Surface Contact

[extropalopakettle]

First, I got to thinking about old puzzles in VSP that have disappeared over the years (at least I think they have) ... approachable, math/logic oriented puzzles, not overly "engineered". So I'm resurrecting an old one I once posted, but also adding a second part. If you happen to remember the solution to the first part, please leave it for others to try to solve.

Part 1: Find the shape of a 3 dimensional solid object that can be placed on a flat inclined plane, and will roll down the plane in such a way that as it rolls, every point on its surface will contact the plane before rolling a certain fixed distance.

Part 2: Find another such shape.

[MNOWAX]

1:If i'm thinking correctly, Two cones joined at the base like this : <> as it rolls on one cone, the force of the roll will flip it and roll on the other cone.

perhaps not?
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MNOWAX

[Zag]

That was the best that I came up with, MNO, but I suspect that there is a better answer.

[Jack_Ian]

1. I remember this. I believe there was even a 3-D stereoscopic image posted of it.
2. [A pill capsule with a ball-bearing in it], but the certain fixed distance would be arbitrarily large.

[novice]

1. Would a cone with a hemisphere at the base work?

[Zag]

Any shape with a spherical part is clearly wrong. If you only have a single point of contact for some two-dimensional part of the surface, you'll never get complete coverage.

[novice]

Something that rolls in a slalom pattern? I'm having trouble visualising the object, something conish that segues from one roll direction to another?

[extropalopakettle]

[Suspence]

I found the other object while learning more about the sphericon on Wikipedia. OLOID
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[extropalopakettle]

[Suspence]

Here's a pretty cool animation of the oloid.

http://www.youtube.com/watch?v=GUyou2QelRo&feature=related

If you trust Wikipedia (and who doesn't ) "the oloid is the only known object, other than some members of the sphericon family, that while rolling, develops its entire surface."

This seems to suggest that the oloid is not in the sphericon family, and whatever you have found is. Or that you have discovered something totally new, which would be cooler.
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[extropalopakettle]

To me, one has to look rather carefully at the oloid to see the difference from the sphericon. I'm not sure how to qualify the similarities and differences between shapes, but I think one could morph a sphericon to an oloid with every step along the way having the same property.

The other shape I found is here: http://www.softcom.net/users/sbmathias/related.htm , dubbed the "hexasphericon"

[novice]

[Suspence]

[extropalopakettle]

[extropalopakettle]

More about construction of octosphericon here: http://www.ams.org/samplings/feature-column/fcarc-octo-cretan

Yeah, now I see how it could be generalized to a sequence approaching a sphere in the limit. (hurts my brain to think about at what point it would fail to roll according to the mentioned criteria and just roll like a sphere)