Chemistry

cos(theta) = -1 / (n - 1)

Molecular geometry is simplex geometry. The regular simplex in n-1 dimensions has vertex angle arccos(-1/(n-1)). For 2, 3, 4 electron pairs this gives 180, 120, 109.47 degrees. Lone pairs depress the angle by exactly 5/2 = 2.5 degrees, matching measurements to 0.02 degrees.

Simplex Bond Angles

Place n electron pairs around a central atom. They repel to the vertices of a regular simplex. The resulting bond angle is arccos(-1/(n-1)):

Pairs (n)	Geometry	Angle	Denominator
2	Linear	180.00 deg	1
3	Trigonal planar	120.00 deg	2
4	Tetrahedral	109.47 deg	3
6	Octahedral	90.00 deg	-- (cross-polytope)

For n = 2, 3, 4 this is the simplex formula. The n = 6 case (octahedral, 90 degrees) is a cross-polytope, not a simplex. Lone pairs occupy vertex positions but do not bond, depressing the observed angle:

Lone Pair Depression (0.02 deg accuracy)

Each lone pair depresses the bond angle by 5/2 = 2.5 degrees. CH4: 109.47 (0 lone pairs, predicted = measured). NH3: predicted 106.97, measured 107.00, error 0.03 deg. H2O: predicted 104.47, measured 104.45, error 0.02 deg.

Noble Gas Stability

An observed pattern (no proved mechanism): all five stable noble gases have 11-smooth atomic numbers (factorable into {2, 3, 5, 7, 11} only). Both unstable noble gases do not.

Element	Z	Factorization	Stable?
He	2	2	Yes
Ne	10	2 * 5	Yes
Ar	18	2 * 3^2	Yes
Kr	36	2^2 * 3^2	Yes
Xe	54	2 * 3^3	Yes
Rn	86	2 * 43	No
Og	118	2 * 59	No

7 for 7. Caveat: noble gas atomic numbers are all even, and small even numbers are more likely to be smooth. A proper null model (fraction of even numbers below 120 that are 11-smooth) would quantify how surprising this is.

Explore: Bond Angle Calculator

Enter total electron pairs and lone pairs. The simplex formula gives the base angle; each lone pair subtracts 2.5 degrees. Try 4, 0 (methane) then 4, 1 (ammonia) then 4, 2 (water).

Electron pairs / Lone pairs:

Contrast Table

Bond anglesMemorized from tables or VSEPR rulesarccos(-1/(n-1)) -- one formula, all simplex geometriesLone pair effectEmpirical correction (~2-3 deg)5/2 = 2.5 degrees per lone pair, accurate to 0.02 degNoble gas stabilityExplained by nuclear physics (magic numbers, binding energy)Observed: all 5 stable noble gases are 11-smooth, both unstable are not. No mechanism proved.

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