Thursday, November 19, 2015

Introducing xi (11.71875 or 187.5/16) : A trans-dimensional constant that links pi and e ?




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
## *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.  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 3.2 there is no bigint by default,however there are 22 digits of accuracy. 
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


# 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
 t1 - exp(1)
#[1] 3.1410931715409549

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