Pi and Units
Sean B. Palmer
that a line doesn't have a square root. You have to break a line up
into equal length segments, let's call them units, in order for it to
have a square root.
So think of a line that is broken into 25 units. The square root of
that is 5, which is one fifth of that line. But if the line were
broken up into 9 units, then the square root would be three. That's a
third of the line, so you can see that the square root of the line
depends on how many units you break it into.
The reason behind this is something to do with packing. You can pack a
25 unit line into a spiral square shape, something that I took fun in
working out one day. But you can't pack a 26 unit line into a square
evenly, you get this bit left over. Anyway, the point is that the line
without the segments doesn't have anything to do with numbers.
The other day I learned another surprising thing, though this time I
didn't work it out myself. Richard Feynman had said somewhere that
whenever you find pi, there's a circle involved. What a great
observation! He found this out because he was obsessed with pi and
went looking for it everywhere.
There is one place though that he said he found pi and couldn't work
out where the circle was. This was in the formula for the frequency of
a resonant circuit, which is f = 1/2 pi LC, where L is the inductance
and C the capacitance. At the time of writing some talk that he was
giving about it, he still didn't know where the circle was.
Anyway, I got thinking about this and I realised that you can mix
these two observations together. The circumference of a circle only
comes out to be pi when you use the diameter of the circle as the base
unit. Otherwise a circle could be 5 units round, or 18, or whatever
you like.
One of the big things that I don't get about pi is why it's a little
more than three. If you got me to look at a circle and guess how many
units round the thing is when you use the diameter, I'd probably say a
bit less than three if anything. But the fact that it's not a whole
number seems a bit weird.
My guess is that it has something to do with a similar kind of packing
to the square root arrangement. In other words I think there must be
some kind of packing arrangement, though it may be more complex than
that, where like the spiral square made from a 26 unit line, you get
this bit left over.
What you could possibly do to figure this out is to start off with
another shape like a triangle, especially considering the special
relationship that circles and equilateral triangles have,[1] or a
square, and gradually put more kinks in it until it turned into a
circle. That way you'd see how the weird curve shape that you have
going round a circle's segments is made up of all these little steps.
That might not work, though, for the same reason that a hypotenuse is
always the same length less than a stepped hypotenuse no matter how
many steps you put in the stepped hypotenuse. When you decrease the
size of the steps you have to do it in the same proportion to how many
steps you need to add. Similarly, perhaps you can't find out how a
circle works using lots of little steps, because you need that smooth
curve?
[1] http://groups.google.com/group/whits/t/1324122c9785c3ce
--
Sean B. Palmer, http://inamidst.com/sbp/
Dave Pawson
> One of the most interesting things that I discovered about numbers is
> that a line doesn't have a square root.
Different ... somethings Sean? Square roots are defined for numbers.
A line isn't in the domain of numbers, just has a property which
may be represented by a number and unit of length?
> There is one place though that he said he found pi and couldn't work
> out where the circle was. This was in the formula for the frequency of
> a resonant circuit, which is f = 1/2 pi LC, where L is the inductance
> and C the capacitance. At the time of writing some talk that he was
> giving about it, he still didn't know where the circle was.
1/2* pi * sqrt(LC) I think, from ... long time ago.
>
> Anyway, I got thinking about this and I realised that you can mix
> these two observations together. The circumference of a circle only
> comes out to be pi when you use the diameter of the circle as the base
> unit. Otherwise a circle could be 5 units round, or 18, or whatever
> you like.
but then surely pi has a representation in your 'units'? perhaps not....
No, 'cos pi is a ratio I think.
>
> One of the big things that I don't get about pi is why it's a little
> more than three. If you got me to look at a circle and guess how many
> units round the thing is when you use the diameter, I'd probably say a
> bit less than three if anything. But the fact that it's not a whole
> number seems a bit weird.
Not weird, just... a bit of a giggle, as is a large part of maths!
Often makes me smile!!
>
> My guess is that it has something to do with a similar kind of packing
> to the square root arrangement. In other words I think there must be
> some kind of packing arrangement, though it may be more complex than
> that, where like the spiral square made from a 26 unit line, you get
> this bit left over.
How does the 'little bit' relate to the line length for different numbers?
Any patterns there?
Hey, you have been having fun Sean!!!
regards
--
Dave Pawson
XSLT XSL-FO FAQ.
Docbook FAQ.
http://www.dpawson.co.uk
Sean B. Palmer
> roots are defined for numbers.
Yeah, though in this case it's not so interesting to me what the right
answer is, as much as why I picked a wrong answer.
Why should I be under the delusion that lines could have square roots?
Probably because we're so used to breaking them up into units!
At some point, I didn't even realise at first that just because a line
can be measured into units, doesn't mean they have the same properties
when they are not broken into units.
> 1/2* pi * sqrt(LC) I think, from ... long time ago.
http://en.wikipedia.org/wiki/LC_circuit#Resonance_effect
According to that, you're right. The document that I copied it from
presumably lost the square root due to not being able to represent it,
perhaps a copy and paste of a formula which had an image for the
square root sign with no alt text. There was no square root in what I
copied from.
> but then surely pi has a representation in your 'units'?
Yeah, but then the diameter will have a representation in the units
too. So for example if we say that pi is five, that we break the
circumference of our circle into five units. Then the diameter would
be 5/3.1415926 etc., which is about 1.59154943.
So as you say, the ratio is more important? But I think the point is,
however you fiddle the units, you're just going to shift the pi about
somewhere. So to work out pi from the diameter is to do it one way,
and pi is then 3.1415926 etc. But if pi were five, well then to get
the diameter we'd have to use 3.1415926 etc. in that calculation.
> Not weird, just... a bit of a giggle
At the moment it seems weird to me, just like it seemed weird that you
should be able to pack exactly six circles around another circle. Why
six? That seemed so weird. But when I figured it out, it was no longer
weird. Surprisingly, it was beautiful, and exciting, and... well, a
bit of a giggle like you say!
But I wouldn't have bothered to try to figure it out if it wasn't
weird first of all, just like I probably wouldn't be bothering with
the weird value of pi. I was interested not least because Wittgenstein
says that pi is a construction. If it's a construction, I was
thinking, then how is it constructed?
Another way to look at the problem might be to figure out what kind of
non-euclidean curvature would be required to make pi three. A while
ago on Swhack I asked people what things would be like if pi were,
say, exactly 3. People started saying that it's a silly question,
impossible, a category error, won't teach you anything, but eventually
I learned that pi is exactly 3 in certain non-euclidean systems:
http://swhack.com/logs/2008-11-15#T10-26-33
So then the question became, can we imagine pi being 3 in a euclidean
system? Now *that* might be a silly question, a category error, like
people were saying, but we didn't get much further on that.
For the purposes of why pi is this strange non whole number, it might
as I say be interesting to consider how far you have to push the
centre of the circle away from the flat plane in order to make the
diameter fit into the circumference three times. But perhaps that's
just a fudge that doesn't tell you anything?
This way of doing maths, by the way, may be called Lockhartine
mathematics, after Lockhart's lament:
http://www.maa.org/devlin/LockhartsLament.pdf
> How does the 'little bit' relate to the line length for different numbers?
> Any patterns there?
If you use whole numbers, there are obviously not going to be any
patterns because pi repeats itself. But if you chose sqrt(2) or e or
something like that, then I wonder. Note that in hyperbolic geometry,
"because distances are measured differently, the points that are
equally far away from our point c still form a circle, but c is no
longer at what looks like its centre. The circumference of this
hyperbolic circle is proportional not to its radius, but to e **
radius, where e is the base of the natural logarithm and is roughly
equal to 2.718."
Sean B. Palmer
square numbers and unit packing with why pi is three and a bit, I was
unaware of the Basel problem.
http://en.wikipedia.org/wiki/Basel_problem
The Basel problem quite simply asks what is the sum of 1/1 + 1/4 + 1/9
+ 1/16 + 1/25 + 1/36 and so on through all the square numbers. Euler
discovered that the answer is pi squared over six.
What's even stranger is that the reciprocal of this, six over pi
squared, turns out to be the probability that any two randomly chosen
numbers are both prime. This seems to suggest a deep link between pi
and the distribution of prime numbers, according to Wikipedia.
Dave Pawson
>> [Dave Pawson wrote:] Different ... somethings Sean? Square
>> roots are defined for numbers.
>
> Yeah, though in this case it's not so interesting to me what the right
> answer is, as much as why I picked a wrong answer.
>
> Why should I be under the delusion that lines could have square roots?
> Probably because we're so used to breaking them up into units!
Sarky answser... 'cos you're not thinking like a mathematician?
More likely,
Because lines have lengths, hence length units, hence I
can take a root of this 'length'?
>
> At some point, I didn't even realise at first that just because a line
> can be measured into units, doesn't mean they have the same properties
> when they are not broken into units.
>> but then surely pi has a representation in your 'units'?
>
> Yeah, but then the diameter will have a representation in the units
> too. So for example if we say that pi is five, that we break the
> circumference of our circle into five units. Then the diameter would
> be 5/3.1415926 etc., which is about 1.59154943.
>
> So as you say, the ratio is more important? But I think the point is,
> however you fiddle the units, you're just going to shift the pi about
> somewhere. So to work out pi from the diameter is to do it one way,
> and pi is then 3.1415926 etc. But if pi were five, well then to get
> the diameter we'd have to use 3.1415926 etc. in that calculation.
When I started learning about electronics I was given a magic number,
the permittivity of free space as 8.854 x 10 - 12. I simply refused to
accept it, being given it as a blanket fact. Of course the lecturer couldn't
source it there and then and eventually I accepted it.
Perhaps it's just one of those Sean?
Bit hard to get past, but necessary?
>
>> Not weird, just... a bit of a giggle
>
> At the moment it seems weird to me, just like it seemed weird that you
> should be able to pack exactly six circles around another circle. Why
> six? That seemed so weird. But when I figured it out, it was no longer
> weird. Surprisingly, it was beautiful, and exciting, and... well, a
> bit of a giggle like you say!
YES (very loud shout). Maths is ... beautiful.
Or can be when well presented.
My best was with the OU. One piece of work on proofs took
me about 7 attempts, each 4 or 5 sheets long.. The end game?
1 = 1.
I jumped for joy when I got it and grinned for days.
My starting point was an equation that took 4 lines to write out.
>
> But I wouldn't have bothered to try to figure it out if it wasn't
> weird first of all, just like I probably wouldn't be bothering with
^^^^
Interesting? rather than weird?
> the weird value of pi. I was interested not least because Wittgenstein
> says that pi is a construction. If it's a construction, I was
> thinking, then how is it constructed?
By geometers?
>
> Another way to look at the problem might be to figure out what kind of
> non-euclidean curvature would be required to make pi three. A while
> ago on Swhack I asked people what things would be like if pi were,
> say, exactly 3. People started saying that it's a silly question,
> impossible, a category error, won't teach you anything, but eventually
> I learned that pi is exactly 3 in certain non-euclidean systems:
>
> http://swhack.com/logs/2008-11-15#T10-26-33
That's where I back off, to admire people that can grock n dimensions,
even dream of other geometries etc.
> For the purposes of why pi is this strange non whole number, it might
> as I say be interesting to consider how far you have to push the
> centre of the circle away from the flat plane in order to make the
> diameter fit into the circumference three times. But perhaps that's
> just a fudge that doesn't tell you anything?
No, it's playing with number systems, planes and geometry.
I immediately had a 3d image of the centre of the circle rising
and the circumference/diameter shrinking towards a ratio that
was wanted. Almost realistic.
>> How does the 'little bit' relate to the line length for different numbers?
>> Any patterns there?
>
> If you use whole numbers, there are obviously not going to be any
> patterns because pi repeats itself. But if you chose sqrt(2) or e or
> something like that, then I wonder. Note that in hyperbolic geometry,
> "because distances are measured differently, the points that are
> equally far away from our point c still form a circle, but c is no
> longer at what looks like its centre. The circumference of this
> hyperbolic circle is proportional not to its radius, but to e **
> radius, where e is the base of the natural logarithm and is roughly
> equal to 2.718."
Sigh. DP. Lost, but curious.
Dave Pawson
> When I sent my original message relating, loosely, the problem of
> square numbers and unit packing with why pi is three and a bit, I was
> unaware of the Basel problem.
>
> http://en.wikipedia.org/wiki/Basel_problem
>
> The Basel problem quite simply asks what is the sum of 1/1 + 1/4 + 1/9
> + 1/16 + 1/25 + 1/36 and so on through all the square numbers. Euler
> discovered that the answer is pi squared over six.
But what a brain to even start to look!!!!
Tends towards .... something, but then to ask
how is this number related to .... math constants!
that takes some special kind of grey matter.
>
> What's even stranger is that the reciprocal of this, six over pi
> squared, turns out to be the probability that any two randomly chosen
> numbers are both prime. This seems to suggest a deep link between pi
> and the distribution of prime numbers, according to Wikipedia.
Spooky rather than weird?
Things like that get me to thinking that Hawkins might just be right
to link the huge with the infinitesimally small.
Given the right maths, it all makes simple sense?
Jannis Andrija Schnitzer
> the permittivity of free space as 8.854 x 10 - 12. I simply refused to
> accept it, being given it as a blanket fact. Of course the lecturer couldn't
> source it there and then and eventually I accepted it.
> Perhaps it's just one of those Sean?
> Bit hard to get past, but necessary?
There *usually* is an explanation to these numbers. You might choose to accept them if understanding them is not necessary for your work, but I believe there'll be an explanation to be found at some point for every "magic number". As long as we still have unexplained constants in physics, we still have something that isn't understood. That's probably not different for maths, either. (As for your example, there are people thinking about it, observable at [1], for instance.)
[1] http://en.wikipedia.org/wiki/Permittivity#Quantum-mechanical_interpretation
Dave Pawson
Oh yes Jannis, I agree.
I was 17 at the time, a bit bolshie, bit rebellious and the lecturer
simply wasn't sure of himself, hence we had this 'incident'.
It just took me back that I was given this number without explanation.
I'm sure a bit of Googling would help, but I'm talking of .... 40
years pre Google?
Kevin Reid
> One of the most interesting things that I discovered about numbers is
> that a line doesn't have a square root. You have to break a line up
> into equal length segments, let's call them units, in order for it to
> have a square root.
>
> So think of a line that is broken into 25 units. The square root of
> that is 5, which is one fifth of that line. But if the line were
> broken up into 9 units, then the square root would be three. That's a
> third of the line, so you can see that the square root of the line
> depends on how many units you break it into.
(I'm going to assume you, or the reader, knows no more about the topic
than is clearly evident in the quoted text; even if you have some or
all of this knowledge, I think that some others might benefit from
this explanation. Should I offend, I will refrain from doing it again,
if so informed.)
Firstly, some established terminology. What is at issue is not “a
line”, but “a length”. Length is a _dimension_ (as are time, mass,
charge, etc.). Your “units” are _units_; a more typical example of
different units for the same dimension would be the foot and the yard.
In your particular case, we have some line (presumably the
International Prototype Line Segment kept in Paris), whose length can
be measured as either 25 alphas or 9 betas. That establishes an
equivalence: 25 alpha = 9 beta. Under the right formalism, this is a
mathematically true statement; we can think of “25 alpha”, “4.5
alpha”, or “1 alpha”, as a number-with-units-on, or as the product of
a number and an algebraic variable: 25 · alpha.
Everything we (may) know about basic algebra works for this. For
example, it is a common technique to verify the proper use of unit
conversions by algebra:
25 alpha = 9 beta
Divide both sides by "9 beta", and simplify fractions in the usual
way:
25 alpha 9 beta
-------- = ------
9 beta 9 beta
25 alpha 1
-------- = ---
9 beta 1
25 alpha
-------- = 1
9 beta
Therefore, every such “conversion factor” is equal to 1; multiplying
or dividing by it doesn't “change the value” but can be used as a
device for checking one's work while doing unit conversion.
Now that I've said a bit about the basics, let’s look at square roots.
What is the square root of a length?
√(length-of-line) = √(25 alpha)
Now, as long as we're not thinking of complex numbers, it is true that
√(ab) = √(a)√(b) (for example, √(9·25) = 15 = 3·5 = √(3)√(5)), and
equally true of our-numbers-with-units-on:
√(length-of-line) = √(25 alpha) = √(25)√(alpha) = 5 √(alpha)
But what, you ask, is the square root of alpha? What is the square
root of length? Well...don't worry about that. All that matters is
that it works perfectly fine as an intermediate result; as soon as you
calculate something meaningful using it, carrying through the units,
you'll find the result is correct and is the same (in actual value, as
opposed to units it's expressed in) no matter what units you started
with.
Let's do that again with beta:
√(length-of-line) = √(9 beta) = √(9)√(beta) = 3 √(beta)
So instead of having the answer that the square root of the length is
either 3 or 5, we have that it is either 3 √(beta) or 5 √(alpha). And
that makes all the difference: all we have to do is use our conversion
factor!
25 alpha
-------- = 1
9 beta
Take the square root of both sides of this equation (noting that the
square root of 1 is 1) and it's still true:
/ 25 alpha
/ -------- = 1
√ 9 beta
Given √(ab) = √(a)√(b) we can move the actual numbers outside of the
square root; we just have to take the square root of them. And
similarly, √(a/b) = √(a)/√(b).
5 / alpha
--- / ----- = 1
3 √ beta
5 √(alpha)
--- -------- = 1
3 √(beta)
(For those not familiar with this sort of thinking, this may look like
a whole mess of algebraic steps, and as I'm presenting it it is; but
it's possible to just see it all at once with a little practice. As
long as there's only multiplication, division, and roots/exponents,
you can rearrange the factors (refactoring, heh heh) these expressions
any way you like without worrying too much about the specific
procedure; all that matters is that numerators stay on top,
denominators stay on the bottom, and anything that eventually got its
square root (or square or cube...) taken still does.)
Since this value is still 1, we can multiply by this all we like.
Let's do that to one of our measurements:
5 √(alpha)
3 √(beta) · --- --------
3 √(beta)
Then rearrange some factors and cancel in fractions in the usual
fashion:
5 √(alpha) 5 √(alpha)
3 √(beta) · --- -------- = 3 · --- · √(beta) · -------- = 5
√(alpha)
3 √(beta) 3 √(beta)
There you go. 3 √(beta) = 5 √(alpha). The square root of (the length
of) your line is equal no matter what units you measure it in; you
just have to not throw out the units!
This system of using units with numbers is formally known as “quantity
calculus”, but is quite often presented as the way to work with
physical measurements (length, time, speed, mass, charge, etc.)
without giving it any specific name or considering it a distinct
mathematical system from the algebra one already knows. A related
field is “dimensional analysis”, which is where you look at how
quantities can be related based on what dimensions they have.
Oh, and π fails to have obvious useful results from this system,
because (if you consider it as about circles), it is a ratio of
lengths, and therefore the units and dimensions cancel; π is a
_dimensionless quantity_.
(Please do let me know whether this is or isn't helpful, clear,
interesting, etc. I don't do this sort of thing as much as it might
seem I think I do.)
Sean B. Palmer
> Length is a _dimension_ (as are time, mass, charge, etc.). Your
> “units” are _units_; a more typical example of different units for the
> same dimension would be the foot and the yard.
So you say that "25 alpha" is a length, and "alpha" is a unit...
> But what, you ask, is the square root of alpha? What is the square
> root of length? Well...don't worry about that.
...but now you say that alpha is not a unit, but length.
So why did you say "What is the square root of length?" rather than
"What is the square root of a unit of measurement?", or at least "What
is the square root of an algebraic variable?" even though in quantity
calculus the units are not algebraic variables?
My current opinion on why you may have said this is that since these
questions are hypothetical questions from the reader, you may be
thinking that they would now be asking what the square root of a
length is, despite the square root being taken of the unit of
measurement. But I thought I'd best check!
Dave Pawson
> √(length-of-line) = √(25 alpha) = √(25)√(alpha) = 5 √(alpha)
Weakness in logic here surely?
5 x 4 = 20
5 feet isn't a multiplication: It's a number and it's unit,
so the rule you gave above doesn't apply?
Kevin Reid
> On 19 October 2010 18:32, Kevin Reid <kpr...@switchb.org> wrote:
>
>> √(length-of-line) = √(25 alpha) = √(25)√(alpha) = 5
>> √(alpha)
>
> Weakness in logic here surely?
>
> 5 x 4 = 20
>
> 5 feet isn't a multiplication: It's a number and it's unit,
> so the rule you gave above doesn't apply?
The point of quantity calculus is that we declare that units may be
treated in such ways.
5 feet = 5 · (1 foot) = 5 · foot
If you like, think of units as algebraic variables, where obviously
√(25a) = 5√(a). Or not; whatever works better for you.
--
Kevin Reid <http://switchb.org/kpreid/>
Kevin Reid
On Oct 19, 2010, at 13:56, Sean B. Palmer wrote:
> Kevin Reid wrote:
>
>> Length is a _dimension_ (as are time, mass, charge, etc.). Your
>> “units” are _units_; a more typical example of different units
>> for the
>> same dimension would be the foot and the yard.
>
> So you say that "25 alpha" is a length, and "alpha" is a unit...
>
>> But what, you ask, is the square root of alpha? What is the square
>> root of length? Well...don't worry about that.
>
> ...but now you say that alpha is not a unit, but length.
Sorry, is that "but length is a unit" or that "alpha is length"?
> So why did you say "What is the square root of length?" rather than
> "What is the square root of a unit of measurement?", or at least "What
> is the square root of an algebraic variable?"
Well, the square root of an algebraic variable is a perfectly sensible
thing to think about. And so is the square root of a unit of
measurement, in some cases: "square centimeters" is a unit, and the
square root of that is obviously "centimeters".
> even though in quantity calculus the units are not algebraic
> variables?
Ah, but they might as well be. In my very short research (I learned
this stuff without exposure to any formal statement of it, as I
alluded to), I find it is often put as something like:
The value of a physical quantity can be expressed as the product
of a numerical value and a unit.
Note that this is not saying that a “physical quantity” is an
object which has a numerical value and a unit; it specifically is the
mathematical product of a value and a unit. Therefore we can apply all
rules for products, including √(a·b) = √(a)·√(b).
> My current opinion on why you may have said this is that since these
> questions are hypothetical questions from the reader, you may be
> thinking that they would now be asking what the square root of a
> length is, despite the square root being taken of the unit of
> measurement. But I thought I'd best check!
Indeed it was a hypothetical question. To put it differently:
"OK, I have a measurement of 5 square-root-of-inches", now what the
heck does square-root-of-inches *mean*?"
Sean B. Palmer
> Indeed it was a hypothetical question.
Okay, thanks!
What you say is very interesting, but I don't believe that it relates
to the packing problem. The introduction to my first message stated
that a line doesn't have a square root, but that when you break it
into units then obviously you can take a square root of the numerical
value part of the length of the line. But what's the point of this?
The point was that the relationship between the square root of the
numerical part and the line itself which is broken into these unit
segments, or "given a length" depending on your conceptualisation,
turns out geometrically to be a very interesting one. You can pack the
line into a square shape in order to geometrically find the square
root of the number of segments, or the numerical part of the length.
In other words, by twisting a segmented line into a square, you can
find the square root of the number of segments that you have without
having to know in advance even what number of segments that you have.
Note that if you use "given a length" as a conceptualisation, this is
not so obvious intuitively. But if you have nodes which can bend, and
unit length segments which can't, then you have the more or less
obvious conceptual ability to be able to do such things.
What you were saying is that there is a tool for working with lengths
where at some point you can to do the peculiar thing of taking the
square root of a unit of measurement. This is different from taking
the square root of a number, in our case the numeric value component
of a length, which is what I had to talk about for the purpose of my
original message.
So your contribution was very interesting, but as far as I can tell it
is only very loosely related. Or did you have some other point behind
mentioning quantity calculus that relates it more strongly to the
original post? I am assuming not because you said that I'm "thinking
of this wrong". If you look, however, at the development of my
original post, you'll see that the whole point of it was to try to
come up with a (probably) geometrical explanation of why pi is
three-and-a-bit, which was partially dependent on my explanation about
a geometric explanation of square numbers.
In fact, a very tantalising thing that happened subsequent to my
original post was finding out that due to the Basel problem, square
numbers and pi seem to be actually related on some level. I spent a
while tinkering with that, but I haven't really got anywhere with it.
If you'd like to consider further what I meant, perhaps for a bit more
insight into why I don't think quantity calculus is relevant, perhaps
you'd like to consider what form an elementary geometric proof of the
solution to the Basel problem would take?
Kevin Reid
> The point was that the relationship between the square root of the
> numerical part and the line itself which is broken into these unit
> segments, or "given a length" depending on your conceptualisation,
> turns out geometrically to be a very interesting one. You can pack the
> line into a square shape in order to geometrically find the square
> root of the number of segments, or the numerical part of the length.
>
> In other words, by twisting a segmented line into a square, you can
> find the square root of the number of segments that you have without
> having to know in advance even what number of segments that you have.
You're assuming a certain packing density widthwise of the lines;
specifically, that it is 1 segment wide. This makes sense in quantity
calculus:
Length · Length = Area
25 segments · 1 segment = 25 segment^2
√(25 segment^2) = 5 segment
So quantity calculus doesn't actually say you have to end up with the
square root of a length unit: properly stating your problem
straightforwardly (that is, without additional unit conversions) gives
the answer you're looking for.
> Note that if you use "given a length" as a conceptualisation, this is
> not so obvious intuitively. But if you have nodes which can bend, and
> unit length segments which can't, then you have the more or less
> obvious conceptual ability to be able to do such things.
True. In fact, quantity calculus is more often used with continuous
values, not discrete ones (counts).
> What you were saying is that there is a tool for working with lengths
> where at some point you can to do the peculiar thing of taking the
> square root of a unit of measurement. This is different from taking
> the square root of a number, in our case the numeric value component
> of a length, which is what I had to talk about for the purpose of my
> original message.
I was particularly objecting to the notion that (the length of) a line
doesn't have a square root, as your first sentence stated. The square
root of a length can be well-defined, and does not require "breaking
up the line into equal length segments"; there has to be a defined
unit length to measure and calculate with, but the line need not have
an integer multiple of that length.
I could also take issue with “the line without the segments doesn't
have anything to do with numbers”. It is true that there is no
particular real or integer number associated with a given line, but if
the space includes some form of length reference, we can certainly
associate the quantity <some constant>·<length of reference> with that
line, and the result does a pretty good job of being something you can
call a “number”, even if it's not in ℝ. That said, I suspect the
consensus of mathematicians is against calling this a “number”, so.
> So your contribution was very interesting, but as far as I can tell it
> is only very loosely related. Or did you have some other point behind
> mentioning quantity calculus that relates it more strongly to the
> original post? I am assuming not because you said that I'm "thinking
> of this wrong".
That was an exaggeration intended for humorous effect.
My point, I think (now), was that when thinking about things like
lengths I feel it is best to *start* by working in quantity calculus
rather than unadorned reals (or integers), at least until you obtain
something that is dimensionless (or has a dimension that is
particularly of interest, as in "5 segments" above).
> If you look, however, at the development of my
> original post, you'll see that the whole point of it was to try to
> come up with a (probably) geometrical explanation of why pi is
> three-and-a-bit, which was partially dependent on my explanation about
> a geometric explanation of square numbers.
Hm.
As I mention above, “packing” brings in the notion of area. For a
circle,
A = πr²
and that seems like a good place to start. Wikipedia mentions that
“π can be defined as the ratio of a circle's area A to the area of a
square whose side is equal to the radius r of the circle: π = A/r²”
and offers the diagram <http://en.wikipedia.org/wiki/File:Circle_Area.svg
>, showing a square (of area r²) covering one quarter of the circle.
(If the square were duplicated over all quadrants, then it would cover
more area than the circle; it would cover the area 4r², which is
obviously just a bit greater than πr², as it ought to be.)
We wish to consider the diameter, not the area; but these formulas can
be related by, for example, integral calculus. The area of a circle
can be seen as the region swept out by the line of a circle as its
radius (call it s) varies from 0 to r. That is:
A = ∫{0 to r} 2πs ds
= 2π ∫{0 to r} s ds
= 2π (1/2 r² - 1/2 0²)
= π r²
So these two formulas are “the same” occurrence of π. (For the
curious, the exact same thing works for spheres, and I have a stalled
project to write up the derivations of all of the formulas for area
and volume of circles and spheres solely in terms of integration in
polar coordinates. Which a Real Mathematician would probably tell me
is somewhere a circular (ahem) argument.)
Um, I was going somewhere with this, having to do with calculating the
area of a circle in terms of integration and how π arises in that, but
I'm failing to think efficiently about it and don't really want to
spend the additional time on it. Sorry.
> In fact, a very tantalising thing that happened subsequent to my
> original post was finding out that due to the Basel problem, square
> numbers and pi seem to be actually related on some level. I spent a
> while tinkering with that, but I haven't really got anywhere with it.
Related because of the n² in ∑{n=1 to ∞} 1/n²? I don't think
that's likely to be significant. Squaring occurs all over the place,
and infinite series can produce transcedental numbers. I see no
particular justification for it to be the *same* (except-for-squaring-
and-dividing-by-six) transcendental number though.
Here's another infinite series that might be of interest:
http://en.wikipedia.org/wiki/Leibniz_formula_for_pi
Dave Pawson
>
> "OK, I have a measurement of 5 square-root-of-inches", now what the heck
> does square-root-of-inches *mean*?"
There's a whole branch / area of mathematics about 'balancing'
units. Durned if I can think of its name though.
I tend to thing of the unit of measure as being a property of the number
or characteristic, it most certainly isn't another number.
HTH
Kevin Reid
On Oct 20, 2010, at 2:30, Dave Pawson wrote:
>> Indeed it was a hypothetical question. To put it differently:
>>
>> "OK, I have a measurement of 5 square-root-of-inches", now what the
>> heck
>> does square-root-of-inches *mean*?"
>
> There's a whole branch / area of mathematics about 'balancing'
> units. Durned if I can think of its name though.
Dimensional analysis.
Dave Pawson
>
> On Oct 20, 2010, at 2:30, Dave Pawson wrote:
>
>>> Indeed it was a hypothetical question. To put it differently:
>>>
>>> "OK, I have a measurement of 5 square-root-of-inches", now what the heck
>>> does square-root-of-inches *mean*?"
>>
>> There's a whole branch / area of mathematics about 'balancing'
>> units. Durned if I can think of its name though.
>
> Dimensional analysis.
You got it.
How come you're trying to take the sqrt of a dimension?
regards
Noah Slater
On 19 Oct 2010, at 15:44, Sean B. Palmer wrote:
> The Basel problem quite simply asks what is the sum of 1/1 + 1/4 + 1/9
> + 1/16 + 1/25 + 1/36 and so on through all the square numbers. Euler
> discovered that the answer is pi squared over six.
I have very little (useful or interesting stuff) to add to this discussion, except to say that particular mention reminded me of the Madhava–Leibniz series which I only learnt about a few weeks ago on some history programme about maths. What an elegant way of calculating pi! They never taught us anything like that at school. I was additionally re-assured this comment was relevant enough after finding how close the mentions of both things are on the Wikipedia article.
Dave Pawson
>
> On 19 Oct 2010, at 15:44, Sean B. Palmer wrote:
>
>> The Basel problem quite simply asks what is the sum of 1/1 + 1/4 + 1/9
>> + 1/16 + 1/25 + 1/36 and so on through all the square numbers. Euler
>> discovered that the answer is pi squared over six.
>
> I have very little (useful or interesting stuff) to add to this discussion, except to say that particular mention reminded me of the Madhava–Leibniz series which I only learnt about a few weeks ago on some history programme about maths. What an elegant way of calculating pi! They never taught us anything like that at school.
don't you think that is a real pity? I *think* even young children
would appreciate
that elegance? It's that sort of thing that makes me chuckle and enjoy maths.
Ulises
> would appreciate
> that elegance? It's that sort of thing that makes me chuckle and enjoy maths.
I would take it even further. I am listening to a series of podcasts
originally transmitted on Radio 4 about the history of mathematics (a
brief history of). I have to admit that I am finding it very very
enjoyable and I wish my math teachers (at all levels) had intertwined
the mathematical knowledge with bits and bobs of history.
There is a famous Argentinian (famous within Argentina and possible
some other places) called Adrian Paenza. He wrote a series of books
called "Mathematics, are you there?" which introduce mathematical
concepts through narrative and applications. They are very good. His
view is that mathematics are considered to be boring/dull mainly
because of how they are taught and I couldn't agree more.
U
Noah Slater
On 20 Oct 2010, at 23:52, Noah Slater wrote:
> I have very little (useful or interesting stuff) to add to this discussion, except to say that particular mention reminded me of the Madhava–Leibniz series which I only learnt about a few weeks ago on some history programme about maths. What an elegant way of calculating pi! They never taught us anything like that at school. I was additionally re-assured this comment was relevant enough after finding how close the mentions of both things are on the Wikipedia article.
I woke up with the idea of a Madhava-Leibniz spiral this morning.
The Madhava–Leibniz series looks like:
4/1
4/1 - 4/3
4/1 - 4/3 + 4/7
4/1 - 4/3 + 4/7 - 4/9
4/1 - 4/3 + 4/7 - 4/9 + 4/11
Each successive step is a closer approximation to pi. Notice how the values oscillate between overestimation and underestimation. I figured that you could plot these points on a graph, and get a spiral towards pi. I did a few checks, but can't find any examples of this on the web.
Jason Davies mocked up a rough approximation to one:
http://www.jasondavies.com/leibniz-spiral/
http://github.com/jasondavies/toys/blob/master/src/leibniz-spiral.html
I find it really fascinating that you can find pi by following a spiral that moves inwards from 4. That it should start at 4, being the number of sides a square has, and then move inwards at steps of fractional 4 — seems like it might be quite significant. But all of this is a little beyond me.
Dave Pawson
> I woke up with the idea of a Madhava-Leibniz spiral this morning.
As you do!!!
>
> The Madhava–Leibniz series looks like:
>
> 4/1
>
> 4/1 - 4/3
>
> 4/1 - 4/3 + 4/7
>
> 4/1 - 4/3 + 4/7 - 4/9
>
> 4/1 - 4/3 + 4/7 - 4/9 + 4/11
To the pattern recognition bit of my brain
something smells wrong here?
denominator is 1 3 7 9 .... What happened to 5?
http://en.wikipedia.org/wiki/Leibniz_formula_for_pi gives pi/4...
Phew. I like pretty!
4/1 -4/3 + 4/5 - 4/7 ....
The idea of approximation series (or some such titles)
came up in one of my maths lectures. Sort of makes sense
since it's the sort of thing that we do? Guess, refine our guess
(possibly based on some error indication) and review again,
until 'we're happy'... by some definition!
Pity we can't design software that way.
Noah Slater
On 21 Oct 2010, at 13:28, Dave Pawson wrote:
> To the pattern recognition bit of my brain
> something smells wrong here?
Aye, my mistake.
There should have been a 4/5 in there.
> http://en.wikipedia.org/wiki/Leibniz_formula_for_pi gives pi/4...
Yeah, it's a bit muddy to be honest.
My first introduction to the formula was through the works of Madhava, who discovered it hundreds of years before the western mathematicians. I think it's kinda funny how many other people have tried to attach their name to it in the time after his discovery.
Anyway, the formulation I originally saw was 4/1, etc — which I think makes a lot more sense, as the final product is pi, and not pi/4. When you structure it like that, it's also a lot clearer what's going on. We're starting with an over estimation at 4, and then moving backwards and forwards by fractional values of 4.
Dave Pawson
> My first introduction to the formula was through the works of Madhava, who discovered it hundreds of years before the western mathematicians. I think it's kinda funny how many other people have tried to attach their name to it in the time after his discovery.
That's another weird one IMHO. Continents apart, people are working on
the same problem
and find a solution within ... not quite minutes, but weeks/months of
each other?
The time is right for a solution? Fine and understandable with todays comms, but
200 years ago? That gets to be quite spooky!
Noah Slater
On 21 Oct 2010, at 13:47, Dave Pawson wrote:
> That's another weird one IMHO. Continents apart, people are working on
> the same problem
> and find a solution within ... not quite minutes, but weeks/months of
> each other?
> The time is right for a solution? Fine and understandable with todays comms, but
> 200 years ago? That gets to be quite spooky!
They were centuries apart!
It's all a bit confusing with this one really.
Dates of birth:
Madhava of Sangamagrama (1350 – 1425)
James Gregory (1638 – 1675)
Gottfried Wilhelm Leibniz (1646 – 1716)
Wikipedia sez:
"Gregory's series, also known as Leibniz's series, is a mathematical series that James Gregory published in 1668. It is now known as the Madhava-Gregory Series. The Indian mathematician Madhava of Sangamagrama invented this series centuries before James Gregory."
Wikipedia sez:
"The infinite series above is called the Leibniz series. It is also called the Gregory–Leibniz series, recognizing the work of James Gregory. The formula may have been discovered by Madhava of Sangamagrama and so is sometimes called the Madhava–Leibniz series."
So, Wikipedia lists it as either:
Gregory's series
Leibniz's series
Madhava-Gregory Series
Gregory–Leibniz series
Madhava–Leibniz series
Although, to be honest, I much prefer:
Madhava's series
Hehe.
Dave Pawson
> So, Wikipedia lists it as either:
>
> Gregory's series
>
> Leibniz's series
>
> Madhava-Gregory Series
>
> Gregory–Leibniz series
>
> Madhava–Leibniz series
>
> Although, to be honest, I much prefer:
>
> Madhava's series
>
> Hehe.
What's that Tom Lehrer song? Nikolai Evanovitch Lobotchevski is his name?
All about plagiarism?
Noah Slater
On 21 Oct 2010, at 14:06, Dave Pawson wrote:
> What's that Tom Lehrer song? Nikolai Evanovitch Lobotchevski is his name?
> All about plagiarism?
'Lobachevsky is the subject of songwriter/mathematician Tom Lehrer's humorous song Lobachevsky from his Songs by Tom Lehrer album. In the song, Lehrer portrays a Russian mathematician who sings about how Lobachevksy influenced him: "And who made me a big success / and brought me wealth and fame? / Nikolai Ivanovich Lobachevsky is his name." Lobachevsky's secret to mathematical success is given as"Plagiarize!", as long as one is always careful to call it "research". According to Lehrer, the song is "not intended as a slur on [Lobachevsky's] character" and the name was chosen "solely for prosodic reasons".'
— http://en.wikipedia.org/wiki/Nikolai_Lobachevsky
I'd not heard about this dude before!
Noah Slater
During my search for a Madhava spiral this morning, I found:
http://en.wikipedia.org/wiki/Ulam_spiral
http://en.wikipedia.org/wiki/Sacks_spiral
Both of them pretty fascinating.
Sean B. Palmer
> Anyway, the formulation I originally saw was 4/1, etc — which I think
> makes a lot more sense, as the final product is pi, and not pi/4.
No, 1 - 1/3 + 1/5 - 1/7 is okay too!
The only problem is that the even series doesn't involve pi:
http://en.wikipedia.org/wiki/User:Sbp/Maths#Even_Madhava_series
What a shame it doesn't come out to be 2pi/3!
Dave Pawson
> — http://en.wikipedia.org/wiki/Nikolai_Lobachevsky
>
> I'd not heard about this dude before!
I have two of his records.. vinyl round black thingies :-)
Listen if you ever get a chance. V. humorous.
Dave Pawson
> Noah Slater wrote:
>
>> Anyway, the formulation I originally saw was 4/1, etc — which I think
>> makes a lot more sense, as the final product is pi, and not pi/4.
>
> No, 1 - 1/3 + 1/5 - 1/7 is okay too!
>
> The only problem is that the even series doesn't involve pi:
>
> http://en.wikipedia.org/wiki/User:Sbp/Maths#Even_Madhava_series
>
> What a shame it doesn't come out to be 2pi/3!
Instead it's that equally weird e number again?
I'm sure there is a relationship between e and pi? Somewhere
in this little universe of ours.
Sean B. Palmer
> Both of them pretty fascinating.
Yes, I knew about the Ulam spiral. That was one of the reasons I
thought that the square root packing spiral might be involved
somewhere.
What's most interesting about your discovery if the Madhava spiral is
that it makes a spiral inside the original unit of the reciprobits
using the reciprobits themselves. But of course it doesn't make a
spiral which is consistent with any of the individual grid/spirals
that the reciprobits come from.
What's interesting is that the way they stack, anyway is related to
pi. Why is this interesting? Because the harmonic series, with the
linear denominators, doesn't even converge! But here, with an
alternation of just the odd denominators, it does converge, and it
converges to a quarter of pi. And then when we square the
denominators, we find pi involved again, though this time we don't
have to alternate, and also we find that pi is squared.
Sean B. Palmer
> I'm sure there is a relationship between e and pi?
Yes, e ** (i * pi) + 1 = 0.
http://en.wikipedia.org/wiki/Euler's_identity
Sean B. Palmer
http://en.wikipedia.org/wiki/User:Sbp/Maths
About extending the even Madhava series along the same lines as the
Dirichlet beta function, to form a function that I call gamma though
it possibly already has a name.
Noah Slater
On 21 Oct 2010, at 14:33, Sean B. Palmer wrote:
> What's most interesting about your discovery if the Madhava spiral is
> that it makes a spiral inside the original unit of the reciprobits
> using the reciprobits themselves. But of course it doesn't make a
> spiral which is consistent with any of the individual grid/spirals
> that the reciprobits come from.
I'm really struggling to keep up with this discussion. The only contribution I made so far which seems to be of any interest came to me during my sleep, so I can hardly take credit for it. Sorry to be a burden, but is there any way you could explain this paragraph in a little more detail?
Sean B. Palmer
> About extending the even Madhava series along the same lines
> as the Dirichlet beta function, to form a function that I call gamma
> though it possibly already has a name.
Though I didn't find a name for it, I did find a very simple
relationship between gamma and the Dirichlet eta function (not the
beta function), which is to do with the alternating harmonic series
and is therefore interesting in itself.
Dave Pawson
>
> Though I didn't find a name for it, I did find a very simple
> relationship between gamma and the Dirichlet eta function (not the
> beta function), which is to do with the alternating harmonic series
> and is therefore interesting in itself.
Did you know that if you add odd harmonics you get a triangular
waveform, add the even ones you get a square one.
I'm almost sure it's that way round, but I am recalling
stuff from a while back.
In case that fits in with your series... square/triangular
shapes that you are using to wrap your lines around...
Or perhaps I'm just adding to the confusion.
Sean B. Palmer
> I'm really struggling to keep up with this discussion.
Yeah sorry, I've discovered quite a lot. For example I found out that
your series has an interesting relationship to the Basel problem. Your
series was for linear denominators, and comes out as pi/4 as you
remember. But the Basel problem is about square denominators.
So what happens when you take the series you found, and use square
denominators instead? Well you come out with this constant called
Catalan's constant, and nobody even knows if it's rational or
irrational. Presumably it's irrational.
Anyway, Catalan's constant can be defined as:
G = 1/8 pi**2 - Z(2, 3/4)
Which is interesting because Z is the Hurwitz zeta function:
http://en.wikipedia.org/wiki/Hurwitz_zeta_function
The Basel problem can be defined as Z(2, 1), and the answer to the
Basel problem is 1/6 pi**2. So you see why it's interesting that G
comes out to be 1/8 pi**2, minus Z(2, 3/4). This is especially
interesting since the series you found comes out as pi/4 (the missing
quarter?).
Sean B. Palmer
http://purl.org/net/latex/%5Cfrac{pi^2}{6}
By which I mean, a nice redirect to an existing image service...
Sean B. Palmer
but I found out that Z(2, 1/2) is pi**2/2. Since Z(2, 1) is obviously
pi**2/6 as we've been discussing, I wondered what value q for Z(2, q)
gives pi**2. It turns out to be close to 1/3, but a little bigger. I
don't recognise the number, and haven't been able to link it to any of
the other numbers that tend to be involved with this sort of thing.
Again, Z is the Hurwitz zeta function:
http://en.wikipedia.org/wiki/Hurwitz_zeta_function
--