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mma
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Nobody mentions that Penrose stairs is possible on the torus. I wonder Why. isn't this obvious?
It's not obvious to me. The idea is that stairs go up and down, to overcome gravity. Under this condition a Penrose stairs would be a free energy device. Just roll a ball down the stairs.mma said:Nobody mentions that Penrose stairs is possible on the torus. I wonder Why. isn't this obvious?
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.
mma said:@jim mcnamaraStill matematicians sometimes take it seriously, see for example here. I would regard these two properties mentioned there by anon:
Hornbein said:Or perhaps there is some imaginary mathematical world within which Penrose stairs are consistent.
Hornbein said:Under this condition a Penrose stairs would be a free energy device.
What do you mean "always rising"? How do you define "up"?mma said:The curve [itex] \mathbb R\to S^1\times S^1: t\mapsto (f(t),f(t))[/itex] is a walk on the torus that is always rising and returns to a point.
Any such geometry would not have a consistent measure of distance in the "up" direction.Hornbein said:Or perhaps there is some imaginary mathematical world within which Penrose stairs are consistent.
MrAnchovy said: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.Hornbein said: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.Hornbein said:I feel sure it is possible to come up with some fantasy world where it works after a fashion.
MrAnchovy said:Either find a flaw in my argument or a counter-example.
Penrose stairs on the torus, also known as the Penrose staircase or Penrose triangle, is a visual illusion created by mathematician Roger Penrose in the 1950s. It is a two-dimensional representation of a staircase that appears to continuously loop and climb in a never-ending cycle.
The illusion is created by using a specific arrangement of angles and proportions in the design of the stairs. When viewed from a certain perspective, the steps seem to seamlessly connect and create the appearance of a never-ending loop.
No, the Penrose staircase is an impossible object and cannot exist in the physical world. It is a mathematical concept that can only be represented in two dimensions. However, it has been recreated in various forms of art and architecture as a visual illusion.
The Penrose staircase is significant in the field of mathematics and art as it challenges our perception and understanding of reality. It also demonstrates the concept of impossible objects and how our brains can be easily deceived by optical illusions.
While the Penrose staircase cannot be physically built, there are real-life examples of the illusion being recreated in art and architecture. For example, the Penrose stairs have been depicted in the artwork of M.C. Escher and have been used in building designs such as the Penrose stairs at the Rochester Institute of Technology in New York.