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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 or something like that? Did you ever wonder what that came from? It is . And I will now show you why:
If circumradii are drawn between the center of an -simplex and its vertices, the angle between these segments is .
Some formulas from this page:
- The height of a regular -simplex of side is
The circumradius, which is the difference between the height and the apothem, is:
Now consider any two circumradii. They go to two different vertices, which must be joined by an edge of the -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:
Here, , and .
The angle we want is between and . Since cosine is one-to-one in that range, the angle is uniquely determined.
This explains, among other things, the angles between hybridized orbitals:
|hybridization||dimensions||angle between orbitals|
|sp||n = 1|
|sp2||n = 2|
|sp3||n = 3|
That's all the simplices that can fit in 3 dimensions, folks; but you see the pattern...
|AX2E0 linear||AlO2- BO2-||CO2 SiO2 MO2||NO2+|
|AX2.5E0||Si2O52-||N2O5 P2O5 As2O5 M2O5|
|AX3E0 trigonal planar||BO33-||CO32- SiO32- SnO32- PbO32-||NO3- VO3-||SO3 O4 SeO3 MO3|
|AX2E1 bent <120||SnO22- PbO22-||NO2-||SO2 O3 SeO2|
|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
|UC Berkeleyfirstname.lastname@example.org||@cal||Lifetime forwarding|
|UCLAemail@example.com||BOL email forwarding||Lifetime forwarding|
|* UCLA Engineering Schoolfirstname.lastname@example.org||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||N/A||malloc'd array||N/A|
|Go||array||N/A||slice value||mutable slice variable|
|Swift||N/A||Array value||N/A||mutable Array variable|