Magnifying glass in hyperbolic space












14












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My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such thing as homothety. So, what a person living in a hyperbolic space would do to improve poor vision?






geometry hyperbolic-geometry







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    14












    $begingroup$


    My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such thing as homothety. So, what a person living in a hyperbolic space would do to improve poor vision?






    geometry hyperbolic-geometry







    share|cite|improve this question









    $endgroup$















      14












      14








      14


      1



      $begingroup$


      My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such thing as homothety. So, what a person living in a hyperbolic space would do to improve poor vision?






      geometry hyperbolic-geometry







      share|cite|improve this question









      $endgroup$




      My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such thing as homothety. So, what a person living in a hyperbolic space would do to improve poor vision?






      geometry hyperbolic-geometry




      geometry hyperbolic-geometry






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      share|cite|improve this question











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      asked 9 hours ago









      liaombroliaombro

      359210




      359210






















          2 Answers
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          11












          $begingroup$

          What you say will still be true: a magnifying glass will still scale the picture by some factor. Let us say that the scale factor is $ell > 1$.



          The difference will be that the scaled picture will no longer be a picture of the old familiar hyperbolic space in which the sectional curvature is $-1$. Instead, it will be a picture of hyperbolic space with curvature $-frac{1}{ell^2}$; I'm using here that the units of curvature are basically $1/text{(length)}^2$.



          So, for example, a really powerful magnifying glass with scale factor $ell >!!> 1$ will present a picture of a hyperbolic space whose curvature is nearly zero, being pretty much indistinguishable from Euclidean space.






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          • 2




            $begingroup$
            I believe, your link currently does not work as intended
            $endgroup$
            – WorldSEnder
            3 hours ago



















          3












          $begingroup$

          Even though a magnifying glass appears to scale the plane picture you're looking at uniformly, the actual image that forms on your retina lives on (the inside of) a sphere. So that actual image cannot actually be scaled uniformly.



          Really, our immediate visual sensations lives not in our 3D space, but in the space of directions emanating from our eye. And that space is a sphere no matter whether the eye itself is embedded in hyperbolic or Euclidean space. More precisely, a sphere in hyperbolic space is always isometric to a sphere in Euclidean space; they just embed differently.



          So a hyperbolic creature's retina might well have the same intrinsic geometry as ours does, and so it is completely conceivable that his magnifying glass might transform his visual sensation in the same (imperfect) way that our magnifying glasses do for our eyes.






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            2 Answers
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            11












            $begingroup$

            What you say will still be true: a magnifying glass will still scale the picture by some factor. Let us say that the scale factor is $ell > 1$.



            The difference will be that the scaled picture will no longer be a picture of the old familiar hyperbolic space in which the sectional curvature is $-1$. Instead, it will be a picture of hyperbolic space with curvature $-frac{1}{ell^2}$; I'm using here that the units of curvature are basically $1/text{(length)}^2$.



            So, for example, a really powerful magnifying glass with scale factor $ell >!!> 1$ will present a picture of a hyperbolic space whose curvature is nearly zero, being pretty much indistinguishable from Euclidean space.






            share|cite|improve this answer









            $endgroup$









            • 2




              $begingroup$
              I believe, your link currently does not work as intended
              $endgroup$
              – WorldSEnder
              3 hours ago
















            11












            $begingroup$

            What you say will still be true: a magnifying glass will still scale the picture by some factor. Let us say that the scale factor is $ell > 1$.



            The difference will be that the scaled picture will no longer be a picture of the old familiar hyperbolic space in which the sectional curvature is $-1$. Instead, it will be a picture of hyperbolic space with curvature $-frac{1}{ell^2}$; I'm using here that the units of curvature are basically $1/text{(length)}^2$.



            So, for example, a really powerful magnifying glass with scale factor $ell >!!> 1$ will present a picture of a hyperbolic space whose curvature is nearly zero, being pretty much indistinguishable from Euclidean space.






            share|cite|improve this answer









            $endgroup$









            • 2




              $begingroup$
              I believe, your link currently does not work as intended
              $endgroup$
              – WorldSEnder
              3 hours ago














            11












            11








            11





            $begingroup$

            What you say will still be true: a magnifying glass will still scale the picture by some factor. Let us say that the scale factor is $ell > 1$.



            The difference will be that the scaled picture will no longer be a picture of the old familiar hyperbolic space in which the sectional curvature is $-1$. Instead, it will be a picture of hyperbolic space with curvature $-frac{1}{ell^2}$; I'm using here that the units of curvature are basically $1/text{(length)}^2$.



            So, for example, a really powerful magnifying glass with scale factor $ell >!!> 1$ will present a picture of a hyperbolic space whose curvature is nearly zero, being pretty much indistinguishable from Euclidean space.






            share|cite|improve this answer









            $endgroup$



            What you say will still be true: a magnifying glass will still scale the picture by some factor. Let us say that the scale factor is $ell > 1$.



            The difference will be that the scaled picture will no longer be a picture of the old familiar hyperbolic space in which the sectional curvature is $-1$. Instead, it will be a picture of hyperbolic space with curvature $-frac{1}{ell^2}$; I'm using here that the units of curvature are basically $1/text{(length)}^2$.



            So, for example, a really powerful magnifying glass with scale factor $ell >!!> 1$ will present a picture of a hyperbolic space whose curvature is nearly zero, being pretty much indistinguishable from Euclidean space.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 9 hours ago









            Lee MosherLee Mosher

            50.9k33888




            50.9k33888








            • 2




              $begingroup$
              I believe, your link currently does not work as intended
              $endgroup$
              – WorldSEnder
              3 hours ago














            • 2




              $begingroup$
              I believe, your link currently does not work as intended
              $endgroup$
              – WorldSEnder
              3 hours ago








            2




            2




            $begingroup$
            I believe, your link currently does not work as intended
            $endgroup$
            – WorldSEnder
            3 hours ago




            $begingroup$
            I believe, your link currently does not work as intended
            $endgroup$
            – WorldSEnder
            3 hours ago











            3












            $begingroup$

            Even though a magnifying glass appears to scale the plane picture you're looking at uniformly, the actual image that forms on your retina lives on (the inside of) a sphere. So that actual image cannot actually be scaled uniformly.



            Really, our immediate visual sensations lives not in our 3D space, but in the space of directions emanating from our eye. And that space is a sphere no matter whether the eye itself is embedded in hyperbolic or Euclidean space. More precisely, a sphere in hyperbolic space is always isometric to a sphere in Euclidean space; they just embed differently.



            So a hyperbolic creature's retina might well have the same intrinsic geometry as ours does, and so it is completely conceivable that his magnifying glass might transform his visual sensation in the same (imperfect) way that our magnifying glasses do for our eyes.






            share|cite|improve this answer









            $endgroup$


















              3












              $begingroup$

              Even though a magnifying glass appears to scale the plane picture you're looking at uniformly, the actual image that forms on your retina lives on (the inside of) a sphere. So that actual image cannot actually be scaled uniformly.



              Really, our immediate visual sensations lives not in our 3D space, but in the space of directions emanating from our eye. And that space is a sphere no matter whether the eye itself is embedded in hyperbolic or Euclidean space. More precisely, a sphere in hyperbolic space is always isometric to a sphere in Euclidean space; they just embed differently.



              So a hyperbolic creature's retina might well have the same intrinsic geometry as ours does, and so it is completely conceivable that his magnifying glass might transform his visual sensation in the same (imperfect) way that our magnifying glasses do for our eyes.






              share|cite|improve this answer









              $endgroup$
















                3












                3








                3





                $begingroup$

                Even though a magnifying glass appears to scale the plane picture you're looking at uniformly, the actual image that forms on your retina lives on (the inside of) a sphere. So that actual image cannot actually be scaled uniformly.



                Really, our immediate visual sensations lives not in our 3D space, but in the space of directions emanating from our eye. And that space is a sphere no matter whether the eye itself is embedded in hyperbolic or Euclidean space. More precisely, a sphere in hyperbolic space is always isometric to a sphere in Euclidean space; they just embed differently.



                So a hyperbolic creature's retina might well have the same intrinsic geometry as ours does, and so it is completely conceivable that his magnifying glass might transform his visual sensation in the same (imperfect) way that our magnifying glasses do for our eyes.






                share|cite|improve this answer









                $endgroup$



                Even though a magnifying glass appears to scale the plane picture you're looking at uniformly, the actual image that forms on your retina lives on (the inside of) a sphere. So that actual image cannot actually be scaled uniformly.



                Really, our immediate visual sensations lives not in our 3D space, but in the space of directions emanating from our eye. And that space is a sphere no matter whether the eye itself is embedded in hyperbolic or Euclidean space. More precisely, a sphere in hyperbolic space is always isometric to a sphere in Euclidean space; they just embed differently.



                So a hyperbolic creature's retina might well have the same intrinsic geometry as ours does, and so it is completely conceivable that his magnifying glass might transform his visual sensation in the same (imperfect) way that our magnifying glasses do for our eyes.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 5 hours ago









                Henning MakholmHenning Makholm

                242k17308550




                242k17308550






























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