Do you think "impossible" objects from mathematics are "real," even if they can't be constructed in our world?
Don't try to build a Lego version of one of M.C. Escher's staircases … it would be impossible, as Robert Ghrist points out in this article about an impossible object that he and a colleague invented. The object fits into the category of visual paradoxes, and in our three-dimensional world it couldn't exist. They started with a variant of a Penrose staircase: "A ladybug walking along a Penrose staircase, for example, will feel like it has climbed a full set of stairs, yet it will have returned to the same height it was at when it started." Would you consider these types of objects real even if they can't exist in our world?
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not as objects but yes technically
Yes I see them all the time. Thru polarized glasses looking thru ceramic tinted window. We unknowingly live in a 3d blocked world. Some see in 2d 3d some see in 3d and 4d. It's a beautiful chaotic place. Enjoy the ride ...
First, what is reality? Are only the things that we can see and touch real? Or are things like thoughts and ideas real too? Are things like love, beauty, justice real? Are numbers, like zero for example, real?
Is a circle real? If I draw a circle on the whiteboard, you would indeed say that the circle is real, would you not? Even if it were not a perfect circle, but still "close enough" to be practically called a circle? So it is called a circle because it resembles, to some degree, the idea of a circle. There are some who might say it is not quite a real circle because of its imperfections, or in other words the more exact in circularity it is the more it can rightly be called a real circle. So do we declare the mere shadow of an idea real because it resembles an unreal idea and grade realness based on comparison to the unreal? That would be preposterous. Certainly, I think, ideas must be real too, or else we can't be certain that anything is real. I'm convinced Escher's staircases and Klein's bottles are every bit as real in the realm of ideas as circles and squares, if not difficult to express materially.
The shapes are correct, but they are not viewed in perspective. To make a physical representation of the graphic, correct and account for perspective, the shapes would not be equal in size and not equally spaced.
It's a product of the common representation of 3D space in two dimensions. It is artifact.
Actually it's just a question. Isn't this a mathematical explanation for an optical illusion?
I am a professor emeritus of Mathematical Sciences, University of Memphis, TN. In my early career, 1969-1970s) I frequently taught "math for liberal arts" courses and tology courses and assigned the (attempted) construction of such objects as homework. An excellent example is Lewis' Carrol's construction of a projective plane: take three pocket handkerchiefs, sew two together to make a mobius strip, sew the four edges of the third to the four edges of the mobius strip. Class discussion: why could you not finish the sewing? Would it be a klein bottle or a projective plane if completed? In one case a student sewed one edge in the wrong orientation and it took two professors with Ph.D.s in topology an hour to determine what object it would be if completed (we tried to triangulate it). I still keep that souvenir in my desk, over 50 years later.)
