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#1 2005-09-02 11:01:54

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,552

tetrahedron: angle from center

When I was learning about angles of atoms in crystals in Chem 1 at WPI in Worcester, Massachusetts in 1986, the
tetrahedron was one of the 3-d configurations.  I don't remember why, but they wanted us to figure out the
angle from the center of the tetrahedron to any two of the vertices.  So I spent all weekend on that
and used complicated navigation formulas to get an answer.  Then I tried to figure out if this number
for the answer could be expressed simpler, and finally I found out the easy way.

I'll post the answer sometime later to give folks a chance to try it out themselves.  So to restate the problem, it
is the same as the angle between any legs on a tripod, if the tripod and a vertical rod form a tetrahedron.
Remember all four sides of a tetrahedron are equilateral triangles.

Last edited by John E. Franklin (2005-09-02 11:26:23)


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#2 2005-09-08 04:52:11

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,552

Re: tetrahedron: angle from center


igloo myrtilles fourmis

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#3 2005-09-08 09:22:05

MathsIsFun
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Registered: 2005-01-21
Posts: 7,529

Re: tetrahedron: angle from center

Take one for a spin


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#4 2005-09-14 10:15:45

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,552

Re: tetrahedron: angle from center

I've decided to give one more hint... 
If the measurement from a pointy corner to the center is one meter,
then how tall is the tetrahedron?


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#5 2005-10-04 04:35:05

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,552

Re: tetrahedron: angle from center

Well, now I'll give it away.
It turns out due to symmetry, that if you had four jet engines on the tetrahedron, one
on each vertice, then it doesn't move (in space, no gravity).  If you place the tetrahedron
so one corner is pointed exactly upward, and the bottom triangle is flat, then you can
say that the downward force caused by the top jet is equal to the upward force of the
bottom 3 jets.  Hence the upward components of the bottom three are one-third that of
the top one.  It then follows that the distance from the center of tetrahedron to the bottom
is 1/3 of a meter, if the distance from center to top is one meter.
Now I don't know how to prove the distances and forces are the same like I just mentioned,
but it turns out to be true.  Anyway, the rest is just a right triangle, two sides you know the
length of.  Then add the angle you get to 90 degrees to get the answer.  If anyone wants
a diagram, just ask.


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#6 2005-10-04 10:19:44

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,529

Re: tetrahedron: angle from center

The tetrahedron is so cool!

Maybe I could add that "jet engine" thing to my page on the tetrahedron. Bit of a challenge to sketch it, though.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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