Very neat! Does the rolling actually leave the height of the center of mass invariant? The video seems to show some (two? four?) slightly-stable equilibria, so I'm guessing not quite.
In the first version I made of this I had completely ignored the height of the center of mass, and it didn't work at all! I spent quite some time making tiny affine transformations to try to get it as near constant height as I could. I suppose one could try analysing it properly to get a true solution rather than just an approximation.
Have you tried working out the integral equation for completely height invariant center of mass? I bet it's not too hard to express, and certainly at least numerically solvable.
Actually, it's just a differential equation plus a global integral constraint. So given an arclength parameterization, there's one scalar ODE for rollability, one scalar ODE for center of mass height invariance, and one further degree of freedom per point.
First define a parametric curve mapping into the unit sphere in R^4 by (1/sqrt(2))(cos(2t), sin(2t), cos(3t), sin(3t)). Then map using stereographic projection to R^3. I think this is by (x,y,z) = (X/(1-W), Y/(1-W), Z/(1-W)), where (X,Y,Z,W) are the coordinates in R^4, but I would have to check, it may have been stereographic projection from a point other than (X,Y,Z,W) = (0,0,0,1). Lastly, I scaled the curve in the x,y and z directions by hand to get the center of mass at constant height.
A previous version of this I made is too short in one direction, meaning that the center of mass rises and falls a few centimeters as it rolls along. It comes to rest in one of two distinct positions where the center of mass is lowest. The same is also true of this one, although the rise and fall is much shorter so it rolls better. But the answer to your question is "two".
@Donald King there is usually some mathematics behind the model that is usually not due to me, but the actual designs are due to my collaborators and me.
New thing: Rolling trefoil. The trefoil knot can be arranged in space so that (if made rigid) it will smoothly roll across the table!
Very neat! Does the rolling actually leave the height of the center of mass invariant? The video seems to show some (two? four?) slightly-stable equilibria, so I'm guessing not quite.
In the first version I made of this I had completely ignored the height of the center of mass, and it didn't work at all! I spent quite some time making tiny affine transformations to try to get it as near constant height as I could. I suppose one could try analysing it properly to get a true solution rather than just an approximation.
Have you tried working out the integral equation for completely height invariant center of mass? I bet it's not too hard to express, and certainly at least numerically solvable.
Actually, it's just a differential equation plus a global integral constraint. So given an arclength parameterization, there's one scalar ODE for rollability, one scalar ODE for center of mass height invariance, and one further degree of freedom per point.
Geoffrey Irving: Sounds like a nice little project! It would be nice to get it updated to have constant height center of mass.
First define a parametric curve mapping into the unit sphere in R^4 by (1/sqrt(2))(cos(2t), sin(2t), cos(3t), sin(3t)). Then map using stereographic projection to R^3. I think this is by (x,y,z) = (X/(1-W), Y/(1-W), Z/(1-W)), where (X,Y,Z,W) are the coordinates in R^4, but I would have to check, it may have been stereographic projection from a point other than (X,Y,Z,W) = (0,0,0,1). Lastly, I scaled the curve in the x,y and z directions by hand to get the center of mass at constant height.
Do you bring this with you in bars as discussion opener with girls? 8-) "See? This trefoil is rolling in a very pleasant way!"
A previous version of this I made is too short in one direction, meaning that the center of mass rises and falls a few centimeters as it rolls along. It comes to rest in one of two distinct positions where the center of mass is lowest. The same is also true of this one, although the rise and fall is much shorter so it rolls better. But the answer to your question is "two".
It really is quite pleasing (and intriguing)!
This is the coolest thing I've ever seen!!!!!!
At roughly constant height that should be! It isn't perfect, there are some small "dips" it likes to roll into.
ViHart also likes this.
I'm surprised there aren't more comments about that
@Donald King there is usually some mathematics behind the model that is usually not due to me, but the actual designs are due to my collaborators and me.
Awesome. Great ability for spatial thinking man. Are these your designs or are they based on existing models (actual or theoretical)?
This is easy to come up with because two circles attached perpendicularly with sqrt2 *r overlap will roll like an ooloid.
Actually, trying it again just now, I think there are two more just barely stable positions.
You can get the same effect with two rings weld together. This feels like a fancier version.
the coolest oloid
No, I get them printed by Shapeways.
This looks like a one-piece sphericon.
Is this almost an anti oloid? Or a kind of knotted anti oloid
Nice
thanks so much!!
If this shape was a die, how many sides would it have/ how many distinct resting positions are there?
ah clever
Dude, do you fabricate these objects yourself?
this is just a skeletised oliod shape
How didi get here :o
What is the equation that describes this knot?
What is the equation that describes this knot?