Questions related to polyhedra and their properties.
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What is the minimal isoperimetric ratio of a polyhedron with $5$ vertices?
I'm asking and answering this question to provide a partial answer to this question and a comment on this answer at MO.
The isoperimetric ratio $\mu$ of a solid is the ratio $A^3/V^2$, where $A$ is ...
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35 views
Maximal volume for given surface area of an $n$-hedron
Is there a term for a polyhedron with $n$ faces (or, similarly, $n$ vertices) that maximises the enclosed volume for a given surface area (equivalently, minimises the surface area for a given volume)?
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Distance to a convex polyhedron: about different approaches
I know there are a lot of litterature out there about convex polyhedra and distance computation, but I don't quite catch which one has the best computational complexity in practice and in theory. I ...
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Which polyhedra have an even number of faces touching each vertex?
A two-coloring of the faces of a polyhedron is always possible when an even number of faces meet at each vertex.
http://www.georgehart.com/virtual-polyhedra/colorings.html
Is there a name for ...
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Sphere containment problem inside a rational convex polytope of general dimensions.
Given a positive number $r$ and a rational convex polytope (bounded polyhedra) described by its set of half-planes (system of linear inequalities: $A\cdot x \leq b$, where $A\in\mathbb{R}^{m\times ...
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79 views
Is There a Formalization of Cauchy's $F - E+V = 2$ proof?
Can anyone provide, or direct me to a formalized version of Cauchy's proof that for any convex polyhedron with $F$ faces, $E$ edges and $V$ vertices that $F - E + V = 2$. I am willing to accept the ...
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Representing point of Polyhedra
is there a result which says that any point in convex polyhedra in dimension $N$ can be uniquely represented by $N+1$ points of polyhedra? What is the name of this result?
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Mappings preserving convex polyhedra
It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra.
Can you give a characterization of the class of mappings that preserve convex polyhedra?
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35 views
Convex Polyhedra with Largest Constant Dihedral Angle for Given Number of Faces
Which $20$ faced convex polyhedron has the largest constant dihedral angle?
Which $24$ faced convex polyhedron has the largest constant dihedral angle?
Also, what about $30$ or $32$ faced polyhedra? ...
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2answers
210 views
Calculate Spherical Distance between points
I have googled this and not come up with an answer yet, but basically, I'm trying to find out the distance between each point or vertice on a sphere (all points are evenly spaced).
I already know this ...
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70 views
Is an unit-cube polyhedron? What about other platonic solids?
Definitions
According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
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23 views
Polyhedral metric
How do I prove that the metric determined by shortest paths on a convex polyhedron is a polyhedral metric but the metric determined by great circular arcs on a sphere is not polyhedral?
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27 views
Maximum Number of Divisions in Octahedron into congruent parts?
I am trying to divide octahedron into congruent parts. I found octahedron inside tetrahedron sided by four smaller tetrahedrons. I found some division here to 12 congruent parts. I can divide ...
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20 views
Smallest amount of planes to enclose a closed space in extended projective geometry $\mathbb R^3_{\pm\infty}$
The smallest amount of planes to enclose a polyhedron is 4 in the euclidean $\mathbb R^3$ where it encloses a tetrahedron. What is the smallest amount of planes to enclose a closed space in extended ...
