Magnifying glass in hyperbolic spaceIs it possible to deduce a model for hyperbolic geometry from a synthetic set of axioms a la Euclid/Hilbert/Tarski?Symbolic coordinates for a hyperbolic grid?Hyperbolic (and related) structures on open unit diskWhat is the volume of the sphere in hyperbolic space?Non-equivalent metrics on $PSL_2(mathbbR)$Is there a relationship between the Cantor set and hyperbolic geometry?Translation in Poincare disc modelProve that a loxodromic transformation has an attractor and a repeller as fixed pointsSpheres in hyperbolic spacesExplicit isomorphisms between the hyperbolic plane and surfaces of constant negative curvature
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Magnifying glass in hyperbolic space
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Magnifying glass in hyperbolic space
Is it possible to deduce a model for hyperbolic geometry from a synthetic set of axioms a la Euclid/Hilbert/Tarski?Symbolic coordinates for a hyperbolic grid?Hyperbolic (and related) structures on open unit diskWhat is the volume of the sphere in hyperbolic space?Non-equivalent metrics on $PSL_2(mathbbR)$Is there a relationship between the Cantor set and hyperbolic geometry?Translation in Poincare disc modelProve that a loxodromic transformation has an attractor and a repeller as fixed pointsSpheres in hyperbolic spacesExplicit isomorphisms between the hyperbolic plane and surfaces of constant negative curvature
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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 2 hours ago
liaombroliaombro
32428
$endgroup$
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 2 hours ago
liaombroliaombro
32428
$endgroup$
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 2 hours ago
liaombroliaombro
32428
$endgroup$
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 2 hours ago
liaombroliaombro
32428
asked 2 hours ago
liaombroliaombro
32428
asked 2 hours ago
liaombroliaombro
32428
asked 2 hours ago
liaombroliaombro
32428
asked 2 hours ago
liaombroliaombro
32428
32428
add a comment |
add a comment |
1 Answer
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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 $-frac1ell^2$; 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 2 hours ago
Lee MosherLee Mosher
50.8k33787
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
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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 $-frac1ell^2$; 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 2 hours ago
Lee MosherLee Mosher
50.8k33787
$endgroup$
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 $-frac1ell^2$; 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 2 hours ago
Lee MosherLee Mosher
50.8k33787
$endgroup$
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 $-frac1ell^2$; 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 2 hours ago
Lee MosherLee Mosher
50.8k33787
$endgroup$
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 $-frac1ell^2$; 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 2 hours ago
Lee MosherLee Mosher
50.8k33787
answered 2 hours ago
Lee MosherLee Mosher
50.8k33787
answered 2 hours ago
Lee MosherLee Mosher
50.8k33787
answered 2 hours ago
Lee MosherLee Mosher
50.8k33787
50.8k33787
add a comment |
add a comment |
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