DanAsimov> I'm afraid I missed the original mention of this. Because: What are the constraints on the four dihedrals? (Maybe the sum of their vertex angles?) --Dan Sorry, my bad -- terminology. I've been saying dihedral and meaning vertex angle! On Mon, Sep 16, 2013 at 3:09 AM, Bill Gosper <billgosper@gmail.com> wrote:
[...] rwg>So, is the solid angle formed by four [vertex angle]s maximal when it inscribes
in a circular cone? (Actually, I should still have the max constraints.
But I think the cone part was way messy.) --rwg
[...]
maxSolidAngle[a_, b_, c_, d_] := 2*ArcCos[((Cos[d] + Cos[c] + Cos[b] + Cos[a])/(4*Cos[a/2]*Cos[b/2]* Cos[c/2]*Cos[d/2])) - Tan[a/2]*Tan[b/2]*Tan[c/2]*Tan[d/2]]
Browsing Carr's synopsis http://archive.org/stream/asynopsiselemen00carrgoog#page/n216/mode/2up to see if that's where Ramanujan got his Bernoulli notation, I found two more formulæ (appended) for the solid angle given the vertex angles: {ArcCos[-1+(1+Cos[a]+Cos[b]+Cos[c])^2/((1+Cos[a]) (1+Cos[b]) (1+Cos[c]))], 2 ArcCos[1/4 (1+Cos[a]+Cos[b]+Cos[c]) Sec[a/2] Sec[b/2] Sec[c/2]], 2 ArcSin[(Sqrt[-1-Cos[2 a]-Cos[2 b]+4 Cos[a] Cos[b] Cos[c]-Cos[2 c]] Sec[a/2] Sec[b/2] Sec[c/2])/(4 Sqrt[2])], 4 ArcTan[Sqrt[Tan[1/4 (a+b-c)] Tan[1/4 (a-b+c)] Tan[1/4 (-a+b+c)] Tan[1/4 (a+b+c)]]]}
The equivalence of these gives FullSimplify fits, even for {a,b,c}=π/{2,3,4}.
E.g., In[72]:= FullSimplify[% /. {a -> π/2, b -> π/3, c -> π/4}] Out[72]= {ArcCos[1/6 (4 + Sqrt[2])], 2 ArcCos[((3 + Sqrt[2]) Sin[π/8])/Sqrt[3]], 2 ArcSin[Sin[π/8]/Sqrt[3]], 4 ArcTan[Sqrt[Cot[11 π/48] Tan[π/48] Tan[5 π/48] Tan[7 π/48]]]} In[73]:= N[%] Out[73]= {0.445561, 0.445561, 0.445561, 0.445561} In[74]:= FullSimplify[Rest[%%] - First[%%]] Still running since yesterday. --rwg The last one is http://mathworld.wolfram.com/LHuiliersTheorem.html,
misprinted in Carr as Llhuillier's by an obviously Welsh typesetter. Amazingly, "huilier" means oiler. (Or cruet.) --rwg
Luckily, David Fowler's ancestors didn't drop the F. I didn't find any explicit listing of Bernoulli-like numbers or polynomials in Carr's, e.g., as Faulhaber polynomials or Taylor coefficients of tan. "This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project to make the world’s books discoverable online."
The first page of the Preface to Part I is significantly obscured by at least two other apparently damaged pages folded over it.