[math-fun] simple in/ex-radii & altitude relation
A neat identity relates inradius r_IN , ex-radius r_EX , altitude h , of a simplex with distinguished base and apex --- *** 1/r_EX + 1/r_IN = 2/h *** --- the altitude equals the harmonic mean of the radii ! I have not come across this elsewhere --- reference, anyone? A fellow by name of Ilya Bogdanov on Math Overflow (carefully rinsing mouth with disinfectant) seems well versed in this stuff. There turn out to be a good few Ilya Bogdanovs in Russia ... does anybody have an email address for this one? WFL
Reminder --- r_EX is negative! On 2/20/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A neat identity relates inradius r_IN , ex-radius r_EX , altitude h , of a simplex with distinguished base and apex ---
*** 1/r_EX + 1/r_IN = 2/h ***
--- the altitude equals the harmonic mean of the radii !
I have not come across this elsewhere --- reference, anyone?
A fellow by name of Ilya Bogdanov on Math Overflow (carefully rinsing mouth with disinfectant) seems well versed in this stuff. There turn out to be a good few Ilya Bogdanovs in Russia ... does anybody have an email address for this one?
WFL
I would appreciate a much better reminder: How is r_EX defined? —Dan
On Feb 19, 2016, at 6:09 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Reminder --- r_EX is negative!
On 2/20/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A neat identity relates inradius r_IN , ex-radius r_EX , altitude h , of a simplex with distinguished base and apex ---
*** 1/r_EX + 1/r_IN = 2/h ***
--- the altitude equals the harmonic mean of the radii !
I have not come across this elsewhere --- reference, anyone?
A fellow by name of Ilya Bogdanov on Math Overflow (carefully rinsing mouth with disinfectant) seems well versed in this stuff. There turn out to be a good few Ilya Bogdanovs in Russia ... does anybody have an email address for this one?
The ex-sphere is tangent to the base facet externally, and to the other facets internally; -r_EX denotes its radius. Here's a succinct proof, in the spirit of Klamkin (and Bogdanov). Denote n-space simplex interior content (volume) by V , boundary content (surface area) by U . Then (n V)/r_EX + (n V)/r_IN = 2 F_0 - U + U = 2 F_0 = (n V)2/h ; now cancel n V . QED On 2/20/16, Dan Asimov <dasimov@earthlink.net> wrote:
I would appreciate a much better reminder:
How is r_EX defined?
—Dan
On Feb 19, 2016, at 6:09 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Reminder --- r_EX is negative!
On 2/20/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A neat identity relates inradius r_IN , ex-radius r_EX , altitude h , of a simplex with distinguished base and apex ---
*** 1/r_EX + 1/r_IN = 2/h ***
--- the altitude equals the harmonic mean of the radii !
I have not come across this elsewhere --- reference, anyone?
A fellow by name of Ilya Bogdanov on Math Overflow (carefully rinsing mouth with disinfectant) seems well versed in this stuff. There turn out to be a good few Ilya Bogdanovs in Russia ... does anybody have an email address for this one?
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A little too succinct --- F_0 here denotes base facet content (by a minor abuse of notation from my previous thread); for consistency too, h might better have been styled h_0 . Note r_EX < 0 . On 2/20/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The ex-sphere is tangent to the base facet externally, and to the other facets internally; -r_EX denotes its radius.
Here's a succinct proof, in the spirit of Klamkin (and Bogdanov). Denote n-space simplex interior content (volume) by V , boundary content (surface area) by U . Then (n V)/r_EX + (n V)/r_IN = 2 F_0 - U + U = 2 F_0 = (n V)2/h ; now cancel n V . QED
On 2/20/16, Dan Asimov <dasimov@earthlink.net> wrote:
I would appreciate a much better reminder:
How is r_EX defined?
—Dan
On Feb 19, 2016, at 6:09 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Reminder --- r_EX is negative!
On 2/20/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A neat identity relates inradius r_IN , ex-radius r_EX , altitude h , of a simplex with distinguished base and apex ---
*** 1/r_EX + 1/r_IN = 2/h ***
--- the altitude equals the harmonic mean of the radii !
I have not come across this elsewhere --- reference, anyone?
A fellow by name of Ilya Bogdanov on Math Overflow (carefully rinsing mouth with disinfectant) seems well versed in this stuff. There turn out to be a good few Ilya Bogdanovs in Russia ... does anybody have an email address for this one?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
A couple more, from the same stable: 1/r_IN = SUM 1/h ; and (1-n)/r_IN = SUM 1/r_EX ; where the sums are over all n+1 altitudes and exradii. For the latter, see http://mathoverflow.net/questions/149811/about-the-inscribed-sphere-and-the-... WFL On 2/20/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A little too succinct --- F_0 here denotes base facet content (by a minor abuse of notation from my previous thread); for consistency too, h might better have been styled h_0 . Note r_EX < 0 .
On 2/20/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The ex-sphere is tangent to the base facet externally, and to the other facets internally; -r_EX denotes its radius.
Here's a succinct proof, in the spirit of Klamkin (and Bogdanov). Denote n-space simplex interior content (volume) by V , boundary content (surface area) by U . Then (n V)/r_EX + (n V)/r_IN = 2 F_0 - U + U = 2 F_0 = (n V)2/h ; now cancel n V . QED
On 2/20/16, Dan Asimov <dasimov@earthlink.net> wrote:
I would appreciate a much better reminder:
How is r_EX defined?
—Dan
On Feb 19, 2016, at 6:09 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Reminder --- r_EX is negative!
On 2/20/16, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A neat identity relates inradius r_IN , ex-radius r_EX , altitude h , of a simplex with distinguished base and apex ---
*** 1/r_EX + 1/r_IN = 2/h ***
--- the altitude equals the harmonic mean of the radii !
I have not come across this elsewhere --- reference, anyone?
A fellow by name of Ilya Bogdanov on Math Overflow (carefully rinsing mouth with disinfectant) seems well versed in this stuff. There turn out to be a good few Ilya Bogdanovs in Russia ... does anybody have an email address for this one?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
History, detailed exhumation of which would constitute neither math nor fun. WFL On 2/20/16, Adam P. Goucher <apgoucher@gmx.com> wrote:
WFL wrote:
on Math Overflow (carefully rinsing mouth with disinfectant)
As an avid MathOverflow user myself, I am intrigued as to the reasoning behind that parenthetical remark...
Best wishes,
Adam P. Goucher
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participants (3)
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Adam P. Goucher -
Dan Asimov -
Fred Lunnon