On Thu, Aug 30, 2012 at 1:40 AM, Bill Gosper <billgosper@gmail.com> wrote:
http://en.wikipedia.org/wiki/Spiral_of_Theodorus gives just two terms of the expansion Sum[ArcTan[1/Sqrt[n]], {n, 1, k}] == 1411/(12672*k^(9/2)) - 1147/(8064*k^(7/2)) + 167/(840*k^(5/2)) - 41/(120*k^(3/2)) + 7/(6*Sqrt[k]) + H + 2*Sqrt[k]
with H, Hlawka's Schneckenkonstante ( A105459 - The On-Line Encyclopedia of Integer Sequences™ (OEIS *...* <http://oeis.org/A105459/internal>
), given as thirteen digits of -2.1577829966594462209291427868295777235041...
=-2.1577829966594462209291427868295777235041395986075624551548955508588696467966064814966942989464 ...
I'm surprised that this constant term (which is not the leading term) is the only difficult one to compute.
Actually, this is analogous to the √2π in Stirling's formula.
With encouragement from Corey, Neil, and Mozart,
H==-Sqrt[-B + k] + (-1 + B - k)*ArcCot[Sqrt[-B + k]] + Sum[ArcTan[1/Sqrt[n]], {n, 1, k}]
where k is any positive integer and B is Bernoulli's umbra. I.e., pick, say, k=1, expand around B=0, then Normal[%]/. B^(p_: 1) -> BernoulliB[p]]. Catch: This process only works to about B^d, where d ~ 2πk, and gives about 2πk/(ln 10) digits. Depending on the relative costs of arctan vs Bernoulli terms, it's cheaper to shoot for higher k and lower d.
This would have involved a messy numerical integration but for the minor miracle that ArcTan[x^rational] integrates in closed form. There's a hint in OEIS that H comes out in half-integer zetas, but no hits from ries nor http://isc.carma.newcastle.edu.au/ .
Foo, H==Sum[((-1)^i*Zeta[1/2 + i])/(1 + 2*i),{i, 1, ∞}] + Zeta[1/2] . (This is Finch's K inhttp://www.people.fas.harvard.edu/~sfinch/csolve/th.pdf.) This is the first simple formula I've seen, and is good for 1 bit/term if rewritten -1 + π/4 + Sum[((-1)^n*(-1 + Zeta[(1/2)*(1 + 2*n)]))/ (1 + 2*n), {n, 1,∞}] + Zeta[1/2] --rwg
On Thu, Aug 16, 2012 at 7:41 PM, George Hart <george@georgehart.com>wrote:
On 8/16/2012 4:52 PM, Bill Gosper wrote:
Here's a piecewise linear spiral approximation to a nautilus (shown me by Jack Holloway): http://www.amazon.com/The-**Irrationals-Story-Numbers-** Count/dp/0691143420/ref=pd_**sim_b_25#reader_0691143420<http://www.amazon.com/The-Irrationals-Story-Numbers-Count/dp/0691143420/ref=pd_sim_b_25#reader_0691143420>
Hi Bill,
That's a "Spiral of Theodorus", which closely approximates an Archimedean spiral, not an exponential spiral. See:
http://en.wikipedia.org/wiki/**Spiral_of_Theodorus<http://en.wikipedia.org/wiki/Spiral_of_Theodorus>
If the image on that book cover included another revolution, it would become obvious how it doesn't match the nautilus it is superimposed on. A nautilus grows by a multiplicative factor of roughly three each revolution, while the square root spiral grows by roughly an additive increment of Pi each revolution.
It is very misleading for them to show just one revolution, so it looks like a reasonable model, but I don't think this error is commonly made. Do you know other places where the Spiral of Theodorus is used as a model of the nautilus?
George http://georgehart.com/
The very name Schneckenkonstante! --rwg