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4
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0answers
88 views

Icosahedral symmetry

Is there a clear and correct reference for full icosahedral symmetry that includes a presentation and its action on vertices, edges, and faces? The Wikipedia article for icoshedral symmetry gives ...
0
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0answers
142 views

how to get optimal vector, which is parallel to intersection line of many plane (Least Square way)

My idea is to construct the best optimal 3D line representing the intersection of many 3D planes. (As we know, due to fitting errors or data errors, the fitted planes might not intersect exactly ...
2
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0answers
130 views

Faces on a rotating cube

This one is probably a softball question, but... Fact: A cube has 6 faces Fact (unless my math is wrong): A cube that is cut in half center horizontal can be reoriented to have 18 unique faces. ...
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0answers
126 views

Creating polyhedra with playing cards

In here, George Hart gives some exciting examples on how to create polyhedra by cutting playing cards and sliding them inside one another. I was wondering if such an approach can be generalized to ...
0
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1answer
136 views

producing tetrahedra within a cube

A common math problem involves dividing a cube into regular and irregular tetrahedra, where the points of the cube must also be the points of the tetrahedra. A problem I'm working on seems to be ...
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3answers
284 views

What is the formula for a 3D line?

Just like we have the formula $y=mx+b$ for $\mathbb{R}^{2}$, what would be a formula for $\mathbb{R}^{3}$? Thanks.
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1answer
45 views

Help with school solid geometry

In the triangular pyramid $MABC$ all side edges equals $1$, $\angle AMB = \angle BMC = 60 ^\circ$, $\angle AMC = 45 ^\circ$. Find: 1) square of the $\triangle ABC$; 2) dihedral angle on the $AB$ ...
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0answers
59 views

Following a polyline along the surface of a polygon that is twisted

I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ...
0
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1answer
137 views

least square solution for obtaining a line 3d by intersecting many planes

If I have been given 4 planes and I know only a point (just a point lie on the plane e.g. o1,o2, o3 & o4) and normal vector of each plane (n1, n2, n3 & n4). Then, by intersecting all 4 (or ...
0
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0answers
42 views

Mymultiple image geometry

I have to work with multiple aerial images. the objective is to reconstruct 3d features. For a particular object, i want to find the images which are giving good viewing geometry than others. so ...
0
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2answers
214 views

Analytic geometry section of cone and sphere

How to show that the cone $yz+zx+xy=0$ cuts the sphere $x^2+y^2+z^2=a^2$ in two equal circle ? I understand that the two equations taken together represent the circle. but how to go about finding the ...
2
votes
1answer
96 views

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 ...
0
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1answer
102 views

What is the minimum number of blocks to build this?

A rectangular solid is built using $N$ cubes of a side length of 1cm. When viewed from such an angle such that only 3 of the sides of the rectangular solid are visible, there are 231 $cm^2$ of area ...
2
votes
1answer
99 views

3D Geometry Proof by Contradiction /Contrapositive (high school)

Could someone evaluate my work? A plane perpendicular to one of 2 parallel lines is perpendicular to the other line also. My two column proof so far: Let AB || CD and AB be perpendicular to plane ...
0
votes
1answer
104 views

Steepest slope gradient of a vertical plane

I know the steepest slope gradient (Azimuth) of a 3D plane can be obtained by projecting normal vector onto XY Plane. So, when the plane is slant, the steepest gradient will be a some value. ...
2
votes
1answer
80 views

Is the intersection of a bunch of cylinders a sphere?

Suppose we have a 3-D shape $S$ with a center $C$, so that a point $p$ is in $S$ if and only if for any direction $\vec d$, $p$ is contained within a cylinder of radius $1$, extending infinitely both ...
5
votes
2answers
326 views

Automorphism group and congruences of the cube

I want to prove that the automorphism group of the cube is $\mathbb{Z}_2 \times S_4$, by using information about the congruences of a cube. By the cube, I mean the graph of the platonic solid, i.e. ...
0
votes
1answer
65 views

Tetrahedralize Mesh

Say I have a triangle mesh which forms the shell of an object which may not be convex. For every triangle, I have the vertices and a normal. I want to turn this mesh into a solid. I want to break up ...
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2answers
845 views

How do I find the nearest point on a sphere?

Say I have a sphere of radius $6$ centered at $(3, 4, 5)$. What's the nearest point on the surface of the sphere to point $(1, 2, 3)$, which is within the sphere? I feel that this is a minimization ...
0
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2answers
138 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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1answer
49 views

Coordinates for vertices of the “silver” rhombohedron.

The "silver" rhombohedron (a.k.a the trigonal trapezohedron) is a three-dimensional object with six faces composed of congruent rhombi. You can see it visualised here. I am interested in replicating ...
7
votes
1answer
253 views

What tesselated three-dimensional shape gives the maximum volume with the minimum surface area?

I recently read an article on the future of buildings. I have long been interested in architecture and it seems to me that this article makes some very good points. It got me thinking about the ...
0
votes
1answer
1k views

Volume of a trapezoidal prism

A pile of ore has a rectangular base, 60 feet wide and 500 feet long. If the sides of the pile are all inclined 45degrees to the horizontal, and the ore weighs 110 lb. per cu.ft. Find the number of ...
0
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2answers
118 views

Sixth Platonic solid

A Sixth Platonic solid? [1] Wouldn't gluing a tetrahedron's one triangle to a another tetrahedron's triangle make a platonic solid ? See the picture to see clearly what I mean. Tetrahedron stacked ...
3
votes
1answer
891 views

Volume from revolving $e^x$ around $y$-axis

In school, I had a problem something like this: A region R is bounded by $x$-axis, $y$-axis, $x = 3$, and $y = e^x$. What is the volume of the solid produced by revolving it around the $y$-axis. ...
1
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1answer
57 views

What kind of solid has a face adjacency graph whose spanning trees are not feasible nets

Was reading an introductory graph theory book, and it says that nets of solids can be represented using adjacency graphs, and new nets can be discovered by searching for all the spanning trees of the ...
1
vote
1answer
290 views

Surface area of revolution about $y$-axis in terms of $f(x)$

I need to find a formula for the surface area of a solid of revolution rotated around the $y$-axis. The curve is $f(x)=x^2$ on $[0,1]$. However, my answer must be in terms of $f$, not $f^{-1}$.
0
votes
1answer
133 views

Relationship between angles in tetrahedron

Let's say I have a tetrahedron like this in image: Do angles $CAD$ and $CBD$ equals in general tetrahedron?
1
vote
1answer
160 views

Find volume of region using change of variables

I want to find the volume of the region $R$ that lies between $$z= x^2 + y^2, \quad z= 4(x^2 + y^2), \quad z = 1, \quad z = 4$$ Using the transformation \begin{align} x &= ...
22
votes
1answer
546 views

A problem of J. E. Littlewood

Many years ago I picked up a little book by J. E. Littlewood and was baffled by part of a question he posed: "Is it possible in 3-space for seven infinite circular cylinders of unit radius each to ...
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vote
2answers
127 views

name of this shape [3d solid]

what is the name of this 3d solid please? "faces" of 3 and 4 sides. Thanks!
11
votes
1answer
251 views

What hexahedra have faces with areas of exactly 1, 2, 3, 4, 5, and 6 units?

I tried for a while, not very hard, to construct a polyhedron with exactly six faces, whose areas were respectively 1, 2, 3, 4, 5, and 6 units. I did not meet with any success. Still, it seems that ...
0
votes
1answer
770 views

Parametric and implicit representation of a cone

http://mathworld.wolfram.com/Cone.html shows the parametric and implicit representation of a cone, I am wondering what the equation would look like if we also consider the bottom circle face for the ...
0
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0answers
124 views

Using Laser Distance Meter to calculate surface area of roof with minimum measurements required

Equipped with a Laser Distance Meter I am trying to calculate the total area of roofs (flat surfaces, not necessarily rectangular), and I am wondering how much data I need to collect in order to do ...
4
votes
2answers
316 views

Intersection of a Cone and Sphere

Show that a the cone $xy + yz + xz = 0$ cuts the sphere $x^2 + y^2 + z^2 = r^2$ into two equal circles and find their area. I have been trying to substitute one of the variables, say $z$, from the ...
0
votes
2answers
190 views

Surface of Cone

A cone is cut by a plane ($z=d$). ($0<d<c$) The top point of the cone is $A(a,b,c)$ Button of the Cone is a circle on $z=0$ plane and center of it is $O(0,0,0)$. If $a=b=0$ then ...
6
votes
1answer
175 views

Surfaces of constant projected area

Generalizing the well-known variety of plane curves of constant width, I'm wondering about three-dimensional surfaces of constant projected area. Question: If $A$ is a (bounded) subset of $\mathbb ...
4
votes
1answer
324 views

Volume and surface area of a sphere by polyhedral approximation

Exposition: In two dimensions, there is a (are many) straightforward explanation(s) of the fact that the perimeter (i.e. circumference) and area of a circle relate to the radius by $2\pi r$ and $\pi ...
5
votes
2answers
185 views

What is the volume of this 3d shape?

I'm wondering if there is an equation that represents the volume of an arbitrary 3d primitive matching this description: 1.) Point at center of sphere 2.) Each edge is the length of the radius 3.) 3 ...
7
votes
1answer
601 views

To find the volume of tetrahedron by using all surfaces areas?

I am looking for a formula: $V=f(S_1,S_2,S_3,S_4)$, where $S_1$, $S_2$, $S_3$, and $S_4$ are the areas of the four faces. We know ...
2
votes
1answer
220 views

Find volume of a revolved solid by integrating wedges.

So, lets say that I wanted to find the volume of the solid formed by rotating the area between $f(x)=\sqrt{1-x^2}, 0<x<1$ and the $x$ axis around the $y$ axis. (This example is simply a ...
9
votes
1answer
457 views

How would you make a (physical) dodecahedron with edges instead of faces?

Call this a "math problem disguised as a woodworking problem" or vice versa. Background: You can construct a dodecahedron by cutting 12 identical, regular pentagon faces, beveling all the edges of ...
3
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0answers
99 views

About the Platonic Solids in all dimensions

I am asking about the Platonic solids in all dimension, some reference about the proofs of many of the statement made in here. I would like to here about how to think about higher dimensions mainly ...
3
votes
3answers
395 views

Platonic Solids

It´s a theorem that there exist only five platonic solids ( up to similarity). I was searching some proofs of this, but I could not. I want to see some proof of this, specially one that uses ...
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4answers
1k views

How to cut a cube into an icosahedron?

Edit: Originally I asked this about a using a cube, but it is not a requirement to start with a cube, just how to end up with an icosahedron as on of the answers showed how to make dodecahedron a ...
9
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2answers
292 views

How to prove there are exactly eight convex deltahedra?

A deltahedron is a polyhedron whose faces are equilateral triangles. It is well-known that there are exactly eight convex deltahedra, and it is easy to find out that this was first proved by ...
0
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1answer
151 views

Polyhedron with $11$ faces

Show that there is no polyhedron with exactly $11$ faces such that each face is a polygon having an odd number of sides.
0
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1answer
2k views

Volume of a parabolic solid

We intend to find the volume of a solid described as follows: The $X$, $Y$ and $Z$ axes are such that the base of the solid is in the $XY$-plane and the vertical direction is parallel to the ...
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2answers
188 views

Shape/volume of this solid (don't know the name)

Here is the construction of the solid. Take an ellipse, make a copy of it, and put it on top of the original ellipse. Now turn the top ellipse by $90^\circ$ (quarter turn). Glue the two boundaries. I ...
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1answer
2k views

Ordering vertices in counter-clockwise manner in 3D space.

This is my first question in math and if I cannot get it right for the first time, please forgive me. I'm working on a simulation and I need to order vertices of a triangle in counter-clockwise ...