Magnifying glass in hyperbolic space
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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
asked yesterday
liaombroliaombro
369210
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add a comment |
$begingroup$
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
asked yesterday
liaombroliaombro
369210
$endgroup$
$begingroup$
They would move out of hyperbolic space into the ambient euclidean space where magnifying glasses scale things properly.
$endgroup$
– user21820
22 hours ago
add a comment |
$begingroup$
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
asked yesterday
liaombroliaombro
369210
$endgroup$
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
asked yesterday
liaombroliaombro
369210
asked yesterday
liaombroliaombro
369210
asked yesterday
liaombroliaombro
369210
asked yesterday
liaombroliaombro
369210
asked yesterday
liaombroliaombro
369210
369210
$begingroup$
They would move out of hyperbolic space into the ambient euclidean space where magnifying glasses scale things properly.
$endgroup$
– user21820
22 hours ago
add a comment |
$begingroup$
They would move out of hyperbolic space into the ambient euclidean space where magnifying glasses scale things properly.
$endgroup$
– user21820
22 hours ago
$begingroup$
They would move out of hyperbolic space into the ambient euclidean space where magnifying glasses scale things properly.
$endgroup$
– user21820
22 hours ago
$begingroup$
They would move out of hyperbolic space into the ambient euclidean space where magnifying glasses scale things properly.
$endgroup$
– user21820
22 hours ago
add a comment |
2 Answers
2
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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.
answered yesterday
Lee MosherLee Mosher
50.9k33888
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$begingroup$
Thanks, fixed. It's just a wikipedia link.
$endgroup$
– Lee Mosher
13 hours ago
add a comment |
$begingroup$
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.
answered yesterday
Henning MakholmHenning Makholm
242k17308550
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add a comment |
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2 Answers
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2 Answers
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$begingroup$
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.
answered yesterday
Lee MosherLee Mosher
50.9k33888
$endgroup$
$begingroup$
Thanks, fixed. It's just a wikipedia link.
$endgroup$
– Lee Mosher
13 hours ago
add a comment |
$begingroup$
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.
answered yesterday
Lee MosherLee Mosher
50.9k33888
$endgroup$
$begingroup$
Thanks, fixed. It's just a wikipedia link.
$endgroup$
– Lee Mosher
13 hours ago
add a comment |
$begingroup$
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.
answered yesterday
Lee MosherLee Mosher
50.9k33888
$endgroup$
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.
answered yesterday
Lee MosherLee Mosher
50.9k33888
edited 13 hours ago
answered yesterday
Lee MosherLee Mosher
50.9k33888
answered yesterday
Lee MosherLee Mosher
50.9k33888
answered yesterday
Lee MosherLee Mosher
50.9k33888
50.9k33888
$begingroup$
Thanks, fixed. It's just a wikipedia link.
$endgroup$
– Lee Mosher
13 hours ago
add a comment |
$begingroup$
Thanks, fixed. It's just a wikipedia link.
$endgroup$
– Lee Mosher
13 hours ago
$begingroup$
Thanks, fixed. It's just a wikipedia link.
$endgroup$
– Lee Mosher
13 hours ago
$begingroup$
Thanks, fixed. It's just a wikipedia link.
$endgroup$
– Lee Mosher
13 hours ago
add a comment |
$begingroup$
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.
answered yesterday
Henning MakholmHenning Makholm
242k17308550
$endgroup$
add a comment |
$begingroup$
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.
answered yesterday
Henning MakholmHenning Makholm
242k17308550
$endgroup$
add a comment |
$begingroup$
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.
answered yesterday
Henning MakholmHenning Makholm
242k17308550
$endgroup$
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.
answered yesterday
Henning MakholmHenning Makholm
242k17308550
answered yesterday
Henning MakholmHenning Makholm
242k17308550
answered yesterday
Henning MakholmHenning Makholm
242k17308550
answered yesterday
Henning MakholmHenning Makholm
242k17308550
242k17308550
add a comment |
add a comment |
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$begingroup$
They would move out of hyperbolic space into the ambient euclidean space where magnifying glasses scale things properly.
$endgroup$
– user21820
22 hours ago