Mixing
![Responsive DIVs in CesiumViewer [CesiumJS]](https://lh3.googleusercontent.com/-QmlSrDOA86U/AAAAAAAAAAI/AAAAAAAAAAA/j48MTtxnuCo/s72-c/photo.jpg?sz=32)
What is the name of a $n$-dimensional space where each dimension contains exactly 3 values?
$begingroup$
Can someone tell me whether this space has a name?
Each element of the space is an $n$-dimensional point (tuple) (vector, but not sure it's a vector space?) where each element of the tuple may be either $-1$, $0$, or $1$?
It seems to me more or less the corners of a $n$-dimensional hyper cube plus all the midpoints of the edges and $(n-j)$-dimensional faces, for $j=1,ldots,(n-1)$.
I will eventually want to talk about densities of sets of such points in the space, and also about measuring the homogeneity or clustering of a distribution of such a set of points.
general-topology group-theory vector-spaces
edited Feb 1 at 14:32
Ernie060
2,940719
asked Feb 1 at 9:24
Jim NewtonJim Newton
444
$endgroup$
add a comment |
$begingroup$
Can someone tell me whether this space has a name?
Each element of the space is an $n$-dimensional point (tuple) (vector, but not sure it's a vector space?) where each element of the tuple may be either $-1$, $0$, or $1$?
It seems to me more or less the corners of a $n$-dimensional hyper cube plus all the midpoints of the edges and $(n-j)$-dimensional faces, for $j=1,ldots,(n-1)$.
I will eventually want to talk about densities of sets of such points in the space, and also about measuring the homogeneity or clustering of a distribution of such a set of points.
general-topology group-theory vector-spaces
edited Feb 1 at 14:32
Ernie060
2,940719
asked Feb 1 at 9:24
Jim NewtonJim Newton
444
$endgroup$
1
$begingroup$
It's a bit unclear from your description but it sounds like you're talking about $(Bbb Z/3Bbb Z)^n$
$endgroup$
– Alessandro Codenotti
Feb 1 at 9:39
$begingroup$
I guess you simply mean ${-1,0,1}^n$ as subset of $Bbb R^n$. This doesn't have any individual name, as far as I know..
$endgroup$
– Berci
Feb 1 at 10:04
add a comment |
$begingroup$
Can someone tell me whether this space has a name?
Each element of the space is an $n$-dimensional point (tuple) (vector, but not sure it's a vector space?) where each element of the tuple may be either $-1$, $0$, or $1$?
It seems to me more or less the corners of a $n$-dimensional hyper cube plus all the midpoints of the edges and $(n-j)$-dimensional faces, for $j=1,ldots,(n-1)$.
I will eventually want to talk about densities of sets of such points in the space, and also about measuring the homogeneity or clustering of a distribution of such a set of points.
general-topology group-theory vector-spaces
edited Feb 1 at 14:32
Ernie060
2,940719
asked Feb 1 at 9:24
Jim NewtonJim Newton
444
$endgroup$
Can someone tell me whether this space has a name?
Each element of the space is an $n$-dimensional point (tuple) (vector, but not sure it's a vector space?) where each element of the tuple may be either $-1$, $0$, or $1$?
It seems to me more or less the corners of a $n$-dimensional hyper cube plus all the midpoints of the edges and $(n-j)$-dimensional faces, for $j=1,ldots,(n-1)$.
I will eventually want to talk about densities of sets of such points in the space, and also about measuring the homogeneity or clustering of a distribution of such a set of points.
general-topology group-theory vector-spaces
general-topology group-theory vector-spaces
edited Feb 1 at 14:32
Ernie060
2,940719
asked Feb 1 at 9:24
Jim NewtonJim Newton
444
edited Feb 1 at 14:32
Ernie060
2,940719
asked Feb 1 at 9:24
Jim NewtonJim Newton
444
edited Feb 1 at 14:32
Ernie060
2,940719
edited Feb 1 at 14:32
Ernie060
2,940719
edited Feb 1 at 14:32
Ernie060
2,940719
2,940719
asked Feb 1 at 9:24
Jim NewtonJim Newton
444
asked Feb 1 at 9:24
Jim NewtonJim Newton
444
asked Feb 1 at 9:24
Jim NewtonJim Newton
444
444
1
$begingroup$
It's a bit unclear from your description but it sounds like you're talking about $(Bbb Z/3Bbb Z)^n$
$endgroup$
– Alessandro Codenotti
Feb 1 at 9:39
$begingroup$
I guess you simply mean ${-1,0,1}^n$ as subset of $Bbb R^n$. This doesn't have any individual name, as far as I know..
$endgroup$
– Berci
Feb 1 at 10:04
add a comment |
1
$begingroup$
It's a bit unclear from your description but it sounds like you're talking about $(Bbb Z/3Bbb Z)^n$
$endgroup$
– Alessandro Codenotti
Feb 1 at 9:39
$begingroup$
I guess you simply mean ${-1,0,1}^n$ as subset of $Bbb R^n$. This doesn't have any individual name, as far as I know..
$endgroup$
– Berci
Feb 1 at 10:04
1
1
$begingroup$
It's a bit unclear from your description but it sounds like you're talking about $(Bbb Z/3Bbb Z)^n$
$endgroup$
– Alessandro Codenotti
Feb 1 at 9:39
$begingroup$
It's a bit unclear from your description but it sounds like you're talking about $(Bbb Z/3Bbb Z)^n$
$endgroup$
– Alessandro Codenotti
Feb 1 at 9:39
$begingroup$
I guess you simply mean ${-1,0,1}^n$ as subset of $Bbb R^n$. This doesn't have any individual name, as far as I know..
$endgroup$
– Berci
Feb 1 at 10:04
$begingroup$
I guess you simply mean ${-1,0,1}^n$ as subset of $Bbb R^n$. This doesn't have any individual name, as far as I know..
$endgroup$
– Berci
Feb 1 at 10:04
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
So you have a set of $3^n$ tuples $(x_1, x_2, dots, x_n)$ with $x_i in {-1,0,1}$. You can regard these as representing a set $S_n$ of $3^n$ points in $mathbb R^n$ or as representing the vector space $(mathbb{Z}/3mathbb{Z})^n$ of dimension $n$ over the field $mathbb{Z}/3mathbb{Z}$.
Note that these are not the same thing. For example, in $mathbb R^2$ the points in $S_2$ that lie on the line $sum x_i=1$ are the two points $(1,0)$ and $(0,1)$. But in $(mathbb{Z}/3mathbb{Z})^2$ the line $sum x_i=1$ also includes a third point $(-1,-1)$.
edited Feb 1 at 15:12
J. W. Tanner
4,6621420
answered Feb 1 at 10:22
gandalf61gandalf61
9,232825
$endgroup$
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%2f3096024%2fwhat-is-the-name-of-a-n-dimensional-space-where-each-dimension-contains-exactl%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
So you have a set of $3^n$ tuples $(x_1, x_2, dots, x_n)$ with $x_i in {-1,0,1}$. You can regard these as representing a set $S_n$ of $3^n$ points in $mathbb R^n$ or as representing the vector space $(mathbb{Z}/3mathbb{Z})^n$ of dimension $n$ over the field $mathbb{Z}/3mathbb{Z}$.
Note that these are not the same thing. For example, in $mathbb R^2$ the points in $S_2$ that lie on the line $sum x_i=1$ are the two points $(1,0)$ and $(0,1)$. But in $(mathbb{Z}/3mathbb{Z})^2$ the line $sum x_i=1$ also includes a third point $(-1,-1)$.
edited Feb 1 at 15:12
J. W. Tanner
4,6621420
answered Feb 1 at 10:22
gandalf61gandalf61
9,232825
$endgroup$
add a comment |
$begingroup$
So you have a set of $3^n$ tuples $(x_1, x_2, dots, x_n)$ with $x_i in {-1,0,1}$. You can regard these as representing a set $S_n$ of $3^n$ points in $mathbb R^n$ or as representing the vector space $(mathbb{Z}/3mathbb{Z})^n$ of dimension $n$ over the field $mathbb{Z}/3mathbb{Z}$.
Note that these are not the same thing. For example, in $mathbb R^2$ the points in $S_2$ that lie on the line $sum x_i=1$ are the two points $(1,0)$ and $(0,1)$. But in $(mathbb{Z}/3mathbb{Z})^2$ the line $sum x_i=1$ also includes a third point $(-1,-1)$.
edited Feb 1 at 15:12
J. W. Tanner
4,6621420
answered Feb 1 at 10:22
gandalf61gandalf61
9,232825
$endgroup$
add a comment |
$begingroup$
So you have a set of $3^n$ tuples $(x_1, x_2, dots, x_n)$ with $x_i in {-1,0,1}$. You can regard these as representing a set $S_n$ of $3^n$ points in $mathbb R^n$ or as representing the vector space $(mathbb{Z}/3mathbb{Z})^n$ of dimension $n$ over the field $mathbb{Z}/3mathbb{Z}$.
Note that these are not the same thing. For example, in $mathbb R^2$ the points in $S_2$ that lie on the line $sum x_i=1$ are the two points $(1,0)$ and $(0,1)$. But in $(mathbb{Z}/3mathbb{Z})^2$ the line $sum x_i=1$ also includes a third point $(-1,-1)$.
edited Feb 1 at 15:12
J. W. Tanner
4,6621420
answered Feb 1 at 10:22
gandalf61gandalf61
9,232825
$endgroup$
So you have a set of $3^n$ tuples $(x_1, x_2, dots, x_n)$ with $x_i in {-1,0,1}$. You can regard these as representing a set $S_n$ of $3^n$ points in $mathbb R^n$ or as representing the vector space $(mathbb{Z}/3mathbb{Z})^n$ of dimension $n$ over the field $mathbb{Z}/3mathbb{Z}$.
Note that these are not the same thing. For example, in $mathbb R^2$ the points in $S_2$ that lie on the line $sum x_i=1$ are the two points $(1,0)$ and $(0,1)$. But in $(mathbb{Z}/3mathbb{Z})^2$ the line $sum x_i=1$ also includes a third point $(-1,-1)$.
edited Feb 1 at 15:12
J. W. Tanner
4,6621420
answered Feb 1 at 10:22
gandalf61gandalf61
9,232825
edited Feb 1 at 15:12
J. W. Tanner
4,6621420
edited Feb 1 at 15:12
J. W. Tanner
4,6621420
edited Feb 1 at 15:12
J. W. Tanner
4,6621420
4,6621420
answered Feb 1 at 10:22
gandalf61gandalf61
9,232825
answered Feb 1 at 10:22
gandalf61gandalf61
9,232825
answered Feb 1 at 10:22
gandalf61gandalf61
9,232825
9,232825
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%2f3096024%2fwhat-is-the-name-of-a-n-dimensional-space-where-each-dimension-contains-exactl%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
1
$begingroup$
It's a bit unclear from your description but it sounds like you're talking about $(Bbb Z/3Bbb Z)^n$
$endgroup$
– Alessandro Codenotti
Feb 1 at 9:39
$begingroup$
I guess you simply mean ${-1,0,1}^n$ as subset of $Bbb R^n$. This doesn't have any individual name, as far as I know..
$endgroup$
– Berci
Feb 1 at 10:04