In this code, I introduce 'xi' (11.71875) a constant with strong relationships to both pi and e . 'Xi' (11.71875 or (187.5/16)) is a 'translator constant' between powers of 2 and 16. More than simply being an efficient means to compute pi or e (e.g. 'Eulers number'), 'Xi' may 'interpolate' between pi and e, giving science the means to measure logarithmic growth patterns of increasing radii. This may make it useful for gravitational and rotational calculations. It is possible that the 187.5 cm-3 pc represents some fixed attenuation limit (see http://arxiv.org/abs/1503.05245) for radio wave signals that require either boosting or chirping. The relationships between pi, e, and xi (187.5./16 = 11.71875) may represent fixed constants descriptive of all EM signaling. In addition, the constants that describe the circumscribing of an octahedron (sqrt(1/2)) and (sqrt(1/6)) can also be used to derive xi with a factor close to 9/16. I am neither mathematician, scientist, or statistician. So all of this code is supposition and a work in progress. -RMF
## Ryan Matthew Ferris 10:26 PM Thursday, November 19, 2015, 2015 Bellingham, WA
## Updated 7/10/2018 with some additional work with the golden ratio far below
## *Notes on 187.5/16 or 11.71875*
##
# In this code, I introduce 'xi' (11.71875) a constant with strong relationships to both pi and # exp(1).
# 'Xi' (11.71875 or (187.5/16)) is a 'translator constant' between powers of 2 and 16.
# Search google for "11.71875mhz" and you can see this. 11.71875mhz is a widely used frequency in 'crystal resonators'.
# More than simply being an efficient means to compute pi or exp(1) (e.g. 'Eulers number'), 'xi' may
# 'interpolate' between pi and e, giving science the means to measure logarithmic growth patterns of
# increasing radii. This may make it useful for gravitational and rotational calculations. It is
# possible that the 187.5 cm-3 pc represents some fixed attenuation limit (see
# http://arxiv.org/abs/1503.05245) for radio waves signal that require either boosting or
# chirping. The relationships between pi, e, and xi(187.5./16 = 11.71875) may represent fixed
# constants descriptive of all EM signaling. In addition, the constants that describe the
# circumscription of an octahedron (sqrt(1/2) and sqrt(1/6) can also be used to derive xi with a
# factor close to 9/16.
## A number of constants and function names are used. Some are reused:
# [1] "almost2" "aR" "c2" "cR" "eR" "F_pe" "F_pi" "findpi" "i" "j" "k" "k1"
# [13] "k2" "l" "l1" "mm1" "p1" "p2" "piR" "pR" "pX" "s4" "seqe" "seqpi"
# [25] "seqpi_e" "t1" "t2" "t3" "Vx" "x" "xi" "xi_sqrt" "xif" "y"
## Set your graphics device to record history if desired. 86 graphs are produced.
## You may want par(mfrow=c(1,2))
## NB : In R 3.5 there is no bigint by default,however there are 22 digits of accuracy.
## See my usage of BIGINT in Powershell
## For this work I used Microsoft Open R 3.5 with Intel's MKL (Math Kernel Library)
options(digits=22)
# main constants
pi
#[1] 3.1415926535897931
exp(1) # Eulers number
#[1] 2.7182818284590451
xi <- 187.5/16 # transdimensional constant
# [1] 11.71875
# xi with pi and exp(1)
pi + exp(1)
# [1] 5.8598744820488378
2 * (pi + exp(1)) # very close to xi
# [1] 11.719748964097676
(pi + exp(1)) / 11.71875 # very close .5
# [1] 0.5000426224681674414185
xi / (pi + exp(1)) # very close to 2
# [1] 1.9998295246594895
((xi /(pi + exp(1))) * (pi + exp(1)))
# [1] 11.71875
# xi without pi or exp(1)
# from regular octahedron constants for finding circumscribed sphere
sqrt(1/6)
# [1] 0.40824829046386302
sqrt(1/2)
# [1] 0.70710678118654757
904319/1600000 # something a little different than 9/16 ???
# 9/16 - 904319/1600000 = -0.0026993749999999483
((10 * (sqrt(1/2) + sqrt(1/6))) + 904319/1600000) # close to xi
# [1] 11.718750091504107
xi_sqrt <- ((10 * (sqrt(1/2) + sqrt(1/6))) + 904319/1600000)
xi_sqrt
# [1] 11.718750091504107
((xi_sqrt /(pi + exp(1))) * (pi + exp(1)))
# [1] 11.718750091504107
# one way of deriving almost pi with exp(1) and xi
xi /(pi + exp(1))
# [1] 1.9998295246594895
almost2 <- xi /(pi + exp(1)) # almost 2
((xi/2) - exp(1)) # two digits right of decimal of pi
# [1] 3.1410931715409549
((xi/almost2) - exp(1)) # 14 digits right of decimal point of pi
# [1] 3.1415926535897927
pi
# [1] 3.1415926535897931
# one way of deriving almost exp(1) with pi and xi
((xi/2) - pi) # two digits right of decimal of exp(1)
#[1] 2.7177823464102069
exp(1)
# [1] 2.7182818284590451
((xi/almost2) - pi) # 14 digits right of decimal point of exp(1)
# [1] 2.7182818284590446
exp(1)
# [1] 2.7182818284590451
# 187.5 is the first fractional member of a series that returns 11.71875 as
# the quotient of what appears to be integer multiples of (3,5,6,10) by powers of 2 (2^(1:16)).
# 11.71875 pops up in google queries for decimal translation of bytes. All of the results below
# equal 11.71875:
cat('
12000/1024
6000/512
3000/256
1500/128
750/64
375/32
187.5/16
46.875/4
23.4375/2
')
# 11.71875 can be seen as a type of hexadecimal 'translator' ratio between binary and decimal:
12000/1024
#[1] 11.71875
6000/512
#[1] 11.71875
3000/256
#[1] 11.71875
1500/128
#[1] 11.71875
750/(187.5/16)
#[1] 64
12000/(187.5/16)
#[1] 1024
6000/(187.5/16)
#[1] 512
3000/(187.5/16)
#[1] 256
1500/(187.5/16)
#[1] 128
750/(187.5/16)
#[1] 64
# However, I can also come up with something more transcendental:
options(digits=22)
(18.75/2)/10
#[1] 0.9375
(.9375/2)/10
#[1] 0.046875
(.046875/2)/10
#[1] 0.0023437499999999999
(.002343749999999999/2)/10
# 0.00011718749999999996
#or
(23.43749999999999/2)
# [1] 11.718749999999995
# ...
# Multiples of 187.5 can be produced by selecting out the products of vector (1:100) multiplication of the
# quotient of 187.5./16. By sixteen,eight,four:
data.frame((187.5/16 * 1:100)[c(seq(0,100,16))])
cat('
1 187.5
2 375.0
3 562.5
4 750.0
5 937.5
6 1125.0
')
# By eight:
cat('
data.frame((187.5/16 * 1:100)[c(seq(0,100,8))])
1 93.75
2 187.50
3 281.25
4 375.00
5 468.75
6 562.50
7 656.25
8 750.00
9 843.75
10 937.50
11 1031.25
12 1125.00
')
# By four:
data.frame((187.5/16 * 1:100)[c(seq(0,100,4))])
cat('
1 46.875
2 93.750
3 140.625
4 187.500
5 234.375
6 281.250
7 328.125
')
# This series or ladder is significant, but I am not quite sure why it accurately generates the range (1:1/10) to five digits.
l <- {};for(i in 1:10) {l <- rbind(l,(exp(pi^-11.71875)/i))};l;plot(l)
cat(' [,1]
[1,] 1.00000149288121576 #1
[2,] 0.50000074644060788 #1/2
[3,] 0.33333383096040525 #1/3
[4,] 0.25000037322030394 #1/4
[5,] 0.20000029857624316 #1/5
[6,] 0.16666691548020263 #1/6
[7,] 0.14285735612588796 #1/7
[8,] 0.12500018661015197 #1/8
[9,] 0.11111127698680175 #1/9
[10,] 0.10000014928812158 #1/10
')
pi # 22 digits R 3.2.2 constant for pi
#[1] 3.1415926535897931
exp(1) # Eulers number in R 3.2.2
#[1] 2.7182818284590451
# 'xi' a 'transdimensional' constant
187.5/16
#[1] 11.71875
xi <- 187.5/16 # 11.71875 - greater than pi or e
k1 <- (spline(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
k2 <- (sort(c(xi,exp(1),pi,-pi,-exp(1),-xi)))
splinefun(diff(k2))
plot(splinefun(diff(k2)))
plot(spline(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
plot(splinefun(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
xi <- (187.5/16)/10 # 1.171875 - less than pi or e
k1 <- (spline(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
k2 <- (sort(c(xi,exp(1),pi,-pi,-exp(1),-xi)))
splinefun(diff(k2))
plot(splinefun(diff(k2)))
plot(spline(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
plot(splinefun(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
pi #pi
#[1] 3.1415926535897931
exp(1) #e
#[1] 2.7182818284590451
187.5/16 #xi
#[1] 11.71875
((11.71875/2) - exp(1))
#[1] 3.1410931715409549 #close to pie
((11.71875/2) - pi)
#[1] 2.7177823464102069 #close to e
((11.71875/2) - exp(1)) + ((11.71875/2) - pi)
#[1] 5.8588755179511622 # close to .5(xi)
(((11.71875/2) - exp(1)) + ((11.71875/2) - pi)) * 2
#[1] 11.717751035902324 # close to xi
11.71875 - 11.717751035902324
exp(11.71875 - 11.717751035902324)
#[1] 1.0009994632285004 # close to 1
(23.43749999999999/2)
exp(11.71875 - 11.718749999999995)
#[1] 1.0000000000000053 # closer to 1
# Since (pi + exp(1)) * 2 ~= (187.5/16) it is possible to use 11.71875 and some
# adjustment to arrive at either pi or e. (See notes and functions seqpi,seqe,seqpi_e (far)below.)
# The products of pi and e doubled roughly equal (187.5/16) or 11.71875:
(((pi + exp(1)) * 2) /(187.5/16))
#[1] 1.0000852449363349 # nearly one again. irrational?
(((pi + exp(1))) /(187.5/16))
#[1] 0.50004262246816744 # nearly five. irrational?
# 11.71875 can be successfully used to calculate pi or e within 3 digits right of the decimal point without any compensation:
((187.5/16)/2) - pi # almost e...
#[1] 2.7177823464102069 # should be 2.7182818284590451
((187.5/16)/2) - exp(1) # almost pi
#[1] 3.1410931715409549 # should be 3.1415926535897931
# Not quite pi or e:
pi - (((187.5/16)/2) - exp(1))
#[1] 0.00049948204883820679
exp(1) - (((187.5/16)/2) - pi)
#[1] 0.00049948204883820679
# The adjustment ( tanh(11.71875) / 2000 ) gives us 5 digits of accuracy for pi or e:
tanh(11.71875) / 2000
[1] 0.00049999999993374431
eR <- ((11.71875/2) - pi + tanh(11.71875) / 2000 )
format(eR - exp(1),scientific=FALSE)
# [1] "0.0000005179510953468025"
eR
#[1] 2.7182823464101404
exp(1)
#[1] 2.7182818284590451
pR <- ((11.71875/2) - exp(1) + tanh(11.71875) / 2000 )
format(pR - pi,scientific=FALSE)
# [1] "0.0000005179510953468025"
pR
# [1] 3.1415931715408885
pi
# [1] 3.1415926535897931
# using -digamma(1)
xi - (gr/exp(1) + -digamma(1))
[1] -0.00058210447924356
# DOES xif makes xi a 'transdimenesional transcendant'?
(18.75/2)/10
#[1] 0.9375
(.9375/2)/10
#[1] 0.046875
(.046875/2)/10
#[1] 0.0023437499999999999
(.002343749999999999/2)/10
# 0.00011718749999999996
(18.75/2)/10
(.9375/2)/10
(.046875/2)/10
(.002343749999999999/2)/10
xif <- (.002343749999999999/2)/10
xif <- (xif * 100000)
# Powershell [BIGINT] will take me this far:
cat('
PS [bigint](.002343749999999999/2 * [MATH]::POW(10,32))
117187499999999960548787617792
PS [bigint](.002343749999999999/2 * [MATH]::POW(10,64))
11718749999999995682656043607972093960765263440537027795222528
PS[bigint](.002343749999999999/2 * [MATH]::POW(10,128))
117187499999999952444488198957480104184273770534709180501147236632873797389362215023609307534290876647973955371666657662468096
PS [bigint](.002343749999999999/2 * [MATH]::POW(10,256))
11718749999999995794845523228309099783974204074624886274076630993851149729639928280300079885632128025211313970818427611098306439684539273451270347793350886454282383635
265648724318526179750550535267831033255464007490486648205861676480655926487782004883456
')
tanh(11.71875) # helps get us close to 1
#[1] 0.99999999986748866
# so does
xif/xi
#[1] 9.9999999999999947
xi/xif
#[1] 0.10000000000000005
eR <- ((11.71875/2) - pi + tanh(11.71875) / 2000 )
format(eR - exp(1),scientific=FALSE)
# [1] "0.0000005179510953468025"
eR
# [1] 2.7182823464101404
exp(1)
#[1] 2.7182818284590451
pR <- ((11.71875/2) - exp(1) + tanh(11.71875) / 2000 )
format(pR - pi,scientific=FALSE)
# [1] "0.0000005179510953468025"
pR
# [1] 3.1415931715408885
pi
# [1] 3.1415926535897931
(18.75/2)/10
(.9375/2)/10
(.046875/2)/10
(.002343749999999999/2)/10
xif <- (.002343749999999999/2)/10
xif <- (xif * 100000)
eR <- ((xif/2) - pi + tanh(xif) / 2000 )
format(eR - exp(1),scientific=FALSE)
eR
exp(1)
pR <- ((xif/2) - exp(1) + tanh(xif) / 2000 )
format(pR - pi,scientific=FALSE)
pR
pi
## By selecting from an array we can get much more accurate pi and e from xi.
((11.71875/2) - pi + tanh((11.71875/4096) * 1:4096) / 2000 )[c(1322)]
#[1] 2.718281828165519
exp(1)
#[1] 2.7182818284590451
((11.71875/2) - exp(1) + tanh((11.71875/4096) * 1:4096) / 2000 )[c(1322)]
#[1] 3.141592653296267
pi
#[1] 3.1415926535897931
## It is then possible for pi or e to converge from series generated from xi (11.71875)
## This series will converge to pi :
c2 <- as.matrix(c(
(((11.71875/2) - exp(1)) + (tanh((11.71875/8) * 1:8)) / 2000 )[3],
(((11.71875/2) - exp(1)) + (tanh((11.71875/16) * 1:16)) / 2000 )[5],
(((11.71875/2) - exp(1)) + (tanh((11.71875/32) * 1:32)) / 2000 )[11],
(((11.71875/2) - exp(1)) + (tanh((11.71875/64) * 1:64)) / 2000 )[21],
(((11.71875/2) - exp(1)) + (tanh((11.71875/128) * 1:128)) / 2000 )[41],
(((11.71875/2) - exp(1)) + (tanh((11.71875/256) * 1:256)) / 2000 )[83],
(((11.71875/2) - exp(1)) + (tanh((11.71875/512) * 1:512)) / 2000 )[165],
(((11.71875/2) - exp(1)) + (tanh((11.71875/1024) * 1:1024)) / 2000 )[331],
(((11.71875/2) - exp(1)) + (tanh((11.71875/2048) * 1:2048)) / 2000 )[661],
(((11.71875/2) - exp(1)) + (tanh((11.71875/4096) * 1:4096)) / 2000 )[1322],
(((11.71875/2) - exp(1)) + (tanh((11.71875/8192) * 1:8192)) / 2000 )[2644],
(((11.71875/2) - exp(1)) + (tanh((11.71875/16384) * 1:16384)) / 2000 )[5288],
(((11.71875/2) - exp(1)) + (tanh((11.71875/32768) * 1:32768)) / 2000 )[10576],
(((11.71875/2) - exp(1)) + (tanh((11.71875/65536) * 1:65536)) / 2000 )[21154]
))
c2
cat(' [,1]
[1,] 3.1415930191734072
[2,] 3.1415925126008446
[3,] 3.1415928546514991
[4,] 3.1415927145681901
[5,] 3.1415926227939854
[6,] 3.1415926707780644
[7,] 3.1415926473344875
[8,] 3.1415926591902781
[9,] 3.1415926532962670
[10,] 3.1415926532962670
[11,] 3.1415926532962670
[12,] 3.1415926532962670
[13,] 3.1415926532962670
[14,] 3.1415926536666201
')
pi
# [1] 3.1415926535897931
# So we can generate series that converge around pi and e :
options(digits=22)
# These print very long sequences/series
# for(x in (2^seq(0,16,1))) {print(format(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000))}
# for(x in (2^seq(0,16,1))) {print(format(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,digits=22,scientific=TRUE))}
# for(x in (2^seq(0,16,1))) {print(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000)}
for(x in (2^seq(0,16,1))) {(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000 )}
for(x in (2^seq(0,16,1))) {plot(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / x ,pty=.25,col="blue",type="p")};abline(h=pi,col="red")
for(x in (2^seq(0,16,1))) {plot(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000 ,pty=.25,col="blue",type="p")};abline(h=pi,col="red")
for(x in (2^seq(0,16,1))) {plot(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,ylim=c(pi - (pi *.000001),pi + (pi *.000001)),lwd=1,col="blue",type="l")};abline(h=pi,col="red")
for(x in (2^seq(8,12,1))) {plot(grep(3.141592,((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,value=TRUE),lwd=1,col="blue",type="l")};abline(h=pi,col="red")
for(x in (2^seq(8,14,1))) {plot(grep(3.141592,((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,value=TRUE),lwd=1,col="blue",type="l")};abline(h=pi,col="red")
findpi <- function(x,y) {for(i in (2^seq(x,y,1))) {p1 <- rbind(p1,(grep(pi,(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE)))}}
findpi <- function(x,y) {for(i in (2^seq(x,y,1))) {p1 <- rbind(p1,(grep(pi,(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE)))};plot(pi)}
p1 <- NULL; p1 <- rbind(p1,cbind(grep(3.1415,(11.71875/2) - exp(1) + (tanh((11.71875/8) * 1:8)) / 2000,value=TRUE)))
# template function code A:
x <- 1
y <- 16
options(digits=(x - 1))
p1 <- NULL
for(i in (2^seq(x,y,1)))
{p1 <- rbind(p1,cbind(grep(strtrim(pi,x),(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE)))}
# options(digits = (x + 1))
plot(p1,lwd=1,col="blue",type="l")
abline(h=pi,col="red")
# template function code B:
x <- 8
y <- 16
options(digits=(22))
p1 <- NULL
plot.default(x,y,type="n", xlim=c(0,15000),ylim=c(3.1415,3.1416))
for(i in (2^seq(x,y,1)))
{
lines(
rbind(p1,cbind(grep(3.1415,(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE))),
lwd=1,col=rgb(runif(1),runif(1),runif(1))
)
}
abline(h=pi,lwd=2,col=rgb(1,0,0,.5))
# These functions below (seqpi,seqe,seqpi_e) need three arguments like:
cat('
seqpi(1,1,8)
seqe(1,1,8)
seqpi_e(1,1,8)
seqpi(0,16,16)
seqe(0,16,16)
seqpi_e(0,16,16)
')
seqpi <- function(x,y,z) {
options(digits=(22))
xi <- 11.71875
#trimpi <- 7
p1 <- NULL
plot.default(x,y,type="n", xlim=c(0,z),ylim=c(3.1415,3.1416))
for(i in (2^seq(x,y,1))) {
lines(
rbind(p1,cbind(grep(3.1415,(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE))),
lwd=2,col=rgb(runif(1),runif(1),runif(1)) )}
abline(h=pi,lwd=6,col=rgb(.5,0,0,.25))
}
seqe <- function(x,y,z) {
options(digits=(22))
xi <- 11.71875
p1 <- NULL
e1 <- NULL
plot.default(x,y,type="n", xlim=c(0,z),ylim=c(2.7182,2.7183))
for(i in (2^seq(x,y,1))) {
lines(rbind(e1,cbind(grep(2.7182,(11.71875/2) - pi + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE))),
lwd=2,col=rgb(runif(1),runif(1),runif(1)))
}
abline(h=exp(1),lwd=6,col=rgb(0,0,.5,.25))
}
seqpi_e <- function(x,y,z) {
options(digits=(22))
xi <- 11.71875
p1 <- NULL
e1 <- NULL
plot.default(x,y,type="n", xlim=c(0,z),ylim=c(2.7182,3.1416))
for(i in (2^seq(x,y,1))) {
lines(rbind(p1,cbind(grep(3.1415,(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE))),
lwd=2,col=rgb(runif(1),runif(1),runif(1)));
lines(rbind(e1,cbind(grep(2.7182,(11.71875/2) - pi + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE))),
lwd=2,col=rgb(runif(1),runif(1),runif(1)))
}
abline(h=pi,lwd=6,col=rgb(.5,0,0,.25))
abline(h=exp(1),lwd=6,col=rgb(0,0,.5,.25))
}
# plotting these graphs with numbers bigger than 16 requires more CPU and memory than I got...
seqpi(1,1,8)
seqe(1,1,8)
seqpi_e(1,1,8)
seqpi(0,16,16)
seqe(0,16,16)
seqpi_e(0,16,16)
# for(x in (2^seq(0,16,1))) {pi + (min((((11.71875/2) - exp(1)) + (tanh((11.71875/8) * 1:8)) / 2000 ) - pi))}
for(x in (2^seq(0,4,1))) {match(3.141592,((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000 )}
for(x in (2^seq(0,8,1))) {print(grep(3.1415926535,((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,value=TRUE))}
for(x in (2^seq(0,8,1))) {plot(grep(3.14159,((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,value=TRUE))}
plot(c2,ylim=c(pi - (pi *.000001),pi + (pi *.000001)),lwd=1,col="blue",type="l");abline(h=pi,col="red")
# A magic series??
l1 <- c(3,5,11,21,41,83,165,331,661,1322,2644,5288,10576,21154)
as.matrix(l1 / (xi * .1))
cat(' [,1]
[1,] 2.5600000000000001
[2,] 4.2666666666666666
[3,] 9.3866666666666667
[4,] 17.9200000000000017
[5,] 34.9866666666666646
[6,] 70.8266666666666680
[7,] 140.8000000000000114
[8,] 282.4533333333333189
[9,] 564.0533333333332848
[10,] 1128.1066666666665697
[11,] 2256.2133333333331393
[12,] 4512.4266666666662786
[13,] 9024.8533333333325572
[14,] 18051.4133333333338669
')
plot(as.matrix(l1 / (xi * .1)))
plot(as.matrix(l1 / (xi * .1)),type="b")
l1 <- c(3,5,11,21,41,83,165,331,661,1322,2644,5288,10576,21154)
mm1 <- data.frame(cbind("Pxi" = l1))
mm1 <- data.frame(mm1,"P2" = 2^(3:16))
for(i in 1:10) {print(((11.71875/2) - pi) + (tanh((11.71875/(mm1$P2[i])) * 1:mm1$P2[i]) / 2000 )[mm1$Pxi[i]])}
F_pi <- function(x) {((11.71875/2) - exp(1)) + (tanh((11.71875/(mm1$P2[x])) * 1:mm1$P2[x]) / 2000 )[mm1$Pxi[x]]}
F_pe <- function(x) {((11.71875/2) - pi) + (tanh((11.71875/(mm1$P2[x])) * 1:mm1$P2[x]) / 2000 )[mm1$Pxi[x]]}
as.matrix(sapply(1:14, F_pi))
cat(' [,1]
[1,] 3.1415930191734072
[2,] 3.1415925126008446
[3,] 3.1415928546514991
[4,] 3.1415927145681901
[5,] 3.1415926227939854
[6,] 3.1415926707780644
[7,] 3.1415926473344875
[8,] 3.1415926591902781
[9,] 3.1415926532962670
[10,] 3.1415926532962670
[11,] 3.1415926532962670
[12,] 3.1415926532962670
[13,] 3.1415926532962670
[14,] 3.1415926536666201
')
as.matrix(mm1$Pxi /mm1$P2) / .1171875
cat(' [,1]
[1,] 3.2000000000000002
[2,] 2.6666666666666665
[3,] 2.9333333333333331
[4,] 2.7999999999999998
[5,] 2.7333333333333334
[6,] 2.7666666666666666
[7,] 2.7500000000000000
[8,] 2.7583333333333333
[9,] 2.7541666666666669
[10,] 2.7541666666666669
[11,] 2.7541666666666669
[12,] 2.7541666666666669
[13,] 2.7541666666666669
[14,] 2.7544270833333333
')
as.matrix(mm1$Pxi /mm1$P2) / 11.71875
cat(' [,1]
[1,] 0.032000000000000001
[2,] 0.026666666666666668
[3,] 0.029333333333333333
[4,] 0.028000000000000001
[5,] 0.027333333333333334
[6,] 0.027666666666666666
[7,] 0.027500000000000000
[8,] 0.027583333333333335
[9,] 0.027541666666666666
[10,] 0.027541666666666666
[11,] 0.027541666666666666
[12,] 0.027541666666666666
[13,] 0.027541666666666666
[14,] 0.027544270833333332
')
# These are some earlier notes in process:
# A reasonable approximation of pi can be made with with aR below
# cR is a correction
((11.71875/2) - exp(1) + exp(-11.71875/(2^14)) / 2000)
format(pi - ((11.71875/2) - exp(1) + exp(-11.71875/(2^14)) / 2000),scientific=FALSE)
#[1] "-0.00000016045116035812157"
aR <- 11.71875/2 - exp(1)
aR
cR <- exp(-11.71875/(2^14)) / 2000
cR
piR <- aR + cR
pi
piR
piR - pi
format(piR - pi, scientific=FALSE)
aR <- 11.71875/2 - exp(1)
aR
#[1] 3.1410931715409549
cR <- exp(-11.71875/(2^14)) / 2000
cR
#[1] 0.00049964249999855234
piR <- aR + cR
pi
#[1] 3.1415926535897931
piR
#[1] 3.1415928140409535
piR - pi
#[1] 1.6045116035812157e-07
format(piR - pi, scientific=FALSE)
#[1] "0.00000016045116035812157"
eR <- ((11.71875/2) - pi + tanh(11.71875) / 2000 )
format(eR - exp(1),scientific=FALSE)
Vx <- as.data.frame(cbind(as.matrix(1 / 1:10),as.matrix(exp(pi^-11.71875) / 1:10)))
Vx$V3 <- with(Vx,cbind(V2 - V1))
Vx$V4 <- with(Vx,cbind(V1 / V2))
Vx$V5 <- with(Vx,cbind(V2 / V1))
Vx
cat(' V1 V2 V3 V4 V5
1 1.00000000000000000 1.00000149288121576 1.4928812157588567e-06 0.99999850712101290 1.0000014928812158
2 0.50000000000000000 0.50000074644060788 7.4644060787942834e-07 0.99999850712101290 1.0000014928812158
3 0.33333333333333331 0.33333383096040525 4.9762707193812261e-07 0.99999850712101290 1.0000014928812158
4 0.25000000000000000 0.25000037322030394 3.7322030393971417e-07 0.99999850712101290 1.0000014928812158
5 0.20000000000000001 0.20000029857624316 2.9857624314622022e-07 0.99999850712101301 1.0000014928812158
6 0.16666666666666666 0.16666691548020263 2.4881353596906131e-07 0.99999850712101290 1.0000014928812158
7 0.14285714285714285 0.14285735612588796 2.1326874510840810e-07 0.99999850712101290 1.0000014928812158
8 0.12500000000000000 0.12500018661015197 1.8661015196985709e-07 0.99999850712101290 1.0000014928812158
9 0.11111111111111110 0.11111127698680175 1.6587569064141494e-07 0.99999850712101290 1.0000014928812158
10 0.10000000000000001 0.10000014928812158 1.4928812157311011e-07 0.99999850712101301 1.0000014928812158
')
# Other Sequences/Series/Prodcuts of interest
23.4375/2
#[1] 11.71875
(23.4375 * 5^1)/1
#[1] 117.1875
(23.4375 * 5^2)/.5
#[1] 1171.875
(23.4375 * 5^3)/.25
#[1] 11718.75
(23.4375 * 5^4)/.125
#[1] 117187.5
(23.4375 * 5^5)/.0625
#[1] 1171875
(23.4375 * 5^6)/.03125
#[1] 11718750
11.71875/2
#[1] 5.859375
(11.71875 * 5^1)/1
#[1] 58.59375
(11.71875 * 5^2)/.5
#[1] 585.9375
(11.71875 * 5^3)/.25
#[1] 5859.375
(11.71875 * 5^4)/.125
#[1] 58593.75
(11.71875 * 5^5)/.0625
#[1] 585937.5
(11.71875 * 5^6)/.03125
#[1] 5859375
exp(pi^-11.71875)
#[1] 1.0000014928812158
exp(pi^-11.71875)/2
#[1] 0.50000074644060788
exp(pi^-11.71875)/3
#[1] 0.33333383096040525
exp(pi^-11.71875)/4
#[1] 0.25000037322030394
exp(pi^-11.71875)/5
#[1] 0.20000029857624316
exp(pi^-11.71875)/6
#[1] 0.16666691548020263
exp(pi^-11.71875)/7
#[1] 0.14285735612588796
exp(pi^-11.71875)/8
#[1] 0.12500018661015197
exp(pi^-11.71875)/9
#[1] 0.11111127698680175
1/9
#[1] 0.1111111111111111
exp(pi^-11.71875)/10
#[1] 0.10000014928812158
p2 <- 2^(1:16)
pX <- data.frame(cbind(p2,(187.5/16) * p2)) # becomes V2
pX$V3 <- cbind((((pi + exp(1)) * 2) * p2))
pX$V4 <- with(pX,cbind(V3 - V2))
pX$V5 <- with(pX,cbind(V4 * 1000))
pX$V6 <- with(pX,cbind(p2/V5))
plot(pX$p2,type="p",col="red");par(new=T);plot(pX$V5,type="l",col="blue");par(new=F);
pX
cat(' p2 V2 V3 V4 V5 V6
1 2 23.4375 23.439497928195351 0.0019979281953510508 1.9979281953510508 1.0010369765308733
2 4 46.8750 46.878995856390702 0.0039958563907021016 3.9958563907021016 1.0010369765308733
3 8 93.7500 93.757991712781404 0.0079917127814042033 7.9917127814042033 1.0010369765308733
4 16 187.5000 187.515983425562808 0.0159834255628084065 15.9834255628084065 1.0010369765308733
5 32 375.0000 375.031966851125617 0.0319668511256168131 31.9668511256168131 1.0010369765308733
6 64 750.0000 750.063933702251234 0.0639337022512336262 63.9337022512336262 1.0010369765308733
7 128 1500.0000 1500.127867404502467 0.1278674045024672523 127.8674045024672523 1.0010369765308733
8 256 3000.0000 3000.255734809004935 0.2557348090049345046 255.7348090049345046 1.0010369765308733
9 512 6000.0000 6000.511469618009869 0.5114696180098690093 511.4696180098690093 1.0010369765308733
10 1024 12000.0000 12001.022939236019738 1.0229392360197380185 1022.9392360197380185 1.0010369765308733
11 2048 24000.0000 24002.045878472039476 2.0458784720394760370 2045.8784720394760370 1.0010369765308733
12 4096 48000.0000 48004.091756944078952 4.0917569440789520741 4091.7569440789520741 1.0010369765308733
13 8192 96000.0000 96008.183513888157904 8.1835138881579041481 8183.5138881579041481 1.0010369765308733
14 16384 192000.0000 192016.367027776315808 16.3670277763158082962 16367.0277763158082962 1.0010369765308733
15 32768 384000.0000 384032.734055552631617 32.7340555526316165924 32734.0555526316165924 1.0010369765308733
16 65536 768000.0000 768065.468111105263233 65.4681111052632331848 65468.1111052632331848 1.0010369765308733
')
t1 <- ((pi + exp(1)) /(187.5/16))^-(1:16)
t2 <- 2^(1:16)
t3 <- t1/t2
s4 <- cbind(t1,t2,t3)
plot(s4[,3],type="l")
s4
cat(' t1 t2 t3
[1,] 1.9998295246594897 2 0.99991476232974486
[2,] 3.9993181276998007 4 0.99982953192495017
[3,] 7.9979544702799732 8 0.99974430878499665
[4,] 15.9945454865482386 16 0.99965909290926491
[5,] 31.9863642975083522 32 0.99957388429713601
[6,] 63.9672757086713943 64 0.99948868294799054
[7,] 127.9236465742348372 128 0.99940348886120967
[8,] 255.8254853212606292 256 0.99931830203617433
[9,] 511.6073587057998679 512 0.99923312247226537
[10,] 1023.1275009729168914 1024 0.99914795016886415
[11,] 2046.0805839367201315 2048 0.99906278512535163
[12,] 4091.8123615891813643 4096 0.99897762734110873
[13,] 8182.9271700727167627 8192 0.99889247681551718
[14,] 16364.4593528497443913 16384 0.99880733354795803
[15,] 32726.1289689190489298 32768 0.99872219753781277
[16,] 65446.6789398585315212 65536 0.99863706878446246
')
t1 <- ((pi + exp(1)) /(187.5/16))^-(1:1023)
t2 <- 2^(1:1023)
t3 <- t1/t2
s4 <- cbind(t1,t2,t3)
plot(s4[,3],type="l")
#print(s4) #
print("Table that shows a relationship of 2^1:1023 produced by the dividend of (pi + e) / xi^-(1:1023):")
cat(' t1 t2 t3
[1,] 1.9998295246594897e+00 2.0000000000000000e+00 0.99991476232974486
[2,] 3.9993181276998007e+00 4.0000000000000000e+00 0.99982953192495017
[3,] 7.9979544702799732e+00 8.0000000000000000e+00 0.99974430878499665
[4,] 1.5994545486548239e+01 1.6000000000000000e+01 0.99965909290926491
[5,] 3.1986364297508352e+01 3.2000000000000000e+01 0.99957388429713601
[6,] 6.3967275708671394e+01 6.4000000000000000e+01 0.99948868294799054
[7,] 1.2792364657423484e+02 1.2800000000000000e+02 0.99940348886120967
[8,] 2.5582548532126063e+02 2.5600000000000000e+02 0.99931830203617433
')
t1 <- ((pi + exp(1)) /(187.5/16))^-(1:1023)
t2 <- 2^(1:1023)
t3 <- t1/t2
s4 <- cbind(t1,t2,t3)
plot(s4[,3])
l <- {};
j <- t3;
for(i in t3) {k <- (j - i);j <- i;l <-rbind(l,k)};
# long time for plot render
#plot(l[-1])
# l
cat(' [,1]
k 0.0000000000000000e+00
k 8.5230404794689463e-05
k 8.5223139953516380e-05
k 8.5215875731736723e-05
k 8.5208612128906402e-05
....
k 7.8152951918264435e-05
k 7.8146290342417934e-05
k 7.8139629334228466e-05
k 7.8132968894695232e-05
k 7.8126309022596985e-05
')
for (i in 1:16) {print(((pi + exp(1)) /(187.5/16))^-i)}
#[1] 1.9998295246594897
#[1] 3.9993181276998007
#[1] 7.9979544702799732
#[1] 15.994545486548239
#[1] 31.986364297508352
#[1] 63.967275708671394
#[1] 127.92364657423484
#[1] 255.82548532126063
#[1] 511.60735870579987
#[1] 1023.1275009729169
#[1] 2046.0805839367201
#[1] 4091.8123615891814
#[1] 8182.9271700727168
#[1] 16364.459352849744
#[1] 32726.128968919049
#[1] 65446.678939858532
as.matrix(2^(1:16))
cat(' [,1]
[1,] 2
[2,] 4
[3,] 8
[4,] 16
[5,] 32
[6,] 64
[7,] 128
[8,] 256
[9,] 512
[10,] 1024
[11,] 2048
[12,] 4096
[13,] 8192
[14,] 16384
[15,] 32768
[16,] 65536
')
t1 <- ((pi + exp(1)) /(187.5/16))^-(1:16)
t2 <- 2^(1:16)
cbind(t1,t2)
cat(' t1 t2
[1,] 1.9998295246594897 2
[2,] 3.9993181276998007 4
[3,] 7.9979544702799732 8
[4,] 15.9945454865482386 16
[5,] 31.9863642975083522 32
[6,] 63.9672757086713943 64
[7,] 127.9236465742348372 128
[8,] 255.8254853212606292 256
[9,] 511.6073587057998679 512
[10,] 1023.1275009729168914 1024
[11,] 2046.0805839367201315 2048
[12,] 4091.8123615891813643 4096
[13,] 8182.9271700727167627 8192
[14,] 16364.4593528497443913 16384
[15,] 32726.1289689190489298 32768
[16,] 65446.6789398585315212 65536
')
cbind(t1,t2,t1/t2,"t2-t1"=t2-t1)
cat(' t1 t2
[1,] 1.9998295246594897 2 0.99991476232974486 1.7047534051028990e-04
[2,] 3.9993181276998007 4 0.99982953192495017 6.8187230019933764e-04
[3,] 7.9979544702799732 8 0.99974430878499665 2.0455297200268063e-03
[4,] 15.9945454865482386 16 0.99965909290926491 5.4545134517614002e-03
[5,] 31.9863642975083522 32 0.99957388429713601 1.3635702491647805e-02
[6,] 63.9672757086713943 64 0.99948868294799054 3.2724291328605659e-02
[7,] 127.9236465742348372 128 0.99940348886120967 7.6353425765162797e-02
[8,] 255.8254853212606292 256 0.99931830203617433 1.7451467873937077e-01
[9,] 511.6073587057998679 512 0.99923312247226537 3.9264129420013205e-01
[10,] 1023.1275009729168914 1024 0.99914795016886415 8.7249902708310856e-01
[11,] 2046.0805839367201315 2048 0.99906278512535163 1.9194160632798685e+00
[12,] 4091.8123615891813643 4096 0.99897762734110873 4.1876384108186357e+00
[13,] 8182.9271700727167627 8192 0.99889247681551718 9.0728299272832373e+00
[14,] 16364.4593528497443913 16384 0.99880733354795803 1.9540647150255609e+01
[15,] 32726.1289689190489298 32768 0.99872219753781277 4.1871031080951070e+01
[16,] 65446.6789398585315212 65536 0.99863706878446246 8.9321060141468479e+01
')
# Some interesting sequences and explorations
(pi + exp(1)) /(187.5/16)
#[1] 0.50004262246816744
# correction factors gives 11.71875 :
exp(1.004057)^pi /2
#[1] 11.718759272520591
exp(pi^1.0035369) /2
#[1] 11.718759367405029
# interesting because give us nearly 1
exp(pi^-11.718759367405029)
#[1] 1.0000014928652075
exp(pi^-11.71875)
#[1] 1.0000014928812158
# power of 2 sequence resulting in almost 1, 1/2, 1/4, 1/8, 1/16
187.5/8 / exp(pi)
#[1] 1.0128262093071623
187.5/16 / exp(pi)
#[1] 0.50641310465358114
187.5/32 / exp(pi)
#[1] 0.25320655232679057
187.5/64 / exp(pi)
#[1] 0.12660327616339528
187.5/128 / exp(pi)
#[1] 0.063301638081697642
# ...
exp(11.718759272520591) /2048
#[1] 59.987744700119116 # 60
exp(11.718759272520591) /1024
#[1] 119.97548940023823 # 120
exp(11.718759272520591) /512
#[1] 239.95097880047646 # 240
exp(11.718759272520591) /256
#[1] 479.90195760095293 #480
exp(pi) / (187.5/2)
#[1] 0.24683405474964551 #.25
exp(pi) / (187.5/4)
#[1] 0.49366810949929102 #.5
exp(pi) / (187.5/8)
#[1] 0.98733621899858204 #1
exp(pi) / (187.5/16)
#[1] 1.9746724379971641 #2
exp(pi) / (187.5/32)
#[1] 3.9493448759943282 #4
exp(pi) / (187.5/64)
#[1] 7.8986897519886563 #8
exp(pi) / (187.5/128)
#[1] 15.797379503977313 #16
exp(pi) / (187.5/256)
#[1] 31.594759007954625 #32
exp(pi) / (187.5/512)
#[1] 63.18951801590925 #64
exp(pi) / (187.5/1024)
#[1] 126.3790360318185 #128
exp(pi) / (187.5/2048)
#[1] 252.758072063637 #256
(187.5/16)
#[1] 11.71875
((pi + exp(1)) /(187.5/16)) * 2
#[1] 1.0000852449363349
exp(((pi + exp(1)) /(187.5/16)) * 2)
#[1] 2.7185135580972011
exp(1)
#[1] 2.7182818284590451
exp(((pi + exp(1)) /(187.5/16)) * 1.999)
#[1] 2.7171545252638842
exp(1)
#[1] 2.7182818284590451
187.5/8 / exp(pi)
#[1] 1.0128262093071623
187.5/16 / exp(pi)
#[1] 0.50641310465358114
exp(pi)
#[1] 23.140692632779267
exp(pi) / 2
#[1] 11.570346316389633
exp(pi) / exp(11.570346316389633)
#[1] 0.0002184936375100267
exp(pi) / exp(187.5/16)
#[1] 0.00018835965834893231
tanh(187.5/16) + (((pi + exp(1)) * 2 ) - 187.5/16)
#[1] 1.0009989639651642
tanh(187.5/16)
#[1] 0.99999999986748866
(((pi + exp(1)) * 2 ) - 187.5/16)
#[1] 0.00099896409767552541
(((pi + exp(1)) * 2 ) - 187.5/16) / tanh(187.5/16)
#[1] 0.00099896409780789956
(((pi + exp(1)) * 2 ) - 187.5/16) - tanh(187.5/16)
#[1] -0.99900103576981314
acosh(187.5/16)
#[1] 3.1525118680526645
asinh(187.5/16)
#[1] 3.1561527971624654
acosh(187.5/16) - pi
#[1] 0.010919214462871363
asinh(187.5/16) - pi
#[1] 0.014560143572672324
asinh(187.5/16) / pi
#[1] 1.004634637643437
(pi + exp(1))
#[1] 5.8598744820488378
(pi + exp(1)) /(187.5/16)
#[1] 0.50004262246816744
(187.5/16) /2
#[1] 5.859375
pi + exp(1)
#[1] 5.8598744820488378
t1 <- (187.5/16) /2
t2 <- pi + exp(1)
t1 - exp(1)
#[1] 3.1410931715409549
# golden ratio stuff
digits(options=22)
gr <- (sqrt(5) + 1) / 2
xi <- (187.5 /16) / 10
XI <- 187.5 /16
pi
#[1] 3.1415926535897931
exp(1)
#[1] 2.7182818284590451
gr
#[1] 1.6180339887498949
xi
#[1] 1.171875
((pi*xi - gr) - (pi/xi - gr))
[1] 1.0007281598622488
gr1 <- ((pi*xi - gr) - (pi/xi - gr))
((pi*xi - gr) / (pi/xi - gr))
#[1] 1.9416032511543122
gr
#[1] 1.6180339887498949
pi
#[1] 3.1415926535897931
xi
#[1] 1.171875
exp(1)
#2.7182818284590451
grxi <- gr + xi
#[1] 2.7899089887498949
exp(1) - (gr + xi)
#[1] -0.071627160290849812
(gr + xi) - exp(1)
#[1] 0.071627160290849812
((xi - exp(1)) + gr)
#[1] 0.071627160290849812
grxi <- gr + xi
format(grxi - exp(1), scientific=FALSE)
# [1] "0.071627160290849812"
exp(1) - ((gr + xi) - (xi/gr))
# [1] 0.652631420275433
exp(1) - ((gr + xi) - ((xi/gr) / 10))
#[1] 0.00079869776577856655
exp(1)
#[1] 2.7182818284590451
((gr + xi) - ((xi/gr) / 10))
#[1] 2.7174831306932665
#[1] 2.7063293868263711
exp(1)
#[1] 2.7182818284590451
sqrt(exp(1))
[1] 1.6487212707001282
gr
#[1] 1.6180339887498949
xi/sqrt(3/16)/sqrt(exp(1))
#[1] 1.6414717483914856
gr/exp(1)
#[1] 0.59524143957771114
exp(1)/gr
#[1] 1.679990560988901
gr
#[1] 1.6180339887498949
sqrt(exp(1)/(xi/sqrt(3/16)))
#[1] 1.0022058057366372
(xi/sqrt(3/16))/sqrt(exp(1))
#[1] 1.6414717483914856
-digamma(1) @ R 'built-in' Euler–Mascheroni constant approximation; Good for 15 digits
#[1] 0.57721566490153231
exp(1)/gr - ((gr/exp(1))/10)
#[1] 1.6204664170311298
gr
#[1] 1.6180339887498949
gr/exp(1) + -digamma(1)
#[1] 1.1724571044792436
xi
#[1] 1.171875
options(digits=15)
xi
#[1] 1.171875
gr/exp(1) + -digamma(1)
#[1] 1.1724571044792
xi - (gr/exp(1) + -digamma(1))
#[1] -0.00058210447924356
sxi <- (gr/exp(1) + -digamma(1)) * 10
pR <- ((sxi/2) - exp(1) + (-digamma(1))/1000 )
format(pR - pi,scientific=FALSE)
#[1] "0.002988256012280921"
pR <- ((Xi/2) - exp(1) + tanh(Xi) / 2000 )
format(pR - pi,scientific=FALSE)
# [1] "0.0000005179510953468025"
## Updated 7/10/2018 with some additional work with the golden ratio far below
## *Notes on 187.5/16 or 11.71875*
##
# In this code, I introduce 'xi' (11.71875) a constant with strong relationships to both pi and # exp(1).
# 'Xi' (11.71875 or (187.5/16)) is a 'translator constant' between powers of 2 and 16.
# Search google for "11.71875mhz" and you can see this. 11.71875mhz is a widely used frequency in 'crystal resonators'.
# More than simply being an efficient means to compute pi or exp(1) (e.g. 'Eulers number'), 'xi' may
# 'interpolate' between pi and e, giving science the means to measure logarithmic growth patterns of
# increasing radii. This may make it useful for gravitational and rotational calculations. It is
# possible that the 187.5 cm-3 pc represents some fixed attenuation limit (see
# http://arxiv.org/abs/1503.05245) for radio waves signal that require either boosting or
# chirping. The relationships between pi, e, and xi(187.5./16 = 11.71875) may represent fixed
# constants descriptive of all EM signaling. In addition, the constants that describe the
# circumscription of an octahedron (sqrt(1/2) and sqrt(1/6) can also be used to derive xi with a
# factor close to 9/16.
## A number of constants and function names are used. Some are reused:
# [1] "almost2" "aR" "c2" "cR" "eR" "F_pe" "F_pi" "findpi" "i" "j" "k" "k1"
# [13] "k2" "l" "l1" "mm1" "p1" "p2" "piR" "pR" "pX" "s4" "seqe" "seqpi"
# [25] "seqpi_e" "t1" "t2" "t3" "Vx" "x" "xi" "xi_sqrt" "xif" "y"
## Set your graphics device to record history if desired. 86 graphs are produced.
## You may want par(mfrow=c(1,2))
## NB : In R 3.5 there is no bigint by default,however there are 22 digits of accuracy.
## See my usage of BIGINT in Powershell
## For this work I used Microsoft Open R 3.5 with Intel's MKL (Math Kernel Library)
options(digits=22)
# main constants
pi
#[1] 3.1415926535897931
exp(1) # Eulers number
#[1] 2.7182818284590451
xi <- 187.5/16 # transdimensional constant
# [1] 11.71875
# xi with pi and exp(1)
pi + exp(1)
# [1] 5.8598744820488378
2 * (pi + exp(1)) # very close to xi
# [1] 11.719748964097676
(pi + exp(1)) / 11.71875 # very close .5
# [1] 0.5000426224681674414185
xi / (pi + exp(1)) # very close to 2
# [1] 1.9998295246594895
((xi /(pi + exp(1))) * (pi + exp(1)))
# [1] 11.71875
# xi without pi or exp(1)
# from regular octahedron constants for finding circumscribed sphere
sqrt(1/6)
# [1] 0.40824829046386302
sqrt(1/2)
# [1] 0.70710678118654757
904319/1600000 # something a little different than 9/16 ???
# 9/16 - 904319/1600000 = -0.0026993749999999483
((10 * (sqrt(1/2) + sqrt(1/6))) + 904319/1600000) # close to xi
# [1] 11.718750091504107
xi_sqrt <- ((10 * (sqrt(1/2) + sqrt(1/6))) + 904319/1600000)
xi_sqrt
# [1] 11.718750091504107
((xi_sqrt /(pi + exp(1))) * (pi + exp(1)))
# [1] 11.718750091504107
# one way of deriving almost pi with exp(1) and xi
xi /(pi + exp(1))
# [1] 1.9998295246594895
almost2 <- xi /(pi + exp(1)) # almost 2
((xi/2) - exp(1)) # two digits right of decimal of pi
# [1] 3.1410931715409549
((xi/almost2) - exp(1)) # 14 digits right of decimal point of pi
# [1] 3.1415926535897927
pi
# [1] 3.1415926535897931
# one way of deriving almost exp(1) with pi and xi
((xi/2) - pi) # two digits right of decimal of exp(1)
#[1] 2.7177823464102069
exp(1)
# [1] 2.7182818284590451
((xi/almost2) - pi) # 14 digits right of decimal point of exp(1)
# [1] 2.7182818284590446
exp(1)
# [1] 2.7182818284590451
# 187.5 is the first fractional member of a series that returns 11.71875 as
# the quotient of what appears to be integer multiples of (3,5,6,10) by powers of 2 (2^(1:16)).
# 11.71875 pops up in google queries for decimal translation of bytes. All of the results below
# equal 11.71875:
cat('
12000/1024
6000/512
3000/256
1500/128
750/64
375/32
187.5/16
46.875/4
23.4375/2
')
# 11.71875 can be seen as a type of hexadecimal 'translator' ratio between binary and decimal:
12000/1024
#[1] 11.71875
6000/512
#[1] 11.71875
3000/256
#[1] 11.71875
1500/128
#[1] 11.71875
750/(187.5/16)
#[1] 64
12000/(187.5/16)
#[1] 1024
6000/(187.5/16)
#[1] 512
3000/(187.5/16)
#[1] 256
1500/(187.5/16)
#[1] 128
750/(187.5/16)
#[1] 64
# However, I can also come up with something more transcendental:
options(digits=22)
(18.75/2)/10
#[1] 0.9375
(.9375/2)/10
#[1] 0.046875
(.046875/2)/10
#[1] 0.0023437499999999999
(.002343749999999999/2)/10
# 0.00011718749999999996
#or
(23.43749999999999/2)
# [1] 11.718749999999995
# ...
# Multiples of 187.5 can be produced by selecting out the products of vector (1:100) multiplication of the
# quotient of 187.5./16. By sixteen,eight,four:
data.frame((187.5/16 * 1:100)[c(seq(0,100,16))])
cat('
1 187.5
2 375.0
3 562.5
4 750.0
5 937.5
6 1125.0
')
# By eight:
cat('
data.frame((187.5/16 * 1:100)[c(seq(0,100,8))])
1 93.75
2 187.50
3 281.25
4 375.00
5 468.75
6 562.50
7 656.25
8 750.00
9 843.75
10 937.50
11 1031.25
12 1125.00
')
# By four:
data.frame((187.5/16 * 1:100)[c(seq(0,100,4))])
cat('
1 46.875
2 93.750
3 140.625
4 187.500
5 234.375
6 281.250
7 328.125
')
# This series or ladder is significant, but I am not quite sure why it accurately generates the range (1:1/10) to five digits.
l <- {};for(i in 1:10) {l <- rbind(l,(exp(pi^-11.71875)/i))};l;plot(l)
cat(' [,1]
[1,] 1.00000149288121576 #1
[2,] 0.50000074644060788 #1/2
[3,] 0.33333383096040525 #1/3
[4,] 0.25000037322030394 #1/4
[5,] 0.20000029857624316 #1/5
[6,] 0.16666691548020263 #1/6
[7,] 0.14285735612588796 #1/7
[8,] 0.12500018661015197 #1/8
[9,] 0.11111127698680175 #1/9
[10,] 0.10000014928812158 #1/10
')
pi # 22 digits R 3.2.2 constant for pi
#[1] 3.1415926535897931
exp(1) # Eulers number in R 3.2.2
#[1] 2.7182818284590451
# 'xi' a 'transdimensional' constant
187.5/16
#[1] 11.71875
xi <- 187.5/16 # 11.71875 - greater than pi or e
k1 <- (spline(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
k2 <- (sort(c(xi,exp(1),pi,-pi,-exp(1),-xi)))
splinefun(diff(k2))
plot(splinefun(diff(k2)))
plot(spline(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
plot(splinefun(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
xi <- (187.5/16)/10 # 1.171875 - less than pi or e
k1 <- (spline(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
k2 <- (sort(c(xi,exp(1),pi,-pi,-exp(1),-xi)))
splinefun(diff(k2))
plot(splinefun(diff(k2)))
plot(spline(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
plot(splinefun(sort(c(xi,exp(1),pi,-pi,-exp(1),-xi))))
pi #pi
#[1] 3.1415926535897931
exp(1) #e
#[1] 2.7182818284590451
187.5/16 #xi
#[1] 11.71875
((11.71875/2) - exp(1))
#[1] 3.1410931715409549 #close to pie
((11.71875/2) - pi)
#[1] 2.7177823464102069 #close to e
((11.71875/2) - exp(1)) + ((11.71875/2) - pi)
#[1] 5.8588755179511622 # close to .5(xi)
(((11.71875/2) - exp(1)) + ((11.71875/2) - pi)) * 2
#[1] 11.717751035902324 # close to xi
11.71875 - 11.717751035902324
exp(11.71875 - 11.717751035902324)
#[1] 1.0009994632285004 # close to 1
(23.43749999999999/2)
exp(11.71875 - 11.718749999999995)
#[1] 1.0000000000000053 # closer to 1
# Since (pi + exp(1)) * 2 ~= (187.5/16) it is possible to use 11.71875 and some
# adjustment to arrive at either pi or e. (See notes and functions seqpi,seqe,seqpi_e (far)below.)
# The products of pi and e doubled roughly equal (187.5/16) or 11.71875:
(((pi + exp(1)) * 2) /(187.5/16))
#[1] 1.0000852449363349 # nearly one again. irrational?
(((pi + exp(1))) /(187.5/16))
#[1] 0.50004262246816744 # nearly five. irrational?
# 11.71875 can be successfully used to calculate pi or e within 3 digits right of the decimal point without any compensation:
((187.5/16)/2) - pi # almost e...
#[1] 2.7177823464102069 # should be 2.7182818284590451
((187.5/16)/2) - exp(1) # almost pi
#[1] 3.1410931715409549 # should be 3.1415926535897931
# Not quite pi or e:
pi - (((187.5/16)/2) - exp(1))
#[1] 0.00049948204883820679
exp(1) - (((187.5/16)/2) - pi)
#[1] 0.00049948204883820679
# The adjustment ( tanh(11.71875) / 2000 ) gives us 5 digits of accuracy for pi or e:
tanh(11.71875) / 2000
[1] 0.00049999999993374431
eR <- ((11.71875/2) - pi + tanh(11.71875) / 2000 )
format(eR - exp(1),scientific=FALSE)
# [1] "0.0000005179510953468025"
eR
#[1] 2.7182823464101404
exp(1)
#[1] 2.7182818284590451
pR <- ((11.71875/2) - exp(1) + tanh(11.71875) / 2000 )
format(pR - pi,scientific=FALSE)
# [1] "0.0000005179510953468025"
pR
# [1] 3.1415931715408885
pi
# [1] 3.1415926535897931
# using -digamma(1)
xi - (gr/exp(1) + -digamma(1))
[1] -0.00058210447924356
# DOES xif makes xi a 'transdimenesional transcendant'?
(18.75/2)/10
#[1] 0.9375
(.9375/2)/10
#[1] 0.046875
(.046875/2)/10
#[1] 0.0023437499999999999
(.002343749999999999/2)/10
# 0.00011718749999999996
(18.75/2)/10
(.9375/2)/10
(.046875/2)/10
(.002343749999999999/2)/10
xif <- (.002343749999999999/2)/10
xif <- (xif * 100000)
# Powershell [BIGINT] will take me this far:
cat('
PS [bigint](.002343749999999999/2 * [MATH]::POW(10,32))
117187499999999960548787617792
PS [bigint](.002343749999999999/2 * [MATH]::POW(10,64))
11718749999999995682656043607972093960765263440537027795222528
PS[bigint](.002343749999999999/2 * [MATH]::POW(10,128))
117187499999999952444488198957480104184273770534709180501147236632873797389362215023609307534290876647973955371666657662468096
PS [bigint](.002343749999999999/2 * [MATH]::POW(10,256))
11718749999999995794845523228309099783974204074624886274076630993851149729639928280300079885632128025211313970818427611098306439684539273451270347793350886454282383635
265648724318526179750550535267831033255464007490486648205861676480655926487782004883456
')
tanh(11.71875) # helps get us close to 1
#[1] 0.99999999986748866
# so does
xif/xi
#[1] 9.9999999999999947
xi/xif
#[1] 0.10000000000000005
eR <- ((11.71875/2) - pi + tanh(11.71875) / 2000 )
format(eR - exp(1),scientific=FALSE)
# [1] "0.0000005179510953468025"
eR
# [1] 2.7182823464101404
exp(1)
#[1] 2.7182818284590451
pR <- ((11.71875/2) - exp(1) + tanh(11.71875) / 2000 )
format(pR - pi,scientific=FALSE)
# [1] "0.0000005179510953468025"
pR
# [1] 3.1415931715408885
pi
# [1] 3.1415926535897931
(18.75/2)/10
(.9375/2)/10
(.046875/2)/10
(.002343749999999999/2)/10
xif <- (.002343749999999999/2)/10
xif <- (xif * 100000)
eR <- ((xif/2) - pi + tanh(xif) / 2000 )
format(eR - exp(1),scientific=FALSE)
eR
exp(1)
pR <- ((xif/2) - exp(1) + tanh(xif) / 2000 )
format(pR - pi,scientific=FALSE)
pR
pi
## By selecting from an array we can get much more accurate pi and e from xi.
((11.71875/2) - pi + tanh((11.71875/4096) * 1:4096) / 2000 )[c(1322)]
#[1] 2.718281828165519
exp(1)
#[1] 2.7182818284590451
((11.71875/2) - exp(1) + tanh((11.71875/4096) * 1:4096) / 2000 )[c(1322)]
#[1] 3.141592653296267
pi
#[1] 3.1415926535897931
## It is then possible for pi or e to converge from series generated from xi (11.71875)
## This series will converge to pi :
c2 <- as.matrix(c(
(((11.71875/2) - exp(1)) + (tanh((11.71875/8) * 1:8)) / 2000 )[3],
(((11.71875/2) - exp(1)) + (tanh((11.71875/16) * 1:16)) / 2000 )[5],
(((11.71875/2) - exp(1)) + (tanh((11.71875/32) * 1:32)) / 2000 )[11],
(((11.71875/2) - exp(1)) + (tanh((11.71875/64) * 1:64)) / 2000 )[21],
(((11.71875/2) - exp(1)) + (tanh((11.71875/128) * 1:128)) / 2000 )[41],
(((11.71875/2) - exp(1)) + (tanh((11.71875/256) * 1:256)) / 2000 )[83],
(((11.71875/2) - exp(1)) + (tanh((11.71875/512) * 1:512)) / 2000 )[165],
(((11.71875/2) - exp(1)) + (tanh((11.71875/1024) * 1:1024)) / 2000 )[331],
(((11.71875/2) - exp(1)) + (tanh((11.71875/2048) * 1:2048)) / 2000 )[661],
(((11.71875/2) - exp(1)) + (tanh((11.71875/4096) * 1:4096)) / 2000 )[1322],
(((11.71875/2) - exp(1)) + (tanh((11.71875/8192) * 1:8192)) / 2000 )[2644],
(((11.71875/2) - exp(1)) + (tanh((11.71875/16384) * 1:16384)) / 2000 )[5288],
(((11.71875/2) - exp(1)) + (tanh((11.71875/32768) * 1:32768)) / 2000 )[10576],
(((11.71875/2) - exp(1)) + (tanh((11.71875/65536) * 1:65536)) / 2000 )[21154]
))
c2
cat(' [,1]
[1,] 3.1415930191734072
[2,] 3.1415925126008446
[3,] 3.1415928546514991
[4,] 3.1415927145681901
[5,] 3.1415926227939854
[6,] 3.1415926707780644
[7,] 3.1415926473344875
[8,] 3.1415926591902781
[9,] 3.1415926532962670
[10,] 3.1415926532962670
[11,] 3.1415926532962670
[12,] 3.1415926532962670
[13,] 3.1415926532962670
[14,] 3.1415926536666201
')
pi
# [1] 3.1415926535897931
# So we can generate series that converge around pi and e :
options(digits=22)
# These print very long sequences/series
# for(x in (2^seq(0,16,1))) {print(format(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000))}
# for(x in (2^seq(0,16,1))) {print(format(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,digits=22,scientific=TRUE))}
# for(x in (2^seq(0,16,1))) {print(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000)}
for(x in (2^seq(0,16,1))) {(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000 )}
for(x in (2^seq(0,16,1))) {plot(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / x ,pty=.25,col="blue",type="p")};abline(h=pi,col="red")
for(x in (2^seq(0,16,1))) {plot(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000 ,pty=.25,col="blue",type="p")};abline(h=pi,col="red")
for(x in (2^seq(0,16,1))) {plot(((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,ylim=c(pi - (pi *.000001),pi + (pi *.000001)),lwd=1,col="blue",type="l")};abline(h=pi,col="red")
for(x in (2^seq(8,12,1))) {plot(grep(3.141592,((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,value=TRUE),lwd=1,col="blue",type="l")};abline(h=pi,col="red")
for(x in (2^seq(8,14,1))) {plot(grep(3.141592,((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,value=TRUE),lwd=1,col="blue",type="l")};abline(h=pi,col="red")
findpi <- function(x,y) {for(i in (2^seq(x,y,1))) {p1 <- rbind(p1,(grep(pi,(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE)))}}
findpi <- function(x,y) {for(i in (2^seq(x,y,1))) {p1 <- rbind(p1,(grep(pi,(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE)))};plot(pi)}
p1 <- NULL; p1 <- rbind(p1,cbind(grep(3.1415,(11.71875/2) - exp(1) + (tanh((11.71875/8) * 1:8)) / 2000,value=TRUE)))
# template function code A:
x <- 1
y <- 16
options(digits=(x - 1))
p1 <- NULL
for(i in (2^seq(x,y,1)))
{p1 <- rbind(p1,cbind(grep(strtrim(pi,x),(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE)))}
# options(digits = (x + 1))
plot(p1,lwd=1,col="blue",type="l")
abline(h=pi,col="red")
# template function code B:
x <- 8
y <- 16
options(digits=(22))
p1 <- NULL
plot.default(x,y,type="n", xlim=c(0,15000),ylim=c(3.1415,3.1416))
for(i in (2^seq(x,y,1)))
{
lines(
rbind(p1,cbind(grep(3.1415,(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE))),
lwd=1,col=rgb(runif(1),runif(1),runif(1))
)
}
abline(h=pi,lwd=2,col=rgb(1,0,0,.5))
# These functions below (seqpi,seqe,seqpi_e) need three arguments like:
cat('
seqpi(1,1,8)
seqe(1,1,8)
seqpi_e(1,1,8)
seqpi(0,16,16)
seqe(0,16,16)
seqpi_e(0,16,16)
')
seqpi <- function(x,y,z) {
options(digits=(22))
xi <- 11.71875
#trimpi <- 7
p1 <- NULL
plot.default(x,y,type="n", xlim=c(0,z),ylim=c(3.1415,3.1416))
for(i in (2^seq(x,y,1))) {
lines(
rbind(p1,cbind(grep(3.1415,(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE))),
lwd=2,col=rgb(runif(1),runif(1),runif(1)) )}
abline(h=pi,lwd=6,col=rgb(.5,0,0,.25))
}
seqe <- function(x,y,z) {
options(digits=(22))
xi <- 11.71875
p1 <- NULL
e1 <- NULL
plot.default(x,y,type="n", xlim=c(0,z),ylim=c(2.7182,2.7183))
for(i in (2^seq(x,y,1))) {
lines(rbind(e1,cbind(grep(2.7182,(11.71875/2) - pi + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE))),
lwd=2,col=rgb(runif(1),runif(1),runif(1)))
}
abline(h=exp(1),lwd=6,col=rgb(0,0,.5,.25))
}
seqpi_e <- function(x,y,z) {
options(digits=(22))
xi <- 11.71875
p1 <- NULL
e1 <- NULL
plot.default(x,y,type="n", xlim=c(0,z),ylim=c(2.7182,3.1416))
for(i in (2^seq(x,y,1))) {
lines(rbind(p1,cbind(grep(3.1415,(11.71875/2) - exp(1) + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE))),
lwd=2,col=rgb(runif(1),runif(1),runif(1)));
lines(rbind(e1,cbind(grep(2.7182,(11.71875/2) - pi + (tanh((11.71875/i) * 1:i)) / 2000,value=TRUE))),
lwd=2,col=rgb(runif(1),runif(1),runif(1)))
}
abline(h=pi,lwd=6,col=rgb(.5,0,0,.25))
abline(h=exp(1),lwd=6,col=rgb(0,0,.5,.25))
}
# plotting these graphs with numbers bigger than 16 requires more CPU and memory than I got...
seqpi(1,1,8)
seqe(1,1,8)
seqpi_e(1,1,8)
seqpi(0,16,16)
seqe(0,16,16)
seqpi_e(0,16,16)
# for(x in (2^seq(0,16,1))) {pi + (min((((11.71875/2) - exp(1)) + (tanh((11.71875/8) * 1:8)) / 2000 ) - pi))}
for(x in (2^seq(0,4,1))) {match(3.141592,((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000 )}
for(x in (2^seq(0,8,1))) {print(grep(3.1415926535,((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,value=TRUE))}
for(x in (2^seq(0,8,1))) {plot(grep(3.14159,((11.71875/2) - exp(1)) + (tanh((11.71875/x) * 1:x)) / 2000,value=TRUE))}
plot(c2,ylim=c(pi - (pi *.000001),pi + (pi *.000001)),lwd=1,col="blue",type="l");abline(h=pi,col="red")
# A magic series??
l1 <- c(3,5,11,21,41,83,165,331,661,1322,2644,5288,10576,21154)
as.matrix(l1 / (xi * .1))
cat(' [,1]
[1,] 2.5600000000000001
[2,] 4.2666666666666666
[3,] 9.3866666666666667
[4,] 17.9200000000000017
[5,] 34.9866666666666646
[6,] 70.8266666666666680
[7,] 140.8000000000000114
[8,] 282.4533333333333189
[9,] 564.0533333333332848
[10,] 1128.1066666666665697
[11,] 2256.2133333333331393
[12,] 4512.4266666666662786
[13,] 9024.8533333333325572
[14,] 18051.4133333333338669
')
plot(as.matrix(l1 / (xi * .1)))
plot(as.matrix(l1 / (xi * .1)),type="b")
l1 <- c(3,5,11,21,41,83,165,331,661,1322,2644,5288,10576,21154)
mm1 <- data.frame(cbind("Pxi" = l1))
mm1 <- data.frame(mm1,"P2" = 2^(3:16))
for(i in 1:10) {print(((11.71875/2) - pi) + (tanh((11.71875/(mm1$P2[i])) * 1:mm1$P2[i]) / 2000 )[mm1$Pxi[i]])}
F_pi <- function(x) {((11.71875/2) - exp(1)) + (tanh((11.71875/(mm1$P2[x])) * 1:mm1$P2[x]) / 2000 )[mm1$Pxi[x]]}
F_pe <- function(x) {((11.71875/2) - pi) + (tanh((11.71875/(mm1$P2[x])) * 1:mm1$P2[x]) / 2000 )[mm1$Pxi[x]]}
as.matrix(sapply(1:14, F_pi))
cat(' [,1]
[1,] 3.1415930191734072
[2,] 3.1415925126008446
[3,] 3.1415928546514991
[4,] 3.1415927145681901
[5,] 3.1415926227939854
[6,] 3.1415926707780644
[7,] 3.1415926473344875
[8,] 3.1415926591902781
[9,] 3.1415926532962670
[10,] 3.1415926532962670
[11,] 3.1415926532962670
[12,] 3.1415926532962670
[13,] 3.1415926532962670
[14,] 3.1415926536666201
')
as.matrix(mm1$Pxi /mm1$P2) / .1171875
cat(' [,1]
[1,] 3.2000000000000002
[2,] 2.6666666666666665
[3,] 2.9333333333333331
[4,] 2.7999999999999998
[5,] 2.7333333333333334
[6,] 2.7666666666666666
[7,] 2.7500000000000000
[8,] 2.7583333333333333
[9,] 2.7541666666666669
[10,] 2.7541666666666669
[11,] 2.7541666666666669
[12,] 2.7541666666666669
[13,] 2.7541666666666669
[14,] 2.7544270833333333
')
as.matrix(mm1$Pxi /mm1$P2) / 11.71875
cat(' [,1]
[1,] 0.032000000000000001
[2,] 0.026666666666666668
[3,] 0.029333333333333333
[4,] 0.028000000000000001
[5,] 0.027333333333333334
[6,] 0.027666666666666666
[7,] 0.027500000000000000
[8,] 0.027583333333333335
[9,] 0.027541666666666666
[10,] 0.027541666666666666
[11,] 0.027541666666666666
[12,] 0.027541666666666666
[13,] 0.027541666666666666
[14,] 0.027544270833333332
')
# These are some earlier notes in process:
# A reasonable approximation of pi can be made with with aR below
# cR is a correction
((11.71875/2) - exp(1) + exp(-11.71875/(2^14)) / 2000)
format(pi - ((11.71875/2) - exp(1) + exp(-11.71875/(2^14)) / 2000),scientific=FALSE)
#[1] "-0.00000016045116035812157"
aR <- 11.71875/2 - exp(1)
aR
cR <- exp(-11.71875/(2^14)) / 2000
cR
piR <- aR + cR
pi
piR
piR - pi
format(piR - pi, scientific=FALSE)
aR <- 11.71875/2 - exp(1)
aR
#[1] 3.1410931715409549
cR <- exp(-11.71875/(2^14)) / 2000
cR
#[1] 0.00049964249999855234
piR <- aR + cR
pi
#[1] 3.1415926535897931
piR
#[1] 3.1415928140409535
piR - pi
#[1] 1.6045116035812157e-07
format(piR - pi, scientific=FALSE)
#[1] "0.00000016045116035812157"
eR <- ((11.71875/2) - pi + tanh(11.71875) / 2000 )
format(eR - exp(1),scientific=FALSE)
Vx <- as.data.frame(cbind(as.matrix(1 / 1:10),as.matrix(exp(pi^-11.71875) / 1:10)))
Vx$V3 <- with(Vx,cbind(V2 - V1))
Vx$V4 <- with(Vx,cbind(V1 / V2))
Vx$V5 <- with(Vx,cbind(V2 / V1))
Vx
cat(' V1 V2 V3 V4 V5
1 1.00000000000000000 1.00000149288121576 1.4928812157588567e-06 0.99999850712101290 1.0000014928812158
2 0.50000000000000000 0.50000074644060788 7.4644060787942834e-07 0.99999850712101290 1.0000014928812158
3 0.33333333333333331 0.33333383096040525 4.9762707193812261e-07 0.99999850712101290 1.0000014928812158
4 0.25000000000000000 0.25000037322030394 3.7322030393971417e-07 0.99999850712101290 1.0000014928812158
5 0.20000000000000001 0.20000029857624316 2.9857624314622022e-07 0.99999850712101301 1.0000014928812158
6 0.16666666666666666 0.16666691548020263 2.4881353596906131e-07 0.99999850712101290 1.0000014928812158
7 0.14285714285714285 0.14285735612588796 2.1326874510840810e-07 0.99999850712101290 1.0000014928812158
8 0.12500000000000000 0.12500018661015197 1.8661015196985709e-07 0.99999850712101290 1.0000014928812158
9 0.11111111111111110 0.11111127698680175 1.6587569064141494e-07 0.99999850712101290 1.0000014928812158
10 0.10000000000000001 0.10000014928812158 1.4928812157311011e-07 0.99999850712101301 1.0000014928812158
')
# Other Sequences/Series/Prodcuts of interest
23.4375/2
#[1] 11.71875
(23.4375 * 5^1)/1
#[1] 117.1875
(23.4375 * 5^2)/.5
#[1] 1171.875
(23.4375 * 5^3)/.25
#[1] 11718.75
(23.4375 * 5^4)/.125
#[1] 117187.5
(23.4375 * 5^5)/.0625
#[1] 1171875
(23.4375 * 5^6)/.03125
#[1] 11718750
11.71875/2
#[1] 5.859375
(11.71875 * 5^1)/1
#[1] 58.59375
(11.71875 * 5^2)/.5
#[1] 585.9375
(11.71875 * 5^3)/.25
#[1] 5859.375
(11.71875 * 5^4)/.125
#[1] 58593.75
(11.71875 * 5^5)/.0625
#[1] 585937.5
(11.71875 * 5^6)/.03125
#[1] 5859375
exp(pi^-11.71875)
#[1] 1.0000014928812158
exp(pi^-11.71875)/2
#[1] 0.50000074644060788
exp(pi^-11.71875)/3
#[1] 0.33333383096040525
exp(pi^-11.71875)/4
#[1] 0.25000037322030394
exp(pi^-11.71875)/5
#[1] 0.20000029857624316
exp(pi^-11.71875)/6
#[1] 0.16666691548020263
exp(pi^-11.71875)/7
#[1] 0.14285735612588796
exp(pi^-11.71875)/8
#[1] 0.12500018661015197
exp(pi^-11.71875)/9
#[1] 0.11111127698680175
1/9
#[1] 0.1111111111111111
exp(pi^-11.71875)/10
#[1] 0.10000014928812158
p2 <- 2^(1:16)
pX <- data.frame(cbind(p2,(187.5/16) * p2)) # becomes V2
pX$V3 <- cbind((((pi + exp(1)) * 2) * p2))
pX$V4 <- with(pX,cbind(V3 - V2))
pX$V5 <- with(pX,cbind(V4 * 1000))
pX$V6 <- with(pX,cbind(p2/V5))
plot(pX$p2,type="p",col="red");par(new=T);plot(pX$V5,type="l",col="blue");par(new=F);
pX
cat(' p2 V2 V3 V4 V5 V6
1 2 23.4375 23.439497928195351 0.0019979281953510508 1.9979281953510508 1.0010369765308733
2 4 46.8750 46.878995856390702 0.0039958563907021016 3.9958563907021016 1.0010369765308733
3 8 93.7500 93.757991712781404 0.0079917127814042033 7.9917127814042033 1.0010369765308733
4 16 187.5000 187.515983425562808 0.0159834255628084065 15.9834255628084065 1.0010369765308733
5 32 375.0000 375.031966851125617 0.0319668511256168131 31.9668511256168131 1.0010369765308733
6 64 750.0000 750.063933702251234 0.0639337022512336262 63.9337022512336262 1.0010369765308733
7 128 1500.0000 1500.127867404502467 0.1278674045024672523 127.8674045024672523 1.0010369765308733
8 256 3000.0000 3000.255734809004935 0.2557348090049345046 255.7348090049345046 1.0010369765308733
9 512 6000.0000 6000.511469618009869 0.5114696180098690093 511.4696180098690093 1.0010369765308733
10 1024 12000.0000 12001.022939236019738 1.0229392360197380185 1022.9392360197380185 1.0010369765308733
11 2048 24000.0000 24002.045878472039476 2.0458784720394760370 2045.8784720394760370 1.0010369765308733
12 4096 48000.0000 48004.091756944078952 4.0917569440789520741 4091.7569440789520741 1.0010369765308733
13 8192 96000.0000 96008.183513888157904 8.1835138881579041481 8183.5138881579041481 1.0010369765308733
14 16384 192000.0000 192016.367027776315808 16.3670277763158082962 16367.0277763158082962 1.0010369765308733
15 32768 384000.0000 384032.734055552631617 32.7340555526316165924 32734.0555526316165924 1.0010369765308733
16 65536 768000.0000 768065.468111105263233 65.4681111052632331848 65468.1111052632331848 1.0010369765308733
')
t1 <- ((pi + exp(1)) /(187.5/16))^-(1:16)
t2 <- 2^(1:16)
t3 <- t1/t2
s4 <- cbind(t1,t2,t3)
plot(s4[,3],type="l")
s4
cat(' t1 t2 t3
[1,] 1.9998295246594897 2 0.99991476232974486
[2,] 3.9993181276998007 4 0.99982953192495017
[3,] 7.9979544702799732 8 0.99974430878499665
[4,] 15.9945454865482386 16 0.99965909290926491
[5,] 31.9863642975083522 32 0.99957388429713601
[6,] 63.9672757086713943 64 0.99948868294799054
[7,] 127.9236465742348372 128 0.99940348886120967
[8,] 255.8254853212606292 256 0.99931830203617433
[9,] 511.6073587057998679 512 0.99923312247226537
[10,] 1023.1275009729168914 1024 0.99914795016886415
[11,] 2046.0805839367201315 2048 0.99906278512535163
[12,] 4091.8123615891813643 4096 0.99897762734110873
[13,] 8182.9271700727167627 8192 0.99889247681551718
[14,] 16364.4593528497443913 16384 0.99880733354795803
[15,] 32726.1289689190489298 32768 0.99872219753781277
[16,] 65446.6789398585315212 65536 0.99863706878446246
')
t1 <- ((pi + exp(1)) /(187.5/16))^-(1:1023)
t2 <- 2^(1:1023)
t3 <- t1/t2
s4 <- cbind(t1,t2,t3)
plot(s4[,3],type="l")
#print(s4) #
print("Table that shows a relationship of 2^1:1023 produced by the dividend of (pi + e) / xi^-(1:1023):")
cat(' t1 t2 t3
[1,] 1.9998295246594897e+00 2.0000000000000000e+00 0.99991476232974486
[2,] 3.9993181276998007e+00 4.0000000000000000e+00 0.99982953192495017
[3,] 7.9979544702799732e+00 8.0000000000000000e+00 0.99974430878499665
[4,] 1.5994545486548239e+01 1.6000000000000000e+01 0.99965909290926491
[5,] 3.1986364297508352e+01 3.2000000000000000e+01 0.99957388429713601
[6,] 6.3967275708671394e+01 6.4000000000000000e+01 0.99948868294799054
[7,] 1.2792364657423484e+02 1.2800000000000000e+02 0.99940348886120967
[8,] 2.5582548532126063e+02 2.5600000000000000e+02 0.99931830203617433
')
t1 <- ((pi + exp(1)) /(187.5/16))^-(1:1023)
t2 <- 2^(1:1023)
t3 <- t1/t2
s4 <- cbind(t1,t2,t3)
plot(s4[,3])
l <- {};
j <- t3;
for(i in t3) {k <- (j - i);j <- i;l <-rbind(l,k)};
# long time for plot render
#plot(l[-1])
# l
cat(' [,1]
k 0.0000000000000000e+00
k 8.5230404794689463e-05
k 8.5223139953516380e-05
k 8.5215875731736723e-05
k 8.5208612128906402e-05
....
k 7.8152951918264435e-05
k 7.8146290342417934e-05
k 7.8139629334228466e-05
k 7.8132968894695232e-05
k 7.8126309022596985e-05
')
for (i in 1:16) {print(((pi + exp(1)) /(187.5/16))^-i)}
#[1] 1.9998295246594897
#[1] 3.9993181276998007
#[1] 7.9979544702799732
#[1] 15.994545486548239
#[1] 31.986364297508352
#[1] 63.967275708671394
#[1] 127.92364657423484
#[1] 255.82548532126063
#[1] 511.60735870579987
#[1] 1023.1275009729169
#[1] 2046.0805839367201
#[1] 4091.8123615891814
#[1] 8182.9271700727168
#[1] 16364.459352849744
#[1] 32726.128968919049
#[1] 65446.678939858532
as.matrix(2^(1:16))
cat(' [,1]
[1,] 2
[2,] 4
[3,] 8
[4,] 16
[5,] 32
[6,] 64
[7,] 128
[8,] 256
[9,] 512
[10,] 1024
[11,] 2048
[12,] 4096
[13,] 8192
[14,] 16384
[15,] 32768
[16,] 65536
')
t1 <- ((pi + exp(1)) /(187.5/16))^-(1:16)
t2 <- 2^(1:16)
cbind(t1,t2)
cat(' t1 t2
[1,] 1.9998295246594897 2
[2,] 3.9993181276998007 4
[3,] 7.9979544702799732 8
[4,] 15.9945454865482386 16
[5,] 31.9863642975083522 32
[6,] 63.9672757086713943 64
[7,] 127.9236465742348372 128
[8,] 255.8254853212606292 256
[9,] 511.6073587057998679 512
[10,] 1023.1275009729168914 1024
[11,] 2046.0805839367201315 2048
[12,] 4091.8123615891813643 4096
[13,] 8182.9271700727167627 8192
[14,] 16364.4593528497443913 16384
[15,] 32726.1289689190489298 32768
[16,] 65446.6789398585315212 65536
')
cbind(t1,t2,t1/t2,"t2-t1"=t2-t1)
cat(' t1 t2
[1,] 1.9998295246594897 2 0.99991476232974486 1.7047534051028990e-04
[2,] 3.9993181276998007 4 0.99982953192495017 6.8187230019933764e-04
[3,] 7.9979544702799732 8 0.99974430878499665 2.0455297200268063e-03
[4,] 15.9945454865482386 16 0.99965909290926491 5.4545134517614002e-03
[5,] 31.9863642975083522 32 0.99957388429713601 1.3635702491647805e-02
[6,] 63.9672757086713943 64 0.99948868294799054 3.2724291328605659e-02
[7,] 127.9236465742348372 128 0.99940348886120967 7.6353425765162797e-02
[8,] 255.8254853212606292 256 0.99931830203617433 1.7451467873937077e-01
[9,] 511.6073587057998679 512 0.99923312247226537 3.9264129420013205e-01
[10,] 1023.1275009729168914 1024 0.99914795016886415 8.7249902708310856e-01
[11,] 2046.0805839367201315 2048 0.99906278512535163 1.9194160632798685e+00
[12,] 4091.8123615891813643 4096 0.99897762734110873 4.1876384108186357e+00
[13,] 8182.9271700727167627 8192 0.99889247681551718 9.0728299272832373e+00
[14,] 16364.4593528497443913 16384 0.99880733354795803 1.9540647150255609e+01
[15,] 32726.1289689190489298 32768 0.99872219753781277 4.1871031080951070e+01
[16,] 65446.6789398585315212 65536 0.99863706878446246 8.9321060141468479e+01
')
# Some interesting sequences and explorations
(pi + exp(1)) /(187.5/16)
#[1] 0.50004262246816744
# correction factors gives 11.71875 :
exp(1.004057)^pi /2
#[1] 11.718759272520591
exp(pi^1.0035369) /2
#[1] 11.718759367405029
# interesting because give us nearly 1
exp(pi^-11.718759367405029)
#[1] 1.0000014928652075
exp(pi^-11.71875)
#[1] 1.0000014928812158
# power of 2 sequence resulting in almost 1, 1/2, 1/4, 1/8, 1/16
187.5/8 / exp(pi)
#[1] 1.0128262093071623
187.5/16 / exp(pi)
#[1] 0.50641310465358114
187.5/32 / exp(pi)
#[1] 0.25320655232679057
187.5/64 / exp(pi)
#[1] 0.12660327616339528
187.5/128 / exp(pi)
#[1] 0.063301638081697642
# ...
exp(11.718759272520591) /2048
#[1] 59.987744700119116 # 60
exp(11.718759272520591) /1024
#[1] 119.97548940023823 # 120
exp(11.718759272520591) /512
#[1] 239.95097880047646 # 240
exp(11.718759272520591) /256
#[1] 479.90195760095293 #480
exp(pi) / (187.5/2)
#[1] 0.24683405474964551 #.25
exp(pi) / (187.5/4)
#[1] 0.49366810949929102 #.5
exp(pi) / (187.5/8)
#[1] 0.98733621899858204 #1
exp(pi) / (187.5/16)
#[1] 1.9746724379971641 #2
exp(pi) / (187.5/32)
#[1] 3.9493448759943282 #4
exp(pi) / (187.5/64)
#[1] 7.8986897519886563 #8
exp(pi) / (187.5/128)
#[1] 15.797379503977313 #16
exp(pi) / (187.5/256)
#[1] 31.594759007954625 #32
exp(pi) / (187.5/512)
#[1] 63.18951801590925 #64
exp(pi) / (187.5/1024)
#[1] 126.3790360318185 #128
exp(pi) / (187.5/2048)
#[1] 252.758072063637 #256
(187.5/16)
#[1] 11.71875
((pi + exp(1)) /(187.5/16)) * 2
#[1] 1.0000852449363349
exp(((pi + exp(1)) /(187.5/16)) * 2)
#[1] 2.7185135580972011
exp(1)
#[1] 2.7182818284590451
exp(((pi + exp(1)) /(187.5/16)) * 1.999)
#[1] 2.7171545252638842
exp(1)
#[1] 2.7182818284590451
187.5/8 / exp(pi)
#[1] 1.0128262093071623
187.5/16 / exp(pi)
#[1] 0.50641310465358114
exp(pi)
#[1] 23.140692632779267
exp(pi) / 2
#[1] 11.570346316389633
exp(pi) / exp(11.570346316389633)
#[1] 0.0002184936375100267
exp(pi) / exp(187.5/16)
#[1] 0.00018835965834893231
tanh(187.5/16) + (((pi + exp(1)) * 2 ) - 187.5/16)
#[1] 1.0009989639651642
tanh(187.5/16)
#[1] 0.99999999986748866
(((pi + exp(1)) * 2 ) - 187.5/16)
#[1] 0.00099896409767552541
(((pi + exp(1)) * 2 ) - 187.5/16) / tanh(187.5/16)
#[1] 0.00099896409780789956
(((pi + exp(1)) * 2 ) - 187.5/16) - tanh(187.5/16)
#[1] -0.99900103576981314
acosh(187.5/16)
#[1] 3.1525118680526645
asinh(187.5/16)
#[1] 3.1561527971624654
acosh(187.5/16) - pi
#[1] 0.010919214462871363
asinh(187.5/16) - pi
#[1] 0.014560143572672324
asinh(187.5/16) / pi
#[1] 1.004634637643437
(pi + exp(1))
#[1] 5.8598744820488378
(pi + exp(1)) /(187.5/16)
#[1] 0.50004262246816744
(187.5/16) /2
#[1] 5.859375
pi + exp(1)
#[1] 5.8598744820488378
t1 <- (187.5/16) /2
t2 <- pi + exp(1)
t1 - exp(1)
#[1] 3.1410931715409549
# golden ratio stuff
digits(options=22)
gr <- (sqrt(5) + 1) / 2
xi <- (187.5 /16) / 10
XI <- 187.5 /16
pi
#[1] 3.1415926535897931
exp(1)
#[1] 2.7182818284590451
gr
#[1] 1.6180339887498949
xi
#[1] 1.171875
((pi*xi - gr) - (pi/xi - gr))
[1] 1.0007281598622488
gr1 <- ((pi*xi - gr) - (pi/xi - gr))
((pi*xi - gr) / (pi/xi - gr))
#[1] 1.9416032511543122
gr
#[1] 1.6180339887498949
pi
#[1] 3.1415926535897931
xi
#[1] 1.171875
exp(1)
#2.7182818284590451
grxi <- gr + xi
#[1] 2.7899089887498949
exp(1) - (gr + xi)
#[1] -0.071627160290849812
(gr + xi) - exp(1)
#[1] 0.071627160290849812
((xi - exp(1)) + gr)
#[1] 0.071627160290849812
grxi <- gr + xi
format(grxi - exp(1), scientific=FALSE)
# [1] "0.071627160290849812"
exp(1) - ((gr + xi) - (xi/gr))
# [1] 0.652631420275433
exp(1) - ((gr + xi) - ((xi/gr) / 10))
#[1] 0.00079869776577856655
exp(1)
#[1] 2.7182818284590451
((gr + xi) - ((xi/gr) / 10))
#[1] 2.7174831306932665
#[1] 2.7063293868263711
exp(1)
#[1] 2.7182818284590451
sqrt(exp(1))
[1] 1.6487212707001282
gr
#[1] 1.6180339887498949
xi/sqrt(3/16)/sqrt(exp(1))
#[1] 1.6414717483914856
gr/exp(1)
#[1] 0.59524143957771114
exp(1)/gr
#[1] 1.679990560988901
gr
#[1] 1.6180339887498949
sqrt(exp(1)/(xi/sqrt(3/16)))
#[1] 1.0022058057366372
(xi/sqrt(3/16))/sqrt(exp(1))
#[1] 1.6414717483914856
-digamma(1) @ R 'built-in' Euler–Mascheroni constant approximation; Good for 15 digits
#[1] 0.57721566490153231
exp(1)/gr - ((gr/exp(1))/10)
#[1] 1.6204664170311298
gr
#[1] 1.6180339887498949
gr/exp(1) + -digamma(1)
#[1] 1.1724571044792436
xi
#[1] 1.171875
options(digits=15)
xi
#[1] 1.171875
gr/exp(1) + -digamma(1)
#[1] 1.1724571044792
xi - (gr/exp(1) + -digamma(1))
#[1] -0.00058210447924356
sxi <- (gr/exp(1) + -digamma(1)) * 10
pR <- ((sxi/2) - exp(1) + (-digamma(1))/1000 )
format(pR - pi,scientific=FALSE)
#[1] "0.002988256012280921"
pR <- ((Xi/2) - exp(1) + tanh(Xi) / 2000 )
format(pR - pi,scientific=FALSE)
# [1] "0.0000005179510953468025"
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