I obviously did not originate this observation, I just supplied a bit of exposition. Eric Weisstein's formulas on A074455 and A074457 states the relationship more succinctly. I glean from Weisstein's comments that if d = A074457 constant e = A074455 constant ψ = digamma function then ψ(d/2) = log(π) d > 0 e = d - 2 uniquely determine the values of d and e. I would not hold out hope that d has any nice mathematical properties.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Daniel Asimov Sent: Tuesday, December 23, 2014 11:15 PM To: math-fun Subject: Re: [math-fun] Dimension where unit ball has maximum content
After years of thinking the maxima differed by only a near-integer, and today noticing the numerical virtual identity with Mac Grapher and then Mma, I verified it (easy) and then found our very own David Wilson's entry in the OEIS from 2007, where he had made that observation. See A074455 <http://oeis.org/A074455>.
--Dan
On Dec 23, 2014, at 4:08 PM, Bill Gosper <billgosper@gmail.com <mailto:billgosper@gmail.com>> wrote:
This exposes a bug in the terminology "unit ball", which really ought
to mean "unit diameter ball". There is no local content maximum when
the ball is inscribed in a unit cube. Question: Is that Amax-Vmax = 2
observation original? Did you use Area = d Volume/dr? --rwg
DanA> Pretending that spheres, balls, and Euclidean spaces can have real dimensions:
* let d_Amax := the real dimension d where the formula
A(d) = 2 pi^(d/2) / Gamma(d/2)
for the (d-1)-dimensional content of the unit (d-1)-sphere in R^d takes its maximum.
-and-
* let d_Vmax := the real dimension d where the formula
V(d) = pi^(d/2) / Gamma(d/2 + 1)
for the d-dimensional content of the unit d-ball in R^d takes its maximum
Then d_Amax = 7.256946404860576780132838388690769236619+ and d_Vmax = 5.256946404860576780132838388690769236619+.
In particular they have the same fractional part:
upsilon := 0.256946404860576780132838388690769236619+
QUESTION: Is anything known about the number theoretic properties of upsilon?
Is it known to be irrational or transcendental? Or related to other numbers, like Euler gamma, whose number-theoretic properties are unknown?
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <mailto:math- fun@mailman.xmission.com> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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