Re: [math-fun] There must be something wrong with Mathematica's Round function--
----- Original Message ----- From: Eugene Salamin Sent: 12/06/13 07:05 PM To: math-fun Subject: Re: [math-fun] There must be something wrong with Mathematica's Round function--
Every system of rounding has some arbitrary convention for dealing with
(The first time I pasted in the Subject field, GMail padded it with internal blanks, like an ancient text justfier!) For http://gosper.org/Article.pdf Julian made the "obvious" observation that balanced ternary can represent the reals using 0 and ±any (nonzero) digit you want! On p6 he used ±2 to derive the Fourier series for Sierpinski's gasket. Then on p20, I used ordinary balanced ternary to compute a peculiar, three-valued function that draws the gasket rather directly, as well as appearing in the preceding alternate Fourier derivation. --rwg Date: 2013-12-06 11:13 From: "Adam P. Goucher" <apgoucher@gmx.com> To: "Eugene Salamin" <gene_salamin@yahoo.com>, "math-fun" < math-fun@mailman.xmission.com> Nice. These problems are circumvented by the use of an odd radix, such as ternary (base-3). Balanced ternary has the beautiful property that rounding and truncation are precisely the same operation, and it's very easy to represent and compute negative numbers. I seem to recall that the Russians even had a balanced ternary computer in the mid-20th-century, called Setun or something like that. Sincerely, Adam P. Goucher 5. I like the convention that was used in numerical tables, in which a least significant digit of 5 that was rounded upward had an overbar over the 5 to signal that the number should be rounded downward in a further rounding. For example, the number 0.12349 in a 4-place table is written as 0.1235 with an overbar over the 5. Then if someone wants a 3-place value, they know to use 0.123 rather than 0.124.
-- Gene
________________________________ From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, December 6, 2013 10:44 AM Subject: Re: [math-fun] There must be something wrong with Mathematica's
Round function--
OK, that would explain Mathematica's reasoning.
But why? According to what system of rounding?
--Dan
On 2013-12-06, at 9:36 AM, Adam P. Goucher wrote:
Half-integers are rounded to the nearest even integer.
Connection with (alleged) topic is on the tenuous side ... Nonetheless, fig. 15 is (seriously) intriguing! WFL On 12/7/13, Bill Gosper <billgosper@gmail.com> wrote: > (The first time I pasted in the Subject field, GMail padded it with > internal blanks, > like an ancient text justfier!) > > For http://gosper.org/Article.pdf > Julian made the "obvious" observation that balanced ternary can represent > the reals > using 0 and ±any (nonzero) digit you want! On p6 he used ±2 to derive the > Fourier > series for Sierpinski's gasket. > > Then on p20, I used ordinary balanced ternary to compute a peculiar, > three-valued > function that draws the gasket rather directly, as well as appearing in the > preceding > alternate Fourier derivation. > --rwg > > Date: 2013-12-06 11:13 > From: "Adam P. Goucher" <apgoucher@gmx.com> > To: "Eugene Salamin" <gene_salamin@yahoo.com>, "math-fun" < > math-fun@mailman.xmission.com> > > > Nice. These problems are circumvented by the use of an odd radix, such as > ternary (base-3). Balanced ternary has the beautiful property that rounding > and truncation are precisely the same operation, and it's very easy to > represent and compute negative numbers. I seem to recall that the Russians > even had a balanced ternary computer in the mid-20th-century, called Setun > or something like that. > > > Sincerely, > > > Adam P. Goucher > >> ----- Original Message ----- >> From: Eugene Salamin >> Sent: 12/06/13 07:05 PM >> To: math-fun >> Subject: Re: [math-fun] There must be something wrong with Mathematica's > Round function-- >> >> Every system of rounding has some arbitrary convention for dealing with > 5. I like the convention that was used in numerical tables, in which a > least significant digit of 5 that was rounded upward had an overbar over > the 5 to signal that the number should be rounded downward in a further > rounding. For example, the number 0.12349 in a 4-place table is written as > 0.1235 with an overbar over the 5. Then if someone wants a 3-place value, > they know to use 0.123 rather than 0.124. >> >> -- Gene >> >> >> >________________________________ >> > From: Dan Asimov <dasimov@earthlink.net> >> >To: math-fun <math-fun@mailman.xmission.com> >> >Sent: Friday, December 6, 2013 10:44 AM >> >Subject: Re: [math-fun] There must be something wrong with Mathematica's > Round function-- >> > >> > >> >OK, that would explain Mathematica's reasoning. >> > >> >But why? According to what system of rounding? >> > >> >--Dan >> > >> > >> > >> >On 2013-12-06, at 9:36 AM, Adam P. Goucher wrote: >> > >> >> Half-integers are rounded to the nearest even integer. >> > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
* Fred Lunnon <fred.lunnon@gmail.com> [Dec 08. 2013 15:43]:
Connection with (alleged) topic is on the tenuous side ...
Nonetheless, fig. 15 is (seriously) intriguing! WFL
[...]
Have you looked at http://jjj.de/tmp-r19/ ? Every file not named R19*curve*.pdf or R19*dragon*.pdf shows a tile for the 3-grid. All pdfs rolled into one file (for convenient viewing) is http://jjj.de/tmp-r19/00-all-r19.pdf I can generate all such tiles if anyone cares. A _seriously_ neat tile can be given by 3 copies of http://jjj.de/tmp-extra/R48-8421-dragon-thinlines.pdf (a challenge for every pdf-viewer, rather print using 1200 dpi resolution, 600 dpi if you have to). This is one of the 1271 shapes of (such) symmetric L-systems of order 48. If the R48-8421-dragon is too much for eyes or software then look at http://jjj.de/tmp-r19/R19-17-* for the same "theme" in less excessive resolution. Best, jj
participants (3)
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Bill Gosper -
Fred Lunnon -
Joerg Arndt