Polyhedra with coplanar non-adjacent faces
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Two non-adjacent faces of a polyhedron are called $textit{buddies}$ if they lie on the same plane. Call a polyhedron $textit{nice}$ if every face has a buddy. What is the smallest $textit{nice}$ polyhedron?
Any ideas?
I tried the case for a $textit{nice}$ polygon and got the smallest polygon as the star (10 sides). This serves as a lower bound for the smallest $textit{nice}$ polyhedron, as any cross-section of this should give a $textit{nice}$ polygon.
recreational-mathematics solid-geometry platonic-solids
asked Dec 19 '18 at 21:21
Sudheesh SurendranathSudheesh Surendranath
18518
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add a comment |
$begingroup$
Two non-adjacent faces of a polyhedron are called $textit{buddies}$ if they lie on the same plane. Call a polyhedron $textit{nice}$ if every face has a buddy. What is the smallest $textit{nice}$ polyhedron?
Any ideas?
I tried the case for a $textit{nice}$ polygon and got the smallest polygon as the star (10 sides). This serves as a lower bound for the smallest $textit{nice}$ polyhedron, as any cross-section of this should give a $textit{nice}$ polygon.
recreational-mathematics solid-geometry platonic-solids
asked Dec 19 '18 at 21:21
Sudheesh SurendranathSudheesh Surendranath
18518
$endgroup$
1
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"Any cross-section should give a nice polygon?" If a plane cuts the polyhedron close to a vertex, it needn't intersect the polyhedron anywhere except close to the vertex, and then we'd have an ordinary convex polygon, wouldn't we? I'm having trouble understanding the quoted sentence. Probably I don't know what you mean by cross-section.
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– saulspatz
Dec 19 '18 at 23:07
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A non-intersecting stella octangula has 24 faces. But one can probably do better.
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– Aretino
Dec 20 '18 at 21:52
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You should probably give your definition of a polyhedron, since with plenty of definitions (e.g. convex hull of finitely many points, or the intersection of finitely many half-spaces) your desired conditions are impossible.
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– Kundor
Feb 13 at 19:50
add a comment |
$begingroup$
Two non-adjacent faces of a polyhedron are called $textit{buddies}$ if they lie on the same plane. Call a polyhedron $textit{nice}$ if every face has a buddy. What is the smallest $textit{nice}$ polyhedron?
Any ideas?
I tried the case for a $textit{nice}$ polygon and got the smallest polygon as the star (10 sides). This serves as a lower bound for the smallest $textit{nice}$ polyhedron, as any cross-section of this should give a $textit{nice}$ polygon.
recreational-mathematics solid-geometry platonic-solids
asked Dec 19 '18 at 21:21
Sudheesh SurendranathSudheesh Surendranath
18518
$endgroup$
Two non-adjacent faces of a polyhedron are called $textit{buddies}$ if they lie on the same plane. Call a polyhedron $textit{nice}$ if every face has a buddy. What is the smallest $textit{nice}$ polyhedron?
Any ideas?
I tried the case for a $textit{nice}$ polygon and got the smallest polygon as the star (10 sides). This serves as a lower bound for the smallest $textit{nice}$ polyhedron, as any cross-section of this should give a $textit{nice}$ polygon.
recreational-mathematics solid-geometry platonic-solids
recreational-mathematics solid-geometry platonic-solids
asked Dec 19 '18 at 21:21
Sudheesh SurendranathSudheesh Surendranath
18518
asked Dec 19 '18 at 21:21
Sudheesh SurendranathSudheesh Surendranath
18518
asked Dec 19 '18 at 21:21
Sudheesh SurendranathSudheesh Surendranath
18518
asked Dec 19 '18 at 21:21
Sudheesh SurendranathSudheesh Surendranath
18518
asked Dec 19 '18 at 21:21
Sudheesh SurendranathSudheesh Surendranath
18518
18518
1
$begingroup$
"Any cross-section should give a nice polygon?" If a plane cuts the polyhedron close to a vertex, it needn't intersect the polyhedron anywhere except close to the vertex, and then we'd have an ordinary convex polygon, wouldn't we? I'm having trouble understanding the quoted sentence. Probably I don't know what you mean by cross-section.
$endgroup$
– saulspatz
Dec 19 '18 at 23:07
$begingroup$
A non-intersecting stella octangula has 24 faces. But one can probably do better.
$endgroup$
– Aretino
Dec 20 '18 at 21:52
$begingroup$
You should probably give your definition of a polyhedron, since with plenty of definitions (e.g. convex hull of finitely many points, or the intersection of finitely many half-spaces) your desired conditions are impossible.
$endgroup$
– Kundor
Feb 13 at 19:50
add a comment |
1
$begingroup$
"Any cross-section should give a nice polygon?" If a plane cuts the polyhedron close to a vertex, it needn't intersect the polyhedron anywhere except close to the vertex, and then we'd have an ordinary convex polygon, wouldn't we? I'm having trouble understanding the quoted sentence. Probably I don't know what you mean by cross-section.
$endgroup$
– saulspatz
Dec 19 '18 at 23:07
$begingroup$
A non-intersecting stella octangula has 24 faces. But one can probably do better.
$endgroup$
– Aretino
Dec 20 '18 at 21:52
$begingroup$
You should probably give your definition of a polyhedron, since with plenty of definitions (e.g. convex hull of finitely many points, or the intersection of finitely many half-spaces) your desired conditions are impossible.
$endgroup$
– Kundor
Feb 13 at 19:50
1
1
$begingroup$
"Any cross-section should give a nice polygon?" If a plane cuts the polyhedron close to a vertex, it needn't intersect the polyhedron anywhere except close to the vertex, and then we'd have an ordinary convex polygon, wouldn't we? I'm having trouble understanding the quoted sentence. Probably I don't know what you mean by cross-section.
$endgroup$
– saulspatz
Dec 19 '18 at 23:07
$begingroup$
"Any cross-section should give a nice polygon?" If a plane cuts the polyhedron close to a vertex, it needn't intersect the polyhedron anywhere except close to the vertex, and then we'd have an ordinary convex polygon, wouldn't we? I'm having trouble understanding the quoted sentence. Probably I don't know what you mean by cross-section.
$endgroup$
– saulspatz
Dec 19 '18 at 23:07
$begingroup$
A non-intersecting stella octangula has 24 faces. But one can probably do better.
$endgroup$
– Aretino
Dec 20 '18 at 21:52
$begingroup$
A non-intersecting stella octangula has 24 faces. But one can probably do better.
$endgroup$
– Aretino
Dec 20 '18 at 21:52
$begingroup$
You should probably give your definition of a polyhedron, since with plenty of definitions (e.g. convex hull of finitely many points, or the intersection of finitely many half-spaces) your desired conditions are impossible.
$endgroup$
– Kundor
Feb 13 at 19:50
$begingroup$
You should probably give your definition of a polyhedron, since with plenty of definitions (e.g. convex hull of finitely many points, or the intersection of finitely many half-spaces) your desired conditions are impossible.
$endgroup$
– Kundor
Feb 13 at 19:50
add a comment |
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1
$begingroup$
"Any cross-section should give a nice polygon?" If a plane cuts the polyhedron close to a vertex, it needn't intersect the polyhedron anywhere except close to the vertex, and then we'd have an ordinary convex polygon, wouldn't we? I'm having trouble understanding the quoted sentence. Probably I don't know what you mean by cross-section.
$endgroup$
– saulspatz
Dec 19 '18 at 23:07
$begingroup$
A non-intersecting stella octangula has 24 faces. But one can probably do better.
$endgroup$
– Aretino
Dec 20 '18 at 21:52
$begingroup$
You should probably give your definition of a polyhedron, since with plenty of definitions (e.g. convex hull of finitely many points, or the intersection of finitely many half-spaces) your desired conditions are impossible.
$endgroup$
– Kundor
Feb 13 at 19:50