plot(data,char,size,clr)

50aabf4c-6388-47fe-8458-85a2efc4e75e 70 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

π

x f 41 x 2^+ / :

https://mathworld.wolfram.com/PiContinuedFraction.html This document does not include the proof of 'π'. The proof is the Isoperimeters [Schwab 1813 ... Nancy]. Célèbres Problèmes Mathématiques...E. Callendrau p.205

355113 / 3.14159292

4 appVersion 0.98.6179.21440 ≡

Perfectly suited  Simpson [100]
MCD-π ≡ 0.000000000000000 π x f x 01 int 21 mat 3.14159265358979 3.14159265358979 21 mat

Simon Plouffe π  'n' incremental algorithm ... 1999 n pi 116 k^/ 48 k * 1 + / 28 k * 4 + / - 18 k * 5 + / - 18 k * 6 + / - (* (k 0 n sum eval :

12 pi 3.14159265358979

π 3.14159265358979

π 3.14159265358979

π ... 40 D pi 3.141592653589793238462643383279502884197 :

Mathcad 3.1*10^-19 pi 12 pi - 0.000000000000000

40 decimals 29 pi 3.14159265358979

x f 1 4 1 1 - 2 2 mat 2 1 sys

10 π * 31.4159265358979

10 pi * 31.4159265358979

Mathcad 11  recognizes the integral as π x f x 01 int 3.141592653589793238462643383279502884197 ≡

Filling the π f(x) region

t0 1 time :

Prepare the project

Construct the project

x f 41 x 2^+ / :

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

N 2000 :

xmin 0 : xmax 1 : 12 mat ymin 0 : ymax x f : 12 mat 21 line

space_x xmax xmin - (: space_y ymax ymin - (: 21 line

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y x f ≤ (x xmax ≤ (* :

PDF x 01 0.0025 range : i 1 x rows range y i el x i el f eval : for x y augment 31 line :

A routine that finds an array of N random 
pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

web color

a b c spec . 8 #808080 13 mat :

fill Pts a b c spec pts :

Area in there is ... 'π'

PDF 1 4 1 0.5 - 2 2 mat fill 3 1 sys

1 f 2

π

1 time t0 - 60 s *

plot(data,char,size,clr)

Universal version  [Smath 6179]. It supports "data" sets from 
source ... data sets subject to be decimated from auxilliary
module Decimate(data,n)... reasonably fast on primitive PC. data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Regular Polygon Area/Perimeter

n 10001 :

'b' vertices b n 1 - :

t0 1 time :

j 1 n range circle j 1 el j b / 2 * π * cos eval : circle j 2 el j b / 2 * π * sin eval : 21 line for circle 21 line

NUMERIC

1 time t0 - min 0.4

vertices 2 circle * . 10 black plot :

circle 2 1 sys

x circle 1 col : y circle 2 col : 12 mat Area 12 / x i el y i 1 + el * x i 1 + el y i el * - (i 1 x rows 1 - sum * eval : Perimeter x i 1 + el x i el - (2^y i 1 + el y i el - (2^+ sqrt i 1 x rows 1 - sum eval : 31 sys

NUMERIC

'n' 10001 6 D ...
NOT candidate  'π' Area Perimeter 21 mat 3.141592 6.283185 21 mat

π 2 π * 21 mat 3.141593 6.283185 21 mat

maple bug ? π float 15 convert maple 3.141592654

π 3.14159265358979

maple bug ? π float 4 convert maple 3.141592654

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Regular Polygons ... demo

N 5 :

n 13 :

m 4 :

'a' vertices a N 1 - :

'b' vertices b n 1 - :

'c' vertices c m 1 - :

j 1 N range sqr j 1 el j a / 2 * π * cos : sqr j 2 el j a / 2 * π * sin : 21 line for sqr 21 line

j 1 n range Circle j 1 el j b / 2 * π * cos : Circle j 2 el j b / 2 * π * sin : 21 line for Circle 21 line

j 1 m range tria j 1 el j c / 2 * π * cos : tria j 2 el j c / 2 * π * sin : 21 line for tria 21 line

θ rot θ cos θ sin - θ sin θ cos 22 mat :

vertices 2 Circle * . 10 black plot :

θ π 2 / : 1.5708

NOT commutative Triangle tria θ rot * :

plot sqr Circle 2 * Triangle 2 * vertices 41 sys :

plot

N 5000 : X 2 N 1 Random (* 1 - : Y 2 N 1 Random (* 1 - : 31 line

Fill the circle exclusive of the square Circle i 1 N range U i el X i el X i el (2^Y i el (2^+ 1 < 0 cases : V i el Y i el X i el (2^Y i el (2^+ 1 < 0 cases : 21 line for U V augment 21 line :

Fill the square exclusive of the circle Square i 1 N range X i el 0 X i el (2^Y i el (2^+ 1 < X i el cases : Y i el 0 X i el (2^Y i el (2^+ 1 < Y i el cases : 21 line for X Y augment 21 line :

Monte Carlo 'Pi'

Square Circle 2 1 sys

i 1 N range U i el X i el X i el (2^Y i el (2^+ 1 < 0 cases : for 11 line

collect My_π
in 'V' ...
collect from
recalculate F9 n U i el X i el ≡ (i 1 N sum : My_π 4 n N / * : 21 sys 3912 3.1296 21 sys

Collected 10 random
No matter population
My_π  only ± close. V 3.176 3.1472 3.156 3.1408 3.1456 3.1408 3.1336 3.1448 3.1328 3.1232 101 mat :

V sum V rows / 3.1441

The zillions continued fraction ... curiosity

up to 99

cf 31632^652^672^692^6112^6132^6152^6172^6192^6212^6232^6252^6272^6292^6312^6332^6352^6372^6392^6412^6432^6452^6472^6492^6512^6532^6552^6572^6592^6612^6632^6652^6672^6692^6712^6732^6752^6772^6792^6812^6832^6852^6872^6892^6912^6932^6952^6972^6992^+ / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + :

cf 3.141591

Two more bricks in the wall

has integral curve ▼  worth 2π at ∞ x f 41 x 2^+ / :

indefinite integral x Curve x f x int maple : 4 x atan *

∞ Curve 2 π * - 0.000000000000000

∞ Curve 2 / π - 0.000000000000000

Continued fraction ... Expand optimiz NONE x g 4 x atan * confrac x 44 convert maple :

1 g π - 0.000000000000000

Infinite random Not candidate for π

N 1000 : X 2 N 1 Random (* 1 - : Y 2 N 1 Random (* 1 - : 31 line

V 3.092 3.184 3.172 3.168 3.128 3.192 3.096 3.068 3.092 3.184 101 mat :

i 1 N range U i el X i el X i el (2^Y i el (2^+ 1 < 0 cases : for 11 line

collect My_π
in 'V' ...
collect from
recalculate F9 My_π 4 U i el X i el ≡ (i 1 N sum N / * : 11 sys 3.132

V sum V rows / 3.1376

Wallis continued fraction

πWallis 41134597169251136134915641781191002112123144251692719629225312563328935324373613940041 / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / + / :

πWallis 3.14159265358979

π 3.14159265358979

πWallis π - 51016 -^*

πWallis 18855713597030460019600489849 /