User:Spoon!

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=Central angle in simplices between radii to vertices= I know that many people have figured this out long ago, but I like to share it anyhow, because I have wondered about it for a long time when I was in high school...

You know how in high school they told you that the angle between two bonds in methane was about $$109.47^\circ$$ or something like that? Did you ever wonder what that came from? It is $$\cos^{-1} (-\frac{1}{3})$$. And I will now show you why:

Theorem
If circumradii are drawn between the center of an $$n$$-simplex and its vertices, the angle between these segments is $$\cos^{-1} (-\frac{1}{n})$$.

Proof
Some formulas from this page:

http://www.math.rutgers.edu/~erowland/polytopes.html#sectionII


 * The height of a regular $$n$$-simplex of side $$s$$ is
 * $$ h_n = \sqrt{ \frac{n+1}{2n} } s $$


 * The apothem (inradius) of a regular $$n$$-simplex of side $$s$$ is
 * $$ a_n = \frac{h_n}{n+1} $$

The circumradius, which is the difference between the height and the apothem, is:
 * $$ R = h_n - a_n = \frac{n}{n+1} h_n = \sqrt{ \frac{n}{2(n+1)} } s $$

Now consider any two circumradii. They go to two different vertices, which must be joined by an edge of the $$n$$-simplex, forming a triangle. Because we know the lengths of all sides of this triangle, we can find the angle between the circumradii using the law of cosines:


 * $$ \cos{\gamma} = \frac{a^2 + b^2 - c^2}{2ab} $$

Here, $$a = b = R$$, and $$c = s$$.

$$\begin{align} \cos{\gamma} &=& \frac{R^2 + R^2 - s^2}{2R^2} \\ &=& 1 - \frac{1}{2} \left(\frac{s}{R}\right)^2 \\ &=& 1 - \frac{1}{2} \cdot \frac{2(n+1)}{n} \\ &=& 1 - \frac{n+1}{n} \\ &=& -\frac{1}{n} \\ \gamma &=& \cos^{-1} (-\frac{1}{n}) \\ \end{align}$$

The angle we want is between $$0$$ and $$180^\circ$$. Since cosine is one-to-one in that range, the angle is uniquely determined.

Q.E.D.

Conclusion
This explains, among other things, the angles between hybridized orbitals:

That's all the simplices that can fit in 3 dimensions, folks; but you see the pattern...

=Oxyanion chart=

=List of free alumni email services= See also:
 * http://www.thesquare.com/ts/signup/alumni-email.html

=Array Types Cross-Reference=