
The Mystery of 355/113 - ColinWright
http://davidbau.com/archives/2010/03/14/the_mystery_of_355113.html
======
CoreDumpling
In Chinese the fraction 355/113 has been known as the 密率 ("detailed ratio")
and 22/7 as the 约率 ("approximate ratio"). It is documented in the Book of Sui,
volume 16:

宋末，南徐州從事史祖沖之，更開密法，以圓徑一億為一丈，圓周盈數三丈一尺四寸一分五厘九毫二秒七忽，朒數三丈一尺四寸一分五厘九毫二秒六忽，正數在盈朒二限之間。密率，圓徑一百一十三，圓周三百五十五。約率，圓徑七，週二十二。
[1]

While it suggests that the value was calculated around the end of the Liu Song
Dynasty (~479 CE) by Zu Chongzhi to have an upper bound of 3.1415927 and lower
bound of 3.1415926, the 密法 method is not explained.

It has been thought that Zu may have used a "method of averaging days" to
calculate an intermediate value between two prior known approximations of 22/7
and 157/50 [2]. Not a very satisfying result for anyone who wants to find
something special in the numbers 355 and 113.

[1]
[https://zh.wikisource.org/wiki/%E9%9A%8B%E6%9B%B8/%E5%8D%B71...](https://zh.wikisource.org/wiki/%E9%9A%8B%E6%9B%B8/%E5%8D%B716)

[2] [http://books.google.com/books?id=QlbzjN_5pDoC&lpg=PA34&#...</a>

------
StefanKarpinski
There is no mystery here. If you pick a random real number and find the best
quality approximation of it — as calculated by log_b 1/|x-a/b| — then the
median quality of the best such approximations within a tolerance of 1/1024 is
around 3.19. So with respect to the measure this article is talking about, pi
is completely normal and 355/113 is a completely unexceptional best
approximation with a denominator of at most nine binary digits (about 3
decimal digits). 355/113 isn't even the best quality approximation of pi by
the given criterion — that would be 22/7, which, as mentioned in the article,
has a quality measure of 3.429288337281781.

Julia code to figure this out can be found here:

<https://gist.github.com/3170899>

~~~
fragsworth
These kinds of articles about pi, or numbers, where they try to summon up some
mystery and wonder around it, come across to me as not very genuine. I get the
impression it gets upvoted because people want to feel like they're smarter,
or more geeky and hip for having looked at it.

~~~
raverbashing
One of the most "mind boggling" areas of math is number theory

Math dealing with real/complex numbers has been vastly explored, and there are
very powerful tools, and computers are very good at these kind of problems
(Calculus, etc)

With number theory, progress is much harder, things work in a completely
different logic (think for example that 5+3 can be 0 for example)

One of the places where these mysterious sides of math comes together is this:
<http://en.wikipedia.org/wiki/Riemann_zeta_function>

~~~
duaneb
>With number theory, progress is much harder, things work in a completely
different logic (think for example that 5+3 can be 0 for example)

I wasn't aware that the modulo operation had been raised to a field of
mathematics.

~~~
raverbashing
Modulo operations is part of "number theory 101"

And ok, addition is nice and fun. Then you go to multiplication

Then you end up with Galois Fields.

Never underestimate the amount of discussion that goes into things like 1+1=2

------
cperciva
FWIW, all of the convergents from the continued fraction expansion of a number
have "quality" greater than 2. Conversely, the Thue–Siegel–Roth theorem says
that any _algebraic_ number and any epsilon > 0, the algebraic number has only
finitely many approximations with quality more than 2 + epsilon; so the
"quality" values of successive best approximations are a series which
converges to 2 from above.

This was how the first known transcendental numbers -- the Liouville numbers
-- were constructed: They have approximations with unbounded quality, thus
they cannot be algebraic. In practice, however, this isn't a very useful
method: "Almost all" transcendental numbers also obey the Thue–Siegel–Roth
theorem.

~~~
michaelf
Is it also the case that all transcendental numbers have approximations of
unbounded quality? Or only that Liouville numbers have approximations of
unbounded quality, and therefore cannot be algebraic?

~~~
cperciva
Liouville numbers have approximations of unbounded quality and thus cannot be
algebraic. "Most" transcendental numbers only have finitely many
approximations of quality higher than 2 + epsilon for any epsilon.

------
Xcelerate
Pi has the continued fraction expansion:

3; 7, 15, 1, 292, 1, ...

That 355/113 is one of the best possible rational approximations to PI, and it
occurs right after the 292. In general, a best rational approximation cut off
at a very large term in the CF expansion will provide more accuracy than you
would think.

~~~
walrus
355/113 is actually [3,7,15,1]. The reason it's particularly good compared to
its length is because it falls right _before_ the 292 term (since 1/292 is a
much smaller contribution than 1/1).

~~~
caf
So, as noted in one of the comments to the original article, the question of
why 355/113 is particularly good can be restated as a question of why the 292
term in that expansion is so large.

------
fredley
I wrote a quick script to walk through Q, spitting out fractions with high
"quality". Here are some early values:

    
    
      22/7: 3.42928833728
      355/113: 3.20195874233
      710/226: 2.79251047296
    

A quick search through 1/1 - 100,000/100,000 shows no other interesting
values. My algorithm is extremely naive though (brute force), a faster search
could potentially be used to find other high quality values much more quickly.

 _Update_ \- A slightly optimized algorithm, and nothing special found with
denominators up to the hundreds of millions. I know this means nothing wrt
actual mathematics though...

~~~
daurnimator
I had a quick go too:

    
    
        num     den     difference to pi
        1	1	2.1415926535898
        3	1	0.14159265358979
        22	7	0.0012644892673497
        333	106	8.3219627529107e-05
        355	113	2.6676418940497e-07
        103993	33102	5.7789062424263e-10
        104348	33215	3.3162805834763e-10
        208341	66317	1.2235634727631e-10
        312689	99532	2.914335439641e-11
        833719	265381	8.7152507433075e-12
        1146408	364913	1.6107115641262e-12
        4272943	1360120	4.0412118096356e-13
        5419351	1725033	2.2204460492503e-14
        80143857	25510582	4.4408920985006e-16
    

Lua script to generate (dirty):

    
    
        local max = 100000000
        local c = {}
        local min , j = math.huge
        for i=1,max do
            local a = math.pi-math.abs(math.cos(i))
            if a < min then
                min = a
                j = i
                table.insert ( c , j )
            end
        end
    
        local min , j = math.huge
        for k=1,#c do
            for i=1,max do
                local a = math.abs(math.pi-c[k]/i)
                if a < min then
                    min = a
                    j = i
                    print(c[k],j,min)
                end
            end
        end

~~~
ColinWright
It's not actually the difference to pi that matters, it's the "quality" of the
approximation, as described in the article. You can always get better
approximations by using larger denominators, it's just that some
approximations are, given the size of the denominator, _unreasonably_ good.

355/113 is one of those.

~~~
daurnimator
True, I didn't do the quality measure they talk of. I was just fascinated by:

"For example, use any scientific calculator to compute cos(355) in radians.
The oddball result is due to the freakish closeness of 355/113 to pi."

And wanted to see what else would fit this criteria...

------
cheshirecat
This is called Diophantine approximation.

<http://en.wikipedia.org/wiki/Diophantine_approximation>

p.s. Take a look at the "Liouville's result" part.

------
darkstalker
355/113 has good enough precision to aproximate pi on single precision
floating point

~~~
ajross
Not quite, but close. I get the delta as one part in 2^21.8. The mantissa of a
IEEE float is 24 bits (including the leading one).

------
bikenaga
355/113 is a convergent of the continued fraction for pi. The convergents of a
continued fraction (for an irrational number) give the best rational
approximations to the number. I'm teaching this stuff this summer; here's a
link to my notes on this:

[http://www.millersville.edu/~bikenaga/number-
theory/approxim...](http://www.millersville.edu/~bikenaga/number-
theory/approximation-by-rationals/approximation-by-rationals.html)

For instance, 278/125 is the best rational approximation to the cube root of
11 having denominator less than 155. The proof is pretty easy --- you don't
have to do it by trial and error.

As for why 355/113 doesn't come up in other expressions for pi --- I'm not
sure why it should. The fact that two expressions converge to pi is no reason
that I can see to expect that they should involve the same integer
coefficients, say. There are infinitely many integers. There are infinitely
many expressions for pi.

------
rplnt
> and it causes various oddities elsewhere in math.

Could someone elaborate? First, explain the one example from the article
(cos(355)) and perhaps show some other oddities?

~~~
_delirium
If p/q is a close approximation to π, then cos(p), taken in radians, will be
close to +1 or -1, because it implies that p is close to an integer multiple
of π, and therefore cos(p) is close to the cosine of an integer multiple of π:

    
    
       p/q =~ π 
       p =~ qπ
       cos(p) =~ cos(qπ)
    

In turn, the cosine of an integer multiple of π is +1 or -1 because the period
of cosine (as a trigonometric function) is 2π radians, with maxima at 0, 2π,
4π, ... and minima at π, 3π, 5π, ...

------
lkbm
I've never been compelled to use any fractional approximation of pi. Either
I'm doing it in my head and pi is ~3, I'm doing it on paper and pi is ~3.14,
or I'm using a calculator and pi is as many digits as I feel like.

We already have the decimal representation memorized for more digits than this
approximates, so unless your diameter is given to you in 113ths, is there any
practical application here?

------
coreygoodie
Okay, I'll bite - why does this matter?

~~~
ColinWright
[http://www.smbc-comics.com/index.php?db=comics&id=2674](http://www.smbc-
comics.com/index.php?db=comics&id=2674)

~~~
ufo
You are kind of stretching though. Why do we have to assume that 355/113 is
"magical" in order to advance mathematics:

~~~
ColinWright
Firstly, you do realise that I'm not the author, I just submitted it, don't
you?

Secondly, there are some curious things about _pi_. It does turn up in places
that apparently have nothing to do with circles. It's a bit like _e_ in that
regard.

Next, we don't have to assume it's "magical," but some of the properties are
noteworthy. The fact that 355/113 is such a good approximation, and yet it
doesn't appear to turn up naturally in any of the proven convergences is a bit
odd. _Why_ does it not turn up? The only place it _does_ turn up is when you
write down the _ad hoc_ continued fraction to express the value you already
know. That seems unnatural, and immediately leads to a desire for further
investigation.

And finally, mathematicians have a feel for things that are "natural," and
that's what they end up exploring. Often it leads nowhere interesting, but
sometimes it leads to unexpected connections, and occasionally to equally
unexpected applications. But in all, some questions just feel right for
exploration, and some properties of _pi_ fall into that category. You never
really know exactly _what_ will advance math - we only have intuition to guide
us in deciding what is an "interesting question."

------
peteri
Very useful when performing integer mathematics... First met it in Brodies
Starting Forth I always remember it as 113355 (split it in half)

Having gone away to check my memory I notice that he also provides: 1068966896
/ 340262731 which is suppose to be accurate to E-18

------
emmelaich
Very interesting stuff.

Lovers of pi might like Eve Andersson's Pi Land:
<http://www.eveandersson.com/pi/>

As for sequences, I particularly liked pimasterfromhk's.

------
K2h
if one wanted to find the 'best' converging sequence, you might go about it by
brute force. Seems lots of people beat me to it.
<http://pastebin.com/iujCgUSu>

    
    
      my $lasterror = 3;
      for (1..10000){ my $d = $_;
          for (3..40000){ my $n = $_;
             if (abs(($n/$d) - pi) < $lasterror){
             $lasterror = abs(($n/$d) - pi);
             print "\n $n / $d: error $lasterror";}}}
      
      #  output
      # 3 / 1: error 0.141592653589793
      # 13 / 4: error 0.108407346410207
      ...
      # 355 / 113: error 2.66764189404967e-007

~~~
eridius
That's not a converging series. That's just a sequence of ratios that have
successively smaller errors. But there's nothing to relate each ration to the
next one.

~~~
K2h
correct - one of observations in the article was that many of the series
identified so far lack the magical 355/113 - I was curious what a converging
series would look like, not the equation that describes such a series.

I would be interested in identifying this next super series in terms of an
equation but given the few minutes on break, and that total it is unlikely
someone as bad at math, such as myself, could come up with this series in a
few minutes - I opted for the fun programming route.

~~~
eridius
You still seem confused. What you have is not a converging series, period. You
just have a list of successively-closer ratios. A series (in math) is the sum
of the terms in a sequence (which itself is a discrete function). You don't
have a discrete function, and you definitely don't have the sum of anything.

~~~
ColinWright
He does have a converging _sequence,_ though. Many people don't understand the
technical (but obvious to mathematicians) distinction between _sequence_ and
_series._

~~~
K2h
Thanks Colin, _sequence_ \- that was the term I needed, and my intention.

------
abc_lisper
Try fourth root of (9^2 + 19^2/22)

------
criveros
I just saw all the numbers and bailed.

