plot(data,char,size,clr)

50aabf4c-6388-47fe-8458-85a2efc4e75e 84 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

Adrien Métius 355113 / 3.141593

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

m 1014^:

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

m π * 3.14159265358979 1014^*

m pi * 3.14159265358979 1014^*

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 - 57 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

3 continued fractions . . .

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1. The Lange continued fraction [1999]

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

2. The symbolic continued fraction ...

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

1 g π - 0.000000000000000

3. The Wallis continued fraction ...

Wallis

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

πWallis 3.14159265358979

π 3.14159265358979

πWallis π - 51016 -^*

πWallis 18855713597030460019600489849 /

Infinite random Not candidate for π

govern while loop n 100 :

t0 1 time :

iter n : j 1 : j iter ≤ N 5 n * : X 2 N 1 Random (* 1 - : Y 2 N 1 Random (* 1 - : 31 line i 1 N range U i el X i el X i el (2^Y i el (2^+ 1 < 0 cases : for V j el 4 U i el X i el ≡ (i 1 N sum N / * : j j 1 + : 21 line partial π collected in V 41 line while V 41 line

No matter how big [n, N], random circle does not converge ► π, simply deceptive.

V 3.176 3.104 3.248 3.152 3.208 3.2 3.184 3.152 3.088 3.152 3.048 111 mat

pi V sum V rows / : 3.1449

1 time t0 - min 0.9

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

π ... 40 D pi 3.141592653589793238462643383279502884197 :

Smath takes the 14 significant digits of 'pi' m 1014^:

m π * 314159265358979.000000000000000

m pi * 314159265358979.000000000000000

x mantissa x x Floor - :

π mantissa 0.141592653589793

pi mantissa 0.14159265358979

The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment.