Having a cube, with a point at its center. What are the points that are equidistant from the center point to...
$begingroup$
Having a cube, with a point at its center.
What shape do the points wich are equidistant between the center and the cubes vertices make?
The source of why I had this question is the following photo
What shape is resultant from this composition of equidistant points?
Thank you very much.
geometry
New contributor
$endgroup$
add a comment |
$begingroup$
Having a cube, with a point at its center.
What shape do the points wich are equidistant between the center and the cubes vertices make?
The source of why I had this question is the following photo
What shape is resultant from this composition of equidistant points?
Thank you very much.
geometry
New contributor
$endgroup$
$begingroup$
What do you mean by shape? Should the connectivity be preserved?
$endgroup$
– lightxbulb
2 hours ago
$begingroup$
They form the vertices of a cube.
$endgroup$
– TonyK
2 hours ago
1
$begingroup$
@lightxbulb If you look at the original question, you'll see that what's actually being asked for is the volume of the set of all points that are closer to the center of a cube than to any of its vertices. OP wishes to understand the boundary surface of this region.
$endgroup$
– MJD
2 hours ago
$begingroup$
I never expected to learn any math when I watched Assasination Classroom, but seing that problem, and the elegant solution to finding the volume was cool. Actually depicting math in progress (dramaticized though that sequence was) is sadly a rarity in popular culture.
$endgroup$
– Arthur
2 hours ago
$begingroup$
@TonyK Eight of them do, but I think we're after the whole surface . . .
$endgroup$
– timtfj
2 hours ago
add a comment |
$begingroup$
Having a cube, with a point at its center.
What shape do the points wich are equidistant between the center and the cubes vertices make?
The source of why I had this question is the following photo
What shape is resultant from this composition of equidistant points?
Thank you very much.
geometry
New contributor
$endgroup$
Having a cube, with a point at its center.
What shape do the points wich are equidistant between the center and the cubes vertices make?
The source of why I had this question is the following photo
What shape is resultant from this composition of equidistant points?
Thank you very much.
geometry
geometry
New contributor
New contributor
edited 2 hours ago
Blue
47.8k870152
47.8k870152
New contributor
asked 2 hours ago
Duero CuadrilleroDuero Cuadrillero
111
111
New contributor
New contributor
$begingroup$
What do you mean by shape? Should the connectivity be preserved?
$endgroup$
– lightxbulb
2 hours ago
$begingroup$
They form the vertices of a cube.
$endgroup$
– TonyK
2 hours ago
1
$begingroup$
@lightxbulb If you look at the original question, you'll see that what's actually being asked for is the volume of the set of all points that are closer to the center of a cube than to any of its vertices. OP wishes to understand the boundary surface of this region.
$endgroup$
– MJD
2 hours ago
$begingroup$
I never expected to learn any math when I watched Assasination Classroom, but seing that problem, and the elegant solution to finding the volume was cool. Actually depicting math in progress (dramaticized though that sequence was) is sadly a rarity in popular culture.
$endgroup$
– Arthur
2 hours ago
$begingroup$
@TonyK Eight of them do, but I think we're after the whole surface . . .
$endgroup$
– timtfj
2 hours ago
add a comment |
$begingroup$
What do you mean by shape? Should the connectivity be preserved?
$endgroup$
– lightxbulb
2 hours ago
$begingroup$
They form the vertices of a cube.
$endgroup$
– TonyK
2 hours ago
1
$begingroup$
@lightxbulb If you look at the original question, you'll see that what's actually being asked for is the volume of the set of all points that are closer to the center of a cube than to any of its vertices. OP wishes to understand the boundary surface of this region.
$endgroup$
– MJD
2 hours ago
$begingroup$
I never expected to learn any math when I watched Assasination Classroom, but seing that problem, and the elegant solution to finding the volume was cool. Actually depicting math in progress (dramaticized though that sequence was) is sadly a rarity in popular culture.
$endgroup$
– Arthur
2 hours ago
$begingroup$
@TonyK Eight of them do, but I think we're after the whole surface . . .
$endgroup$
– timtfj
2 hours ago
$begingroup$
What do you mean by shape? Should the connectivity be preserved?
$endgroup$
– lightxbulb
2 hours ago
$begingroup$
What do you mean by shape? Should the connectivity be preserved?
$endgroup$
– lightxbulb
2 hours ago
$begingroup$
They form the vertices of a cube.
$endgroup$
– TonyK
2 hours ago
$begingroup$
They form the vertices of a cube.
$endgroup$
– TonyK
2 hours ago
1
1
$begingroup$
@lightxbulb If you look at the original question, you'll see that what's actually being asked for is the volume of the set of all points that are closer to the center of a cube than to any of its vertices. OP wishes to understand the boundary surface of this region.
$endgroup$
– MJD
2 hours ago
$begingroup$
@lightxbulb If you look at the original question, you'll see that what's actually being asked for is the volume of the set of all points that are closer to the center of a cube than to any of its vertices. OP wishes to understand the boundary surface of this region.
$endgroup$
– MJD
2 hours ago
$begingroup$
I never expected to learn any math when I watched Assasination Classroom, but seing that problem, and the elegant solution to finding the volume was cool. Actually depicting math in progress (dramaticized though that sequence was) is sadly a rarity in popular culture.
$endgroup$
– Arthur
2 hours ago
$begingroup$
I never expected to learn any math when I watched Assasination Classroom, but seing that problem, and the elegant solution to finding the volume was cool. Actually depicting math in progress (dramaticized though that sequence was) is sadly a rarity in popular culture.
$endgroup$
– Arthur
2 hours ago
$begingroup$
@TonyK Eight of them do, but I think we're after the whole surface . . .
$endgroup$
– timtfj
2 hours ago
$begingroup$
@TonyK Eight of them do, but I think we're after the whole surface . . .
$endgroup$
– timtfj
2 hours ago
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
The answer to the question from screenshot is a truncated octahedron. If you consider center of the cube and a vertex, the locus of equidistant points is a perpendicular bisector plane. 8 of those planes create an octahedron. However, if you consider centers of other cubes, locus of equidistant points with them will give the initial cube itself. So the answer is the intersection of cube and octahedron, which is a truncated octahedron.
$endgroup$
add a comment |
$begingroup$
The shape you get is called a truncated octahedron:
The resulting tesselation (tiling) pattern is called a bitruncated cubic honeycomb:
(Both images taken from the Wikipedia article on the truncated octahedron.)
$endgroup$
add a comment |
$begingroup$
Just to add some further notions to the already given answers: Those describe the Voronoi domain resp. Voronoi complex of the body centered cubical (bcc) lattice. - Within crystallography the Voronoi domain also is called Brillouin zone.
--- rk
$endgroup$
add a comment |
$begingroup$
Within the pic of OP there is the quest for the volume of $D(0)$ as well. $D(0)$ was already being mentioned to be the truncated octahedron. Thus the searched for volume is
$$V=8sqrt 2 s^3$$
where $s$ is the edge length of that polyhedron. That length in turn can be seen from that very pic to be (in units of the lattice constant $a$)
$$s=frac 12 sqrt 2 a$$
Thus the searched for total value would become
$$V=4 a^3$$
--- rk
$endgroup$
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
});
}
});
Duero Cuadrillero is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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%2f3078296%2fhaving-a-cube-with-a-point-at-its-center-what-are-the-points-that-are-equidist%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The answer to the question from screenshot is a truncated octahedron. If you consider center of the cube and a vertex, the locus of equidistant points is a perpendicular bisector plane. 8 of those planes create an octahedron. However, if you consider centers of other cubes, locus of equidistant points with them will give the initial cube itself. So the answer is the intersection of cube and octahedron, which is a truncated octahedron.
$endgroup$
add a comment |
$begingroup$
The answer to the question from screenshot is a truncated octahedron. If you consider center of the cube and a vertex, the locus of equidistant points is a perpendicular bisector plane. 8 of those planes create an octahedron. However, if you consider centers of other cubes, locus of equidistant points with them will give the initial cube itself. So the answer is the intersection of cube and octahedron, which is a truncated octahedron.
$endgroup$
add a comment |
$begingroup$
The answer to the question from screenshot is a truncated octahedron. If you consider center of the cube and a vertex, the locus of equidistant points is a perpendicular bisector plane. 8 of those planes create an octahedron. However, if you consider centers of other cubes, locus of equidistant points with them will give the initial cube itself. So the answer is the intersection of cube and octahedron, which is a truncated octahedron.
$endgroup$
The answer to the question from screenshot is a truncated octahedron. If you consider center of the cube and a vertex, the locus of equidistant points is a perpendicular bisector plane. 8 of those planes create an octahedron. However, if you consider centers of other cubes, locus of equidistant points with them will give the initial cube itself. So the answer is the intersection of cube and octahedron, which is a truncated octahedron.
answered 2 hours ago
Vasily MitchVasily Mitch
1,51337
1,51337
add a comment |
add a comment |
$begingroup$
The shape you get is called a truncated octahedron:
The resulting tesselation (tiling) pattern is called a bitruncated cubic honeycomb:
(Both images taken from the Wikipedia article on the truncated octahedron.)
$endgroup$
add a comment |
$begingroup$
The shape you get is called a truncated octahedron:
The resulting tesselation (tiling) pattern is called a bitruncated cubic honeycomb:
(Both images taken from the Wikipedia article on the truncated octahedron.)
$endgroup$
add a comment |
$begingroup$
The shape you get is called a truncated octahedron:
The resulting tesselation (tiling) pattern is called a bitruncated cubic honeycomb:
(Both images taken from the Wikipedia article on the truncated octahedron.)
$endgroup$
The shape you get is called a truncated octahedron:
The resulting tesselation (tiling) pattern is called a bitruncated cubic honeycomb:
(Both images taken from the Wikipedia article on the truncated octahedron.)
edited 2 hours ago
answered 2 hours ago
ArthurArthur
112k7109191
112k7109191
add a comment |
add a comment |
$begingroup$
Just to add some further notions to the already given answers: Those describe the Voronoi domain resp. Voronoi complex of the body centered cubical (bcc) lattice. - Within crystallography the Voronoi domain also is called Brillouin zone.
--- rk
$endgroup$
add a comment |
$begingroup$
Just to add some further notions to the already given answers: Those describe the Voronoi domain resp. Voronoi complex of the body centered cubical (bcc) lattice. - Within crystallography the Voronoi domain also is called Brillouin zone.
--- rk
$endgroup$
add a comment |
$begingroup$
Just to add some further notions to the already given answers: Those describe the Voronoi domain resp. Voronoi complex of the body centered cubical (bcc) lattice. - Within crystallography the Voronoi domain also is called Brillouin zone.
--- rk
$endgroup$
Just to add some further notions to the already given answers: Those describe the Voronoi domain resp. Voronoi complex of the body centered cubical (bcc) lattice. - Within crystallography the Voronoi domain also is called Brillouin zone.
--- rk
answered 1 hour ago
Dr. Richard KlitzingDr. Richard Klitzing
1,54616
1,54616
add a comment |
add a comment |
$begingroup$
Within the pic of OP there is the quest for the volume of $D(0)$ as well. $D(0)$ was already being mentioned to be the truncated octahedron. Thus the searched for volume is
$$V=8sqrt 2 s^3$$
where $s$ is the edge length of that polyhedron. That length in turn can be seen from that very pic to be (in units of the lattice constant $a$)
$$s=frac 12 sqrt 2 a$$
Thus the searched for total value would become
$$V=4 a^3$$
--- rk
$endgroup$
add a comment |
$begingroup$
Within the pic of OP there is the quest for the volume of $D(0)$ as well. $D(0)$ was already being mentioned to be the truncated octahedron. Thus the searched for volume is
$$V=8sqrt 2 s^3$$
where $s$ is the edge length of that polyhedron. That length in turn can be seen from that very pic to be (in units of the lattice constant $a$)
$$s=frac 12 sqrt 2 a$$
Thus the searched for total value would become
$$V=4 a^3$$
--- rk
$endgroup$
add a comment |
$begingroup$
Within the pic of OP there is the quest for the volume of $D(0)$ as well. $D(0)$ was already being mentioned to be the truncated octahedron. Thus the searched for volume is
$$V=8sqrt 2 s^3$$
where $s$ is the edge length of that polyhedron. That length in turn can be seen from that very pic to be (in units of the lattice constant $a$)
$$s=frac 12 sqrt 2 a$$
Thus the searched for total value would become
$$V=4 a^3$$
--- rk
$endgroup$
Within the pic of OP there is the quest for the volume of $D(0)$ as well. $D(0)$ was already being mentioned to be the truncated octahedron. Thus the searched for volume is
$$V=8sqrt 2 s^3$$
where $s$ is the edge length of that polyhedron. That length in turn can be seen from that very pic to be (in units of the lattice constant $a$)
$$s=frac 12 sqrt 2 a$$
Thus the searched for total value would become
$$V=4 a^3$$
--- rk
answered 48 mins ago
Dr. Richard KlitzingDr. Richard Klitzing
1,54616
1,54616
add a comment |
add a comment |
Duero Cuadrillero is a new contributor. Be nice, and check out our Code of Conduct.
Duero Cuadrillero is a new contributor. Be nice, and check out our Code of Conduct.
Duero Cuadrillero is a new contributor. Be nice, and check out our Code of Conduct.
Duero Cuadrillero is a new contributor. Be nice, and check out our Code of Conduct.
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');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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%2f3078296%2fhaving-a-cube-with-a-point-at-its-center-what-are-the-points-that-are-equidist%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');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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');
var $window = $(window),
onScroll = function(e) {
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom)) {
StackExchange.using('gps', function() { StackExchange.gps.track('embedded_signup_form.view', { location: 'question_page' }); });
$window.unbind('scroll', onScroll);
}
};
$window.on('scroll', onScroll);
});
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
$begingroup$
What do you mean by shape? Should the connectivity be preserved?
$endgroup$
– lightxbulb
2 hours ago
$begingroup$
They form the vertices of a cube.
$endgroup$
– TonyK
2 hours ago
1
$begingroup$
@lightxbulb If you look at the original question, you'll see that what's actually being asked for is the volume of the set of all points that are closer to the center of a cube than to any of its vertices. OP wishes to understand the boundary surface of this region.
$endgroup$
– MJD
2 hours ago
$begingroup$
I never expected to learn any math when I watched Assasination Classroom, but seing that problem, and the elegant solution to finding the volume was cool. Actually depicting math in progress (dramaticized though that sequence was) is sadly a rarity in popular culture.
$endgroup$
– Arthur
2 hours ago
$begingroup$
@TonyK Eight of them do, but I think we're after the whole surface . . .
$endgroup$
– timtfj
2 hours ago