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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 ${\displaystyle 109.47^{\circ }}$ or something like that? Did you ever wonder what that came from? It is ${\displaystyle \cos ^{-1}(-{\frac {1}{3}})}$. And I will now show you why:

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

Some formulas from this page:

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

• The height of a regular ${\displaystyle n}$-simplex of side ${\displaystyle s}$ is

${\displaystyle h_{n}={\sqrt {\frac {n+1}{2n}}}s}$

• The apothem (inradius) of a regular ${\displaystyle n}$-simplex of side ${\displaystyle s}$ is

${\displaystyle a_{n}={\frac {h_{n}}{n+1}}}$

The circumradius, which is the difference between the height and the apothem, is:

${\displaystyle R=h_{n}-a_{n}={\frac {n}{n+1}}h_{n}={\sqrt {\frac {n}{2(n+1)}}}s}$

Now consider any two circumradii. They go to two different vertices, which must be joined by an edge of the ${\displaystyle n}$-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:

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

Here, ${\displaystyle a=b=R}$, and ${\displaystyle c=s}$.

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

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

Q.E.D.

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

hybridization	dimensions	angle between orbitals
sp	n = 1	${\displaystyle \cos ^{-1}(-{\frac {1}{1}})=180^{\circ }}$
sp2	n = 2	${\displaystyle \cos ^{-1}(-{\frac {1}{2}})=120^{\circ }}$
sp3	n = 3	${\displaystyle \cos ^{-1}(-{\frac {1}{3}})\approx 109.47^{\circ }}$

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

See also:

• http://www.thesquare.com/ts/signup/alumni-email.html