[math-fun] Dimension where unit ball has maximum content
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
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
Clearly I didn't discover this relationship, I just elaborated. Eric Weisstein made the same observation in the formula. It also appears in the formula for A074457.
-----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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Well, that didn't come out like I sent it. The phi should be a psi. The eth should be a pi.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of David Wilson Sent: Wednesday, December 24, 2014 9:24 AM To: 'math-fun' Subject: Re: [math-fun] Dimension where unit ball has maximum content
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 ø = digamma function
then
ø(d/2) = log(ð) 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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Dec 23, 2014, at 8:15 PM, Daniel Asimov <asimov@msri.org <mailto:asimov@msri.org>> 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.
Is that right? Using diameter = 1 instead of radius = 1, I'm getting for A(d) / 2^(d-1) a maximum at d = 2.4765825823060852952076157688588232403016455151805+, and for V(d) / 2^d a maximum at d = 0.4765825823060852952076157688588232403016455151805+ . --Dan
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+.
participants (3)
-
Bill Gosper -
Daniel Asimov -
David Wilson