But do note: never does a circle, cylinder, or sphere actually exist -- in reality, all contours are discretely determined and finite, while being dynamic rather than rigid. Straight and smooth and solid and fixed are each approximate abstractions.
You can still roll a polygonal shape on flat ground ! Here we have the case where the mobile remains at the same height wrt to the road ! no need for suspension springs !
It's interesting how time and motion can relate polygons to curves. It's also interesting, that as the number of sides increase the basic shape of the curve produced doesn't change, it just scales.
Sean Walker It does change. With more sides you get a smaller slice of the sine wave. The angle at which edges of the valleys meet becomes larger the more sides you have and as the number of sides approach infinity the angle approaches 180 deg or a straight line. But I guess you could look at it as being stretched/squashed at some ratio involving pi in the different dimensions. :)
Anders Öhlund Yes, my point was that although it changes, it appears to do so as a linear transformation of the basic sine curve shape. I'm not sure, but it does look as though the vertical compression is somewhat greater than the horizontal. It would be interesting to work out the variance between the vertical and horizontal compression as a function of the number of sides of polygon. I mean to help my son through his high school and university math so I should try working this out myself... I have been considering purchasing Mathematica - perhaps this could be my first application of it. :)
btw it's interesting to see a graph of rotational speed vs displacement for the triangles in the gif -- or, for that matter, rotational speed vs. angular displacement for non-round gearsets. There's some beautiful work being done with non-round surfaces.
So, what comes below triangles? A star polygon, ..?
ReplyDeleteSo the number of sides correlates with the curve steepness?
ReplyDeleteWell...smooth straight-line ride not possible on triangular wheels. Wedges would cut into ground. ;)
ReplyDeleteSquare wheels can work after all..
ReplyDeleteI think also the wheels can spin in counter direction from one another and object treads in place.
ReplyDeleteHehe...very much so Eszter Mariczáné Nagy ! Some roads in E Europe are horrible =)
ReplyDeleteWith a tire, one gets something on the order of Avogadro's number of angles.
ReplyDeleteOr heli ;))
ReplyDeleteBut do note: never does a circle, cylinder, or sphere actually exist -- in reality, all contours are discretely determined and finite, while being dynamic rather than rigid. Straight and smooth and solid and fixed are each approximate abstractions.
ReplyDeleteYou can still roll a polygonal shape on flat ground ! Here we have the case where the mobile remains at the same height wrt to the road ! no need for suspension springs !
ReplyDeleteHere is a real world implementation.
ReplyDeleteSquare-Wheeled Tricycle
You can also get a smooth ride on flat surface using solids of constant width and some clever suspension work :)
triangle bicycle
It's interesting how time and motion can relate polygons to curves. It's also interesting, that as the number of sides increase the basic shape of the curve produced doesn't change, it just scales.
ReplyDeleteSean Walker It does change. With more sides you get a smaller slice of the sine wave. The angle at which edges of the valleys meet becomes larger the more sides you have and as the number of sides approach infinity the angle approaches 180 deg or a straight line.
ReplyDeleteBut I guess you could look at it as being stretched/squashed at some ratio involving pi in the different dimensions. :)
Anders Öhlund Yes, my point was that although it changes, it appears to do so as a linear transformation of the basic sine curve shape. I'm not sure, but it does look as though the vertical compression is somewhat greater than the horizontal. It would be interesting to work out the variance between the vertical and horizontal compression as a function of the number of sides of polygon.
ReplyDeleteI mean to help my son through his high school and university math so I should try working this out myself... I have been considering purchasing Mathematica - perhaps this could be my first application of it. :)
Corina Marinescu -- you can roll smoothly on flat ground using pointy/non-round wheels, if they're Reuleaux triangles. You just can't have them revolving around standard axles. http://exploreideasdaily.wordpress.com/2013/03/13/impossible-objects-1-how-round-is-your-wheel/
ReplyDeleteJohn Bump those are not "really" triangles in my opinion..same like the fake "doctors" - doctors in literature, anthropologie..etc :D
ReplyDeleteTrue, but they're definitely pointy!
ReplyDeletebtw it's interesting to see a graph of rotational speed vs displacement for the triangles in the gif -- or, for that matter, rotational speed vs. angular displacement for non-round gearsets. There's some beautiful work being done with non-round surfaces.
ReplyDeleteI know what you're saying...but I refuse to call that fatty figure a triangle =)
ReplyDeleteA Rubenesque triangle? A zaftig triangle?
ReplyDeleteHurol Aslan it would follow wouldn't it! lol
ReplyDelete