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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 $c=s$ .

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

The angle we want is between $0$ 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:

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

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

Oxyanion chart

Hybridization	Orbital configuration	III	IV	V	VI	VII	VIII
sp (double)	AX_1.5E₀	B₂O₃ M₂O₃
sp	AX₂E₀ linear	AlO₂^- BO₂^-	CO₂ SiO₂ MO₂	NO₂⁺
sp	AX₁E₁		CO	NO⁺
sp² (double)	AX_2.5E₀		Si₂O₅^2-	N₂O₅ P₂O₅ As₂O₅ M₂O₅
	AX_1.5E₁			N₂O₃ As₂O₃
	AX_0.5E₂			N₂O
sp²	AX₃E₀ trigonal planar	BO₃^3-	CO₃^2- SiO₃^2- SnO₃^2- PbO₃^2-	NO₃^- VO₃^-	SO₃ O₄ SeO₃ MO₃
	AX₂E₁ bent <120		SnO₂^2- PbO₂^2-	NO₂^-	SO₂ O₃ SeO₂
	AX₁E₂				SO O₂
sp³ (double)	AX_3.5E₀		Si₂O₇^6-	P₂O₇^4-	Cr₂O₇^2-	Cl₂O₇ M₂O₇
	AX_2.5E₁
	AX_1.5E₂				S₂O₃^2-
	AX_0.5E₃					Cl₂O
sp³	AX₄E₀ tetrahedral		SiO₄^4-	PO₄^3- AsO₄^3-	SO₄^2- SeO₄^2- TeO₄^2- CrO₄^2- MoO₄^2- WO₄^2-	ClO₄^- BrO₄^- IO₄^- MnO₄^- TcO₄^- ReO₄^-	XeO₄ RuO₄ OsO₄
	AX₃E₁ trigonal pyramidal			PO₃^3- AsO₃^3-	SO₃^2- SeO₃^2- TeO₃^2-	ClO₃^- BrO₃^- IO₃^-	XeO₃
	AX₂E₂ bent <109.5			PO₂^3-	SO₂^2-	ClO₂^- BrO₂^- IO₂^-
	AX₁E₃				O₂^2-	ClO^- BrO^- IO^-

List of free alumni email services

University	Domain	Info	Notes
UC Berkeley	@cal.berkeley.edu	@cal	Lifetime forwarding
UCLA	@ucla.edu	BOL email forwarding	Lifetime forwarding
* UCLA Engineering School	@engineering.ucla.edu	Engineering email forwarding	Lifetime forwarding

Array Types Cross-Reference

Programming Language	Size set at compile time, mutable contents	Size set at runtime creation time, immutable contents	Size set at runtime creation time, mutable contents	Size growable after creation, mutable contents
C	static array	—	malloc'd array	—
C++	static array	—	new'd array	std::vector
Java	—	—	array	ArrayList
Python	—	tuple	—	list
Objective-C Cocoa	—	NSArray	—	NSMutableArray
Go	array	—	slice value	mutable slice variable
Swift	—	Array value	—	mutable Array variable

Spoon!

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