What is Tetrahedron: Definition and 79 Discussions
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex.
The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".
Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces.
Hi, I'm calculating the work done by regular tetrahedron during taking from the water by crane (USING INTEGRALS). I don't know how bad is that solved so if anyone checks my work and gives me some advice or hints I would be very glad.
Everything is written in the PDF file.
There were given...
My approach to this problem is to recognize that the tetrahedron being still means that net torque is zero and net force is zero.
Fd is given
Fa + Fb + Fc = -Fd
Fa X a + Fb X b + Fc X c = <0,0,0>
This can be split up into a series of 6 equations, 2 for each component.
However, this is where I...
Hey! 😊
Let one of the four sides of a tetrahedron be red, one blue, one green and the fourth side painted with all three colours. We consider the following events :
A: = The tetrahedron falls on a side with red colour.
B: = The tetrahedron falls on a side with a blue colour.
C: = The...
Hi, 2 part question trying to get tetrahedron Finite Element shape functions working: 1) How do I properly setup the shape coefficient matrix and 2) How do I build the coefficient quantities in the shape functions properly? ANY tips or corrections may unblock me and would be of much value...
For a tetrahedron with four spin (1/2) particles, I know there are three separate energy levels at $$l=2,l=1,and l=0$$. My question is how I would go about finding the degeneracy of each level. I know that the number of states must be $$2^4$$. Any clues on where to start would be appreciated...
I suck at geometry, but I have this intuitive notion that the points on the corners of a regular tetrahedron are all equidistant. How do I go about proving this true (or false, if I'm wrong)? Note that the highest geometry class I've taken is high school, but I'm okay with any undergraduate...
The pioneering work by G. Lehmann, M. Taut, please see the attached files or download from wiley
On the Numerical Calculation of the Density of States and Related Properties,
http://onlinelibrary.wiley.com/doi/10.1002/pssb.2220540211/abstract
The question is how the middle line of Eq. (3.9)...
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$\textrm{Imagine $3$ unit spheres
(radius equal to 1) with centers at,}\\$
$\textrm{$O(0,0,0)$, $P(\sqrt{3},-1,0)$ and $Q(\sqrt{3},1,0)$.} \\$
$\textrm{Now place another unit sphere symmetrically on top of these spheres with its center at R.} \\$
$\textrm{a Find the...
Hi, I am having trouble understanding why three vectors that lie in the same plane can't form a tetrahedron. If the plane is somewhat vertical or titlted will it not be possible for one vector to higher up than another so that you have a difference in height? Also, for three vectors to form a...
Homework Statement
As part of a longer problem:
"Find necessary and sufficient conditions for the point with positionvector r to lie inside, or on, the tetrahedron formed by the vertices 0, a, b and c."
Homework Equations
I am not sure... vector addtion?
The Attempt at a Solution
I don't...
This question was originally posted by ElConquistador, but in my haste I mistakenly deleted it as a duplicate. My apologies...
For part (a) we can define two cyclic subgroups of order $2$, both normal, $\langle J\rangle$ and $\langle K\rangle$ such that $V=\langle J\rangle \langle K\rangle$...
Homework Statement
Calculate the volume integral of the function $$f(x,y,z)=xyz^2$$
over the tetrahedron with corners at $$(0,0,1) (1,0,0) (0,1,0) (0,0,1)$$
Homework Equations
I was able to solve it mathematically, but still can't figure out why the answer is so small.
I only understand...
Homework Statement
Volume of tetrahedron T.ABC = V
Point P is on the middle of TA, Q is on the expansion of AB making AQ = 2AB
A shape is made through PQ which is parallel to BC so that it cuts the tetahedron into 2 pieces.
What is the volume of the biggest piece?
The Attempt at a Solution
I...
Homework Statement
Let S be the tetrahedron in \mathbb{R}^3 having vertices (0,0,0), (1,2,3), (0,1,2), and (-1,1,1). Evaluate \int_S f where f(x,y,z) = x + 2y - z.
Homework EquationsThe Attempt at a Solution
I just want to confirm that I am setting up the integral properly: Looking at the...
I already found how to calculate the volume of tetrahedron from 4 vertices, i.e. V = 1/6(dot(d1,D), where D = cross(d2,d3).
Could somebody specify the formula or an article for volume of tetrahedron using 5 vertices, A = (x1, y1, z1), B = (x2, y2, z2), C = (x3, y3, z3), D = (x4, y4, z4) and O =...
See the image that I uploaded...
I want to write the surface S (bounded by edges u, v and w) in terms of x, y and z, u, v and w and A, B and C. And I got it!
See:
S(A,B,C) = \sqrt{A^2+B^2+C^2}
S(x,y,z) = \sqrt{\frac{1}{4}( (yz)^2 + (zx)^2 + (xy)^2 )}
S(u,v,w) =...
Homework Statement
What is the largest possible radius of a sphere which is inscribed in a regular tetrahedron
a=10 ( this is the side of the tetrahedron)
r=?
r=5*√6/6
Homework EquationsThe Attempt at a Solution
So first I calculated the Height of pyramid
a2=(2/3*va)2+h2
h=√(a2-(2/3*a*√3/2)2)...
I am new to physics, and my studies have taken me to this question. Mathetmatically, the tetrahedron is essentially the building block of geometries, does this make it then the building block of our universe? Though I understand this hasn't been proven and we haven't seen this, if mathematics is...
How to find out the position vector of the centroid of tetrahedron , the position vectors of whose vertices are a,b,c,d respectively.
I am familiar with the result, namely a+b+c+d/4 but want to know how to derive it without using the 3:1 ratio property.
Any help would be appreciated. Thank you.
Homework Statement
4 ants are arranged in such a way that they make up vertex of a regular tetrahedron, of side length 1m . The ants are named Calvin , Peter , David and Aron. Each ant moves at a speed 1m/s , and moves in such a way that:
Calvin moves toward Peter,
Peter moves toward...
Homework Statement
By using triple integral, find the volume of the tetrahedron bounded by the coordinate planes and the plane 2x+3y+2z=6.Homework Equations
Volume= ∫vdv=∫∫∫dxdydz
The Attempt at a Solution
find intercepts of the plane on the axes,
x-intercept=3
y-intercept=2...
Homework Statement
Prove that if bonding-pair repulsions were maximized in CH3X, then the sum of the bond angles would be 450°.
Homework Equations
In a perfect tetrahedral molecule (e.g. methane), the sum of the bond angles is about 438 degrees (109.5° times 4).
The Attempt at a...
I have a unique problem that I'm struggling with with regards to surveying.
Because my surveying equipment is much more accurate at measuring angles than distances I'd like to find an analytic solution using only the angular measurements.
Let the surveyor sit at the origin of the...
Homework Statement
Four charged particles (A, B, C, D), of mass m and charge q each, are connected by light silk threads of length d forming a tetrahedron floating in outer space. The thread connecting particles A and B suddenly snaps. Find the maximum speed of particle A after that.
The...
Homework Statement
Suppose a point charge is placed on the face of a regular tetrahedron. What is the flux through another one of the faces?
Homework Equations
The Attempt at a Solution
I know that if the charge is placed at the center of the tetrahedron, the flux through any face...
Homework Statement
Set up an integral to find the volume of the tetrahedron with vertices
(0,0,0), (2,1,0), (0,2,0), (0,0,3).Homework Equations
The Attempt at a Solution
My method of solving this involves using a triple integral. The first step is deciding on the bounds of the triple integral...
Problem:
Suppose in a tetrahedron ABCD, AB=1; CD=$\sqrt{3}$; the distance and the angle between the skew lines AB and CD are 2 and $\pi/3$ respectively. Find the volume of tetrahedron.
Attempt:
Let the points A,B,C and D be represented by the vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$...
Homework Statement
An astronaut in the International Space-station attaches himself to the four vertices of a regular tetrahedron shaped frame with 4 springs. The mass of the springs and their rest length are negligible, their spring constants are ##D_1=150 N/m##, ##D_2=250 N/m##, ##D_3=300...
Greetings,
I have taken a probability course a year ago; however my mind is a bit rusty and I cannot recall the concepts. I want to be able to calculate the probability distribution function for the following question:
Suppose you have a tedrahedron with number 1,2,3,4 on respective faces...
Hello,
I made this video with my iphone depicting the Symmetries of a Tetrahedron for a presentation I did recently:
I have been searching and trying to figure out if I have presented it correctly that a Tetrahedron has full S4 symmetry if we could reflect it in a "higher dimension."
I was...
This is my first post and I wish to get help in finding an analytically way to get the coordinate of an irregular tetrahedron.
let ABCD be the 4 vertices of the tetrahedron in 3D, all vertices have different (x,y,z).
the coordinate of vertex D is known (Xd,Yd,Zd), and the 3 angle between...
Here's a picture of an irregular tetrahedron, for reference:
The base triangle (ABC) is completely known (lengths AB, AC, BC, and the angles between them are all known). The 3 angles at vertex P are also known (APB, APC, and BPC).
I believe that is enough information to completely solve the...
Homework Statement
Actually, the problem was addressed in a prior post:
https://www.physicsforums.com/showthread.php?t=178250
Which is closed.Homework EquationsI would like to know how HallsofIvy (or anyone) arrived at the formula for the tetrahedron given the vertices (1,0,0), (0,2,0)...
Homework Statement
I am trying to find the face angles on a tetrahedron. I have only the base edge lengths, the angles connected the base edges and an (approximate) height of the top/non-base vertex. I might be able to extrapolate other information from images of the base of the tetrahedron...
Hello all! (I'm new to the forum)
I'd like to ask if you could give me a simple explanation regarding the 6j symbols: I don't understand their formulation in terms of Clebsch-Gordan coefficients with three angular momenta and the related "tetrahedron rule".
Alternatively, could you please...
Prove that, if (c - b).a = 0 and (c - a).b = 0, then (b - a).c = 0. Show that this can be used to prove the following geometric results:
a. The lines through the vertices of a triangle ABC perpendicular to the opposite sides meet in a point.
b. If the tetrahedron OABC has two pairs of...
Prove that, if (c - b).a = 0 and (c - a).b = 0, then (b - a).c = 0. Show that this can be used to prove the following geometric results:
a. The lines through the vertices of a triangle ABC perpendicular to the opposite sides meet in a point.
b. If the tetrahedron OABC has two pairs of...
A tetrahedron has an equilateral triangle base with 20-cm-long edges and
three equilateral triangle sides. The base is parallel to the ground and a
vertical uniform electric ﬁeld of strength 200 N/C passes upward through
the object.
(a) What is the electric ﬂux through the base?
(b) What is...
Varying Gravitational Field - Invariant Tetrahedron??
Classical Theory of Fields, Landau Lifgarbagez, page 246:
"Strictly speaking, the number of particles should be greater than four. Since we can construct a tetrahedron from any six line segments, we can always, by a suitable definition of...
I'm dealing with a problem that seems (to my uneducated mind) like it should be more or less straightforward, but for some reason I've been unable to find any help on forums that are geared towards high school and college level math. Please forgive me if the solution is obvious.
If I know...
Homework Statement
Calculate the volume integral of the function T = z^2 over the tetrahedron with corners at
(0,0,0), (1,0,0) , (0,1,0), and (0,0,1)
The Attempt at a Solution
z to x (1,0,-1)
z to y (0,1,-1)
Then i crossed them to get (1,1,1)
Found the plane n dot (x-1, y , z) =...
I was skimming through the book "The Divine Proportion a Study in Mathematical Beauty" by H.E. Huntley and found an interesting passage labeled "The Tetrahedron Problem." The problem is stated like this:
The faces of a tetrahedron are all scalene triangles similar to one another, but not all...
Hi,
I have 4 non-planar points
P1 = ( x1 , y1 , z1 )
P2 = ( x2 , y2 , z2 )
P3 = ( x3 , y3 , z3 )
P4 = ( x4 , y4 , z4 )
what is the coordinate of center of mass of
the object ( tetrahedron ) whose vertices
are P1 P2 P3 and P4? ( uniform density )
Thanks
Homework Statement
Evaluate the triple integral \int\int\int^{}_{E} xy dV where E is the tetrahedron (0,0,0),(3,0,0),(0,5,0),(0,0,6).
Is there a simple way to simplify the integration?
Homework Equations
The Attempt at a Solution
\frac{z}{6} + \frac{y}{5} + \frac{x}{3} = 1
z =...
I have to integrate this function:
f(x,y,z)=y+x
Over the region S which is a tetrahedron defined by points (0,0,0), (2,0,0), (0,2,0), (0,0,2).
So after I drew it out I saw that three of the faces were right up against the XZ, YZ, and XY planes. I'm getting stuck on parameterizing the...
Homework Statement
A regular tetrahedron has the vertices of its base A(1,1,0) B(3,1,0) C(2,1+(3^(1/2),0). Find coordinate of vertex S?
Homework Equations
The Attempt at a Solution
If this is a tetrahedron
Then we know the length by caclulating the distance formula, which gives...
Homework Statement
Given a general (not necessarily a rectangular) tetrahedron, let V1, V2, V3, V4 denote vectors whose lengths are equal to the areas of the four faces, and whose directions are perpendicular to these faces and point outward. Show that:
V1 + V2 + V3 + V4 = 0.
The Attempt...
I want to make a tetrahedron in mathematica. For example, suppose:
{a = 0.2, b = 0.5, c = 0.1, d = 0.3}
{a = 0.1, b = 0.2, c = 0.4, d = 0.3}
{a = 0.4, b = 0.3, c = 0.2, d = 0.1}
{a = 0.6, b = 0.2, c = 0.1, d = 0.1}
and I want a tetrahedrom so that the four points are a, b, c and d -...