002, 01:31:19 pm »

Here's a hint...

Gatobrit · « **Reply #3 on:** July 23, 2002, 02:05:26 pm »

Scronch - OK. But why?

sub-24 · « **Reply #4 on:** July 23, 2002, 02:34:46 pm »

Excellent.

Might get some real questions answered.

J-Ro · « **Reply #5 on:** July 23, 2002, 02:37:19 pm »

Cause if it's round and sits over a hole that it is slightly larger than, then there is no way it can fall in said hole.

J-Ro · « **Reply #6 on:** July 23, 2002, 02:50:31 pm »

Sooo, did I get it?

KingSparta · « **Reply #7 on:** July 23, 2002, 02:51:55 pm »

I would Think It Would Be Oval

So The Wokers Beer Belly And Big Ass Can Fit.

Gatobrit · « **Reply #8 on:** July 23, 2002, 02:58:36 pm »

J-Ro is correct as is Scronch although he didn't provide a reason. Round manhole covers will not fall through the hole.

J-Ro - your prize will be in heaven....

gateley · « **Reply #9 on:** July 23, 2002, 05:38:16 pm »

I don't get it. For any shape you could design it with enough extra so that it won't fall in. Doesn't the circular shape minimize the amount of extra material needed?

and besides - they made them circular so it is easier to chocolate-cover them.

j (godot is still waiting on me)

DeathRider · « **Reply #10 on:** July 23, 2002, 06:00:38 pm »

Zevele, What do you hear from your window at night, In the past you have been asked what you see, I would like to find out what you hear, such as birds, or crickets at night the wind blowing through the trees, how about what you smell, are there plants around your place that give off distinctive scents. Discribe the color of the sky during sun set.

Just some ideas for when you are out of ideas.

Yaobing · « **Reply #11 on:** July 23, 2002, 08:08:43 pm »

Doesn't the circular shape minimize the amount of extra material needed?

Yes. There are two reasons:

1. For a given area, circle has the smallest circumference.

2. For a circle as long as the radius of the cover is slightly larger than the radius of the hole, it will not fall in. For a square (that is the next logical shape to consider), the side length of the cover must be more than sqrt(2) times the side length of the hole or it will fall in. That makes the area of cover twice as large as the area of the hole.

gateley · « **Reply #12 on:** July 23, 2002, 08:59:25 pm »

Hi Yaobing,

Can you show that any irregular n-gon requires more area than a regular n-gon? I think I can do the n-gon requires more area than the n|PLS|1-gon, but the irregular stuff is beyond me.

And, of course, all bets are off when they invent quantum polygons.

j

Yaobing · « **Reply #13 on:** July 24, 2002, 07:43:26 pm »

Can you show that any irregular n-gon requires more area than a regular n-gon?

John,

That is beyond me too.

And, of course, all bets are off when they invent quantum polygons

Then you would not need a manhole anymore.

BigMike · « **Reply #14 on:** July 25, 2002, 07:18:54 am »

That's a good question, but the answer is to obvious. Here is a far more difficult one, that at first seems silly, but after some careful thinking, you will understand why some guy at MIT came up with it. Here's the question - If a car went over a cliff, how many apples would you have, true or false?

JimH · « **Reply #15 on:** July 25, 2002, 07:33:25 am »

Mike,
> If a car went over a cliff, how many apples would you have, true or false?

I don't know, but here's where you might be able to find out:

http://www-swiss.ai.mit.edu/htbin/coke

( courtesy of http://www.cs.ucsd.edu/users/bsy/coke.html )

Agonostis · « **Reply #16 on:** July 25, 2002, 10:43:58 am »

Can you show that any irregular n-gon requires more area than a regular n-gon?

That's not true, at least not with the definitions of regular/irregular as I understand them...

-----
Robert
(Mathematics major @ Columbia University)

gateley · « **Reply #17 on:** July 25, 2002, 10:52:42 am »

Cool! Have a counter-example?

j
(math geek from years ago)

Agonostis · « **Reply #18 on:** July 25, 2002, 11:52:17 am »

Cool! Have a counter-example?

Let S be a square with side length l.
Then S is a regular n-gon with n = 4 and area l^2.

Let R be a rhombus with side length l.
Then R is an irregular n-gon with n = 4 and area l^2.

Therefore an irregular n-gon shaped hole does not always require a cover of larger area than a regular n-gon shaped hole.

And remember kids, gons don't kill people, mathematicians bore them to death.

On the subject of math killing people, anybody out there read TEXT Cryptonomicon by Neal Stephenson? Good read if you're up for 1000|PLS| pages of pseudo-history.

-----
Robert

gateley · « **Reply #19 on:** July 25, 2002, 12:29:50 pm »

not quite. Choose a really really skinny rhombus (wlog).
It has a maximum diameter of almost 2l, and so any cover you have must have a minumum diameter of almost 2l plus a little bit.
This is a much bigger cover than is needed for the square (max diameter sqrt(2l^2)). [i can hear the yawns already]

Thanks for the recommendation on the Neal Stephenson book. I'm looking for new stuff to read.

j

Yaobing · « **Reply #20 on:** July 25, 2002, 12:35:08 pm »

Then R is an irregular n-gon with n = 4 and area l^2.

The area of R is not l*l.

zevele1 · « **Reply #21 on:** July 25, 2002, 02:00:49 pm »

As i said on my post:

BUT THAT DOES NOT MIND THAT THIS FORUM WILL RUN IN A MORE COHERENT WAY.....................

KingSparta · « **Reply #22 on:** July 25, 2002, 02:12:27 pm »

High Level Math

1|PLS|1=2

Or

X|PLS|X=2

Or

Mp2|PLS|Mp2=Mp4

Agonostis · « **Reply #23 on:** July 25, 2002, 05:12:52 pm »

Choose a really really skinny rhombus

Sorry, but there's no such thing. A rhombus by definition has four equal sides, any one of them parallel to the one opposite.

Interestingly enough, you made exactly the same mistake I did:

The area of R is not l*l.

Which was forgetting the difference between a rhombus and a parallelogram. Been a long time since middle school geometry...

Anyway, for my counterexample, Yaobing is exactly right, the area of R is in fact sqrt(3)/2*l^2 and not simply l^2. It ends up not making any difference, because sqrt(3)/2*l^2 is always less than l^2, as long as l is positive.
So, unless someone can make me a manhole cover with a negative perimeter, my counter-example still shows that a regular n-gon shaped cover is not always preferable to an irregular one.

The counter to my counter is valid (as long as you replace rhombus with parallelogram), but doesn't make mine any less so, and it just takes one counterexample to disprove a statement.

-----
Robert
(somewhat embarrassed for his "high level" math mistake)

J-Ro · « **Reply #24 on:** July 25, 2002, 05:25:24 pm »

I don't think anyone here has ever lifted a manhole cover and certainly no one has climbed through one.

gateley · « **Reply #25 on:** July 25, 2002, 05:26:38 pm »

Zevele - I've been laughing everytime I load this page and see "more coherent way".

KingSparta: 1|PLS|1=3 for sufficiently large values of 1.
(actually, 1|PLS|1 is whatever you feel like, but that's really off topic).

Agonostis - some defs (to make sure I know what I'm talking about). A rhombus is 4 sides, and each side is parallel to exactly one other, and all sides have the same length. A parallelogram has the parallel requirement, but does not require all 4 sides to be the same length, right? (doesn't matter too much - your counterexample relies on different angles instead of different length sides, I think).

A "really really skinny" rhombus has two angles of about 1 degree, and two angles of about 179 degrees. It looks kind of like a toothpick. If you position it perfectly, it'll cover a hole, but when you pick it up and drop it, it'll probably fall through the hole, so it's not a counterexample. What am I missing? The "mininum diameter" of the rhombus is the distance between the two vertices of the 179 degree angles, and this is the "maximum diameter" of the largest hole the rhombus can cover, which is a lot smaller than the largest hole a square of the same area could cover (without falling in).

Rest of the world: google on manhole and feynmann and you'll find the best answer.

j

Charlemagne 8 · « **Reply #26 on:** July 25, 2002, 05:30:13 pm »

Let S be a square with side length l.
Then S is a regular n-gon with n = 4 and area l^2.

Let R be a rhombus with side length l.
Then R is an irregular n-gon with n = 4 and area l^2.

Therefore an irregular n-gon shaped hole does not always require a cover of larger area than a regular n-gon shaped hole.

Now we're talking the language of the common person.

And I HAVE both lifted manhole covers AND climbed down in them. And back out again as quickly as possible. There's LIVE things down there.

CVIII

JimH · « **Reply #27 on:** July 25, 2002, 05:40:12 pm »

Yeah, I've had one off once. It was really hard to do. And I don't remember what happened after that.

Yaobing · « **Reply #28 on:** July 25, 2002, 08:08:56 pm »

Agonostis,

I think you misunderstood the original question. We are not trying to find the shape with smallest area. We are trying to find a cover that will cover a manhole of given area without falling into the manhole and having smallest extra area (extra area here means material needed in addition to what is needed to bearly cover the hole).

Wile · « **Reply #29 on:** July 25, 2002, 08:56:36 pm »

Been down a manhole before too... didn't stay long... CVIII is right... I wasn't the only thing living down there.

I'm in no rush to visit another.

Scronch · « **Reply #30 on:** July 26, 2002, 11:17:47 am »

>I don't think anyone here has ever lifted a manhole cover and certainly no one has climbed through one.

Speak for yourself. Not all of us were put through school by Mommy and Daddy.

Scronch

Scronch · « **Reply #31 on:** July 28, 2002, 05:32:04 pm »

as is Scronch although he didn't provide a reason. Round manhole covers will not fall through the hole.

Gatobrit - I just realized that you misinterpreted my hint. Look again. I'm not intending to show a round head--I'm intending to show a head with scars from falling manhole covers.

If I gave the reason, I would've ruined the game for others.

Scronch

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Author Topic: of topic-i do not have a off topic post to send today (Read 4286 times)

zevele1

Scronch

gateley

Yaobing

gateley

Yaobing

BigMike

Agonostis

gateley

Agonostis

gateley

Yaobing

zevele1

Agonostis

gateley

Yaobing

Wile

Scronch

Scronch