## Tuesday, June 10, 2008

### Gardner's constant

Gardner's constant is the transcendental number e^{\pi \sqrt{163}}. It is given as the integer 262,537,412,640,768,744 in his April 1975 Scientific American Column. So Lets see what programming langugaes have the accuracy to give this result?

So with these figures what will your programming languages will give Gardner's constant?
e=2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174135966290435729003342952605956307381323286279434907632338298807531952510190115738341879307021540891499348841675092447614606680822648001684774118537423454424371075390777449920695517027618386062613313845830007520449338265602976067371132007093287091274437470472306969772093101416928368190255151086574637721112523897844250569536967707854499699679468644549059879316368892300987931277361782154249992295763514822082698951936680331825288693984964651058209392398294887933203625094431173012381970684161403970198376793206832823764648042953118023287825098194558153017567173613320698112509961818815930416903515988885193458072738667385894228792284998920868058257492796104841984443634632449684875602336248270419786232090021609902353043699418491463140934317381436405462531520961836908887070167683964243781405927145635490613031072085103837505101157477041718986106873969655212671546889570350354021234078498193343210681701210056278802351930332247450158539047304199577770935036604169973297250886876966403555707162268447162560798826517871341951246652010305921236677194325278675398558944896970964097545918569563802363701621120477427228364896134225164450781824423529486363721417402388934412479635743702637552944483379980161254922785092577825620926226483262779333865664816277251640191059004916449982893150566047258027786318641551956532442586982946959308019152987211725563475463964479101459040905862984967912874068705048958586717479854667757573205681288459205413340539220001137863009455606881667400169842055804033637953764520304024322566135278369511778838638744396625322498506549958862342818997077332761717839280349465014345588970719425863987727547109629537415211151368350627526023264847287039207643100595841166120545297030236472549296669381151373227536450988890313602057248176585118063036442812314965507047510254465011727211555194866850800368532281831521960037356252794495158284188294787610852639pi=3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208095146550225231603881930142093762137855956638937787083039069792077346722182562599661501421503068038447734549202605414665925201497442850732518666002132434088190710486331734649651453905796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007230558763176359421873125147120532928191826186125867321579198414848829164470609575270695722091756711672291098169091528017350671274858322287183520935396572512108357915136988209144421006751033467110314126711136990865851639831501970165151168517143765761835155650884909989859982387345528331635507647918535sqrt163=Math.sqrt(163)gard=262537412640768744guess=e**(pi*sqrt163)puts "Gardiners constant is 262537412640768744 \n guess #{guess}\n"

Ruby does not give this result by a fair margin.

Obviously Haskell, matlab, mathematica and such are the proper languages for these problems. The Haskell compiler for OS X seems to require you to blow Satan to get it to work though*.

Your language may give a give some weird answers but are you going to believ someone called Hermite?

*Don Stewart pointed out you don't have to felate beelzbub to get Haskell. So if you don't want to get his scaley member down your gullet try here

## Monday, June 09, 2008

### Apéry's constant using monte Carlo methods

Apéry's Constant (1.20205690315...) "arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics."

I described how to use monte carlo methods to estimate mathematical constants here and here. A passion for mathematics select three positive integers at random the odds of them having no common divisor are 1 in 1.202056....

So her is a ruby program that does not work for calculating this
def findCD(num1, num2, num3, cd) if num1 % cd==0 && num2 % cd==0  && num3 % cd==0 return cd elsif cd < ([num1,num2,num3].max)/2 findCD(num1, num2, num3, cd+1) else return 0 endendi=1 factor =0N=1000while (i < N) x=rand(1000) y=rand(1000) z=rand(1000) ans=findCD(x,y,z,2) if ans >0 factor=factor+1 endendans=(factor).quo(N)puts "Apéry's Constant estimated as #{ans}" ;

The first problem is that my find common divisor algorithm is idiotic. There are all sorts of clever ways of doing this.

The second one is that computers have trouble creating random integers. The integers go up forever but a computers numbers don't. How do you ask a computer to give you one of any infinite one of the integers? rand(1,000,000) will not do it.

## Sunday, June 08, 2008

### Using a Random Number Generator to estimate e

After this post I started to wonder what other constants can be estimated using probabalistic methods.

You can estimate e = 2.7182818 using
monte carlo methods. The sum S = X1 + X2 + ... + Xn > 1 averages as the eth element. So if you rand(0,1)+rand(0,1)+... until they are greater then 1. If you average the number of elements needed it approximates e.

This code is dog ugly and probably bug riddled but it might give you some idea how the calculation is done.
i=0N=1000000number=0total=0  p=0k=1   while k < N        i=i+rand    p=p+1  if i>1     total=total+p     number=number+1     p=0     i=0  end      k=k+1   end   total=total.quo(number)   puts "e is #{total}"

If you know of a better way to use iterators to carry out this calculation please comment.

### Testing Random Number Generators with π

"Imagine that you have a dart board that is 2 units square. It inscribes a circle of unit radius. The center of the circle coincides with the center of the square. Now imagine that you throw darts at that dart board randomly. Then the ratio of the number of darts that fall within the circle to the total number of darts thrown is the same as the ratio of the area of the circle to the area of the square dart board. The area of a circle with unit radius is just π square unit. The area of the dart board is 4 square units. The ratio of the area of the circle to the area of the square is π / 4."

I got the book Digital dice the other day

It is about using monte carlo methods to solve probability problems that are hard to solve analytically. The first example given is how to estimate pi based on the % of darts that fall inside a cirle that is inside a square with the side length of twice the circles radius.

I decided to try out some languages to see how they got on with this problem. The aim is to test
1. Accuracy for mante carlo simulations
2. Ease of writing
3. Speed
in order of importance

If a language cannot accurately estimate π using this method then it indicates its random number generator (RNG) might not be very good. To test the RNG propery you need to do chi square tests and such as described in Seminumerical Algorithms by Knuth but i think its funny you can use π to test a RNG.

Perl
$N=10000000;$p=0;   $k=1; while ($k<$N){$x=rand();    $y=rand(); if (($x*$x)+($y*$y)<1){$p=$p+1; }$k=$k+1; }$ans=((4*$p)/$N);   print "ans".\$ans;

Ruby
N=10000000   p=0   k=1   while k < N    x=rand    y=rand     if (x*x)+(y*y)<1     p=p+1     end    k=k+1   end   ans=(4*p).quo(N)   puts "pi is #{ans}"

Scheme
I cannot figure out how to get random numbers with scheme. There is code to do it here but I am not going to write my own rand function.

C
int main(void){long N=10000000;double x=0;double y=0;double p=0;int k=1;   while (k < N){    x=(   (double)rand() / ((double)(RAND_MAX)+(double)(1)) );    y=(   (double)rand() / ((double)(RAND_MAX)+(double)(1)) );     if ((x*x)+(y*y)<1){     p=p+1;     }    k=k+1;   }      double mypi=((4*p)/N);      printf ("my pi  %f \n",mypi);   return 0;}

(less then sign) is < (thanks Helium).
I should use srand here to make it reproducible. Or more to be really fair run the test a few times and collect the average error. This is how the Digital dice runs the test with matlab. Anyway I wont tell you what language worked out the best. If you have any suggestions or code please comment.

### Sign Language

If you want to speak to someone you need to share a language. Because languages have different phonemes it can be difficult to pronounce a foreign language.
So how about if we all had a base of sign language that we could communicate with? The advantage is simple movements do not have a local accent that localises them.
So how many words do you need to know?

Language word usage follows zipfs law. Which means a suprisingly low number of words cover a high percentage of our speech.

There are 850 words in basic english which implies you can communicate basic information with about this many words.

Newspeak the artificial language in the novel 1984 was simplified to the extent to make coherent political thought impossible. The point of here is to search for a simple universal language is for emergency situations or for providing an anchor for language learning so such a Newspeak nightmare is not a worry.

One way of looking for a basic subset of language is to find the common subset of all languages. This has been studied as the semantic metalanguage. There follows a list of the words (in english) that occur in all known languages and their international sign language equivalant.

Semantic primitives

The English exponents of the 61 Semantic Primitives (addition of LONG is proposed)

substantives
I,
YOU
SOMEONE
PEOPLE
SOMETHING/THING
BODY

mental predicates
THINK
KNOW
WANT
FEEL
SEE
HEAR
BE
speech
SAY
WORD
TRUE
actions, events and movement
DO
HAPPEN
MOVE
PUT
GO
existence and possession
THERE
IS
HAVE
life
TIME
WHEN/TIME
BEFORE
A LONG TIME
A SHORT TIME
FOR SOME TIME
MOMENT
space
WHERE/PLACE
HERE
ABOVE
BELOW
FAR
NEAR
SIDE
INSIDE
TOUCHING

"logical" concepts
NOT
Maybe
CAN
BECAUSE
IF
intensifier
VERY
augmentor
MORE
quantifiers
ONE
TWO
SOME
ALL
MANY/MUCH
evaluators
GOOD
descriptors
BIG, SMALL, (LONG)
taxonomy, partonomy
KIND OF, PART OF;
similarity
LIKE
determiners
THIS
SAME
OTHER