Mixing
3-polytope with 9 vertices and 8 facets
$begingroup$
My guess is that this is a 3-polytope, since the existence of the two facets {357} and {048} rules out the possibility of dimensionality 2 and 4.
How can I go about sketching it?
polytopes discrete-geometry
asked Jan 30 at 12:17
ensbanaensbana
252214
$endgroup$
add a comment |
$begingroup$
My guess is that this is a 3-polytope, since the existence of the two facets {357} and {048} rules out the possibility of dimensionality 2 and 4.
How can I go about sketching it?
polytopes discrete-geometry
asked Jan 30 at 12:17
ensbanaensbana
252214
$endgroup$
add a comment |
$begingroup$
My guess is that this is a 3-polytope, since the existence of the two facets {357} and {048} rules out the possibility of dimensionality 2 and 4.
How can I go about sketching it?
polytopes discrete-geometry
asked Jan 30 at 12:17
ensbanaensbana
252214
$endgroup$
My guess is that this is a 3-polytope, since the existence of the two facets {357} and {048} rules out the possibility of dimensionality 2 and 4.
How can I go about sketching it?
polytopes discrete-geometry
polytopes discrete-geometry
asked Jan 30 at 12:17
ensbanaensbana
252214
asked Jan 30 at 12:17
ensbanaensbana
252214
asked Jan 30 at 12:17
ensbanaensbana
252214
asked Jan 30 at 12:17
ensbanaensbana
252214
asked Jan 30 at 12:17
ensbanaensbana
252214
252214
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
There are many ways of sketching it, but they all are quite similar. I started with the triangle ${3, 5, 7}$:
Then I looked at all facets (faces, really) which shared a side with this one. They are the ones in the top row. I filled them in to get this:
(Note that the top triangle isn't a face; it's "open".) Then I took the remaining three quadrilateral faces, put them on:
And thus the sketch was finished.
answered Jan 30 at 12:38
ArthurArthur
122k7122210
$endgroup$
$begingroup$
So the vertices of a facet does not necessarily follow the order specified in the question? (e.g. {1567} should really be {1657} or {1756}?)
$endgroup$
– ensbana
Jan 30 at 12:43
1
$begingroup$
@ensbana The vertices for each face are listed in increasing order, with no actual geometrical significance, yes.
$endgroup$
– Arthur
Jan 30 at 12:44
add a comment |
$begingroup$
Identify shared edges, and use that to draw a graph of the polytope connections.
For instance, 3-5-7 and 1-2-3-7 have the shared edge 3-7. So, putting 3-5-7 in the center of your graph, you have 3 and 7 next to 1 and 2 in some order. Similarly 3 and 5 are next to 2 and 6, and 5 and 7 ate next to 1 and 6. This combibation is consistent with the graph having 3 directly connected to 2, 5 connected to 6, and 7 connected to 1. You also connect 1, 2 and 6 to each other to close the quadrilateral faces.
Now you have the open edges 1-2, 1-6, and 2-6. Find the faces that share those edges and draw them in using similar logic to that above. Your complete graph now shows the two triangular faces (3-5-7 in the center and 0-4-8 in the exterior region) sandwiched around two layers of quadrilaterals. Can you see now what sort of polyhedron has layers of quadrilaterals capped by triangles at opposite ends?
answered Jan 30 at 13:00
Oscar LanziOscar Lanzi
13.5k12136
$endgroup$
add a comment |
$begingroup$
The sketch of @arthur already shows the shape.
If you'd set up the 6 tetragons as trapezia, you'd recognize this figure to be a bitruncated triagular bipyramid.
--- rk
answered Jan 31 at 12:38
Dr. Richard KlitzingDr. Richard Klitzing
1,77026
$endgroup$
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%2f3093455%2f3-polytope-with-9-vertices-and-8-facets%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
There are many ways of sketching it, but they all are quite similar. I started with the triangle ${3, 5, 7}$:
Then I looked at all facets (faces, really) which shared a side with this one. They are the ones in the top row. I filled them in to get this:
(Note that the top triangle isn't a face; it's "open".) Then I took the remaining three quadrilateral faces, put them on:
And thus the sketch was finished.
answered Jan 30 at 12:38
ArthurArthur
122k7122210
$endgroup$
$begingroup$
So the vertices of a facet does not necessarily follow the order specified in the question? (e.g. {1567} should really be {1657} or {1756}?)
$endgroup$
– ensbana
Jan 30 at 12:43
1
$begingroup$
@ensbana The vertices for each face are listed in increasing order, with no actual geometrical significance, yes.
$endgroup$
– Arthur
Jan 30 at 12:44
add a comment |
$begingroup$
There are many ways of sketching it, but they all are quite similar. I started with the triangle ${3, 5, 7}$:
Then I looked at all facets (faces, really) which shared a side with this one. They are the ones in the top row. I filled them in to get this:
(Note that the top triangle isn't a face; it's "open".) Then I took the remaining three quadrilateral faces, put them on:
And thus the sketch was finished.
answered Jan 30 at 12:38
ArthurArthur
122k7122210
$endgroup$
$begingroup$
So the vertices of a facet does not necessarily follow the order specified in the question? (e.g. {1567} should really be {1657} or {1756}?)
$endgroup$
– ensbana
Jan 30 at 12:43
1
$begingroup$
@ensbana The vertices for each face are listed in increasing order, with no actual geometrical significance, yes.
$endgroup$
– Arthur
Jan 30 at 12:44
add a comment |
$begingroup$
There are many ways of sketching it, but they all are quite similar. I started with the triangle ${3, 5, 7}$:
Then I looked at all facets (faces, really) which shared a side with this one. They are the ones in the top row. I filled them in to get this:
(Note that the top triangle isn't a face; it's "open".) Then I took the remaining three quadrilateral faces, put them on:
And thus the sketch was finished.
answered Jan 30 at 12:38
ArthurArthur
122k7122210
$endgroup$
There are many ways of sketching it, but they all are quite similar. I started with the triangle ${3, 5, 7}$:
Then I looked at all facets (faces, really) which shared a side with this one. They are the ones in the top row. I filled them in to get this:
(Note that the top triangle isn't a face; it's "open".) Then I took the remaining three quadrilateral faces, put them on:
And thus the sketch was finished.
answered Jan 30 at 12:38
ArthurArthur
122k7122210
answered Jan 30 at 12:38
ArthurArthur
122k7122210
answered Jan 30 at 12:38
ArthurArthur
122k7122210
answered Jan 30 at 12:38
ArthurArthur
122k7122210
122k7122210
$begingroup$
So the vertices of a facet does not necessarily follow the order specified in the question? (e.g. {1567} should really be {1657} or {1756}?)
$endgroup$
– ensbana
Jan 30 at 12:43
1
$begingroup$
@ensbana The vertices for each face are listed in increasing order, with no actual geometrical significance, yes.
$endgroup$
– Arthur
Jan 30 at 12:44
add a comment |
$begingroup$
So the vertices of a facet does not necessarily follow the order specified in the question? (e.g. {1567} should really be {1657} or {1756}?)
$endgroup$
– ensbana
Jan 30 at 12:43
1
$begingroup$
@ensbana The vertices for each face are listed in increasing order, with no actual geometrical significance, yes.
$endgroup$
– Arthur
Jan 30 at 12:44
$begingroup$
So the vertices of a facet does not necessarily follow the order specified in the question? (e.g. {1567} should really be {1657} or {1756}?)
$endgroup$
– ensbana
Jan 30 at 12:43
$begingroup$
So the vertices of a facet does not necessarily follow the order specified in the question? (e.g. {1567} should really be {1657} or {1756}?)
$endgroup$
– ensbana
Jan 30 at 12:43
1
1
$begingroup$
@ensbana The vertices for each face are listed in increasing order, with no actual geometrical significance, yes.
$endgroup$
– Arthur
Jan 30 at 12:44
$begingroup$
@ensbana The vertices for each face are listed in increasing order, with no actual geometrical significance, yes.
$endgroup$
– Arthur
Jan 30 at 12:44
add a comment |
$begingroup$
Identify shared edges, and use that to draw a graph of the polytope connections.
For instance, 3-5-7 and 1-2-3-7 have the shared edge 3-7. So, putting 3-5-7 in the center of your graph, you have 3 and 7 next to 1 and 2 in some order. Similarly 3 and 5 are next to 2 and 6, and 5 and 7 ate next to 1 and 6. This combibation is consistent with the graph having 3 directly connected to 2, 5 connected to 6, and 7 connected to 1. You also connect 1, 2 and 6 to each other to close the quadrilateral faces.
Now you have the open edges 1-2, 1-6, and 2-6. Find the faces that share those edges and draw them in using similar logic to that above. Your complete graph now shows the two triangular faces (3-5-7 in the center and 0-4-8 in the exterior region) sandwiched around two layers of quadrilaterals. Can you see now what sort of polyhedron has layers of quadrilaterals capped by triangles at opposite ends?
answered Jan 30 at 13:00
Oscar LanziOscar Lanzi
13.5k12136
$endgroup$
add a comment |
$begingroup$
Identify shared edges, and use that to draw a graph of the polytope connections.
For instance, 3-5-7 and 1-2-3-7 have the shared edge 3-7. So, putting 3-5-7 in the center of your graph, you have 3 and 7 next to 1 and 2 in some order. Similarly 3 and 5 are next to 2 and 6, and 5 and 7 ate next to 1 and 6. This combibation is consistent with the graph having 3 directly connected to 2, 5 connected to 6, and 7 connected to 1. You also connect 1, 2 and 6 to each other to close the quadrilateral faces.
Now you have the open edges 1-2, 1-6, and 2-6. Find the faces that share those edges and draw them in using similar logic to that above. Your complete graph now shows the two triangular faces (3-5-7 in the center and 0-4-8 in the exterior region) sandwiched around two layers of quadrilaterals. Can you see now what sort of polyhedron has layers of quadrilaterals capped by triangles at opposite ends?
answered Jan 30 at 13:00
Oscar LanziOscar Lanzi
13.5k12136
$endgroup$
add a comment |
$begingroup$
Identify shared edges, and use that to draw a graph of the polytope connections.
For instance, 3-5-7 and 1-2-3-7 have the shared edge 3-7. So, putting 3-5-7 in the center of your graph, you have 3 and 7 next to 1 and 2 in some order. Similarly 3 and 5 are next to 2 and 6, and 5 and 7 ate next to 1 and 6. This combibation is consistent with the graph having 3 directly connected to 2, 5 connected to 6, and 7 connected to 1. You also connect 1, 2 and 6 to each other to close the quadrilateral faces.
Now you have the open edges 1-2, 1-6, and 2-6. Find the faces that share those edges and draw them in using similar logic to that above. Your complete graph now shows the two triangular faces (3-5-7 in the center and 0-4-8 in the exterior region) sandwiched around two layers of quadrilaterals. Can you see now what sort of polyhedron has layers of quadrilaterals capped by triangles at opposite ends?
answered Jan 30 at 13:00
Oscar LanziOscar Lanzi
13.5k12136
$endgroup$
Identify shared edges, and use that to draw a graph of the polytope connections.
For instance, 3-5-7 and 1-2-3-7 have the shared edge 3-7. So, putting 3-5-7 in the center of your graph, you have 3 and 7 next to 1 and 2 in some order. Similarly 3 and 5 are next to 2 and 6, and 5 and 7 ate next to 1 and 6. This combibation is consistent with the graph having 3 directly connected to 2, 5 connected to 6, and 7 connected to 1. You also connect 1, 2 and 6 to each other to close the quadrilateral faces.
Now you have the open edges 1-2, 1-6, and 2-6. Find the faces that share those edges and draw them in using similar logic to that above. Your complete graph now shows the two triangular faces (3-5-7 in the center and 0-4-8 in the exterior region) sandwiched around two layers of quadrilaterals. Can you see now what sort of polyhedron has layers of quadrilaterals capped by triangles at opposite ends?
answered Jan 30 at 13:00
Oscar LanziOscar Lanzi
13.5k12136
answered Jan 30 at 13:00
Oscar LanziOscar Lanzi
13.5k12136
answered Jan 30 at 13:00
Oscar LanziOscar Lanzi
13.5k12136
answered Jan 30 at 13:00
Oscar LanziOscar Lanzi
13.5k12136
13.5k12136
add a comment |
add a comment |
$begingroup$
The sketch of @arthur already shows the shape.
If you'd set up the 6 tetragons as trapezia, you'd recognize this figure to be a bitruncated triagular bipyramid.
--- rk
answered Jan 31 at 12:38
Dr. Richard KlitzingDr. Richard Klitzing
1,77026
$endgroup$
add a comment |
$begingroup$
The sketch of @arthur already shows the shape.
If you'd set up the 6 tetragons as trapezia, you'd recognize this figure to be a bitruncated triagular bipyramid.
--- rk
answered Jan 31 at 12:38
Dr. Richard KlitzingDr. Richard Klitzing
1,77026
$endgroup$
add a comment |
$begingroup$
The sketch of @arthur already shows the shape.
If you'd set up the 6 tetragons as trapezia, you'd recognize this figure to be a bitruncated triagular bipyramid.
--- rk
answered Jan 31 at 12:38
Dr. Richard KlitzingDr. Richard Klitzing
1,77026
$endgroup$
The sketch of @arthur already shows the shape.
If you'd set up the 6 tetragons as trapezia, you'd recognize this figure to be a bitruncated triagular bipyramid.
--- rk
answered Jan 31 at 12:38
Dr. Richard KlitzingDr. Richard Klitzing
1,77026
answered Jan 31 at 12:38
Dr. Richard KlitzingDr. Richard Klitzing
1,77026
answered Jan 31 at 12:38
Dr. Richard KlitzingDr. Richard Klitzing
1,77026
answered Jan 31 at 12:38
Dr. Richard KlitzingDr. Richard Klitzing
1,77026
1,77026
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%2f3093455%2f3-polytope-with-9-vertices-and-8-facets%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
