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Main Question or Discussion Point
Nobody mentions that Penrose stairs is possible on the torus. I wonder Why. isn't this obvious?
Nobody mentions that Penrose stairs is possible on the torus. I wonder Why. isn't this obvious?
Still matematicians sometimes take it seriously, see for example here. I would regard these two properties mentioned there by anon:I would seriously consider this to indicate that this construct is an Escher-illusion, not a mathematically definable object.
(1) it wraps back around to itself, and (2) it (at least ostensibly) increases in height along the way.
Seriously in the sense that Penrose feels the illusion is worth studying. Possibly new and compelling illusions would be discovered. Or perhaps a real-world 3D object could be made that gives the illusion of being a Penrose stair when viewed from a certain angle under certain conditions.@jim mcnamara
Still matematicians sometimes take it seriously, see for example here. I would regard these two properties mentioned there by anon:
I meant originnaly exactly this.Or perhaps there is some imaginary mathematical world within which Penrose stairs are consistent.
Or an energy losing device. If we neglect the energy dissipation, then in our world we can call height on equal right the vertial distance measured from a fixed point and the mechanical work done since started from this point (because then [itex] E = mgh[/itex]). But if we take into account also the dissipation (e.g. energy loss by air resistance), then the "heigth" on the energy axis will always increase during a walk on closed loop. So, if we take into account the dissipation, then in our world we cannot use this energy axis as a spatial coordinate line, while in the torus-world we can.Under this condition a Penrose stairs would be a free energy device.
Any such geometry would not have a consistent measure of distance in the "up" direction.Or perhaps there is some imaginary mathematical world within which Penrose stairs are consistent.
Oh, maybe the forces aren't conservative. Maybe the measure decreases when traveling the stairs in the opposite direction. I feel sure it is possible to come up with some fantasy world where it works after a fashion, but I can't say I'm interested in pursuing that.Any such geometry would not have a consistent measure of distance in the "up" direction.
Consider the condition for moving from one stair to the next in a "real" Penrose staircase: you must increase your distance in the vertical direction from the origin. You can never decrease this distance (if you allowed this it would be easy to construct in our world - you simply ramp down the tread of each step). Consider now the condition for completing a circuit of the staircase: you must return to a position which has the same vertical distance from the origin as your start point (because it is your start point). A measure that is always increasing can never return to its starting value.
My argument has nothing to do with forces or direction of travel.Oh, maybe the forces aren't conservative. Maybe the measure decreases when traveling the stairs in the opposite direction.
There is no point in making that statement. Either find a flaw in my argument or a counter-example.I feel sure it is possible to come up with some fantasy world where it works after a fashion.
I have a third option. I can do something more to my liking.Either find a flaw in my argument or a counter-example.