Magnifying glass in hyperbolic space
$begingroup$
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 9 hours ago
liaombroliaombro
359210
$endgroup$
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 9 hours ago
liaombroliaombro
359210
$endgroup$
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 9 hours ago
liaombroliaombro
359210
$endgroup$
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 9 hours ago
liaombroliaombro
359210
asked 9 hours ago
liaombroliaombro
359210
asked 9 hours ago
liaombroliaombro
359210
asked 9 hours ago
liaombroliaombro
359210
asked 9 hours ago
liaombroliaombro
359210
359210
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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 9 hours ago
Lee MosherLee Mosher
50.9k33888
$endgroup$
2
$begingroup$
I believe, your link currently does not work as intended
$endgroup$
– WorldSEnder
3 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 5 hours ago
Henning MakholmHenning Makholm
242k17308550
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3154386%2fmagnifying-glass-in-hyperbolic-space%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 9 hours ago
Lee MosherLee Mosher
50.9k33888
$endgroup$
2
$begingroup$
I believe, your link currently does not work as intended
$endgroup$
– WorldSEnder
3 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 9 hours ago
Lee MosherLee Mosher
50.9k33888
$endgroup$
2
$begingroup$
I believe, your link currently does not work as intended
$endgroup$
– WorldSEnder
3 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 9 hours ago
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 9 hours ago
Lee MosherLee Mosher
50.9k33888
answered 9 hours ago
Lee MosherLee Mosher
50.9k33888
answered 9 hours ago
Lee MosherLee Mosher
50.9k33888
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
add a comment |
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
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 5 hours ago
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 5 hours ago
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 5 hours ago
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 5 hours ago
Henning MakholmHenning Makholm
242k17308550
answered 5 hours ago
Henning MakholmHenning Makholm
242k17308550
answered 5 hours ago
Henning MakholmHenning Makholm
242k17308550
answered 5 hours ago
Henning MakholmHenning Makholm
242k17308550
242k17308550
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3154386%2fmagnifying-glass-in-hyperbolic-space%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
