1292: Pi vs. Tau

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PM 2Ring
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Re: 1292: Pi vs. Tau

Postby PM 2Ring » Fri Nov 22, 2013 10:04 am UTC

Here are the first 1001 octal digits of pau, with groups of 3 or more identical digits colourised. Calculated using bc and formatted using python.

4.
55457 43763 14416 44323 62345 14475 05012 24254 71573 01565
03147 63354 52700 30431 67712 61165 50546 74757 03133 12523
40351 47165 76464 33317 27311 24310 20107 64472 70723 62457
37216 40220 43765 21550 65544 22014 31161 55742 51563 44621
36362 51744 10110 77702 61115 60241 17447 12522 41762 03716
33674 20573 53303 21647 02576 62666 74462 75343 25504 33450
60027 30517 10254 75041 45216 66121 12500 27531 71664 12767
65735 56334 17212 14013 55345 36541 06045 24506 64011 41437
74062 67077 57305 45070 36064 40651 11177 52700 32710 03552
13521 01513 62206 21644 57304 32645 05244 32531 65266 66260
42202 56220 25505 66425 64304 05563 65710 25003 16424 67447
60566 32406 61743 60004 10522 12627 76707 32776 00402 57202
73162 22345 35603 63010 02572 54175 00001 14422 03631 21223
41474 26723 27617 75450 07165 26136 27306 74507 41502 51171
50772 02772 50030 27044 22571 06542 45644 17224 55345 34037
02056 46442 15633 41255 64557 52033 63402 23313 31255 66344
50170 62641 72343 76702 44311 70311 35045 42016 54674 26237
45475 45660 12204 31613 00230 63506 43006 33622 03021 26243
44644 10604 27522 46065 23356 70257 26100 31171 34441 17665
05734 61525 61210 34660 77330 61400 32365 32641 57732 27551

FWIW, setting the environment variable BC_LINE_LENGTH to a huge value is handy for this sort of thing.

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Re: 1292: Pi vs. Tau

Postby Wnderer » Fri Nov 22, 2013 10:05 pm UTC

The only way to solve Pi vs Tau is with a Rock, Paper, Scissors, Lizard, Spock competition. We assign Rock=0, Paper=1, Scissors=2, Lizard=3, and Spock=4. We take each digit of Pi and Tau and divide and truncate so the digits from 0 to 9 are changed to 0 to 4 as follows.
0 -> 0
1 -> 0
2 -> 1
3 -> 1
4 -> 2
5 -> 2
6 -> 3
7 -> 3
8 -> 4
9 -> 4

Then we compare corresponding digits of Pi and Tau. Now the selections of Rock,Paper,Scissors,Lizard,Spock are random but the relationship between Pi and Tau isn't. So is there a winner?

standingwave
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Re: 1292: Pi vs. Tau

Postby standingwave » Fri Nov 22, 2013 11:19 pm UTC

I have no dog in this fight except to say that, as an electrical engineer who deals with 2pi on an almost daily basis, pick a consistent system and stick with it. Either use tau and continue measuring angles based on radii (radians) or use pi and measure angles based on diameter. Dadians? But mixing systems is just ignorant. IMO.

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Re: 1292: Pi vs. Tau

Postby BlueSloth » Sat Nov 23, 2013 5:32 am UTC

Wnderer wrote:We take each digit of Pi and Tau and divide and truncate so the digits from 0 to 9 are changed to 0 to 4

Why not use base 5?

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Re: 1292: Pi vs. Tau

Postby Flumble » Sat Nov 23, 2013 12:31 pm UTC

Wnderer wrote:So is there a winner?

There shouldn't be, as pi and tau are believed to be normal numbers so their wins should even out.

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Re: 1292: Pi vs. Tau

Postby Wnderer » Sat Nov 23, 2013 1:51 pm UTC

BlueSloth wrote:
Wnderer wrote:We take each digit of Pi and Tau and divide and truncate so the digits from 0 to 9 are changed to 0 to 4

Why not use base 5?


Base 5 would work but I was thinking about actually writing the code and trying it out. With the divide and truncate, I could download text files of the 10,000 or so already computed digits and handle each digit individually. I'm not sure how to convert to base 5. I'm sure with Python you just need to import something to convert to base 5, but a divide by 2 and truncate is just an integer bit shift. I'm showing my outdated 68hc705 assembly skills again.

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Re: 1292: Pi vs. Tau

Postby nwm » Sat Nov 23, 2013 8:59 pm UTC

Using base 5, and the more logical order rock, paper, scissors, Spock, lizard, the match-up function looks like

Code: Select all

def rpssl(a, b):
   if a == b: #tie
      return None
   else: #a wins iff:
      return ((a-b)%5)%2 == 1

and the result for the first 10⁵ digits after the comma is:
π wins 39943 times; τ wins 39960 times; 20097 ties


So I guess that is settled then ;-)

EDIT:

π wins 399533 times; τ wins 400213 times; 200254 ties

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Wnderer
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Re: 1292: Pi vs. Tau

Postby Wnderer » Sat Nov 23, 2013 11:23 pm UTC

nwm wrote:Using base 5, and the more logical order rock, paper, scissors, Spock, lizard, the match-up function looks like

Code: Select all

def rpssl(a, b):
   if a == b: #tie
      return None
   else: #a wins iff:
      return ((a-b)%5)%2 == 1

and the result for the first 10⁵ digits after the comma is:
π wins 39943 times; τ wins 39960 times; 20097 ties


So I guess that is settled then ;-)

EDIT:

π wins 399533 times; τ wins 400213 times; 200254 ties


Cool. I thought he meant converting Pi and Tau to base 5 and then comparing digits. I guess this gets you the same thing. I'll have to think about it.

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Re: 1292: Pi vs. Tau

Postby PM 2Ring » Sun Nov 24, 2013 6:00 am UTC

Wnderer wrote:Cool. I thought he meant converting Pi and Tau to base 5 and then comparing digits. I guess this gets you the same thing. I'll have to think about it.


No, using mod 5 on the base 10 digits is not the same as using actual base 5 digits.

It's not that hard to write arbitrary base conversion functions, but handling the "decimal" point is slightly fiddly.

However, if you have the Python mpmath module, you could use pidigits.py, which rapidly calculates large numbers of digits of pi in any base from 2 - 36. It's pretty easy to modify it to calculate Tau (or Pau).

Unfortunately, pidigits.py has a minor bug in its number to string conversion for octal and hex. Of course, that's irrelevant if you're just interested in base 5 output, but I decided to fix it anyway.

Code: Select all

#! /usr/bin/env python

"""
Calculate digits of pi. This module can be run interactively with

    python pidigits.py

    From http://code.google.com/p/mpmath/source/browse/trunk/demo/pidigits.py

    Note: The original version of this program mucks up the conversion in base 8 and 16,
    since numeral() prepends '0' to octal output and '0x' to hex output.
    Repaired by PM 2Ring 2013.11.24
"""

__docformat__ = 'plaintext'

import sys
import math
from time import clock

from mpmath.libmp import bin_to_radix, pi_fixed
from mpmath.libmp.libintmath import numeral_gmpy

#Uncomment the line below if you're on a *nix system and you'll get line editing at the interactive prompts.
#import readline

#Remove the cruft that numeral_gmpy() prepends to octal & hex output
def numeral(n, base=10, size=0, digits='0123456789abcdefghijklmnopqrstuvwxyz'):
    s = numeral_gmpy(n, base, size, digits)
    if base == 8:
        return s[1:]
    elif base == 16:
        return s[2:]
    else:
        return s

def display_fraction(digits, skip=0, colwidth=10, columns=5):
    perline = colwidth * columns
    printed = 0
    for linecount in range((len(digits)-skip) // (colwidth * columns)):
        line = digits[skip+linecount*perline:skip+(linecount+1)*perline]
        for i in range(columns):
            print line[i*colwidth : (i+1)*colwidth],
        print ":", (linecount+1)*perline
        if (linecount+1) % 10 == 0:
            print
        printed += colwidth*columns
    rem = (len(digits)-skip) % (colwidth * columns)
    if rem:
        buf = digits[-rem:]
        s = ""
        for i in range(columns):
            s += buf[:colwidth].ljust(colwidth+1, " ")
            buf = buf[colwidth:]
        print s + ":", printed + colwidth*columns

def calculateit(base, n, tofile):
    intpart = numeral(3, base)
    skip = 1
    if base <= 3:
        skip = 2

    prec = int(n*math.log(base,2))+10

    print "Step 1 of 2: calculating binary value..."
    t = clock()
    a = pi_fixed(prec, verbose=True, verbose_base=base)
    step1_time = clock() - t

    print "Step 2 of 2: converting to specified base..."
    t = clock()
    d = bin_to_radix(a, prec, base, n)
    d = numeral(d, base, n)
    step2_time = clock() - t

    print "\nWriting output...\n"

    if tofile:
        out_ = sys.stdout
        sys.stdout = tofile
    print "%i base-%i digits of pi:\n" % (n, base)
    print intpart, ".\n"

    display_fraction(d, skip, colwidth=10, columns=5)
    if tofile:
        sys.stdout = out_
    print "\nFinished in %f seconds (%f calc, %f convert)" % \
        ((step1_time + step2_time), step1_time, step2_time)

def interactive():
    print "Compute digits of pi with mpmath\n"
    base = input("Which base? (2-36, 10 for decimal) \n> ")
    digits = input("How many digits? (enter a big number, say, 10000)\n> ")
    tofile = raw_input("Output to file? (enter a filename, or just press " \
        "enter\nto print directly to the screen) \n> ")
    if tofile:
        tofile = open(tofile, "w")

    calculateit(base, digits, tofile)
    raw_input("\nPress enter to close this script.")

if __name__ == "__main__":
    interactive()

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Re: 1292: Pi vs. Tau

Postby Flumble » Sun Nov 24, 2013 10:26 am UTC

nwm wrote:
π wins 399533 times; τ wins 400213 times; 200254 ties

Good, good...

Now let us reject our arbitrary choice of taking decimal digits modulo 5 and using a specific RPS-configuration... unless τ wins using the quinary expansion and a specific RPS-configuration.

I think one of the few 'fair' ways (instead of arbitrary rules like 1 > 3 > 2 > 1 or 1 > 2 > 3 > 1) to let them battle is over the binary expansion of π and τ. :roll:

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Re: 1292: Pi vs. Tau

Postby nwm » Sun Nov 24, 2013 11:46 am UTC

I guess you misunderstood me. I did say “using base 5“. So a and b are indeed the digits auf π and τ in base 5. Just to be clear, I made my script use only mpmath:

Code: Select all

import sympy.mpmath as mp
mp.mp.dps = 100000

def get_digits(nr, base, digits):
      fval = nr - int(nr)
      for _ in range(digits):
         fval *= base
         digit = int(fval)
         yield digit
         fval -= digit

pi = get_digits(mp.pi, 5, mp.mp.dps)
tau = get_digits(2*mp.pi, 5, mp.mp.dps)

def rpssl(a, b):
   if a == b: #tie
      return None
   else: #a wins iff:
      return ((a-b)%5)%2 == 1

lst = list(map(rpssl, pi, tau))
print("π wins %d times; τ wins %d times; %d ties" % (len([1 for x in lst if x is True]), len([1 for x in lst if x is False]), len([1 for x in lst if x is None])))

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Re: 1292: Pi vs. Tau

Postby Wnderer » Sun Nov 24, 2013 2:40 pm UTC

nwm wrote:I guess you misunderstood me.

Thanks. Now I understand. But it didn't go on long enough. Now you should set up a website that continuously calculates new digits of Pi and Tau and then has them battle it out in Rock, Paper, Scissors, Spock, Lizard for all eternity. Maybe Sam Kass will let you post it on his site.

http://www.samkass.com/theories/RPSSL.html

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Re: 1292: Pi vs. Tau

Postby joneses » Mon Nov 25, 2013 1:04 pm UTC

This much trouble just to turn pi into a "two-thirds compromise" joke?

Cheeky.

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Re: 1292: Pi vs. Tau

Postby gmalivuk » Mon Nov 25, 2013 5:55 pm UTC

Was there a famous two-thirds compromise? I'm only familiar with the slightly smaller three-fifths compromise.
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Re: 1292: Pi vs. Tau

Postby Philip Thomas » Mon Nov 25, 2013 9:53 pm UTC

I just finished reading a science fiction novel called Zero Tau which is about what happens when a spaceship full of people approaches the speed of light. There's a formula for Tau given there which is the square root of (1-(v^2/c^2)).

I'm guessing that is a completely different tau, since it is always less than pi. But surely the existence of something else called tau is an argument for using pi, since as far as I'm aware pi is unambiguous?
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Re: 1292: Pi vs. Tau

Postby speising » Mon Nov 25, 2013 10:14 pm UTC

Philip Thomas wrote:I just finished reading a science fiction novel called Zero Tau which is about what happens when a spaceship full of people approaches the speed of light. There's a formula for Tau given there which is the square root of (1-(v^2/c^2)).

I'm guessing that is a completely different tau, since it is always less than pi. But surely the existence of something else called tau is an argument for using pi, since as far as I'm aware pi is unambiguous?

you mean Tau Zero, i trust. (hope, because otherwise there'd be a story i haven't read yet :))

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Re: 1292: Pi vs. Tau

Postby Philip Thomas » Mon Nov 25, 2013 10:25 pm UTC

speising wrote:
Philip Thomas wrote:I just finished reading a science fiction novel called Zero Tau which is about what happens when a spaceship full of people approaches the speed of light. There's a formula for Tau given there which is the square root of (1-(v^2/c^2)).

I'm guessing that is a completely different tau, since it is always less than pi. But surely the existence of something else called tau is an argument for using pi, since as far as I'm aware pi is unambiguous?

you mean Tau Zero, i trust. (hope, because otherwise there'd be a story i haven't read yet :))


Tau Zero yes. Although I'm pretty sure there are stories you haven't read yet...
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Re: 1292: Pi vs. Tau

Postby rmsgrey » Mon Nov 25, 2013 11:05 pm UTC

Tau in Tau Zero is being used as a variant of 't' as in "time" - that Tau is a measure of time dilation - how many seconds ship's time pass per second on Earth.

There are other uses of pi - for example, pi(x) is the number of primes up to and including x...

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Re: 1292: Pi vs. Tau

Postby gladiolas » Tue Nov 26, 2013 1:45 am UTC

In my previous post I posted a whole bunch of digits of pi, and something truncated it, so I don't know if people knew what I was getting at. I was suggesting that pi might be equal to...

3.14159265358979323846264338327000000000000000...

gmalivuk wrote:
gladiolas wrote:I know about those algorithms which generate the number pi. But how do we know this isn't the *actual* value of the ratio of a circle's circumference to its diameter?
Some of the algorithms are derived directly from the definition of a (Euclidean) circle, and the rest of them can be proven to converge on the same number.


*slaps head* D'oh! Shoulda thought of that! But...but...but...cosmologists disagree on whether the universe's curvature is positive, negative, or flat, and it seems to be expanding, so it seems plausible that the ratio of the circumference to diameter might be changing ever so slightly. And I think somebody else mentioned the possibility, but I don't remember who.

I read a novel which opens with somebody showing that pi=3. It was the Land and Overland trilogy,
by Bob Shaw. The Ragged Astronauts, The Wooden Spaceships and The Fugitive Worlds

I met his widow at an SF convention, and she said he wasn't trying to work out the resulting equations, he just wanted to show that the novels take place in a different universe.

But it would be fun to work out the results...what if pi=.00001, 1, 4 or 1000? I can think of some of the results--the sun would be either too hot or too cold for life as we know it.

This was one of the what-if questions I sent Randall, but he hasn't used it, so I wondered about posting it as a separate topic for everybody else to natter over.

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Re: 1292: Pi vs. Tau

Postby Klear » Tue Nov 26, 2013 9:16 am UTC

gladiolas wrote:*slaps head* D'oh! Shoulda thought of that! But...but...but...cosmologists disagree on whether the universe's curvature is positive, negative, or flat, and it seems to be expanding, so it seems plausible that the ratio of the circumference to diameter might be changing ever so slightly. And I think somebody else mentioned the possibility, but I don't remember who.


I think Roger Penrose thought it possible our space-time is Lobachevskian rather than euclidean. In any case, pi is referring to euclidean space, so the value is correct, only it may not precisely apply to any real-world circle (which is kinda moot, since there are no perfect circles anyway).

Also, we do know that space-time is curved locally, so yeah, using pi would give you results that are slightly off, if it wasn't for the fact that our measurements and our circles are far from perfect.

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Re: 1292: Pi vs. Tau

Postby PM 2Ring » Tue Nov 26, 2013 9:54 am UTC

A previous discussion on physical v mathematical pi: viewtopic.php?f=17&t=63092

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Re: 1292: Pi vs. Tau

Postby nwm » Tue Nov 26, 2013 10:59 am UTC

Philip Thomas wrote:But surely the existence of something else called tau is an argument for using pi, since as far as I'm aware pi is unambiguous?


Not quite. Just look at http://en.wikipedia.org/wiki/Pi_(disambiguation).
In mathematics π is used in many places. For example in topology it denotes the fundamental groups and also higher homotopy groups. The prime-counting function was already mentioned and in the same context I've seen π being used for the set of all primes.

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Re: 1292: Pi vs. Tau

Postby speising » Tue Nov 26, 2013 1:48 pm UTC

the greek alphabet does not have enough letters.

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Re: 1292: Pi vs. Tau

Postby Mikeski » Tue Nov 26, 2013 2:43 pm UTC

speising wrote:the greek alphabet does not have enough letters.

Thus the suggestion upthread to convert to Chinese.

Assuredly still not enough, though we'd have far fewer hash table collisions.

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Re: 1292: Pi vs. Tau

Postby Quizatzhaderac » Tue Nov 26, 2013 3:24 pm UTC

Time dilation and osmotic pressure use the same units? Unit cancellation is weird. </facetiousness>
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Re: 1292: Pi vs. Tau

Postby gmalivuk » Tue Nov 26, 2013 8:00 pm UTC

gladiolas wrote:In my previous post I posted a whole bunch of digits of pi, and something truncated it, so I don't know if people knew what I was getting at. I was suggesting that pi might be equal to...

3.14159265358979323846264338327000000000000000...

gmalivuk wrote:
gladiolas wrote:I know about those algorithms which generate the number pi. But how do we know this isn't the *actual* value of the ratio of a circle's circumference to its diameter?
Some of the algorithms are derived directly from the definition of a (Euclidean) circle, and the rest of them can be proven to converge on the same number.


*slaps head* D'oh! Shoulda thought of that! But...but...but...cosmologists disagree on whether the universe's curvature is positive, negative, or flat, and it seems to be expanding, so it seems plausible that the ratio of the circumference to diameter might be changing ever so slightly. And I think somebody else mentioned the possibility, but I don't remember who.

I read a novel which opens with somebody showing that pi=3. It was the Land and Overland trilogy,
by Bob Shaw. The Ragged Astronauts, The Wooden Spaceships and The Fugitive Worlds

I met his widow at an SF convention, and she said he wasn't trying to work out the resulting equations, he just wanted to show that the novels take place in a different universe.

But it would be fun to work out the results...what if pi=.00001, 1, 4 or 1000? I can think of some of the results--the sun would be either too hot or too cold for life as we know it.

This was one of the what-if questions I sent Randall, but he hasn't used it, so I wondered about posting it as a separate topic for everybody else to natter over.
As far as I know, Euclidean space is the only one where the circumference:diameter ratio is constant everywhere. Where you have positive curvature, that ratio decreases as the diameter increases, and where you have negative curvature it increases.
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Re: 1292: Pi vs. Tau

Postby nwm » Tue Nov 26, 2013 10:34 pm UTC

gmalivuk wrote:As far as I know, Euclidean space is the only one where the circumference:diameter ratio is constant everywhere. Where you have positive curvature, that ratio decreases as the diameter increases, and where you have negative curvature it increases.


That is a correct observation. The [url]http://en.wikipedia.org/wiki/Bertrand–Diquet–Puiseux_theorem[/url] shows this.
[math]K = \lim_{r\to 0} 3\frac{2\pi r - C(r)}{\pi r^3}[/math]
So if [imath]C(r)/r[/imath] is constant but different from τ, the expression diverges. Also, just like you stated, for positive K it decreases with r and with negative K it increases.

(mmh, this looks strange, am I doing something wrong?)

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Re: 1292: Pi vs. Tau

Postby gmalivuk » Wed Nov 27, 2013 2:11 am UTC

nwm wrote:
gmalivuk wrote:As far as I know, Euclidean space is the only one where the circumference:diameter ratio is constant everywhere. Where you have positive curvature, that ratio decreases as the diameter increases, and where you have negative curvature it increases.


That is a correct observation. The Bertrand-Diquet-Puiseaux theorem shows this.
[; K = \lim_{r\to 0} 3\frac{2\pi r - C(r)}{\pi r^3} ;]
So if [; C(r)/r ;] is constant but different from τ, the expression diverges. Also, just like you stated, for positive K it decreases with r and with negative K it increases.

(mmh, this looks strange, am I doing something wrong?)
[math] tags no longer work, but in my quote it'll now show up correctly if you use the TeX the World plugin. And the forum wasn't liking the dashes in that url of yours.
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Re: 1292: Pi vs. Tau

Postby Quizatzhaderac » Wed Nov 27, 2013 2:35 pm UTC

nwm wrote:(mmh, this looks strange, am I doing something wrong?)
Also, the you put the URLS target when the display text goes. URLs tags are of the form:

Code: Select all

[url=http://site.com]readable display text[/url]
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Re: 1292: Pi vs. Tau

Postby orthogon » Wed Nov 27, 2013 3:12 pm UTC

Flumble wrote:
Wnderer wrote:So is there a winner?

There shouldn't be, as pi and tau are believed to be normal numbers so their wins should even out.

Is their both being normal numbers sufficient for this to be true? If pi is a normal number, would pi+1 also be a normal number? If so, then one of them would win the "units" round and they would tie every subsequent round, so there could be a bias.
xtifr wrote:... and orthogon merely sounds undecided.

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Re: 1292: Pi vs. Tau

Postby Quizatzhaderac » Wed Nov 27, 2013 3:55 pm UTC

What Flumble is implying is that for any finite n: either pi of tau may be the winner, however as n gets larger the net score (i.e. tau's won 6 more than pi) divided by n will approach zero.

What I'm uncertain of is: even if pi & tau are normal, their digits are correlated and only certain mappings between the nth digits of pi and tau are possible.
pin = (taun/2)%1 or (5+ taun/2)%1
taun = pin*2 or 1+pin*2

Assuming pi & tau's normalcy, each of the twenty mappings should appear equally. The question is: does rock paper scissors variant come out even for after those twenty mappings?
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Re: 1292: Pi vs. Tau

Postby orthogon » Wed Nov 27, 2013 4:10 pm UTC

Quizatzhaderac wrote:What Flumble is implying is that for any finite n: either pi of tau may be the winner, however as n gets larger the net score (i.e. tau's won 6 more than pi) divided by n will approach zero.

What I'm uncertain of is: even is pi & tau are normal, their digits are correlated and only certain mappings between the nth digits of pi and tau are possible.
pin = (taun/2)%1 or (5+ taun/2)%1
taun = pin*2 or 1+pin*2

Assuming pi & tau's normalcy, each of the twenty mappings should appear equally. The question is: does rock paper scissors variant come out even for after those twenty mappings?

OK, as a proportion it would tend to zero in my example, although in absolute terms there would be a bias forever. I think you've picked up on what I was getting at, i.e. the two numbers are related so there could be a systematic bias even if they're both normal.

You imply that there is a carry exactly half the time; is that a consequence of multiplying by 2 and would it be different if you multiplied by 3/2? (i.e. might pau beat pi?)
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Re: 1292: Pi vs. Tau

Postby balthasar_s » Wed Nov 27, 2013 4:32 pm UTC

π*21/2 is the ideal compromise. :)
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Re: 1292: Pi vs. Tau

Postby Quizatzhaderac » Wed Nov 27, 2013 4:38 pm UTC

For unrelated normal numbers :Any subsection can be biased. Tau might start winning, and then pi might take the lead once you process 10 more digits. It would wobble back and forth pretty much as if the numbers were random. If you had a meaningful n, you could say one or the other wins, but the varous n s will also balence out wins and losses.

Yes, I'm implying there' is a carry half the time because this is a multiple of 2 and we're assuming each pair of digits is equally frequent in the two numbers. If we were using a multiple of 3/2 there would be a carry 1- (3/2)-1 = 1/3 of the time. There would also be a "downward" carry of 5 half the time, because 1.5 has a value after the decimal.

EDIT- I realized the with the pi-pau relation carried numbers can potentially chain. so my figure of 1/3 is somewhat off.
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Re: 1292: Pi vs. Tau

Postby Flumble » Fri Nov 29, 2013 2:27 am UTC

orthogon wrote:OK, as a proportion it would tend to zero in my example, although in absolute terms there would be a bias forever. I think you've picked up on what I was getting at, i.e. the two numbers are related so there could be a systematic bias even if they're both normal.

Indeed there is some bias.
If you get the binary notation of τ (110.0…) and π (011.0…) and play an endless game of prisoner's delimma (or anything else that has 1,0 as a win for τ, 0,1 as a win for π and no winners when both are 1 or 0), τ is always 1 win ahead or they have a tie. (and tau with a tie is clearly a winner) :D

Though, that's a systematic bias in base 2n. For any other expansion (like base 5 or 10) the effect of the factor 2 should even out. (although that's just my instinct and knowledge of galois fields)

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Re: 1292: Pi vs. Tau

Postby phlip » Fri Nov 29, 2013 4:01 am UTC

orthogon wrote:Is their both being normal numbers sufficient for this to be true? If pi is a normal number, would pi+1 also be a normal number? If so, then one of them would win the "units" round and they would tie every subsequent round, so there could be a bias.

This is true... what you almost want is that the difference between the two numbers is normal (and since that difference is pi, it probably is normal), but not quite, as there's carry digits and all of that so that "the nth digit of τ − π" isn't quite the same thing as "the difference between the nth digit of τ and the nth digit of π"... But maybe there's something which can be worked with there...

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Re: 1292: Pi vs. Tau

Postby Elvish Pillager » Fri Nov 29, 2013 4:33 am UTC

Quick note - If it's sufficiently like a random walk, then it will (not might) cross the "pi and tau are tied" threshold back and forth infinitely many times. On the other hand, if it has any fixed arbitrarily small finite amount of bias, then it will eventually stay on one side of the threshold forever.

I rather suspect it is, but I don't know either part of the math well enough. I suppose I could go learn it, but I think I've done quite enough mathematical puzzles for one day. :P
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Re: 1292: Pi vs. Tau

Postby rmsgrey » Fri Nov 29, 2013 12:31 pm UTC

Elvish Pillager wrote:Quick note - If it's sufficiently like a random walk, then it will (not might) cross the "pi and tau are tied" threshold back and forth infinitely many times. On the other hand, if it has any fixed arbitrarily small finite amount of bias, then it will eventually stay on one side of the threshold forever.

I rather suspect it is, but I don't know either part of the math well enough. I suppose I could go learn it, but I think I've done quite enough mathematical puzzles for one day. :P


Also, if it's sufficiently random walkish, it will spend most of its time favouring one side over the other.

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Re: 1292: Pi vs. Tau

Postby Quizatzhaderac » Fri Nov 29, 2013 4:16 pm UTC

phlip wrote: what you almost want is that the difference between the two numbers is normal (and since that difference is pi, it probably is normal), but not quite, as there's carry digits and all of that so that "the nth digit of τ − π" isn't quite the same thing as "the difference between the nth digit of τ and the nth digit of π"... But maybe there's something which can be worked with there...
Since the comparison is on a per digit basis, I think a better question is: is a number built with alternating digits of pi and tau normal? That way each "round" is determined by a pair of numbers in the interleaved concatenation.

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3 1 4 1 5 9 2 6 5 3 5 9
 6 2 8 3 1 8 5 3 0 7 1 8
3.61248135198256350375198

25% of the time (when the digit originated in pi and there's no carry) 1 is followed by 2
even assuming a 1 from taun is never followed by a 2 from pin+1 (this in not true),
one is followed by 2 more than ten percent of the time, which makes pi-interleaved-tau not normal.
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Re: 1292: Pi vs. Tau

Postby patzer » Fri Nov 29, 2013 4:26 pm UTC

orthogon wrote:
PolakoVoador wrote:
danhaas wrote:Pau means "Penis" in portuguese, so this discussion is quite cringesworthy for me.


It's worse than that. Pau is actually the slang/crude word for it, like dick or cock. Definetly weird to read this discussion.

Ah, so now I know why my Brazillian friends always snigger when I attempt to pronounce pão de queijo. Obviously it comes out sounding like "cheesy dick".

seems like an easy mistake to make
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