h a l f b a k e r yBreakfast of runners-up.
add, search, annotate, link, view, overview, recent, by name, random
news, help, about, links, report a problem
browse anonymously,
or get an account
and write.
register,
|
|
|
If you recall your junior high math classes, a big deal was made
about pi and the fact that it's irrational. Well, maybe so, but only in
isolated theory. Here's the problem, though: Once you throw physics
into the mix, you soon realize that can't possibly be true.
39 digits of pi is reportedly
sufficient to calculate the volume of the
universe to a margin of error of one atom. There must then be some
(surprisingly small) number of digits that allows you to calculate the
diameter of any possible circle to a margin of error of less than one
Planck length. Beyond this, further precision is literally impossible,
as there's no way to measure any length smaller than that. So it's
meaningless to say that pi is any more precise than that, because
there's no way it could be within the confines of our universe.
All we have to do is calculate that number of digits, and fix pi at
whatever that works out to be (maybe throw in a few extra digits to
be safe). And instantly, pi goes from a magical, "irrational" number
to just a plain old constant with a lot of decimal places. Much easier
to grasp for students, much easier to deal with for engineers and
scientists. The only people who are going to have a hissy fit about
this are geeks who love memorizing pi to thousands of decimal
places, and that lobby is not as powerful as it once was.
Oh, and mathematicians, I suppose, but they're the ones who got us
into this mess in the first place. Just hand them a physics textbook
and tell them to get a REAL job.
The bible makes it easy!
http://www.extremel...en/Bible_errors.php
12. Pi equals exactly three. [Klaatu, Jun 16 2012]
Mathematical Infighting Promotes Tau instead of Pi
http://tauday.com/tau-manifesto
"No, really, Pi is wrong!" [zen_tom, Jun 17 2012]
Considering n-dimensional Spheres
http://en.wikipedia...me_and_surface_area
Yup, when working out the volume/circumpherence, surface area etc, pi works in pretty much all the dimensions. [zen_tom, Jun 19 2012]
Reduced Plank Constant = h/2*Pi
http://en.wikipedia...iki/Planck_constant
Hmm, so the plank length itself is defined in terms of Pi! Your move [ytk] [zen_tom, Jun 19 2012]
Attempt made to rationalize pi in 1897
http://www.straight...-saying-pi-equals-3
There is also a way around the biblical link provided by [Klaatu] [AusCan531, Jul 22 2012]
vi hart pie
https://www.youtube...watch?v=5iUh_CSjaSw
[pashute, Jan 21 2016]
vi is still wrong
https://www.youtube...watch?v=jG7vhMMXagQ
[pashute, Jan 21 2016]
Field's metal
https://en.wikipedi...iki/Field%27s_metal
for [Cuit_au_Four] [notexactly, Jan 22 2016]
Please log in.
If you're not logged in,
you can see what this page
looks like, but you will
not be able to add anything.
Destination URL.
E.g., https://www.coffee.com/
Description (displayed with the short name and URL.)
|
| |
Yes, for most practical purposes you only need pi
to so many places (although there are some real-
world functions that involve high powers of pi, for
which you'd need more digits to get a reasonable
accuracy in the result). I am pretty sure that
carpenters, engineers and architects use pi to just
a few digits. |
|
| |
So, what exactly is your point? Is it to define an
approximation of pi which is good enough for
practical porpoises? If so, this is widely baked. |
|
| |
Also, who wants things to be such that e-to-the-i-
pi= 1.000000000000000000000037? |
|
| |
My vast ignorance leads me to ponder inexpertly on the Nyquist-
Shannon sampling theorem and wonder if you need a few more
decimal places, and also that it may not fit exactly into multiples
of ten, thereby leading to some weird recurring decimal or
something. |
|
| |
Wouldn't it be easier to define other numbers as ratios of pi and
make them all approximate? |
|
| |
//within the confines of our universe// |
|
| |
When did we get so provincial? |
|
| |
Not to mention all the silly lopsided circles we'd end up
with. |
|
| |
A mathematician told me yesterday, "Physics and mathematics are fundamentally different. In math we use definitions." Hm... |
|
| |
// e-to-the-i- pi= 1.000000000000000000000037//
I believe you mean e-to-the-i- pi= -1.000000000000000000000037
Or were you simply 'rounding off' the minus sign? |
|
| |
I think I put the minus sign between the e and the
to. |
|
| |
While our Observable Universe (at least what we observe so far) appears to be a large finite object, such that this Idea could be practical as far as real-world uses are concerned, there remains the possibility of some sort of infinite expanse outside the boundaries of what we currently observe. |
|
| |
If we ever obtain some hints that there is more Out There than just the so-far-Observable Universe, especially if there is an infinity of "more", then this Idea becomes wrong. |
|
| |
If your Observable Universe appears to be finite, you've
clearly only been observing it in three dimensions. |
|
| |
Even if you wish to continue hindering yourself in such a
manner, I'd say the fact that we're constantly expanding
the scope of our observation is a pretty solid indicator that
there is indeed more Out There, and our well-established
inability to observe a point of origin speaks to its probable
infiniteness. |
|
| |
Hmmm...I think this new Pi needs a slight name change: |
|
| |
Crusty Pi; maybe Baked Pi. |
|
| |
On another point, if we were to count in Base Pi or something like that*, would the irrational numbers be removed? |
|
| |
* Completely not thought through. |
|
| |
//So, not a rant, then?// |
|
| |
Hardly. What I'm proposing is nothing less than the
unification of the hardworking, proletarian sciences like
physics with the aristocratic, ivory tower discipline of
mathematics, thereby improving both. Math that doesn't
reflect and acknowledge the fundamental reality of our
existence is meaningless and completely arbitrary. |
|
| |
//Also, who wants things to be such that e-to-the-i- pi=
1.000000000000000000000037?// |
|
| |
Simply define e in terms of pi such that the equation works
out nicely. This has the beauty of unifying theory and
practice, as it would define e to the same degree of
precision as pi, which is to say "more than enough". |
|
| |
Never did understand 'what' pi was until I saw the gif about a year ago.
It was just some random number we had to memorize to make the circle math work. |
|
| |
// Never did understand 'what' pi was // |
|
| |
Don't feel bad; Euclid didn't get it, either. |
|
| |
When in doubt, consult the bible. <link> |
|
| |
1 Kings 7:23 gives the absolute value if pi. |
|
| |
//The bible makes it easy!// |
|
| |
Yes, yes, we've all seen this and it's ever so amusing. But it's
easily explainable by the fact that the vat was unlikely to be
a perfect cylinder. If it sloped outward like a tumbler then
the diameter measurement would have been taken at the
top but the circumference would have been taken at the
bottom. In fact, the ancients knew that you could calculate
the circumference of a circle from its diameter and vice-
versa, so the only reason to provide this information would
in fact be to describe the vat's shape as an inverse truncated
cone. |
|
| |
//7:23 gives the absolute value// |
|
| |
7:23 is 3.28571429 - wayy out. |
|
| |
[ytk] either you have both tongues in your left
cheek, or you are missing the point of pi. |
|
| |
One of the beautiful things about pi is that it
expresses something fundamental about circles,
regardless of where they are or even whether they
exist or not. The fact that so simple and
fundamental a number cannot be written down is,
to me, also very beautiful and strange. |
|
| |
What I find even stranger about pi is to ask if it
could be different, in the same way that (say) the
charge on an electron could be different, in a
different universe. |
|
| |
I've always found that 355/113 was close enough for most purposes. |
|
| |
// if it could be different, in the same way that (say) the
charge on an electron could be different, in a different
universe. // |
|
| |
In his novel 'Eon', Greg Bear proposed that changes in the
local value of pi would indicate singularities and
dimensional wormholes. |
|
| |
//changes in the local value of pi would indicate
singularities and dimensional wormholes.// That's
interesting, albeit complete poppywash. Carl Sagan
did something similar in Contact, where data had
been built into the value of pi. |
|
| |
Yes, that's why I made sure to mention the word 'novel'. I
suppose I meant it as some sort of evidence supporting the
natural artistry of pi. Something like that. |
|
| |
//you are missing the point of pi// |
|
| |
It comes right after the "3". |
|
| |
//One of the beautiful things about pi is that it expresses something
fundamental about circles, regardless of where they are or even whether they
exist or not.// |
|
| |
The problem is that this is only actually true if you apply mathematics as
separated from absolutely all other scientific knowledge. So, it's really more
accurate to say that it expresses something fundamental about circles that
couldn't possibly exist. But just like the discovery of quantum physics forced us
to rethink what we knew of the universe based on Newtonian physics, it is time
to revise our understanding of mathematics based on our enhanced knowledge. |
|
| |
In our universe, there is only so much precision possible. Take Zeno's paradox
of Achilles and the tortoise as an example. Prior to the discovery of the Planck
length as the smallest possible measurement of distance, Zeno's conclusion
(that motion was impossible) was nothing more than a reductio ad absurdem of
his entire philosophy. And yet, with our current understanding of quantum
physics, we can say that perhaps he was on to something after all. His paradox
was conditioned on the fact that there is no minimum length. Once we've
shown that there is, in fact, such a minimum distance, the paradox neatly
resolves itself. Since there is no longer an infinite number of points between
Achilles and the tortoise, he /can/ in fact overtake the tortoise in a race.
Without realizing it, Zeno effectively proved the existence of the Planck length
long before anyone knew what it was. And today, we can no longer take for
granted what we once assumed to be fundamental truths about physics. |
|
| |
So it must also go with mathematics. There is a point beyond which one simply
cannot make a more precise measurement. I grant that our perception of the
size and scope of the universe may be flawed, and so it may be impossible to
make an exact calculation based on our limited knowledge, but the principle
still holds that pi /must/ actually be some rational number in our universe.
Simply put, our understanding of mathematics, like Zeno's understanding of
physics, was incomplete. |
|
| |
You make the point yourself when you ask //if [pi] could be different, in the
same way that (say) the charge on an electron could be different, in a different
universe//. It certainly could. In some other universe where the speed of light
or the gravitational constant is different, the Planck length, and therefore pi,
must be different as well. Once we accept that mathematics is not necessarily
multiversal, we have to accept that what we know of mathematics only applies
within our own universe. And we therefore have to apply the rules of our
universe to mathematics. Trying to prove that there are an infinite number of
decimal places to pi is as futile as trying to prove the existence of God. |
|
| |
Pi could be different if certain non-Euclidean geometries applied,
for instance hyperbolic geometry would entail that it was a
variable rather than a constant, although it would still be a limit
of some kind. |
|
| |
Concerning the Bible, this probably reflects the lifestyle and the
resultant approach to mathematics. Decimal fractions were
unknown at the time, so pi could at best only be approximated
as a ratio. The Bible also claims insects have four legs, which is
a similar phenomenon. |
|
| |
//Take Zeno's paradox of Achilles and the
tortoise// |
|
| |
The argument that the Planck length solves this
paradox is not really true. The paradox resolves
even if there is no smallest possible length: it
resolves through pure mathematics, which shows
that the infinite series (1+1/2+1/4...) does not
sum to infinity. |
|
| |
The more general point you raise is whether pi
should represent the mathematical limit of a
circle's circumference, or the nearest physical
value that can be attained in our own (finite and
granular) universe. |
|
| |
However, an approximation to "fit" our reality will
differ on a case by case basis. If my circle has a
diameter of 1 femtometre, then its value of pi will
differ from the abstract pi by quite a large amount
(either more or less than the abstract pi). If it has
a diameter of 1 billion light years, then its value
of pi will be much closer to the abstract value. |
|
| |
In other words, the value of "physical pi" depends
on the size of the circle, whereas the "abstract pi"
does not. |
|
| |
Regarding the dependence of pi on G or on the
speed of light: this applies only to the "physical
pi". (Also, to calculate the dependence of the
"physical pi" on G or C, you need to use the
"abstract pi".) |
|
| |
Regarding [19thly]'s point that non-Euclidean
geometries will change pi, again this applies to a
physical pi but not to an abstract pi, which is
defined for a flat geometry. |
|
| |
Finally, it's worth pointing out that "abstract pi"
crops up in many non-circle contexts (as a simple
example, it's the limit sum of several series). |
|
| |
Anyway, I guess my point is that I don't see your
point. Pi exists as a mathematical construct, just
as the square root of 2 exists as a mathematical
construct. You're arguing that for practical
purposes we just need an approximation (which
might vary depending on circumstances), which is
fine but so what? |
|
| |
// The Bible also claims insects have four legs, which is a similar
phenomenon.// To be fair, spectacles hadn't been invented at
the time. |
|
| |
And, insects generally do have four legs. Plus a couple extra, sure. |
|
| |
// To be fair, spectacles hadn't been invented at the
time.// Yeah, but some pretty big beetles had. |
|
| |
If the bible included just sixteen digits of pi, the atomic numbers of a single element, or an accurate description of earth's place in the universe, I would be more inclined to respect it as a document of divine and transcendent inspiration. Clearly if it is a divine document of super human knowledge, it would have at least a few helpful hints thrown in, for skeptics sake. Also, apparently god needs and editor in the worst kind of way. |
|
| |
//If the bible included just sixteen digits of pi, the
atomic numbers of a single element, or an accurate
description of earth's place in the universe//... |
|
| |
... then Occam's razor would guide one to the
simplest interpretation, namely that humans will
invent time travel one day, and will have written the
bibble as a joke. |
|
| |
look, that's magic enough for me. Time travel is a good enough excuse for a religion. The bible as a the wrote word of an all knowing deity is like kind of time traveling prank that you might get if the pranksters were from a future where all that remained of society was illiterate subsistence shepherding, functional time machines, and complicated unfunny pranks. |
|
| |
//the value of "physical pi" depends on the size of the circle, whereas the
"abstract pi" does not// |
|
| |
Not quite. The ratio of one particular circle's diameter to its circumference
may vary slightly, but the constant pi remains the same throughout the
universe. |
|
| |
//Anyway, I guess my point is that I don't see your point. Pi exists as a
mathematical construct, just as the square root of 2 exists as a
mathematical construct. You're arguing that for practical purposes we just
need an approximation (which might vary depending on circumstances),
which is fine but so what?// |
|
| |
In our universe, there must be some possible largest circle, and we know
there's some smallest measurement. While we can, in theory, calculate
what pi /would/ be if this were not the case, that makes no sense to do in
our universe. It is as meaningless as talking about half a Planck length.
While you can theoretically conceive of such a thing, it simply cannot exist.
So, there must be some fixed, rational value "pi" that is sufficient for use in
any calculation that is possible in the universe. I'm not arguing for an
approximation—this number /is/ pi in our universe. Anything more "precise"
would, in fact, be an approximation, not the other way around. Insisting
that pi is nevertheless infinite as a "mathematical construct" is tantamount
to a denial of current scientific knowledge. It would be like a Newtonian
physicist simply denying the existence of quantum physics. |
|
| |
What's the point of all of this? Well, I believe we should teach our students
with an eye towards a holistic model of the universe rather than supplying
them with building blocks that are inadequate to progress to further levels
of scientific understanding. Ask any grade school student whether the Sun
revolves around the Earth, and you will get an emphatic “No, it's the other
way around!” But this is, of course, wrong: The correct answer is that
without specifying a frame of reference, either view is valid. But this
incorrect “knowledge” has been so deeply embedded in the minds of
students that significant effort is required to unlearn it in order to progress
to a higher level of understanding of physics; most students never do so,
and remain in ignorance throughout their adult lives. |
|
| |
In the same way, telling students that, under our old theoretical
understanding of mathematics, pi was considered irrational, but given our
knowledge of quantum physics we can show pi to be rational, and even
know approximately what that rational value is, is not only more truthful in
that it presents a broader view of the sum total of human scientific
knowledge, but it opens up the door to a greater understanding of the
universe at an earlier age. And all we have to give up is our outdated,
security blanket belief that we can have such things as “mathematical
constructs” apply to a universe that has a physical form. |
|
| |
//this number /is/ pi in our universe// |
|
| |
Well, no it isn't. It would be the practical value of
pi for the largest possible circle in the universe of
the size we currently know it to be. |
|
| |
If our measure of the universe were slightly
wrong, then the value of your pi would be wrong. |
|
| |
If the universe is expanding, your pi would also be
changing. |
|
| |
If the universe is not spherical, then your value of
pi will depend on the orientation of the circle
(since this will determine its diameter). |
|
| |
If you are interested in circles smaller than the
diameter of the universe, your value of pi will
again be different. |
|
| |
In short, your pi will be a variable, not a constant. |
|
| |
If it makes you happier, then of course we can
define a new value of pi (which we might call, say,
chi to avoid confusion), defining chi as the ratio of
the circumference to the diameter of a circle
which just fits into the universe as we understand
it today, where both measures have been rounded
to the nearest planck length before computing
their ratio. Would that do? |
|
| |
Perhaps it would clarify things for me if you could
explain how and where this new value of pi/chi
will be used, by means of an example. |
|
| |
The Greeks had an issue with irrational numbers as well - but that problem only occurs when you cling too strongly to the idea that number is the same thing as cardinality. |
|
| |
The problem with doing this is that you're going to have to provide a whole bunch of ugly standardisation lists for all the other irrational numbers, root2 (and all other non-perfect square roots) e (and an infinite number of other logarithms) - that means you've got an infinite list of irrational numbers to standardise, but then if you're going to avoid any references to infinity, then you're going to need to round off and effectively outlaw an infinite number of numbers, which since you're outlawing the concept of infinity, puts you in somewhat of a sticky predicament. |
|
| |
[Max]: None of that matters. We may not know
exactly what the value of pi is at any
given time. We may not ever know. But we do know
that it is some rational
constant, that depends on the size of the universe.
If the size of the universe
changes (assuming it ever does—we aren't sure about
that yet), the value of pi
changes. But it's not a variable, because its precise
value is entirely dependent on
the size of the universe at any given time. I believe
you are confused about one
thing, though. Pi doesn't change based on the size of
the circle you are working
with. Pi is simply the appropriate value for the
largest circle possible. In real life,
the ratio of a circle's circumference to its diameter
simply isn't precisely tied to pi.
The “mathematical abstraction” is wrong, and it's
even more wrong than the actual
value of pi (although neither one can be exactly
precise all of the time). |
|
| |
So again, the point of this is that we shouldn't teach
people something wrong just
because it's easier than teaching something right.
When we discovered quantum
physics, we didn't just say “Oh, let's just keep piling
on new theories of Newtonian
physics to explain the universe, because classical
physics should remain ‘pure’ in
some way”. Why should we do so with mathematics?
A theory that is
demonstrably wrong (even if we can't necessarily
demonstrate that to be the case
with today's technology, a simple
gedankenexperiment will suffice) has no place or
purpose in our scientific body of knowledge. |
|
| |
Regardless of what it actually is at any given time, or
whether its precise value
changes over time, pi /is/ rational. Believing
otherwise requires stubbornly
clinging onto an outdated view of the universe. Just
like we no longer teach about
the aether, it's time to consign the irrationality of pi
to the dustbin of scientific
history. For what purpose, you ask? The
advancement of knowledge is a goal unto
itself. |
|
| |
//The Greeks had an issue with irrational numbers as
well// |
|
| |
I'm not making any statement here about irrational
numbers in general. Just that pi, as we understand it,
simply /cannot/ be irrational when taken in context
with the totality of human scientific understanding.
There's no reason why root 2 couldn't be irrational,
since it's not based on any geometric property of the
universe. |
|
| |
//There's no reason why root 2 couldn't be irrational, since it's not based on any geometric property of the universe.// |
|
| |
Take a right-angled triangle (i.e. one with one of the angles equal to Pi/2) with the two sides either side of the right-angle with a length 1. The slopey side will then, quite physically, have a length of root 2. |
|
| |
If you fiddle about with one irrational number, you have to be prepared to mess around with all of them. And since there's an infinite number of them, and all of them need to "add up" and continue to interrelate with one another, that's a lot of work you're proposing. By the time we've gotten around to filing off the tail ends of all those numbers, the universe will have been long-gone. |
|
| |
The same thing goes for all the other irrational numbers - there will be a physical analogy for each of them, picking on Pi alone isn't an option. |
|
| |
No, because root 2 is not defined in terms of geometry. If
you perform the division in question in our universe, you
arrive at some rational number that is /approximately/
root 2, which is irrational. It's fine for numbers to be
irrational, except with regards to physical measurements of
distance. Since pi is defined by measurements of distance,
it cannot be irrational. |
|
| |
If the Bible included sixteen digits of pi, Hebrew numerical
notation would have to be completely different. It can't express
irrational numbers and even has a problem with integers
because it avoids accidentally spelling out the name of God with
the letters it uses to write them by expressing fifteen and sixteen
without using the letter for ten. To express sixteen digits worth
of accuracy for pi, it would have to describe it as a ratio in
words, and i'm almost sure it couldn't express numbers that
high. I see its expression of pi as an integer as quite appropriate
since its expression of morality is also wildly inaccurate, at least
in the Old Testament. |
|
| |
Again, this “the Bible says pi is 3” nonsense has been
debunked. I'm no fan of religion, but this gratuitous Bible-
bashing irks me. |
|
| |
//No, because root 2 is not defined in terms of geometry. // |
|
| |
It depends on how you choose to define Pi or Root 2. But if you wished, you could choose to define Pi as the limit of an infinite series without any recourse to geometry. |
|
| |
Similarly, Root 2 could be constructed in a similar way. No geometry required. |
|
| |
They'd still both exist, and they'd still be *applicable* to geometric functions, but they'd still be irrational. |
|
| |
This may be cheating, but you can define Pi as a repeating (alternating) series of root 2's. Even if that doesn't meet the criteria for differentiating between a "physical" thing and an abstract one, it does further the case that these two are very tightly linked together. |
|
| |
Likewise, Euler's formula provides a direct link between Pi and the exponential constant, e. That formula is couched within Complex Numbers, which are (I suppose) inherently geometric in their composition. But again, if you mess with Pi, you also mess with e, and if you alter the value of e, then exponentation goes out the window, and by direct association multiplication, addition and everything else. |
|
| |
More strongly, since each of these things (where "things" means *any* number at all) show that each number (rational or otherwise) are just different ways of talking about the same thing. |
|
| |
Since you can extend this out to any other number, it follows that they *must* have the values that they have, or otherwise 1 doesn't equal 1 anymore. |
|
| |
How about redefining Pi as 1, and then defining a new number, Ip, with a value of 0.318309886.... |
|
| |
Then instead of the circumference being = 2 * Pi * radius, |
|
| |
We could just say that the radius is = Ip * Circumference / 2. |
|
| |
But [bigsleep], why do you need reams of exegesis on a whole load of Iron age mythical texts to lead you to the conclusion "be nice"? |
|
| |
//The bible also claims that insects have four legs.// |
|
| |
I don't seem to recall formal scientific taxonomy as having been invented before biblical writings. |
|
| |
Let pi remain irrational. po, on the other hand, should stay completely rational. We need a sane po. |
|
| |
Leviticus eleven twenty-three says, in the KJV,
"But all other flying creeping things, which have
four feet, shall be an abomination unto you." It
refers to animals which fly and walk on all fours,
shortly after mentioning locusts, beetles and
grasshoppers, and the Hebrew does use a form
derived from its word for "four". To my mind,
that's a reference to
insects. The Israelites may not have had a modern
taxonomy in mind and probably thought of all small
invertebrates as similar, but there are no other
animals which both fly and have more than two
walking limbs. It's sometimes claimed that the
hopping legs are not considered legs, but the text
also refers to beetles. However, i agree that the
idea that the Bible is supposed to be making
scientific assertions is anachronistic, and
presumably the same applies to mathematics.
Concerning the precise nature of the moral code,
it's memetic and probably worked better in that
set of circumstances than it does here and now,
but there were other cultures in similar
circumstances which were, for example, less
opposed to gender role blurring, for instance in
Pre-Columbian North America. |
|
| |
Anyway, circles. Pi makes sense in Euclidean
geometry as a constant, but geometry is not
Euclidean. It's a very important constant but
doesn't quite mean what it seems to mean,
because in this Universe, any circle, even if
perfect, would be slightly off, since the space in
which it is drawn is not flat. |
|
| |
// Pi doesn't change based on the size of the
circle you are working with. Pi is simply the
appropriate value for the largest circle possible// |
|
| |
Ah, OK, so your "pi" is simply a theoretical value
for a circle with an arbitrarily-chosen and
imprecisely-known size. That's obviously better
than a theoretical value which is independent of
the size of the circle and is known with arbitrary
precision, but can you just remind me exactly
why? |
|
| |
// The “mathematical abstraction” is wrong// No,
the mathematical abstraction is right, as a
mathematical abstraction. |
|
| |
But, whatever. If you want to define a new type
of "bricklayer's pi", that's fine, although it would
be churlish to insist on giving it the same Greek
letter as the one everyone already uses for the
mathematical pi. |
|
| |
It would also be worth bearing in mind that you'll
need to define new, pragmatic values for the
square roots of all the non-square numbers. |
|
| |
At the same time, you'd need also to redefine a
lot of the of the rational numbers, based on
commensurability of the two numbers. I'm
thinking of things like "1/3rd", which is really just
a mathematical symbol (like pi) for a number
which cannot be written out. You'd have to divide
the diameter of the universe by 3, and then see
how much of a Planck unit you had left over, and
then... |
|
| |
Moreover, we might find that the diameter of the
universe is not an even number of Planck units, in
which case the "Halfbakery" will have to be given
an inconveniently long name. |
|
| |
One last question: there are plenty of irrational
mathematical constants which have a less direct
relationship to physical objects than pi does. How
do you propose to redefine them? |
|
| |
Irrational numbers exist wherever
one tool for measurement is laid up
against another. Your tool for
measuring circumference is different
from your tool for measuring radius,
yielding an irrational number. This
fact that different tools of
measurement do not perfectly match
one another, cannot perfectly match
one another, is a perfectly valid
observation. It helps, not hinders, the
application of mathematics to reality.
Hate it all you want, 10 planks is a
unit of measure, a 10 plank circle will
appear to have an irrational radius.
Yes, irrational. |
|
| |
Also, while tou is useful it is simply 2pi. So I'm not going to get
my panties in a twist about it. |
|
| |
If it's two*pi, shouldn't it have twice the number of legs rather than half? |
|
| |
Read the manifesto: The legs have a line across the top, so they are in the denominator. Hence half the number of legs. One early paper on this topic proposed a 3 legged pi-like symbol, but adoption seems much less likely if it is a symbols that isn't available in most font packs. |
|
| |
// Trying to prove that there are an infinite number of decimal places to pi is as futile... // I agree completely that it is futile to probe the depths of pi or any of the irrational numbers that we find useful in day-to-day life. I don't think there is anything more of interest to be learned along that avenue, but I completely disagree that we should declare that pi or any of these other values is a rational number. Pi (or Tau) being irrational is not some magical commentary on the ratio between the dimensions of a circle. It just demonstrates one limitation of our very useful system of numerical representation. For practical purposes we overcome this shortcoming by making sure we use enough digits for the task at hand. For theoretical purposes we call it irrational and use a different representation that isn’t as convenient to work with. |
|
| |
If there is some confusion caused by the emphasis placed on irrational numbers in junior-high math classes, that ought to be addressed with better teaching, not by attempting to redefine a concept to hide the limitations of our numerical representation system. |
|
| |
hmmm, so, visually if you roll a circle it will roll past three point one four yadda-yadda other circles of the same size to complete one revolution.
Is there any number of times it can roll to come up with a number of revolutions which give an answer that is a whole number? |
|
| |
The answer will always be irrational. All non incremental
measurements are irrational. That's the point! (Planks be near
damned!) |
|
| |
//Is there any number of times it can roll to come
up with a number of revolutions which give an
answer that is a whole number?// |
|
| |
That is why pi is so beautiful. No matter how
large the circle, you can never make a measuring
stick which will fit a whole number of times into
both the radius and the diameter, no matter how
small a measuring stick you try. |
|
| |
The same is true of the square root of all integers
(except the perfect squares like 1, 4, 9...), and
indeed for most numbers. |
|
| |
//That's obviously better than a theoretical value
which is independent of the size of the circle and is
known with arbitrary precision, but can you just
remind me exactly why?// |
|
| |
Because it's /correct/, that's why. If a theoretical
value cannot possibly exist in our universe, then it's
pointless to talk about it. You can theorize all you
want about God, or the arbitrary precision of pi, but
science as we know it is unequipped to discuss such
things. Simply put, by insisting pi has an infinitely
precise value, or even a more precise value than it
could possibly have, you are being unscientific. Far
be it from me to deny you your religious beliefs about
the nature of pi, though. |
|
| |
[ytk]:
1: You keep going on about measuring, and not being "abstract". Maths is not about measuring; it is abstract - that's the whole point of maths!
2: As [MaxwellBuchanan] has pointed out, pi shows up all over the place completely unrelated to circles, so your "circle measuring" premise is flawed.
3: In using the Planck length as your "unit", the absolute size of the circle would matter, as a smaller circle would have a (slightly) smaller number of steps. (Eg: if I use the radius of a circle as the step "unit", I get a pi value of 3; 6 steps of the radius around the circle, ie: a hexagonal approximation.)
Conclusion: Use an approximation for measuring, but theoretical maths uses the theoretical version, which is irrational. |
|
| |
//insisting pi has an infinitely precise value, or
even a more precise value than it could possibly
have, you are being unscientific.// |
|
| |
Pi is a mathematical abstraction. I'm not insisting
that there's any circle in the universe with any
degree of precision. I am simply observing that
the series 4*(1-1/3+1/5-1/7...) converges toward a
limit, which most people call pi. |
|
| |
I also note that the ratio of a real, physical circle's
circumference to its diameter approximates to pi. |
|
| |
OK, I'll make you a deal. If you want to use the
name 'pi', you can have it. I'll call mine 'Bob'. So,
now we can both be happy. |
|
| |
Now that we've resolved that one, can we start on
phi? I think it's only fair that I get to keep the
name "phi", but I think it would be OK if you called
the bricklayer's version "fi". |
|
| |
Also, can you tell me what value you'd like for a
third? |
|
| |
//If a theoretical value cannot possibly exist in our universe, then it's pointless to talk about it.// |
|
| |
Please Google "computable numbers". [-] |
|
| |
[Chronicles 4:2] Also he made a molten sea of ten
cubits from brim to brim, round in compass, and
five cubits the height thereof; and a line of thirty
cubits did compass it round about. |
|
| |
How do we know that pi is constantly associated
with what we believe to be circles and spheres
throughout the universe? We are beginning to
understand that the universe is possibly a toroidal
(though not in the sense we know a toroid) shape
and may be subject to multi-dimensional
distortions we cannot even visualise, let alone
describe mathematically. |
|
| |
In primary school, through which I passed before
the advent of calculators and personal computers,
we approximated pi with 22/7 = 3.142857
recurring. Near enough for 10 year old kids to use.
Then in high school we used 355/113 = 3.2415929.
Again, near as you'll get without picking all of the
nits off the monkey for something to do. |
|
| |
I'm not sure why people are so fascinated with pi.
1/3 and 2/3 give you infinite decimal progressions
*and* they're ones we can all remember to as many
places as we have time to recite. Knowing pi past
6 places is largely pointless, though I have a
mnemonic to recall it to thirty places. Beyond
that it's pretty much academic. |
|
| |
//How do we know that pi is constantly associated
with what we believe to be circles and spheres ...
the universe is possibly a toroidal// |
|
| |
First of all, me and [ytk] have agreed that pi will
hencefroth be known as Bob, leaving "pi" to be
used for an approximation of Bob. |
|
| |
That said, it doesn't matter what shape the
universe is. Pi is defined for circles in flat, two-
dimensional space, even though we live in three
dimensions. (In fact, mathematicians don't like
the circle business to define pi, and it's instead
defined in terms of angles and cosines, but that's
a hamster of an entirely different lineage.) |
|
| |
//Pi is defined for circles in flat, two- dimensional
space, even though we live in three dimensions// |
|
| |
Exactly. Multiple dimensions and non-Euclidean
planes stuff it up. There are always radians (rad),
which are so much more predictable that pis. |
|
| |
Well, I'm glad we can finally agree to disagree, [Max],
but I still don't see why this is so hard for you to
understand. Finding the actual value of pi is easy.
You simply measure the circumference of the entire
universe in Planck lengths, divide by its diameter in
the same units, and, as they say, (the limit of the
series 4*(1-1/3+1/5-1/7...))'s your uncle. |
|
| |
Are you asserting that the Planck length is somehow
indivisible, [ytk]? |
|
| |
//Are you asserting that the Planck length is somehow
indivisible, [ytk]?// |
|
| |
I'm not asserting it. I'm stating a widely known
scientific fact. |
|
| |
1.616199 × 10-35 metres is the length of a Planck unit. |
|
| |
Therefore 8.080995 x 10-36 metres would be half of a
Planck unit, right? |
|
| |
//Therefore 8.080995 x 10-36 metres would be half of
a Planck unit, right?// |
|
| |
OK, please explain why I'm wrong. |
|
| |
My understanding is that there may be an apparently
insurmountable obstacle in observing processes that
occur within a Planck time at this point but there is
no way of knowing whether a process that occurs
within the elapsement of a single Planck unit is not
possibly divisible into process steps using technology
still to be forwarded (reversed?) to us from my
contacts in the future. |
|
| |
//Wrong// - Wow. Nice flat statement there, [ytk]. |
|
| |
Particularly ironic for being in the context of taking a provable fact and popularly replacing it with a falsehood, just because it is more convenient to teach our children lies. |
|
| |
(Were you to calculate that one of the bones up there was mine, you'd be correct to a very high - nay, even infinite - accuracy.) |
|
| |
I have no real place in a math battle--other than the
observation bunker, perhaps--but this one gets my goat as
well. |
|
| |
Pi is pi. It is inviolate. |
|
| |
Just because we never calcuate it beyond a dozen
decimal places for any practical use does not give us the
right to declare it rational. It's not a theory or a formula or
even a law, it's a _fundamental property_. We have no
more right to do
that than we have to declare the Sun the center of the
Universe or to declare the proton indivisible. |
|
| |
//Wow. Nice flat statement there, [ytk].// |
|
| |
I had hoped that this would prompt [UB] to research
the Planck length on his own, and
discover /why/ his statement was wrong. The
information isn't hard to find, and you can
find better and more thorough explanations of it via
a quick Wikipedia or Google search
than I could possibly offer here. However, in
retrospect, the statement probably came
across as a peremptory dismissal, which is entirely my
fault. |
|
| |
Accordingly, here is an explanation for you, [UB]:
From the Wikipedia article on Planck
length: “…the Planck length is the length scale at
which the structure of spacetime
becomes dominated by quantum effects, and it
would become impossible to determine the
difference between two locations less than one
Planck length apart.”, and “In string
theory, the Planck length is the order of magnitude
of the oscillating strings that form
elementary particles, and shorter lengths do not
make physical sense.” |
|
| |
In other words, it makes no sense to talk about “half
a Planck length”, because if two points
were less than a Planck length apart from each other,
it is physically impossible to
distinguish them from each other. Not really hard,
or beyond our technological capability
to do so—it's impossible, and therefore they're really
the same point. It has no meaning in
our universe for two discrete points to be less than a
Planck length apart. It's one of the
weird things about quantum physics that you just
kind of have to accept because it can't
possibly be otherwise. |
|
| |
So, it's wrong to talk about “8.080995 x 10-36
metres”, because such a distance can't exist.
You can't move half a Planck length, nor can you
measure it, because it doesn't exist. The
only distance less than a single Planck length is 0. |
|
| |
//Just because we never calcuate it beyond a dozen
decimal places for any practical use
does not give us the right to declare it rational.// |
|
| |
If you take any given circle that could possibly exist
in our universe, and calculate the ratio
of its circumference to its diameter, you will get a
rational number. The only way to get an
irrational number is to use a circle that's not bound
by the laws of our universe. Okay, fine,
but then we have to say that mathematics is not
bound by the laws of our universe. But it
is also theoretically possible that mathematics works
differently in some other universe—we
simply don't know enough to be able to say. It would
then follow that mathematics /is/
possibly bound by the laws of the universe it's in. So,
there are only two conclusions we
can draw from this: |
|
| |
1) Mathematics is bound by the laws of our universe,
and therefore pi is rational (though we
may not know precisely what it is).
2) Pi is not rational, and therefore math is not bound
by the laws of our universe, and must
be the same in every universe. |
|
| |
Since concluding 2 would lead to a contradiction
(because math may be different in other
universes), then the only conclusion left is 1, and pi
must be rational. Actually, what we
can conclude is that, if math can be different in
another universe, pi must be rational in
this one. However, since our current thinking points
to the possibility that math is
different in other universes, we have to at least
allow for the possibility that pi is rational in
this one. |
|
| |
//It's not a theory or a formula or even a law, it's a
_fundamental property_.// |
|
| |
So is the Planck length. It seems what we have is a
case of dueling fundamental properties. |
|
| |
//We have no more right to do that than we have to
declare the Sun the center of the
Universe or to declare the proton indivisible.// |
|
| |
Well, depending on what you mean by “the center of
the universe”, we can declare any
point to be its center, including the Sun if we so
choose (this gets back to what I mentioned
earlier regarding the Sun orbiting the Earth or vice
versa). There is no “absolute” frame of
reference that can be said to be the true “center” of
the universe. From the point of view
of the Sun, it is stationary and the entire universe
moves about it. This is true for any
frame of reference. Einstein demonstrated this with
the special theory of relativity. As for
the proton, it may or may not be indivisible. The
Planck length, however, is—and that's the
crux of my argument. |
|
| |
So are you claiming 2, that math must be the same in
every universe? |
|
| |
There are no perfect circles, perfectly flat surfaces and so forth
in physical reality and when we say a real object is spherical or
make other statements of a similar kind, we are always slightly
wrong. An analogous process takes place when we use
numerical notation, in that it takes us away from the actual
numbers and we get bogged down in the patterns we can make
with them, which are often confused with the numbers
themselves. However, those patterns are still mathematically
interesting. Both those issues apply to pi because not only are
there no real objects whose perimeter is exactly pi times any of
their various diameters (since they are not perfect circles
anyway, they have a large number of diameters), but also the
string of symbols we end up with when we try to calculate that
ratio is to some extent an artifact of our system of numerical
notation. Having said that, that string of symbols has various
interesting features such as being arbitrarily close to a "normal",
that is, truly random, sequence. Those features of pi are useful.
That particular feature makes it useful as a source of random
numbers, for instance, though there are doubtless better ones.
That means that pi has a life of its own. Similarly, a rational
approximation of pi might have its own interesting mathematical
features. Both of them are real, incidentally, in a non-
mathematical sense of the word. |
|
| |
// math may be different in other universes// |
|
| |
That statement completely misses the point of
mathematics, and in particular of mathematics
over the last century or so. |
|
| |
Mathematics is a game, like chess. We start by
choosing a set of axioms - a minimal set of
statements; these are equivalent to the starting
positions on the chess board. We also choose a
set of rules (formal logic) by which to manipulate
statements; these are the equivalent of the rules
of chess. |
|
| |
From the axioms, by applying the rules, we derive
theorems. This is equivalent to saying that from
the starting positions on the board, by applying
the rules of chess, we arrive at a position. |
|
| |
The axioms are to some extent arbitrary (there
are other sets of axioms you can choose). The
rules of logic are also somewhat arbitrary.
However, those axioms and rules seem to give
results (games) that we find interesting and
which, in some cases, are useful. So, that is how
we choose to play the game. |
|
| |
So, saying "maths may be different in other
universes" is like saying "chess may be different in
other universes". As long as I define what I mean
by "chess", it's independent of the universe in
which I play it. |
|
| |
If, in another universe, pawns can move three
squares instead of one, then that's fine - it's just
not my version of "chess". |
|
| |
It might not be fine at all, in fact, because it might make the
game unplayable - haven't thought it through, but another
variant - a piece with combined Queen's and Knight's moves -
ruins the game. In the same way, some versions of maths might
ultimately just not work, and we might never know which. |
|
| |
Well, [Max], I just think that by only considering the entire
universe, you're taking a very narrow point of view. |
|
| |
Consider that our universe has a fourth dimension—time—
that can't be mapped relationally to the three physical
dimensions. Pi is useful for computing two dimensional
areas and three dimensional volumes, but not for four
dimensional volume-seconds (at least, not in the time
dimension specifically). So there are some dimensions in
which pi has no meaning. |
|
| |
Consider a universe in which spatial dimensions don't relate
to each other in a linear fashion. In such a universe, pi
would vary based on which way you orient your object. So
in order to define (mathematical) pi, you have to set up a
very narrow set of perfect circumstances based on
selectively and arbitrarily choosing physical (and ignoring
temporal) properties of our universe that are amenable to
your chosen definition. It sounds a little bit like you're
cherry picking to arrive at a predetermined conclusion… |
|
| |
Crooked space in other words. |
|
| |
//but not for four dimensional volume-seconds// <Pantomime> Oh, yes it does! </Pantomime> |
|
| |
Consider it this way, if you want to plot a line where each point on the line is a fixed distance (r)from a point (a,b) you can use a formula in the form
|
|
| |
See all the maths there? The squaring is just multiplying values by themselves, right? Nothing up my sleaves! |
|
| |
And yet, having plotted all the values for which x and y fit the above equation, and r = 1, you get a line who's length you don't need to measure. |
|
| |
If you arbitrarily decided to fix this value at some random amount (decided by intergalactic committee perhaps) then you wouldn't be able to do the algebra quite so cleanly and errors would creep into your calculations. Errors which repeated iteration (such as that which computers do) would quickly turn into whapping great ones! (e.g. Try to determine the angular velocity of some spinning thing by iterating a starting error factor of 0.0000001% - fine for the first couple of turns, but what if you're talking about a satellite that's been spinning for 20 years, how big is your error compounded by then?) |
|
| |
At least if you allow for the transcendental nature of Pi, you don't need to worry about precision until you are finally sure you need an answer that you are going to use for practical purposes, at which you are given the *choice* of deciding at what level of precision you *want* to use. |
|
| |
*Forcing* everyone to conform to some arbitrary value is not only wrong on a precision level, it also smacks of authoritarianism, Nanny Statism and frankly suggests that you may be of a Stalinist bent. Sure you can legislate away people's rights, but there will always be people who, in the cause of *freedom* will never bow down to arbitrary authority, even if it is couched in terms of making life "easy". If we're going to do that, why not make life even easier for all by legislating that we all switch to plastic cutlery and wear padded helmets while we're at it, eh?
I sometimes find it difficult to understand lobsters (they are so very *hard*) let's wipe them out as well! |
|
| |
And another thing - if we must define everything in purely physical terms (poof! there goes kindness, sociology and logic!) then we need to look more closely at this Plank-length thing. |
|
| |
You say that the Plank-length is fundamental, but *how* fundamental is it? The plank-length is itself defined by the Gravitational Constant, the speed of light, and the Plank Constant. |
|
| |
Where l = Plank Length, h = (Reduced) Plank's Constant, G = Gravitational Constant and C = speed of light. |
|
| |
Back to the constants, that's 3 things that we've measured pretty closely here on earth, and which we all accept as being inviolate, static and what have you. But hold on a minute - the speed of light is what, distance over time, right? So our fundamental unit of distance (the Plank Length) is itself defined in circular (pardon the pun) terms. That part aside, fiddling with one of the other constants, and you suddenly get different values for your Plank Length - shakey ground to stand on. |
|
| |
Even more shakey is that pesky square root which means what?... that the Plank Length is itself another irrational number!! Oh no! Congratulations, you've abolised one irrational, only to supplant it with another one, Ooops! |
|
| |
Still, I suppose it's *much* simpler (note heavy sarcasm here) to define your fundamental values in terms that rely on experimentally determined constants, the meanings of which nobody fully understands (not to mention keeping track of the current estimated size of the universe) rather than the tricky notion of the distance that a simplified (read abstracted) wheel will travel in the course of a full revolution. |
|
| |
[edit] See link above - It turns out that the Reduced Plank Constant, one of the constituents of the Plank Length, is itself defined by the value of Pi. So if we are going to redefine Pi in terms of the Plank Length, that means redefining the Plank Length, whcih means redefining the value of Pi, which means...ad inf because The Plank Length relies fundamentally on the value of Pi. Ouch. |
|
| |
//Consider a universe in which...... |
|
| |
...by only considering the entire universe, you're
taking a very narrow
point of view. // |
|
| |
You're still missing the point. We have maths for
any number of
dimensions, for pretty much any shape of space.
Pi (I mean Bob)
happens to apply to circles in flat two-dimensional
space (as well as to
spheres and whatever in higher dimensions),
which is nice. It doesn't
apply (in quite the same way) to circles drawn on
a sphere or on a
saddle, but we have different maths for those. |
|
| |
Maths is a puzzle, a game. It can be played with
various rules (that is,
various axioms and various formal logics). By very
good fortune, one
set of axioms and rules seem to give results that
are analogous to our
own physical universe - so much so that we can
predict the motions of
planets or the breaking stress on a bridge. So, we
sometimes use
those results for practical porpoises. |
|
| |
Now, as regards declaring Pi to be some fixed
rational number, I still
have two questions for you, [ytk]. |
|
| |
(1) What's the point? What I mean is, suppose you
convince everyone
that Pi is 22/7 (or whatever), and that they should
use the name "Bob"
for the irrational number that we used to call Pi -
what happens next?
How does it make my day any better? |
|
| |
(2) I ask again - what's your chosen value for 1/3rd
(which is rational but
happens to be an infinite decimal), or the square
root of 2 (which is
irrational), or e (which is, if I remember correctly,
transcendental like
Pi)? |
|
| |
(3) Finally, if you are only going to allow numbers
that can be
represented by physical values in our own
universe, how do you answer
the question "what would happen if the universe
grew by one planck
unit"? By your reasoning, the answer has no
meaning, because there is
no way to represent the size of a universe bigger
than its current size. |
|
| |
Joking aside, I honestly don't see your point or
aim in all of this. |
|
| |
[Edit - having re-read the idea, I see that your
point is to do away with the 'meaningless'
precision imposed by those naughty
mathematicians. Well, OK, as long as the point is
silly, the argument is irrelevant. You can have
your Pi, me and the mathematicians will make do
with Bob, and everyone's happy.] |
|
| |
I can see a point to using an even lower precision
value of pi to save computing time when accuracy is
less important. For instance, pi needn't have even
five decimal places of accuracy to enable a circle to
be drawn on a monitor. I'm not sure it would save
much time though, because i imagine all that's
worked out in advance or the algorithms for dealing
with trig or whatever don't even use pi or would deal
just as fast with a higher-precision version. |
|
| |
Actually, Parmenides tried to convey that what we live in and what we are a part of is not a physical reality as such but rather more akin to a single thought. In my own folly I tried to convince others and defend my master with some sophistry of my own. |
|
| |
So please, in discussing this, fall back on the teachings of Parmenides and do not blame him for my shortcommings (sp?). |
|
| |
Also it is often forgotten that my example of the turtoise and the hare is not the only one. |
|
| |
Now, even though I will not claim to understand all the science that came after me, it seems to me that one cannot move any faster(or slower) than one planck length in one planck time unit and therefore different speeds are not at all possible. |
|
| |
I state that that none of the above explains accelleration and that nobody has ever explained or calculated it to a point beyond discussion. |
|
| |
Clearly people attribute some magical propensities to what happens within the planck measure of time and space that enables movement of matter, the existence of which depends on the discovery of some definite particle at cern. |
|
| |
So, and here you see I did not veer of topic, it is clear that Bob is what(among other things) defines what the universe is and those of us that live in the real world, in between philosophical musings, are great fans of Pi. |
|
| |
//At least if you allow for the transcendental nature of Pi, you don't need to worry about precision until you are
finally sure you need an answer that you are going to use for practical purposes, at which you are given the *choice*
of deciding at what level of precision you *want* to use.// |
|
| |
I think you misunderstand. I'm not proposing that anyone use some arbitrary, but inaccurate value of pi. I'm saying
that there is clearly some value pi that, in our universe, is accurate enough to calculate any possible value you
might need it for within the margin of error of a Planck length, which is to say no error at all. Beyond that level of
precision, it is impossible to determine a difference in any calculation, and therefore that level of precision /is/ the
exact, rational value of pi in our universe. |
|
| |
//*Forcing* everyone to conform to some arbitrary value is not only wrong on a precision level, it also smacks of
authoritarianism, Nanny Statism and frankly suggests that you may be of a Stalinist bent.// |
|
| |
I ask you, which of us is the communist, the authoritarian? Is it I, who declares that we have no right to fiddle with
the natural order of things, no right to tell Nature that she is incorrect when she clearly teaches us that pi is
rational? Or is it you, who says “No! Nature is wrong! We must correct her according to our established theory!” If
anything, my outlook is far closer to a sort of mathematical Objectivism, where only hard evidence survives. You,
on the other hand, seem to prefer a centrally planned system of mathematics based on your preconceived notions of
what must be "right" for the people. Well, Atlas has finally shrugged, “comrade”! No longer will the great be
constrained by the small! |
|
| |
Well, I mean, I guess it will be, if the “great” is the circle the size of the universe and the “small” is the Planck
length, but you get my point. |
|
| |
//the Reduced Plank Constant, one of the constituents of the Plank Length, is itself defined by the value of Pi// |
|
| |
You have it backwards. The reduced Planck constant is just the Planck constant divided by 2*pi, which is the
standard method of converting between cycles per second and angular frequency (i.e. revolutions per second). The
Planck constant is not itself defined by pi. |
|
| |
//What's the point? What I mean is, suppose you convince everyone that Pi is 22/7 (or whatever), and that they
should use the name "Bob" for the irrational number that we used to call Pi - what happens next? How does it make
my day any better?// |
|
| |
I think the truth is purpose enough. |
|
| |
//I ask again - what's your chosen value for 1/3rd (which is rational but happens to be an infinite decimal), or the
square root of 2 (which is irrational), or e (which is, if I remember correctly, transcendental like Pi)// |
|
| |
Only pi is defined in terms of a real world measurement. As I've said before, irrational numbers in general are just
fine. |
|
| |
//Finally, if you are only going to allow numbers that can be represented by physical values in our own universe// |
|
| |
I'm not doing that. I'm just saying that, like it's meaningless to talk about a distance of half a Planck length, it's
meaningless to talk about the precision of pi past what is usable in our universe. It cannot be observed or tested,
nor can it be applied in any manner. So, if something cannot be seen, and cannot be felt, and cannot possibly
affect anything in any way, in what manner does it exist? That sounds to me like a question for a theologian or a
philosopher, not a scientist. Science is not equipped to deal with things that cannot be tested experientially, and
something with no possible form or presence that has no possible effect on anything cannot be tested, and, as far as
science is concerned, does not exist. |
|
| |
I did a little bit of research on this, and believe it or not I'm not the first person to come up with this theory that pi
is rational. As I understand the argument, there are really two values for pi: the calculated value, and the measured
value as I've described it. In our universe, pi has the measured value, which is rational but constantly changing with
the size of the universe. The measured value of pi is always slightly smaller than the calculated value. As the size
of the universe approaches infinity, the measured value approaches the calculated value. Thus, the calculated
value is really a theoretical value describing what pi /should/ be, and the measured value describes what it actually
is. I don't really think the reasoning is the sticking point here, and we seem to agree (at least somewhat) on that. |
|
| |
So what's the idea then? It's really more a point of principle, since we have no way of knowing precisely what the
value of measured pi is at any given time. Everyone would just have to use an approximation as they do now. But
by teaching that even though pi is theoretically irrational, in our universe it must be rational (and here's a quick
explanation why), we tie mathematics and physics together, make a point about the inadequacy of mathematics
alone to describe our universe, and simultaneously introduce the intriguing concept that what we think of as an
analog world is, in fact, really more digital in nature than we realize. It's a subtle point, I guess, and one that will
probably be lost on most students. But I wish the point had been made to me when I was being taught about pi. To
me, the concept of our universe being divided into discrete quanta is fascinating, and it might have sparked an
interest in further exploration of the sciences that could have changed the entire trajectory of my academic career. |
|
| |
Well, probably not, but I have to blame my miserable grades on /something/, and the Newtonian view of the
universe seems as good a scapegoat as any. |
|
| |
//Only pi is defined in terms of a real world
measurement.// |
|
| |
Actually, modern definitions of Pi are in terms of
cosines, or as the limit of a summed series, rather
than in terms of a circle. |
|
| |
Also, sqrt(2) is definable as the length of the
diagonal of the unit square. |
|
| |
In other words, Pi is no more "real world" than
sqrt(2) or, for that matter, e. |
|
| |
Am I the only one whose goat is thoroughly got by e
not being rational? |
|
| |
Well, of course e is irrational. You'd have to be some sort
of lunatic to argue otherwise. |
|
| |
It just seems like if e is irrational our math must be
wrong. I can't explain it any better than that |
|
| |
Don't worry about it, [Voice]. Someone recently
told me I'm not allowed to imagine half of an
imaginary distance, where both of them are far
too small to ever see. |
|
| |
Apparently we're allowed imagine something
0.000 000 000 000 000 000 000 000 000 000 000 001
616 199
metres long, but not something
0.000 000 000 000 000 000 000 000 000 000 000 000
808 099 500
metres long. |
|
| |
Irrationality is all around us, it seems. |
|
| |
//it's meaningless to talk about a distance of half a
Planck length// It is no less meaningless to talk
about a distance of one Planck length. That you can
have two points or particles exactly one Planck
length apart, but no closer, is utter nonsense. The
Planck length is not a cosmic Snap To Grid. |
|
| |
So, when objects move through space, do they do it by Plank length stairsteps, or is that only a limitation when applied to measuring matter? Forgive me, I'm far out of my architect's ruler comfort zone. |
|
| |
All of which reminds me of another point. If
distance is quantized into Planck lengths, and if I
make a square ten Planks on a side, what's its
diagonal? |
|
| |
Sometimes you meet a maths nut who's as thick as
two Plancks. |
|
| |
<considers entering into a round of physicist puns> |
|
| |
<decides that he'd rutherford a river of molten lava
than sink so low> |
|
| |
I hadron, I lepton him. He certainly has a fermion him. Meson a
new drug, now I wear a pion my head. |
|
| |
//if I make a square ten Planks on a side, what's its diagonal?// |
|
| |
14 plancks and a "titch"(C) |
|
| |
Unless of course, you really meant 'planks' and not 'plancks' in which case I withdraw the above anno. |
|
| |
//Meson a new drug// what happened to your old
trino? And is it true you can no longer get an
electron? |
|
| |
Physics gives me a hadron. |
|
| |
Dyslexia does the same, to me. |
|
| |
//Physics gives me a hadron.// - ha! [marked-for-tagline] |
|
| |
//So, when objects move through space, do they
do it by Plank length stairsteps, or is that only a
limitation when applied to measuring matter? //
Closer to the second one. Stuff does not happen
in Planck lengths or integer multiples thereof. |
|
| |
I think a close analogy is the lower wavelength
limit for water surface waves, where viscosity
overwhelmes inertia. It becomes increasingly
difficult to measure or produce waves as you
approach that limit, and impossible *at or below*
that limit, but it would be silly to speak of water
waves as propagating in steps of that size, and the
same holds for the Planck length. |
|
| |
Similarly, the speed of light could loosely be
described as the upper limit for the velocity of
massive particles, but it is mistake to think of
them as travelling at that velocity. |
|
| |
(Disclaimer: I'm not a proper physicist, and will
gladly be proved wrong if anyone really knows
better) |
|
| |
//If distance is quantized into Planck lengths, and
if I make a square ten Planks on a side, what's its
diagonal?// AFAIK, distance is not quantised into
Planck lengths. It is not possible to construct or
measure such a square, so the question is
meaningless. Or, to put it the other way, your
question is a reductio ad absurdum argument that
length is not so quantised. |
|
| |
It has been thought of doing this before. [link] |
|
| |
I say we round pi to the nearest 10. |
|
| |
You know what I don't get, how can pi not be a rational number? A rational number is a number that can be expressed as a ratio of two integers. |
|
| |
And pi is the ratio of the circumference to the diameter. So it is inherently a ratio. It's just not a ratio that can be expressed in terms of integers... So wait, does that put it outside the realm of rational numbers? |
|
| |
Maybe there is a pair of integers out there that represent pi exactly. |
|
| |
If we are going to define Pi as the ratio of the circumfrence of a universe sized circle and the diameter of the universe, then we run into a problem. |
|
| |
If the planck length is the smallest possible unit, then the universe is not circular, but an n-gon figure with planck length edges. As such, it's circumfrence is n*Lp. Therefore under rational calculations (per YTK) Pi drops out completely, so there is no need to rationalize it. |
|
| |
Stuff and nonsense. 1/7 can indeed be represented
exactly on a computer. To pick just one example, the
(superb) calculator program Pari stores and represents
rational numbers, including 1/7, exactly. Pi, on the other
hand, cannot be represented exactly by any finite means
- the definition enables arbitrarily close approximations
to be found, but not an exact value. |
|
| |
Again, my readiness to assent is incomplete. For
instance, the exact, irrational value (1/7)^(1/2) can
be correctly represented and manipulated with no loss
of precision (until, as you say, you choose to display a
decimal or other approximation). But Pi cannot be
exactly represented, in a computer or otherwise, by a
finite set of arithmetic operations on integers. Hence
Pari, etc. simply store Pi in symbolic form, and
calculate an approximate representation when asked
to. |
|
| |
(Many mainstream programs, such as Microsoft Excel,
insist on representing everything as floating point
numbers, but they're simply mathematically naive) |
|
| |
Regardless of how fractions and pi are represented
by software, there is a fundamental difference. |
|
| |
One seventh can be precisely represented by
"1/7". In base 7 it would be "0.1". Conversely, one
tenth cannot be conveniently represented in base
7, but can be represented as "1/10", or "0.1" which
is the same thing. |
|
| |
Pi, on the other hand, cannot be represented by
any ratio of two numbers. Nor can it be
represented by a decimal in any base (except in
base pi, obviously). |
|
| |
Pi, to my mind, is wonderful and mysterious - a
very simple number which cannot be represented
by any ratio in any base - it's sort of kind of like
the numeric equivalent of a fractal. |
|
| |
And Pi is transcendental, not just irrational. In
other words, it cannot be represented by any
equation. (Root 2, as a counterexample, is
irrational but can be represented as the solution
to the equation X^2=2.) |
|
| |
//And Pi is transcendental, not just irrational. In other
words, it cannot be represented by any equation.// |
|
| |
x = lim n->inf (2^(4n+1)*n!^4) / ((2n+1)(2n)!^2) |
|
| |
My apologies - it is not the solution to any
polynomial equation. |
|
| |
Thanks for the heads-up, [gnome], but I'm not
interested. |
|
| |
Cool. I hadn't seen this before. |
|
| |
...So who's to say that a circle the size of the universe is the most precision that Pi might be needed for? It's easy to come up with more complex geometry that would need further precision, ie what about if you tried to calculate the strand length of a helix of pitch and diameter = 1 metre, who's axis is concentric around said universe-sized cirlce. There's another couple of orders of magnitude of accuracy you'll need to get it right within one planck length. What about if you wanted to calculate the volume of the sphere created by rotating this universe-circle, down to cubic planck-lengths. More precision required. |
|
| |
It's not really about the size of the universe, it's about the most complex problem you're trying to solve, which isn't even arbitrary - it's unknowable. |
|
| |
It could be argued that there is a "fundamental" valule for Pi. That is by using up the entire mass of the universe and storing the value of pi in every bit of information available. |
|
| |
...But then it depends on how you write down the number, and if you use any compression algorithms. |
|
| |
....So this whole debate reduces to the absurd. |
|
| |
Interesting read, though. |
|
| |
Is anyone who is obsessed with Pi known as a Pivert? |
|
| |
No, they're usually just called "fatty" ... |
|
| |
Given that the universe is turning out to be made of
smaller and smaller bits, the precision of pi needs to
get better and better. |
|
| |
Strings are meant to wibble on scales far, far below
Planck lengths, shirley? |
|
| |
And then there are all those other dimensions which
are wrapped up even smaller, and will need their
own pis, which will need to be related to our pi. |
|
| |
Insteady of being called tiny, they'll be called piny. |
|
| |
//Strings are meant to wibble on scales far, far
below Planck lengths,
shirley?// |
|
| |
I acknowledge that I understand very little about
string theory
(although most scientists who study it would probably
say the same
thing). Nevertheless, it seems you have touched on a
paradox here. I
would posit that there are only three possibilities:
1) If the Planck length is indeed the smallest possible
unit of
measurement, the requirement for strings to vibrate
on scales smaller
than that is a reductio ad absurdum of string theory.
2) Or vice versa.
3) Our understanding of both the Planck length and
string theory is
incomplete, such that, like the “circles within
circles” model of
Ptolemaic astronomy, both phenomena are accurate
representations of
the Universe when examined out of context, but
taken together can
only be explained by some other phenomenon (yet
unrealized by us)
that neatly accounts for both. Like the Ptolemaic
model, neither one is
“wrong”—just not the easiest depiction to
comprehend.
//Given that the universe is turning out to be made
of smaller and
smaller bits, the precision of pi needs to get better
and better.// |
|
| |
So what about this: Rather than trying to find a
fixed, universal value
of pi, we can theorize a “local pi” value that
represents the greatest
possible precision that can be attained for the
current calculation, at a
given point in the Universe, under a specified set of
circumstances. We
needn't know exactly what the value of “local pi” is
at any given time
(that's an engineering problem), but knowing that
there is some such
value gives us a precisely defined target to reach in
terms of accuracy.
We may not know exactly what that target is, but we
do know that we
can, at least theoretically, reach perfection. |
|
| |
Thing is, all of these arguments seem to revolve
around the idea that pi is the ratio of some circle's
circumference to its diameter. |
|
| |
It isn't. Pi is many things, including the limit
value of any of several different series which
involve only integers (and powers thereof). What
is remarkable and beautiful is that elementary
operations on integers can give you such a
beautiful number. |
|
| |
So, once again, I fail to see the point of this
proposal to rationalize pi. To begin with, if you
do that, then a whole bunch of other, much more
elementary things come unglued. |
|
| |
For example, if pi can be expressed as a ratio, you
can prove that all the integers are equal to one
another. Yet integers are useful things used by
real people to do helpful things with. |
|
| |
So, help me out here, and explain to me the point
of this idea. |
|
| |
//Thing is, all of these arguments seem to revolve around the idea that
pi is the ratio of some circle's circumference to its diameter.// |
|
| |
Quite the opposite. In fact, your argument seems to be based on
precisely that—that there is some circle whose ratio divided by its
diameter requires an infinite degree of precision of pi to precisely
express. I am saying that there is not, and cannot be such a circle in our
universe. |
|
| |
//Pi is many things, including the limit value of any of several different
series which involve only integers (and powers thereof). What is
remarkable and beautiful is that elementary operations on integers can
give you such a beautiful number.// |
|
| |
Or perhaps those things all happen to sum up to an irrational number
that happens to also coincide with the theoretical limit of what pi in our
universe would be if not for the existence of quantum phenomena. The
fact that A -> B, and C -> B, does not establish that A = C, no matter
how similar A and C may outwardly seem. |
|
| |
//For example, if pi can be expressed as a ratio, you can prove that all
the integers are equal to one another.// |
|
| |
On a quantum scale, all integers *are* equal to one another, in the
sense that an equation can be considered where the solution is the
superposition of all integers. Isn't this simply an expression of the many-worlds interpretation?
Set
up some experiment where the outcome is
dependent on a quantum phenomenon, and the range is the set of all
integers (or even all real numbers). Before the system is observed and a
result determined, there exists a single Universe where the solution
consists of every number superimposed; after it is observed, the
Universe splits off into an infinite number of equally probable Universes,
wherein the solution is some different number. |
|
| |
Can any of those Universes be said to be the “correct” one? None more
so than any other. Therefore, since all solutions are equally correct, all
numbers (and thus all integers) must be equal to each other. Q.E.D. |
|
| |
Of course, it would be a fallacy of logic to use this to prove that pi
must indeed be rational, unless there exists a biconditional relationship
between the rationality of pi and the equivalence of all integers (i.e., pi
is rational if and only if all integers are equal to each other). But the
equivalence of all integers would nevertheless would tend to strongly
suggest that it is indeed the case that pi is rational, or at least that it's
/not/ impossible for it to be so. |
|
| |
// In fact, your argument seems to be based on
precisely that—that there is some circle whose
ratio divided by its diameter requires an infinite
degree of precision of pi to precisely express. // |
|
| |
No, I'm not. I don't care whether pi represents a
circle or not. |
|
| |
//On a quantum scale, all integers *are* equal to
one another// Well, that's a silly thing to say.
Still, at least your grasp of quantum mechanics is
on a par with that of mathematics. |
|
| |
However, setting such points aside, answer me
this. Let us imagine that pi is declared to be only
59 (or only 127, or only 8) digits long. How is my
life improved? |
|
| |
You're of course asked to memorize all of them in primary school, which improves your memory. |
|
| |
Me, I've memorized them to an indeterminate number of digits. I just don't bother putting them into the correct order or worrying about repeats. |
|
| |
I know this is from 2012 spilling over to 2013 and now is
3.1415 years later, but just wanted to note a few
interesting
things: |
|
| |
1. Rabbi Eliyahu of Vilnius noticed 250 years ago something
interesting: |
|
| |
The verse reads (my literal translation): And he made the
'sea' by casting it ten arm from his lip to his lip around, five
in arm his standing, and his line thirty in arm will round it
around. |
|
| |
Hebrew is written without vowels. |
|
| |
Rabbi Eliyahu noticed that the word QaWo' meaning 'and-
his-
line' has a written version QF read today as Kav and in
ancient times as Kow and there's a read version QFE read
today as Kavvoe and in ancient times: Kaawoe. |
|
| |
The numeric values of these Hebrew letters are 5 for the E,
6
for the Wow and 100 for the Quf. |
|
| |
(The first 9 letters are with values of 1-9,
the next ten are with values of 10-90,
and finally the last four are with values of 100-400). |
|
| |
So (QF / QFE) x 30/10 = 106/111 x 3 = 3.141509 |
|
| |
2. Insects explicitly mentioned in the bible and which we
can be sure we understand the terms correctly are ants,
locust, bees, wasps, lice and grasshoppers. |
|
| |
But then there are terms that are NOT clear at all:
There are three similar sounding words (in ancient Hebrew
pronunciation, which can be deduced in many ways)
sometimes translated as dealing with insects: Remmess
Shekqess and Sherress. (These are pronouced in modern
Hebrew a bit differently). |
|
| |
And as a side remark: Just to mention that almost all nouns
in Hebrew can be used as verbs. So people can ride a horse
by horsing it, and wear a shoe by shoing it. In the same way
the Remmess can be Romssed (whatever those terms
mean). |
|
| |
Shekqess undoubtedly means disgust (notice how the
English
and Hebrew words sound alike, and in modern Hebrew even
more so: Shecketz). This is deduced from the many
mentions
of the word in that context, and its continued similar use in
other Semitic languages like Arabic. |
|
| |
The Torah tells us to be disgusted by these Shequess
animals (whatever they are) by using the verb Shaquess.
Something like: "Be disgusted of the dis-guests." |
|
| |
Sherress means either to spawn - to give birth to many live
descendants or to lie in one place (as crocodiles do). |
|
| |
The exact translation of Remmess - is not clear. It could
mean 'slithery things'. The similar noun Romess means
sliding around, or it could also mean 'trampling' things,
squatting down, hiding things, or squashing them. It is
mentioned many times as a term for all those that living
beings inside the ground or close to it. |
|
| |
b. The verse reads: "All Sherress of flight that walks on
four:
Shekqess he for (plural) you." the next verses discuss the
locust, so the claim that Leviticus 11:20 is translated "All
winged insects that walk upon all fours are detestable to
you"
is probably correct (although in any case there's no "all" in
the verse). |
|
| |
The Jewish traditional (Talmudic) explanation prohibits any
flying animal with four OR MORE legs... (so that includes
bats
on the one hand, and ants on the other) |
|
| |
But it doesn't mean that all winged insects walk on four. It
is
a list of types of insects that are allowed or denied. |
|
| |
The next verse reads (my own literal translation): |
|
| |
But this you will eat of all the Sherress of flight that walk
on
four:
Those that (written version: Loa - meaning: no,
read version: Loe - meaning: him) bending-knees above the
legs to jump upon with them on the faces of earth, these of
them (you) eat: The locust to its kind, the Salaam to its
kind,
the Hargoll to its kind, the Haggav to its kind. |
|
| |
Haggav is probably a grasshopper since it looks like people
("There was the land of giants, and we felt like Haggavs in
our own eyes, and so we were in their eyes"). |
|
| |
Hargol has something to do with a large foot. Is it the
praying
mantis? In modern Hebrew the mantis is a Gamal Shlomo -
King Solomon's camel. - sounds similar... |
|
| |
Salaam may have something to do with a rock Sella or with
a
basket Sall. |
|
| |
During Passover in 2013 Adam Matan took pictures and
videos
of the locust infestation in southern Israel including close
ups
of individual locusts on the ground. They were ready to
leap,
and stood and walked on their four legs... |
|
| |
So: The verses may be allowing insects that walk on six
such
as ants and dragonflies, and denying permission to eat only
those that "walk on four", such as most Arcididae (locust
grasshoppers and crickets) which have four walking legs and
two jumping legs, and which on the other hand usually
stand
on the four hind legs holding the two front legs out for
climbing and grabbing, bees which have four long back legs
and two seemingly "hands", and denying the eating of the
praying mantis which walks on four. Or it may be a list of
animals that usually stand on four (although Holech means
walk not stand). It may be that the large skipping legs were
not considered "walking legs" - although all giant
grasshoppers clearly walk with all six. |
|
| |
Enough apologetics for today. |
|
| |
circumference / diameter. That's a ratio. |
|
| |
I'm not convinced dragonflies can walk, although presumably their
nymphs can. What I came here to say, though, is that in practical
terms pi may still need to be considered irrational because there is a
lot of time and I can imagine, for example, two black holes orbiting a
common centre getting out of kilter as the kalpas pass because
people use pi to calculate their period, so basically you never know
when it might come in handy even if it takes a Graham's Number of
years to do so. Therefore even physically it has to remain irrational to
ensure its utility. |
|
| |
Looks like the visible universe is something like 10^70 Planck lengths across. So does that suggest that this idea proposes standardising pi(approx) for daily use, to 10^70 decimal places? |
|
| |
Given 10ˆ6 characters in a normal printed book, and 10ˆ6 books in a modest-sized private library, and 10ˆ9 households in the world, and 10ˆ3 exoplanets so far discovered, that takes us to 10ˆ24 digits. About one third of the way there! |
|
| |
I guess we need to think about using extra-thin paper and very small print. Shouldn't be too much of a problem. |
|
| |
Looks like current world data strage is about 10ˆ21 bytes. So we will be stuck with paper calculations for a few years yet while we build the tech to handle this number electronically. |
|
| |
//So does that suggest that this idea proposes
standardising pi(approx) for daily use, to 10^70
decimal places?// |
|
| |
Uh, no. That would be 70 decimal places. |
|
| |
You would only need 10^70 decimal places if the
universe were 10^(10^70) Planck units across. |
|
| |
Margin of error, old boy. Better too big than too small. |
|
| |
The Planck time needs to be taken into consideration too
though. Supposedly the stelliferous era will last a total
of 10^14 years. Given that the Planck time is of the
order of 10^43 seconds and the length of the stelliferous
era alone is something like 3 x 10^21 seconds, that makes
sixty-four decimal places advisable in terms of the
Universe as we know it, in order to take account of, for
example, two identical iron lumps of equal mass orbiting
a metre from each other. After the stars have gone out,
and after stars created by collisions between red dwarfs
(sp?) have also gone out, there will be occasional leptons
circling at distances of gigaparsecs from each other very
slowly. These will be interacting very slightly
electromagnetically and gravitationally, and
consequently will be describing conic section trajectories
related to the value of pi. If a pair is orbiting in a near-
circle, after 10^70 orbits or so, a discrepancy will occur
between the relationship between the velocity of the
particles and the radius of the orbits predicted by a
rational approximation of pi and that predicted by pi
itself, given a value of seventy decimal places. |
|
| |
Even in the case of something happening in the
stelliferous era, the Planck length exponent and the
Planck time exponent would need to be added together
to ensure that such a discrepancy could not take place in
the observable Universe during this time, so that's
already well over a hundred decimal places required. |
|
| |
Then there's the Nyquist-Shannon sampling theorem to be
taken into consideration, which I feel sure is relevant but
can't quite work out how, but it would probably only add
a few places - sorry to be vague but I can't really grasp
what it means. I imagine some error would accumulate if
this were not taken into consideration. |
|
| |
Regarding the Bible, it surely just comes down to the fact
that Biblical Hebrew lacks a method of expressing that
degree of mathematical precision, doesn't it? |
|
| |
Also, let's suppose I want to fill the universe with
string, the string being one Planck unit in diameter. I
do this by starting with an atom in the middle of the
universe, and wrapping the string around it
repeatedly to build up a ball, until I've filled the
universe. I'm going to need all the digits I can
handle, to work out the length of string I need. |
|
| |
I disagree, in theory at 1p scale everything is minecraft
rules. Your string is cubic, and filling space is cubic and
going diagonally is impossible. Your ball of string will have
to fill the space in a cubic fashion; L x W x H. Or put better,
it will have to be as long as the total resolution of the
universe. |
|
| |
I'm glad that's settled. I'll have a slice. |
|
| |
This discussion is irrational |
|
| |
// The Bible also claims insects have four legs// |
|
| |
Are you sure you're not thinking of Aristotle, [xixly]? |
|
| |
I don't think anyone has ever claimed that Aristotle had four legs. Even Aristotle himself made no such claim, and he was notoriously unreliable. |
|
| |
Here: No Aristotle has 3 legs. All Aristotles have 2 legs
more than no Aristotle. Therefore all Aristotles have 5
legs. And if they have 5, then they also have 4. |
|
| |
Better than two heads, and quite useful for smacking when
someone thinks that pi is only for measuring things. |
|
| |
//at 1p scale everything is minecraft rules. Your string is cubic// |
|
| |
Couldn't everything at that scale be packed into a stack of offset
hex grids, like certain kinds of atomic lattice? On the one hand
you would have some trouble with orthogonality, but on the other
hand you would have room for slightly more quanta of everything,
by wasting less space in the quantum corners. |
|
| |
// Better than two heads // |
|
| |
"Foreheads good, two heads better" ? |
|
| |
George Orwell would be so proud ... |
|
| |