User:Spoon!

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This user attends or attended the
University of California,
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This user exploits the pigeonhole principle.

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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)	AX1.5E0	B2O3 M2O3
sp	AX2E0 linear	AlO2- BO2-	CO2 SiO2 MO2	NO2+
	AX1E1		CO	NO+
sp2 (double)	AX2.5E0		Si2O52-	N2O5 P2O5 As2O5 M2O5
	AX1.5E1			N2O3 As2O3
	AX0.5E2			N2O
sp2	AX3E0 trigonal planar	BO33-	CO32- SiO32- SnO32- PbO32-	NO3- VO3-	SO3 O4 SeO3 MO3
	AX2E1 bent <120		SnO22- PbO22-	NO2-	SO2 O3 SeO2
	AX1E2				SO O2
sp3 (double)	AX3.5E0		Si2O76-	P2O74-	Cr2O72-	Cl2O7 M2O7
	AX2.5E1
	AX1.5E2				S2O32-
	AX0.5E3					Cl2O
sp3	AX4E0 tetrahedral		SiO44-	PO43- AsO43-	SO42- SeO42- TeO42- CrO42- MoO42- WO42-	ClO4- BrO4- IO4- MnO4- TcO4- ReO4-	XeO4 RuO4 OsO4
	AX3E1 trigonal pyramidal			PO33- AsO33-	SO32- SeO32- TeO32-	ClO3- BrO3- IO3-	XeO3
	AX2E2 bent <109.5			PO23-	SO22-	ClO2- BrO2- IO2-
	AX1E3				O22-	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

See also:

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

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++			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