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 Post subject: Solids with constant width
PostPosted: Thu Sep 20, 2012 4:53 pm 
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I'm not really sure where this belongs, I suppose it could be off topic, but it kind of is puzzle like and really since I want to buy some, it could be in marketplace. So mods, feel free to move the topic if you'd like.

So I'm sure some of you are familiar with solids with a constant width. For most people, the sphere would come to mind and probably wouldn't realize that their are others. So I guess the puzzle is to think of others. Bonus points for not searching the internet :)

Anyway, would like to purchase a set of three that all have the same width, but are different shapes. The only place I've seen that comes close is a website called grand-illusions.com. The set they sell is all one shape though. So I would imagine some of you CAD/Shapeways people could probably design some.

Okay, nevermind, I just did a search on Shapeways and it appears someone has already done this. But if you'd like to create some different ones, I think that would be kind of neat as well.

-d


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 Post subject: Re: Solids with constant width
PostPosted: Thu Sep 20, 2012 4:57 pm 
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What do you mean by constant width? Doesn't a book or pencil, or any other regular shape have a constant width?

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 Post subject: Re: Solids with constant width
PostPosted: Thu Sep 20, 2012 5:12 pm 
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The sphere and reuleaux tetrahedron are examples of constant width shapes. If you roll them on a table then the maximum height is constant.

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 Post subject: Re: Solids with constant width
PostPosted: Thu Sep 20, 2012 5:19 pm 
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TomZ wrote:
The sphere and reuleaux tetrahedron are examples of constant width shapes. If you roll them on a table then the maximum height is constant.

Not according to Wikipedia:
Wikipedia wrote:
One can find repeated claims in the mathematical literature that the Reuleaux tetrahedron is analogously a surface of constant width, but it is not true: the two midpoints of opposite edge arcs are separated by a larger distance...
... but I get what you mean. :wink:

EDIT: Apparently Meissner modified the shape so that it is constant width. I wonder how well it actually rolls: does the center of gravity also stay at the same height? Otherwise it will repeatedly accelerate and decelerate as it rolls.

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 Post subject: Re: Solids with constant width
PostPosted: Thu Sep 20, 2012 5:32 pm 
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Ta-da: http://www.shapeways.com/shops/greatgeometers
And also: http://www.shapeways.com/model/115463/r ... s_3cm.html

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 Post subject: Re: Solids with constant width
PostPosted: Fri Sep 21, 2012 12:16 pm 
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The centers of gravity do change, so the acceleration would change, but if you have three of them all of the same height and put a book on it. You could put a bubble level on the book and roll the book all around and the level will stay centered (at least once you stop).

The second link is what I saw, I'll probably order them once I put an order together.

-d


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 Post subject: Re: Solids with constant width
PostPosted: Fri Sep 21, 2012 1:03 pm 
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darryl wrote:
The centers of gravity do change, so the acceleration would change...

So are there any other solids - besides a sphere - with constant width *and* constant center of gravity? Would it be possible to design one, e.g., by adding cavities, springs and/or ball bearings inside a Meissner's tetrahedron, so that it can roll as smoothly as a ball with uniform velocity?

Or how about any other shape with constant center of gravity? Even with variable width it should still roll as smoothly as a ball. Constant center of gravity seems a much more interesting a concept than constant width...

Just imagine modifying a regular cube so that could roll like a ball: Now that would be *very* cool! :D :D :D

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 Post subject: Re: Solids with constant width
PostPosted: Fri Sep 21, 2012 4:51 pm 
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I think it could be done by varying the density. I'm sure Oskar could design one in a matter of minutes. :D

Actually though, a two dimensional one would probably be easier to work on. Like a lopsided wheel, but have the density such that it's center of gravity is still in the middle. That would be kind of neat to watch.

-d


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 Post subject: Re: Solids with constant width
PostPosted: Fri Sep 21, 2012 4:59 pm 
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darryl wrote:
Actually though, a two dimensional one would probably be easier to work on. Like a lopsided wheel, but have the density such that it's center of gravity is still in the middle. That would be kind of neat to watch.

-d

You mean like this:

http://www.youtube.com/watch?v=u-hDEEl67_Y

or this:

http://www.youtube.com/watch?v=i-hnspcmh_M

...but works on a flat surface.

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 Post subject: Re: Solids with constant width
PostPosted: Mon Oct 01, 2012 5:30 pm 
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KelvinS wrote:
Just imagine modifying a regular cube so that could roll like a ball: Now that would be *very* cool! :D :D :D

Did you just re-invent the wheel?
Slightly off topic but it reminds me of those wheels which can roll down stairs but still give a smooth ride.

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 Post subject: Re: Solids with constant width
PostPosted: Mon Oct 01, 2012 5:32 pm 
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Tony Fisher wrote:
KelvinS wrote:
Just imagine modifying a regular cube so that could roll like a ball: Now that would be *very* cool! :D :D :D

Did you just re-invent the wheel?
Slightly off topic but it reminds me of those wheels which can roll down stairs but still give a smooth ride.

You mean like in the video link just above your post? :wink:

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 Post subject: Re: Solids with constant width
PostPosted: Tue Oct 02, 2012 4:55 pm 
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KelvinS wrote:
You mean like in the video link just above your post? :wink:

I actually meant down proper steps.

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