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

==Theorem==
If circumradii are drawn between the center of an <math>n</math>-[[simplex]] and its vertices, the angle between these segments is <math>\cos^{-1} (-\frac{1}{n})</math>.

==Proof==
Some formulas from this page:

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

* The height of a regular <math>n</math>-simplex of side <math>s</math> is
:<math> h_n = \sqrt{ \frac{n+1}{2n} } s </math>

* The ''[[apothem]]'' ([[inradius]]) of a regular <math>n</math>-simplex of side <math>s</math> is
:<math> a_n = \frac{h_n}{n+1} </math>

The ''[[circumradius]]'', which is the difference between the height and the apothem, is:
:<math> R = h_n - a_n = \frac{n}{n+1} h_n = \sqrt{ \frac{n}{2(n+1)} } s </math>

Now consider any two circumradii.  They go to two different vertices, which must be joined by an edge of the <math>n</math>-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:

:<math> \cos{\gamma} = \frac{a^2 + b^2 - c^2}{2ab} </math>

Here, <math>a = b = R</math>, and <math>c = s</math>.

<math>\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}</math>

The angle we want is between <math>0</math> and <math>180^\circ</math>.  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:

{| class=wikitable
!hybridization || dimensions || angle between orbitals
|-
| sp || n = 1 || <math>\cos^{-1} (-\frac{1}{1}) = 180^\circ</math>
|-
| sp<sup>2</sup> || n = 2 || <math>\cos^{-1} (-\frac{1}{2}) = 120^\circ</math>
|-
| sp<sup>3</sup> || n = 3 || <math>\cos^{-1} (-\frac{1}{3}) \approx 109.47^\circ</math>
|}

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


=Oxyanion chart=
{| class=wikitable
!
Hybridization
!
Orbital configuration
!
III
!
IV
!
V
!
VI
!
VII
!
VIII
|- valign=top bgcolor=#ddddff
!
sp (double)
! align=left |AX<sub>1.5</sub>E<sub>0</sub>
|[[boron trioxide|B<sub>2</sub>O<sub>3</sub>]] M<sub>2</sub>O<sub>3</sub>
|- valign=top
! rowspan = 2 |
sp
! align=left |AX<sub>2</sub>E<sub>0</sub> linear
|[[aluminate|AlO<sub>2</sub><sup>-</sup>]] [[borate|BO<sub>2</sub><sup>-</sup>]]
|bgcolor=#ddffdd|[[carbon dioxide|CO<sub>2</sub>]] [[silicon dioxide|SiO<sub>2</sub>]] MO<sub>2</sub>
|[[nitronium ion|NO<sub>2</sub><sup>+</sup>]]
|-
! align=left |AX<sub>1</sub>E<sub>1</sub>
|
|[[carbon monoxide|CO]]
|[[nitrosonium|NO<sup>+</sup>]]
|- valign=top bgcolor=#ddddff
! rowspan = 3 |
sp<sup>2</sup> (double)
! align=left |AX<sub>2.5</sub>E<sub>0</sub>
|
|bgcolor=#ddffdd|[[disilicate|Si<sub>2</sub>O<sub>5</sub><sup>2-</sup>]]
|bgcolor=#ddffdd|[[dinitrogen pentoxide|N<sub>2</sub>O<sub>5</sub>]] [[diphosphorus pentoxide|P<sub>2</sub>O<sub>5</sub>]] [[arsenic pentoxide|As<sub>2</sub>O<sub>5</sub>]] M<sub>2</sub>O<sub>5</sub>
|- bgcolor=#ddddff
! align=left |AX<sub>1.5</sub>E<sub>1</sub>
|
|
|bgcolor=#ddffdd|[[dinitrogen trioxide|N<sub>2</sub>O<sub>3</sub>]] [[arsenic trioxide|As<sub>2</sub>O<sub>3</sub>]]
|- bgcolor=#ddddff
! align=left |AX<sub>0.5</sub>E<sub>2</sub>
|
|
|[[dinitrogen monoxide|N<sub>2</sub>O]]
|- valign=top
! rowspan = 3 |
sp<sup>2</sup>
! align=left |AX<sub>3</sub>E<sub>0</sub> trigonal planar
|[[borate|BO<sub>3</sub><sup>3-</sup>]]
|bgcolor=#ddffdd|[[carbonate|CO<sub>3</sub><sup>2-</sup>]] [[metasilicate|SiO<sub>3</sub><sup>2-</sup>]] [[stannate|SnO<sub>3</sub><sup>2-</sup>]] [[plumbate|PbO<sub>3</sub><sup>2-</sup>]]
|bgcolor=#ddffdd|[[nitrate|NO<sub>3</sub><sup>-</sup>]] [[vanadate|VO<sub>3</sub><sup>-</sup>]]
|bgcolor=#ddffdd|[[sulfur trioxide|SO<sub>3</sub>]] [[tetraoxygen|O<sub>4</sub>]] [[selenium trioxide|SeO<sub>3</sub>]] MO<sub>3</sub>
|-
! align=left |AX<sub>2</sub>E<sub>1</sub> bent <120
|
|[[stannite|SnO<sub>2</sub><sup>2-</sup>]] [[plumbite|PbO<sub>2</sub><sup>2-</sup>]]
|bgcolor=#ddffdd|[[nitrite|NO<sub>2</sub><sup>-</sup>]]
|bgcolor=#ddffdd|[[sulfur dioxide|SO<sub>2</sub>]] [[ozone|O<sub>3</sub>]] [[selenium dioxide|SeO<sub>2</sub>]]
|-
! align=left |AX<sub>1</sub>E<sub>2</sub>
|
|
|
|bgcolor=#ddffdd|[[sulfur monoxide|SO]] [[oxygen|O<sub>2]]</sub>
|- valign=top bgcolor=#ddddff
! rowspan = 4 |
sp<sup>3</sup> (double)
! align=left |AX<sub>3.5</sub>E<sub>0</sub>
|
|bgcolor=#ddffdd|[[pyrosilicate|Si<sub>2</sub>O<sub>7</sub><sup>6-</sup>]]
|bgcolor=#ddffdd|[[pyrophosphate|P<sub>2</sub>O<sub>7</sub><sup>4-</sup>]]
|bgcolor=#ddffdd|[[dichromate|Cr<sub>2</sub>O<sub>7</sub><sup>2-</sup>]]
|bgcolor=#ddffdd|[[dichlorine heptoxide|Cl<sub>2</sub>O<sub>7</sub>]] M<sub>2</sub>O<sub>7</sub>
|- bgcolor=#ddddff
! align=left |AX<sub>2.5</sub>E<sub>1</sub>
|
|
|bgcolor=#ddffdd|
|bgcolor=#ddffdd|
|bgcolor=#ddffdd|
|- bgcolor=#ddddff
! align=left |AX<sub>1.5</sub>E<sub>2</sub>
|
|
|
|bgcolor=#ddffdd|[[thiosulfate|S<sub>2</sub>O<sub>3</sub><sup>2-</sup>]]
|bgcolor=#ddffdd|
|- bgcolor=#ddddff
! align=left |AX<sub>0.5</sub>E<sub>3</sub>
|
|
|
|
|bgcolor=#ddffdd|[[dichlorine monoxide|Cl<sub>2</sub>O]]
|- valign=top
! rowspan = 4 |
sp<sup>3</sup>
! align=left |AX<sub>4</sub>E<sub>0</sub> tetrahedral
|
|bgcolor=#ddffdd|[[orthosilicate|SiO<sub>4</sub><sup>4-</sup>]]
|bgcolor=#ddffdd|[[phosphate|PO<sub>4</sub><sup>3-</sup>]] [[arsenate|AsO<sub>4</sub><sup>3-</sup>]]
|bgcolor=#ddffdd|[[sulfate|SO<sub>4</sub><sup>2-</sup>]] [[selenate|SeO<sub>4</sub><sup>2-</sup>]] [[tellurate|TeO<sub>4</sub><sup>2-</sup>]] [[chromate|CrO<sub>4</sub><sup>2-</sup>]] [[molybdate|MoO<sub>4</sub><sup>2-</sup>]] [[tungstate|WO<sub>4</sub><sup>2-</sup>]]
|bgcolor=#ddffdd|[[perchlorate|ClO<sub>4</sub><sup>-</sup>]] [[perbromate|BrO<sub>4</sub><sup>-</sup>]] [[periodate|IO<sub>4</sub><sup>-</sup>]] [[permanganate|MnO<sub>4</sub><sup>-</sup>]] [[pertechnetate|TcO<sub>4</sub><sup>-</sup>]] [[perrhenate|ReO<sub>4</sub><sup>-</sup>]]
|bgcolor=#ddffdd|[[xenon tetroxide|XeO<sub>4</sub>]] [[ruthenium tetroxide|RuO<sub>4</sub>]] [[osmium(VIII) oxide|OsO<sub>4</sub>]]
|-
! align=left |AX<sub>3</sub>E<sub>1</sub> trigonal pyramidal
|
|
|bgcolor=#ddffdd|[[phosphite|PO<sub>3</sub><sup>3-</sup>]] [[arsenite|AsO<sub>3</sub><sup>3-</sup>]]
|bgcolor=#ddffdd|[[sulfite|SO<sub>3</sub><sup>2-</sup>]] [[selenite (ion)|SeO<sub>3</sub><sup>2-</sup>]] [[tellurite (ion)|TeO<sub>3</sub><sup>2-</sup>]]
|bgcolor=#ddffdd|[[chlorate|ClO<sub>3</sub><sup>-</sup>]] [[bromate|BrO<sub>3</sub><sup>-</sup>]] [[iodate|IO<sub>3</sub><sup>-</sup>]]
|bgcolor=#ddffdd|[[xenon trioxide|XeO<sub>3</sub>]]
|-
! align=left |AX<sub>2</sub>E<sub>2</sub> bent <109.5
|
|
|[[hypophosphite|PO<sub>2</sub><sup>3-</sup>]]
|bgcolor=#ddffdd|[[hyposulfite|SO<sub>2</sub><sup>2-</sup>]]
|bgcolor=#ddffdd|[[chlorite|ClO<sub>2</sub><sup>-</sup>]] [[bromite|BrO<sub>2</sub><sup>-</sup>]] [[iodite|IO<sub>2</sub><sup>-</sup>]]
|-
! align=left |AX<sub>1</sub>E<sub>3</sub>
|
|
|
|[[peroxide|O<sub>2</sub><sup>2-</sup>]]
|style="background:#ddffdd"|[[hypochlorite|ClO<sup>-</sup>]] [[hypobromite|BrO<sup>-</sup>]] [[hypoiodite|IO<sup>-</sup>]]
|}

=List of free alumni email services=
{| class=wikitable
! University
! Domain
! Info
! Notes
|-
|{{rh}}| [[UC Berkeley]]
| @cal.berkeley.edu
| [http://cal.berkeley.edu/ @cal]
| Lifetime forwarding
|-
|{{rh}}| [[UCLA]]
| @ucla.edu
| [https://www.bol.ucla.edu/cgi-ssl/accounts/setforward/ BOL email forwarding]
| Lifetime forwarding
|-
| * UCLA Engineering School
| @engineering.ucla.edu
| [http://forwarding.seas.ucla.edu/ Engineering email forwarding]
| Lifetime forwarding
|}
See also:
* http://www.thesquare.com/ts/signup/alumni-email.html

=Array Types Cross-Reference=
{|class=wikitable
|-
!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
|-
|{{rh}}|[[C (programming language)|C]]
|rowspan=2|static array
|rowspan=2 {{n/a}}
|malloc'd array
|{{n/a}}
|-
|{{rh}}|[[C++]]
|new'd array
|std::vector
|-
|{{rh}}|[[Java (programming language)|Java]]
|{{n/a}}
|{{n/a}}
|array
|ArrayList
|-
|{{rh}}|[[Python (programming language)|Python]]
|{{n/a}}
|tuple
|{{n/a}}
|list
|-
|{{rh}}|[[Objective-C]] [[Cocoa (API)|Cocoa]]
|{{n/a}}
|NSArray
|{{n/a}}
|NSMutableArray
|-
|{{rh}}|[[Go (programming language)|Go]]
|array
|{{n/a}}
|slice value
|mutable slice variable
|-
|{{rh}}|[[Swift (programming language)|Swift]]
|{{n/a}}
|Array value
|{{n/a}}
|mutable Array variable
|-
|}