User:Spoon!

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Central angle in simplices between radii to vertices[edit]

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[edit]

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[edit]

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
 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[edit]

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[edit]

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[edit]

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:

Array Types Cross-Reference[edit]

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 N/A malloc'd array N/A
C++ new'd array std::vector
Java N/A N/A array ArrayList
Python N/A tuple N/A list
Objective-C Cocoa N/A NSArray N/A NSMutableArray
Go array N/A slice value mutable slice variable
Swift N/A Array value N/A mutable Array variable