Questions tagged [solid-geometry]
In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space. (Ref: http://en.m.wikipedia.org/wiki/Solid_geometry)
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Polyhedron with least number of vertices whose diagonal faces enclose an interior solid region
We know in $\mathbb{R}^2$ a polygon with the least number of vertices whose diagonals enclose an interior region is a pentagon.
What about in $\mathbb{R}^3$? The least number of vertices that a ...
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Proving a Claim about four mutually tangent unit spheres
Prove the Claim about four mutually tangent unit spheres :
(1) The centers of each sphere lie at the vertices of a regular tetrahedron of edge length $2$
(2) Their points of tangency lie at the ...
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Prove that the center of the sphere, the centers of two small circles, and their single common point lie in the same plane
Two circles are drawn on a sphere, having a single common point.
Prove that the center of the sphere, the centers of both circles, and their common point lie in the same plane.
This is equivalent to :...
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Probability of rolling a specified face for Archimedean solids
Here it is shown that (for a "suitable" mathematical definition of fairness) there are no fair $n$-sides die with odd $n$. The question originated with fairness being defined as, among other,...
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What is the definition of a subdivision of a surface?
I am reviewing the Oxford Concise Dictionary of Mathematics (6th edition, 2021, edited by Richard Earl and James Nicholson) and I am pondering the entry "subdivision (of a surface)"
This ...
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What are the dihedral angles that an inscribed tetrahedron makes with its cube?
Inscribe a regular tetrahedron in a cube. What dihedral angles do its faces make with the faces of the cube?
Proposed Solution: The angles formed fall into two categories:
Where their intersection ...
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Does such a formula for solid angle exist?
Let there be a cube of side $40$ units whose center is at the origin of Cartesian coordinate system.
Let:
$(a,b,c)$ be any point outside the cube
$s,t,u \in \{ -,+ \}$
$A_{(s,t,u)} = a\ (s)\ 20$
$B_{(...
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Is there a term for rectangular skew lines?
Skew lines can either be "rectangular" (the edges in a cube) or "oblique") (the edges in a non-rectangular parallelepiped).
The difference can be important. Is there a name for ...
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Cross-section of a square pyramid makes what?
Let $ABCD$ be a square, and $PABCD$ be a equilateral pyramid. Cut $PABCD$ with a plane going through point $C$ and perpendicular to $PD$. What resulting shapes do you get?
Partial solution:
Let $Q$ ...
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What can be said about an equilateral pyramid with quadrilateral base?
All edges of the quadrangular pyramid $PABCD$ are equal. What can be said about $ABCD$?
Proposed solution: $ABCD$ is a square.
Proof: $ABCD$ is clearly a rhombus. Let $Q$ be the orthogonal projection ...
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How to develop a cone into a sector using synthetic geometry only?
Consider a right circular cone with radius $r$ and slant height $s$. Its surface area is
$$
A = \pi r s.
$$
Proof: It suffices to show that the cone can be sliced and unwrapped, without deformation, ...
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Linking probability with 3D space filling complete graphs
Problem. Let $t$ hard-core spheres of diameter $D$ be placed i.i.d. uniformly in the unit cube. Every pair is joined by a rigid cylinder of the same diameter $D$. Spheres and cylinders are mutually ...
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Finding the points on a cone where the tangent plane contains the line formed by intersecting two given planes
There are several resources (especially Calculus books) that talk about finding tangent planes to a given (level surface) with certain prescribed conditions. I was trying to make a general question ...
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Planar geometry problems solved by thinking out of the plane
What are some examples of a planar geometry problem with an elegant nonplanar solution?
I will post one example as an answer.
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How to find volume of solid body which is formed by right-handed triplet of vectors in first coordinate octant?
I want to find volume of shape on picture below. Red vector is $\vec{a}$, green is $\vec{b}$ and blue is $\vec{c}$. Vectors are right-handed and located in first coordinate octant. I've suggested that ...
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