Author Topic: Delling's Thread  (Read 36201 times)

Offline Yonkey

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Re: Delling's Thread
« Reply #120 on: March 05, 2009, 06:19:45 AM »
Ah... Now I get it. :P

I remember learning pretty quick in university that trying to get partial marks is hopeless. :P  Most of the time, a question had to be done entirely right to get full marks, or you get a 0.  Especially in first year Calculus, where you're only marked on the final answer, as opposed to the 15-minute step-wise process of getting there. ::)

But yeah, I also remember having a few questions on tests & exams in other years where I'd get stuck and have absolutely no idea how to finish answering them, so I'd just continue on some wild tangent of solving until I ended up with some completely non-sensical answer. XD  Needless to say, I was lucky to get even 1 partial mark on those questions.  :P

I think in general, I always felt that it's better to try to solve something to the best of my ability, than to just give up and leave it blank. 
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Offline tessspoon

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Re: Delling's Thread
« Reply #121 on: March 05, 2009, 06:27:14 AM »
One of my math profs gives partial credit if you write down what you do know if it has something to do with the problem, which is really good since I would have gotten a good bit lower grade in Math Reasoning had it been otherwise. And probably ditto for this Abstract Algebra test I have today :stabs:

Offline Delling

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Re: Delling's Thread
« Reply #122 on: March 05, 2009, 07:30:25 AM »
I have no problem with giving students credit for what they know when what they know is correct, but it baffles me when they do things that... well, that just can't be believed sin(x+y) =/= sin(x) + sin(y) and no amount of wishing can make it so (and this naive guess would be believable if it weren't for the week we spent going over trig identities!).

*is going to get worked up into a fervor again and rant*

Actually, I *try* really hard to make sure that they get some credit on every question provided they've shown their work... but it's hurting my SAN score...
Noli me tangere! Nescio ubi fuisti!
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...resistance was obviously useless against a family that could invent italics.

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Offline Rosella

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Re: Delling's Thread
« Reply #123 on: March 05, 2009, 10:17:34 PM »
That's one thing I love about both my calculus class and my calculus teacher. The class is made up of high school students, so you know we had to be nerds to get here, and my teacher works tirelessly to follow our work so we get partial credit. She really is a wonderful teacher. XD

Incidentally, anyone want to explain Taylor and Maclaurin Series? :P
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Offline Delling

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Re: Delling's Thread
« Reply #124 on: September 08, 2009, 01:45:13 PM »
*rocks back and forth*

The Taylor Series is the general case. It's terms are

(1/n!) dnf(a)/dxn (x - a)n

The MacLaurin Series is the case where we take a = 0, so the sum becomes...

(1/n!) dnf(0)/dxn xn

... and then we sum up to n = infinity.

So, basically, reaching back to the idea that the derivative is a slope:

(df(a)/dx) (x-a) is a good approximation of the value of f at x = a... it turns out that it is only good up to the second derivative...

[(df(a)/dx) (x-a)] + [(1/2) d2f(0)/dx2 x2]

... you can think of needing to multiply the second derivative by x2 as a matter of units (or a matter of order): the derivative of nth order has units of [f/xn]. So, to be able to add them all up (in terms of units) we need to multiply by xn. Basically, we are approximating f so we need to have the same units/order/degree/highest power in x be the same in each term.

So, to actually take a MacLaurin series, all we have to do is evaluate the derivative at 0 for successive derivatives up to whatever degree polynomial we want to find, or you might want to find the general form for the function. Such as,

cos x = SUM( 1/(2n)! (-1)n x2n )

so for this, you just need to recognize that every odd derivative will be sin(0) which allows us to replace all the n's with 2n's. Each of the even derivatives will alternate between -cos x and cos x with x=0 so it's just (-1)n. Tuh-duh... so, I was going to keep going until I derived Euler's equation from the MacLaurin series of sin x and cos x... but it's 1am... so more on this later maybe :P

(Posted on: March 06, 2009, 02:00:13 AM)


*is back to continue said derivation now* :D

So, I need two more things to get to Euler's equation: namely, the Taylor series for the exponential function and the sine function.

Let's do sin(x) now!

sin(0) = 0
(sin x)' = cos x => cos(0) = 1 :D
(cos x)' = -sin x => -sin(0) = 0 :(
-(sin x)' = -cos x => -cos(0) = -1 :D

So, once again, we alternate signs ( (-1)n) but in this case, we pick off the ODD powers instead. So,

sin x = SUM( 1/(2n+1)! (-1)n x2n+1 )

ok, now, for something completely different: exp(x)

I am going to assume that you know [exp(x)]' = exp(x) :D This happy little fact is the crux of so much higher mathematics AND in truth, trigonometry! I know, it's amazing, isn't it?

Anyway, this means that the Taylor series for exp(x) is really simple

exp(x) = SUM( (1/n!) xn)

It turns out (by the chain rule) that I can pick up an alternating sign in my exp(x) sum by taking the Taylor Series of exp(-x) instead.

Now, if sine picks off the odd numbers... and cosine picks off the even...

cos x = [exp(x) + exp(-x)]/2 :D

sin x = [exp(x) - exp(-x)]/2 :D

BUT THIS IS WRONG!!!!!!!!! Seriously, don't believe the two lines above... they are the hyperbolic versions of the sine and cosine. What's missing?

I didn't account for the fact that the cos and sine sums also alternate sign!

The solution is trivial: insert the factor i = sqrt(-1).

Since sin (x) picked off the odd powers, the exp(ix) derivatives now give us also an (i)2n+1... which equals (-1)n*i... which is right within a factor of i, and cos(x) does not have this issue because it picks off the even powers so it in turn only gets the even powers of i which are all real (they're -1 and 1).

That was a fairly handwavy argument, but typing out the expressions like this is fairly awkward. I hope you can do it out yourself... if not, maybe I'll do it out in an equation editor and attach it at some later time.

If you follow/believe the above, you get that

sin(x) = [exp(ix) - exp(-ix))]/2i
and
cos(x) = [exp(ix) + exp(-ix)]/2

So, what does cos(x) + isin(x)?

(Posted on: March 24, 2009, 02:56:12 PM)


*has a story to tell and realizes he needs to start linking up pictures, etc.*

Today in the car, I was telling my dad: "Spanish really missed an opportunity to have inclusive and exclusive 'we' forms: the we-form they have is nosotros,' which is [etymologically at least: nos utros] 'we others.' With a nostodos, they could have had an inclusive form to contrast with it." ...pause in conversation... "They really missed a golden opportunity to improve on something Latin didn't do right."

To which he responded: "But, Brandon, you know how languages work. They don't improve: they decline."

I grunting and groaning admitted: "Yes, all right, and declensions are often the first thing to decline."

(Posted on: August 19, 2009, 09:41:33 AM)




/~Picture Time~\



Handel's keyboard, on which he first performed Handel's Messiah...



I think I took the same picture of Gothic arches extending back and away in every Gothic structure I visited if possible. Both of these pictures are from St. Patrick's in Dublin.



I swear the thought process here was: "Ooh! Swords! :D" ...which now that I think of it... is a strange thought for a belenophobe to be having... oh, well, I'll ponder that later. (This picture is from Enniskillen Castle which has some sort of war museum in it... and pointy objects, cannons, a silver display, etc., etc.)



Galway Bay as seen from the first B&B the tour stayed at in Ballyshannon. I took a walk into town from the B&B. I still think this was the best B&B by far... that probably has something to do with the time spent looking at really old Gaelic schoolbooks. :P



Muckross House was constructed during An tOcras Mor (The Great Hunger, which is what they call it now since Britain had more than enough food to feed the starving dying masses in Ireland... This is why I prefer Grainne Mhaol to Elizabeth I now... that and Good Queen Bess called Gaelic barbaric :P). Anyway, it was constructed to be a sort of Irish home away from home for Queen Victoria, who came and stayed for a couple of days, hated it, and left.

(Posted on: August 30, 2009, 02:02:20 PM)


*skipping to some good stuff* (not that there isn't more good stuff in the Ireland album :P)

York Minster


Tintern Abbey








Noli me tangere! Nescio ubi fuisti!
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...resistance was obviously useless against a family that could invent italics.

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Offline Suzie

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Re: Delling's Thread
« Reply #125 on: September 25, 2009, 02:00:49 AM »
Just as this came up on the recent posts page, I thought it was in the question game thread and was about to come over here and post a whole lot of physics-y, math-y, language-y stuff XD... but as it isn't, I'll just say that if you'd really like to have a physics/astronomy discussion... that'd be a cool use for my thread :P

And so I do. :P


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Offline Delling

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Re: Delling's Thread
« Reply #126 on: September 25, 2009, 11:05:50 AM »
Let's see... secrets of the universe... where to start...

1) 42

2) 3.14159...

3) 2.71828...

4) 6.626... * 10-34 m2 kg/s

5) 6.673... * 10-11 m3 /(kg*s2)

6) 2.998... * 108 m/s

I suppose these aren't so much secrets of the universe as the "open questions of the universe." Why is G G? Why is c c? e e? or π π? or h h? or for that matter what does 42 have to do with anything!? One of the interesting things about these, e and π in particular and 2 for that matter or even say 2n, is that they come up constantly in physics. (One might expect c, h, and G to show up since they are "physical constants" though that really just means that they were obtained by measurement and effectively shown to be the constant of proportionality in some relationship.) ...although an equally annoying question for me is how all these purportedly intelligent people constantly insist on pronouncing Euler as "yu-ler"... when it has, of a necessity, being German, the pronunciation: "oi-ler."

Ok, bonus points will be awarded for recognizing which value is which. :P Also, there are very straightforward reasons why 2 and 2n show up. :P Know any?

Hrmm... I should find some astronomy stuff to talk about... when I get home...

EDIT: in the meantime, pretty picture:



Look! It's the universe!-ish... :P [use view image, not surprisingly, the universe is too huge to post a picture of directly :P (actually, I think this is still just within the supercluster that is home to the Milky Way)]
« Last Edit: September 25, 2009, 06:32:32 PM by Delling »
Noli me tangere! Nescio ubi fuisti!
Don't touch me! I don't know where you've been!

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Duke of Saxony in Her Majesty's Court
Knight of the Swan for Her Imperial Highness

...resistance was obviously useless against a family that could invent italics.

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http://my.ddo.com/referral/Delling87

Offline Yonkey

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Re: Delling's Thread
« Reply #127 on: September 25, 2009, 11:36:36 PM »
Considering how many things are variable in this universe, it's only natural for there to also be a few things that remain constant. :scholar: XB

Also, there are very straightforward reasons why 2 and 2n show up. :P Know any?
I don't know the mathematical reasons, but anything digital boils down to a representation of 2n bits.  In nature, it represents exponential growth, cell division, etc.  As for why 2, as opposed to 1 or 3, I have a theory. XD

Everything in this universe seems to work on the basis of attraction or repulsion: gravity, chemistry, Hollywood celebrities, etc. :P  In order to be attracted or repulsed by something, there must be at least 2 frames of reference.  Physics tell us that opposites attract, yet most people form relationships based on having things in common, rather than differences.  The other thing that's interesting is the strongest materials are made up of densely packed molecules of the same nature.  Yet, a society made up of a single race or culture is completely ignorant to foreigners.  Perfection also falls into the realm of homogeneity, where the symmetry or purity of something is inversely related to the number of flaws or deviations. 

The thing is, nothing in this universe is 100% perfect.  There is never a perfect balance between attraction and repulsion.  As a result, over time this slight imbalance turns into randomness and chaos.  It's not complete chaos, because that would be 100% perfection as well.  Our world mirrors the universe, but just on a more localized scale.  Life itself is completely chaotic and unpredictable, yet as humans we try to maintain order, and use our own unique talents to achieve some form of perfection. 

The question is: if perfection is unattainable and maintaining complete order is impossible, why do we still do it?  The answer goes back to the to the definition.  Between attraction and repulsion, attraction is quantitatively and qualitatively stronger.  People do it because they prefer being attracted to the fantasy, as opposed to being repulsed by reality.  It makes us happy, and in this grand universe of ours, that's all that really matters. :)
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Offline Delling

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Re: Delling's Thread
« Reply #128 on: September 26, 2009, 08:09:54 AM »
Considering how many things are variable in this universe, it's only natural for there to also be a few things that remain constant. :scholar: XB
XD True enough... but the interesting question becomes "which things are constant?" and then "why are these things constant?"

Also, there are very straightforward reasons why 2 and 2n show up. :P Know any?
I don't know the mathematical reasons, but anything digital boils down to a representation of 2n bits.  In nature, it represents exponential growth, cell division, etc.  As for why 2, as opposed to 1 or 3, I have a theory. XD
Actually, we often find exponential growth of different bases than 2 but we usually just leave it in the form of ekx (this however equals (ek)x where ek would be a new base). Cell division is proportional to 2n because when cells split, they split into two: theoretically, there's no real reason for this apart from perhaps Occam's Razor and pragmatics (cell division requires that the cells have some minimum quantity of material to impart to the next cell so that it will survive). So, with say 2x the energy and 1.5x the material, they could split into three (it's 1.5 b/c roughly 0.5 of the initial material stays in the parent cell in normal cell division: it is possible that this is too much material for one cell to house at a time (which would be pragmatics again)).

With bits, it's because switches are treated as two-state systems with only the value 0 or 1. (Funny sidenote: with some tweaking it is possible to get a circuit with two light switches to cease functioning because one switch is stuck in neither ON nor OFF. A friend of mine recently demonstrated that it's possible to get the passenger-side door of my car stuck in a neither (in terms of being locked or unlocked) too (still not sure how he did that: the switch in the car, looked like it was unlocked but he couldn't open the door until I went around and physically switched the lock into locked and then unlocked (there was some physical resistance to trying to turn it into the unlocked position first)))

Everything in this universe seems to work on the basis of attraction or repulsion: gravity, chemistry, Hollywood celebrities, etc. :P  In order to be attracted or repulsed by something, there must be at least 2 frames of reference.
 
Actually, this plus the above is roughly what I was going for: 2 and 2n occur for purely statistical reasons related to counting. This necessitates an important conclusion: numbers are themselves fundamental. (People tend to miss that the important thing here is the essence of the numbers and not the praxis or description of them. Some systems are better and some are worse at representing numbers, but they're still just the same old numbers underneath the description (2 is 2 even if it's 10 or 12 is 12 even if it's B).)

Physics tell us that opposites attract, yet most people form relationships based on having things in common, rather than differences.  The other thing that's interesting is the strongest materials are made up of densely packed molecules of the same nature.
Actually, the principle that opposites attract ONLY holds up in electrodynamics/statics. In gravity, if there were some sort of antimatter which had matter in a negative sense, the gravitational equations say that it would be repelled by normal matter. In a bulk-properties/material science sense, what matters are cohesion and adhesion. People are cohesive for small-n.

  Yet, a society made up of a single race or culture is completely ignorant to foreigners.  Perfection also falls into the realm of homogeneity, where the symmetry or purity of something is inversely related to the number of flaws or deviations.
There are interpretations of perfection (say in ionic crystals) in which absolute homogeneity is impossible, BUT isotropy is not. In a crystal, it should be that if I'm standing on element X, for every X, there must be exactly 1 equivalent set say {A, B, Y, Z} of elements to which I can step over bonds. The structure is not homogeneous on this scale, but on any scale (for a reasonable partition), it is isotropic.

The thing is, nothing in this universe is 100% perfect.  There is never a perfect balance between attraction and repulsion.
Hydrostatic equilibria can be maintained forever. True, this doesn't happen in stars, but that's because they undergo internal changes. Jupiter is in hydrostatic equilibrium (balance of its self-gravity (attraction) with the pressure (repulsion) of its gases).

[the pursuit of perfection] makes us happy, and in this grand universe of ours, that's all that really matters. :)
Happiness makes a poor standard for action though: what makes one man happy may make a million sad or dead. It may make him ecstatically happy and/or even manic, but I doubt anyone here thinks his lone happiness could ever outweigh the attendant losses. (Though this over-valuing of a single ideal such as happiness, the safety of others, etc. seems to be common pitfall in sociological reasoning: I think it arises from the idea that there must be 1 grand unifying principle to explain the formation of groups. I find it far more likely that there are N principles or levels of interaction (probably inherent to each sets' premises) which act to attract or repel.)
Noli me tangere! Nescio ubi fuisti!
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Duke of Saxony in Her Majesty's Court
Knight of the Swan for Her Imperial Highness

...resistance was obviously useless against a family that could invent italics.

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Offline Yonkey

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Re: Delling's Thread
« Reply #129 on: September 26, 2009, 11:57:16 AM »
XD True enough... but the interesting question becomes "which things are constant?" and then "why are these things constant?"
While I (obviously :P) don't know the exact answer, maybe it's due to the basis on which things are defined.  Let's take Pi as an example.  It's supposed to be the ratio of any circle's circumference to its diameter.  But, it only applies on "perfect" circles.  Ellipses are circles as well, yet their ratio of C to d isn't exactly Pi.  Most likely, this is because their diameter isn't constant in all directions as it is in a circle. 

So, my guess is that the things we consider constant are in fact based on perfect or idealistic conditions.  As we've said, while perfection cannot realistically exist, symmetry, balance, equilibrium and isotropy appear in nature (and in circles too :P) quite often.  For whatever reason, we see beauty in these things (a.k.a attraction), and our definition of success as human beings seems to be based on them.


Cell division is proportional to 2n because when cells split, they split into two: theoretically, there's no real reason for this apart from perhaps Occam's Razor and pragmatics (cell division requires that the cells have some minimum quantity of material to impart to the next cell so that it will survive). So, with say 2x the energy and 1.5x the material, they could split into three (it's 1.5 b/c roughly 0.5 of the initial material stays in the parent cell in normal cell division: it is possible that this is too much material for one cell to house at a time (which would be pragmatics again)).
But if a cell contains 1.5x the material, it would mean an error occurred during DNA replication.  Whenever this accelerated form of cell growth happens during cell division, it leads to cancer.  :-\ Which (as we all know) is a fatal disease that negatively impacts the growth of our species.  The treatment, and eventual cure for cancer, is to eradicate the abnormally created cells and return the cell division in the affected areas back to normal levels.  The goal is to return the body to homoeostasis, or a "perfect" balance.


With bits, it's because switches are treated as two-state systems with only the value 0 or 1. (Funny sidenote: with some tweaking it is possible to get a circuit with two light switches to cease functioning because one switch is stuck in neither ON nor OFF. A friend of mine recently demonstrated that it's possible to get the passenger-side door of my car stuck in a neither (in terms of being locked or unlocked) too (still not sure how he did that: the switch in the car, looked like it was unlocked but he couldn't open the door until I went around and physically switched the lock into locked and then unlocked (there was some physical resistance to trying to turn it into the unlocked position first)))
LOL!  That's more just due to wear and tear. :P  The lock itself may have been in an unlocked state, but the door hinge may not. :P  But yeah, not just with switches.  Sound waves get digitally represented by high & low voltages used to represent 1's and 0's.  Peaks and valleys on CD's & DVD's represent the 1's and 0's burned onto them.  Pixels on a screen being a bit-mapped representation of cyan, magenta, yellow and black.  They're all based on binary data.

Some systems are better and some are worse at representing numbers, but they're still just the same old numbers underneath the description (2 is 2 even if it's 10 or 12 is 12 even if it's B).)
For the longest time, I had no idea what you were talking about when you said 12 is B... then it hit me.  Hexadecimal! XD  I was actually thinking you were talking about letters being yet another representation of ASCII or Unicode characters.  Which they are for machines, but not really for humans. :P  Actually, language and communication is another thing that distinguishes us from the rest of the universe, ranking us more "perfect" on the great chain of being, so to speak.

Happiness makes a poor standard for action though: what makes one man happy may make a million sad or dead. It may make him ecstatically happy and/or even manic, but I doubt anyone here thinks his lone happiness could ever outweigh the attendant losses. (Though this over-valuing of a single ideal such as happiness, the safety of others, etc. seems to be common pitfall in sociological reasoning: I think it arises from the idea that there must be 1 grand unifying principle to explain the formation of groups. I find it far more likely that there are N principles or levels of interaction (probably inherent to each sets' premises) which act to attract or repel.)
I agree that one person's definition of happiness is not, and never will be equivalent to another's.  It's also true what you're saying, that any seemingly benevolent act could also be interpreted as malevolent, depending on your frame of reference.  But on an individualistic standpoint, people are attracted to things that make them happy and repel away from things that don't.  At a group-level, I don't know.  Some people may share common beliefs, ideals and ethics, but they're not genetic clones of each other.  Even if you happen to do something which pleases the majority of people, when you dig down to the individual level, their measure of happiness is unlikely to be exactly the same as someone else's.

But to get back to what you're saying... :P Actions are made based on decisions.  Some people make rational and logical decisions, and some people make decisions on what feels like the right thing to do.  While most decisions do not result in personal pleasure, I think most people at least try to choose things which results in the least amount of pain.
"A wish changes nothing. A decision changes everything."

Offline Suzie

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Re: Delling's Thread
« Reply #130 on: September 26, 2009, 12:05:14 PM »
Even though I seem to have started this I haven't quite had the time or mood to enter this discussion as of yet, but the following made me want to jump in.. so here goes.. although I fear getting crushed under the combined weight of your intelligence quotients. :P

I was actually thinking you were talking about letters being yet another representation of ASCII or Unicode characters.  Which they are for machines, but not really for humans. :P  Actually, language and communication is another thing that distinguishes us from the rest of the universe, ranking us more "perfect" on the great chain of being, so to speak.

Communication is not really unique to humans, though language is, atleast on Earth. (Our ancestors may also have developed speech, but this is much debated). But how so does it distinguish us from the rest of the universe? Unless you assume the rest of the universe is a vast empty space in terms of intelligent life other than our own, which, while no evidence suggests it isn't, contrarily no evidence suggests that it is. I also am not sure what you mean by "perfect", or why you seem to think language would make one more valuable than say... a quark? :P
« Last Edit: September 26, 2009, 12:13:19 PM by Suzie »


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Offline Yonkey

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Re: Delling's Thread
« Reply #131 on: September 26, 2009, 05:09:00 PM »
Don't worry about it, just jump on in wherever you like! XD 

Communication is not really unique to humans, though language is, atleast on Earth. (Our ancestors may also have developed speech, but this is much debated). But how so does it distinguish us from the rest of the universe? Unless you assume the rest of the universe is a vast empty space in terms of intelligent life other than our own, which, while no evidence suggests it isn't, contrarily no evidence suggests that it is. I also am not sure what you mean by "perfect", or why you seem to think language would make one more valuable than say... a quark? :P
I think I was more talking about communication in the broader sense.  Like how you can talk to someone on the other side of the planet instantaneously and they fully understand you.  Other species are capable of communicating, but not quite at that same level.  Of course, part of that communication is human understanding and the rest is technology... but that's another topic altogether. :P

And to be honest, considering how vast and unexplored the universe is, I have serious doubts that we're the only forms of intelligent life.  You're right that to date, there hasn't been any evidence to prove there is or there isn't.  The main reason why I think there is is due to how incredibly small our planet is in relation to the universe.  There could easily be another similar planet somewhere far away, but technology hasn't advanced far enough to facility first contact.  In any case, the first step would be to establish some means of communication.

Oh, and when I said "perfect" it was in the sense of people considering human beings to be the most intelligent form of life.  I personally disagree with that, because there are different forms of intelligence, and there are many animals which can easily outwit humans.  The main (and possibly the only) things that make us superior to other animals are our language and our technology.

You also bring up another interesting topic: value.  While language holds a lot of value to humans, they have very little value to any other species.  And to an inorganic object like a particle, you're right, it has no value whatsoever to it.  I don't really know enough about quarks to the power of what a single one can do.  I assume it's more the attraction of many quarks that produces results in things that directly affect us?
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Offline Delling

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Re: Delling's Thread
« Reply #132 on: September 26, 2009, 08:50:13 PM »
For now, let's leave quantum chromodynamics and quarks off the table. :P

While I (obviously :P) don't know the exact answer, maybe it's due to the basis on which things are defined.  Let's take Pi as an example.  It's supposed to be the ratio of any circle's circumference to its diameter.  But, it only applies on "perfect" circles.  Ellipses are circles as well, yet their ratio of C to d isn't exactly Pi.  Most likely, this is because their diameter isn't constant in all directions as it is in a circle. 

There's an error in saying that an ellipse is a circle and vice versa. We say that a circle is a special case of an ellipse (one which has both of its foci at the same point) and that all ellipses by extension are degenerate circles (because rather than a center they have two foci: this degeneracy can be measured and is called the "ellipticity"). In general, if a geometric form is a degenerate version of some other form, it cannot be said to be the other form (rectangles* aren't squares but squares are rectangles; a rhombus is a parallelogram but a parallelogram is not necessarily a rhombus; all of these objects are by necessity quadrilaterals) whereas if it's a special case of a form, all members of that form are a subset of the form for which they are a special case: all circles are ellipses but all ellipses are not circles. (and I still cringe to write that all circles are ellipses... technically true, but I don't want to grade the resulting fall out on Calc3 quizzes/exams...)
(Likewise it is pointless to speak of the "diameter of an ellipse": it doesn't have one. Ellipses have major and minor axes.)
___
*: Notably, this depends on one's definition of a rectangle: some definitions require that the sides be pair-wise of unequal lengths. (Usually, because they've already defined the square. :P) Technically, a rectangle is a quadrilateral whose sides meet at right angles. Under this definition: a rectangle is not necessarily a square but a square is necessarily a rectangle.

But if a cell contains 1.5x the material, it would mean an error occurred during DNA replication.  Whenever this accelerated form of cell growth happens during cell division, it leads to cancer.  :-\ Which (as we all know) is a fatal disease that negatively impacts the growth of our species.  The treatment, and eventual cure for cancer, is to eradicate the abnormally created cells and return the cell division in the affected areas back to normal levels.  The goal is to return the body to homeostasis, or a "perfect" balance.
The example of a cell that splits into three instead of two was hypothetical and was merely attached to the nature of things as growing as 2n (if cells split in that way, they would grow as 3n and would presumably have the DNA to match :P). Cancer cells still grow as 2n: the mechanisms involved there are different from this hypothesis.

So, my guess is that the things we consider constant are in fact based on perfect or idealistic conditions.
Unfortunately, this isn't true. It would be nice if all physical constants could be readily chopped up to geometry, ideals, etc. Unfortunately, things like h (Planck's Constant), c (the speed of light in a "vacuum"), permeability and permittivity of free space, and G (gravitational constant) are based on repeated measurements and statistics (as instrumentation improves we get better and better measurements of these constants). The metaphysical question to ask--the "Why?"--is why should any one of these constants A) be constant? and B) have the value that they have?


As we've said, while perfection cannot realistically exist, symmetry, balance, equilibrium and isotropy appear in nature (and in circles too :P) quite often. For whatever reason, we see beauty in these things (a.k.a attraction), and our definition of success as human beings seems to be based on them.
I somewhat suspect, taking some history of body modification into account and the nature of art in the west, that the valuing of symmetry, balance, and isotropy as beautiful might be somewhat buried in our received epistemology from such things as the Renaissance, Classical Greece, Classical Rome, etc. While say Oriental and African cultures might still value these same properties and even associate them with beauty, they choose to employ different forms and different hierarchies of form. (Though that is a cursory summation at best... *is not overly involved with art*)
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Offline Yonkey

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Re: Delling's Thread
« Reply #133 on: October 06, 2009, 10:09:49 PM »
Sorry I didn't get a chance to reply sooner.  Been pretty busy. :-\

There's an error in saying that an ellipse is a circle and vice versa. We say that a circle is a special case of an ellipse ...
(Likewise it is pointless to speak of the "diameter of an ellipse": it doesn't have one. Ellipses have major and minor axes.)
But that's just it!  While the diameter of a circle remains constant, Pi remains constant.  As soon as you stretch that circle 1 pixel in the major axis, suddenly it's no longer considered a circle, no longer has constant diameter.  When the diameter becomes variable, Pi (i.e. the ratio of the circumference to this "diameter") is no longer a constant.

So, that's why I think constants aren't truly constant.  They're constant only when certain assumptions and conditions are met in order for them to remain true.  The speed of light is 2.998... * 108 m/s, only when it's in a perfect vaccum.  Shine a light underwater, and its speed will be much less (i.e. index of refraction).

The metaphysical question to ask--the "Why?"--is why should any one of these constants A) be constant? and B) have the value that they have?
I think I just answered these above.  When you restrict a constant to only apply under certain conditions, it's no longer that mysterious why they work.  As for why they have the value they have, I would say it's because all other variables in the equation are proportional to each other and can therefore be simplified through a common denominator to that same constant value.

While say Oriental and African cultures might still value these same properties and even associate them with beauty, they choose to employ different forms and different hierarchies of form. (Though that is a cursory summation at best... *is not overly involved with art*)
I'm not entirely sure what you mean here.  You can find symmetrical and basic geometric shapes in any form of art, regardless of the time period or geographic region.  I think when we recognize these shapes and colours in paintings, sculptures and nature, we find them beautiful.  Now obviously, beauty is in the eye of the beholder.  Some people see splattered paint on a wall and find it breathtaking... I personally don't. :P  But, if I was able to interpret it on some deeper level (such as knowing the rationale behind the colours and the frame of mind the artist had), I might find it beautiful in that respect.  That is, seeing beauty beyond the physical.
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Offline Delling

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Re: Delling's Thread
« Reply #134 on: February 16, 2010, 10:49:32 AM »
Sorry I didn't get a chance to reply sooner.  Been pretty busy. :-\

There's an error in saying that an ellipse is a circle and vice versa. We say that a circle is a special case of an ellipse ...
(Likewise it is pointless to speak of the "diameter of an ellipse": it doesn't have one. Ellipses have major and minor axes.)
But that's just it!  While the diameter of a circle remains constant, Pi remains constant.  As soon as you stretch that circle 1 pixel in the major axis, suddenly it's no longer considered a circle, no longer has constant diameter.  When the diameter becomes variable, Pi (i.e. the ratio of the circumference to this "diameter") is no longer a constant.
BUT the value of pi is still fundamental: the area of an ellipse is pi*a*b, the area of the circle (pi*r2) is just pi*a*b when a=b.

So, that's why I think constants aren't truly constant.  They're constant only when certain assumptions and conditions are met in order for them to remain true.  The speed of light is 2.998... * 108 m/s, only when it's in a perfect vaccum.  Shine a light underwater, and its speed will be much less (i.e. index of refraction).

The metaphysical question to ask--the "Why?"--is why should any one of these constants A) be constant? and B) have the value that they have?
I think I just answered these above.  When you restrict a constant to only apply under certain conditions, it's no longer that mysterious why they work.  As for why they have the value they have, I would say it's because all other variables in the equation are proportional to each other and can therefore be simplified through a common denominator to that same constant value.
But there's something not true here. The speed of light changes, but the speed of light in a vacuum does not (in theory... there are still some games you can play with this). When I ask "why?" I'm not interested in the value so much as its properties: indexes of refraction aside, relativity says that c, not the speed of light in anything else, is the maximum speed for physical processes. Likewise, pi and powers of 2 emerge from statistical applications constantly: why pi and why e? These values are special for reasons that may or may not be wholly independent of their geometric interpretations. http://www.dr-mikes-maths.com/eix.html: this link explores the question of "why" Euler's equation is true: in this case, the argument is that these values have the properties they have because it is mathematically and logically consistent (I'm asking more about things like h and c anyway which are purely physical constants. In some sense, Yonkey, your common denominator idea is right here: c emerges from 1/sqrt(epsilon-not*mu-not) both of which are experimentally determined constants and h is likewise measured by experiment. It is however not satisfying.)

While say Oriental and African cultures might still value these same properties and even associate them with beauty, they choose to employ different forms and different hierarchies of form. (Though that is a cursory summation at best... *is not overly involved with art*)
I'm not entirely sure what you mean here.  You can find symmetrical and basic geometric shapes in any form of art, regardless of the time period or geographic region.  I think when we recognize these shapes and colours in paintings, sculptures and nature, we find them beautiful.  Now obviously, beauty is in the eye of the beholder.  Some people see splattered paint on a wall and find it breathtaking... I personally don't. :P  But, if I was able to interpret it on some deeper level (such as knowing the rationale behind the colours and the frame of mind the artist had), I might find it beautiful in that respect.  That is, seeing beauty beyond the physical.
What I'm saying is that the underlying rationale for selecting forms of symmetries (or geometry) as beautiful differ: post-Platonic Europe values circles and degenerate forms of circles at a higher level than something with simple point symmetry but curves that are point symmetric and smooth are in a higher position in the hierarchies of some African art, or something that evokes a feeling or the idea of flowing might be valued higher than a "solid" form (a rectangle, a square, a circle versus the representation of a wave or a piece of cloth billowing in the wind.)

(Posted on: October 07, 2009, 01:44:49 PM)


conversations are fun :)

anywho... I'm studying some astronomy/astrophysics topics that interest me independently in preparation for grad school. This excerpt from a lecture amuses me GREATLY.

Quote
Thus, the average mass-to-light ratio of the entire galaxy is about 10^12 Msun / 10^11 Lsun or 10 Msun/Lsun. For comparison, the mass to light ratio in the solar neighborhood is about 1 Msun/Lsun.
Of course, the mass to light ratio in the solar system is 1! The sun dwarfs everything else in the solar system and is the source of ALL of the luminosity!! ::)
Noli me tangere! Nescio ubi fuisti!
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...resistance was obviously useless against a family that could invent italics.

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Offline Yonkey

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Re: Delling's Thread
« Reply #135 on: April 02, 2010, 11:18:06 AM »
*revives* XD

I happened to see this video today, and I found it incredibly inspiring and thought-provoking.  Anyway, it's kinda geeky, but I figured out of anyone, a Physics dude like you would appreciate it. :P

<a href="http://flash.revver.com/player/1.0/player.swf?mediaId=99898" target="_blank" class="new_win">http://flash.revver.com/player/1.0/player.swf?mediaId=99898</a>

Have a look and let me know your thoughts! XD
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Offline crayauchtin

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Re: Delling's Thread
« Reply #136 on: April 02, 2010, 02:29:49 PM »
I was just reading through this thread for the first time, and I have to interject with something I saw on the first page....

Have you guys told the Oreck guy you don't think his vacuums are perfect???? He will be APPALLED!
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Offline Delling

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Re: Delling's Thread
« Reply #137 on: April 04, 2010, 11:31:02 AM »
Sorry, for the delayed reply. *just got around to watching the video past the 6th or 7th dimension* :P

Basically, their higher dimensions past 5 are not higher dimensions. Each simply involves "traveling" further back in time(4) to reach a point in probability(5) that wasn't available from their current location. As such, we wouldn't want to construct descriptions of points in their perception of the dimensions with a basis set greater than 5 (although we could: you can always construct a description which includes non-orthonormal dimensions, but they will be reducible to information contained in the basis set and therefore uninteresting as well as needless clutter).

To put it another way, a point in their "10" dimensions is described perfectly well by a 5-vector (x,y,z,t,p). Each of their higher dimensions is not orthogonal to an equation of state for the universe given in such a basis U(x,y,z,t,p).

For their 6th dimension, for instance, IF I can travel freely in the 5th dimension, then it doesn't matter that the timeline I'm currently on didn't lead to my inventing something as a child: all that matters is that such a timeline exists for some probability. Their saying that I could travel back in time and trigger off those events is really akin to saying how I could climb a hill to get to a different z-value or I could jump from an airplane to land there. Whether or not I fold or branch, I fold or branch in the 5 dimensions they've already described. Likewise, their possible universes dimension (7, I think) is just changing the initial conditions which have some configuration described by some probability... and so no new dimension is gained there. (Also, I think their 8th dimension (a line between such universes) is precluded by the principles of cosmology anyway.)
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Offline B'rrr

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Re: Delling's Thread
« Reply #138 on: April 04, 2010, 12:19:26 PM »
I kinda got stuck between the 4th and 5th di'ension, thinking that if I can not 'ove back in ti'e (which I can not), then there are no alternative ti'elines, so this all is just theory, 'ade up, like a cute fairytale. unless I a' 'issing so'ething.

It was intresting to watch tho, but I rather watch stardust!

EDIT: 'aybe I got stuck because I still have proble's with the English language fro' ti'e to ti'e *ponders*
« Last Edit: April 04, 2010, 12:29:47 PM by B'rrr »
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Offline Delling

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Re: Delling's Thread
« Reply #139 on: April 04, 2010, 01:32:58 PM »
I kinda got stuck between the 4th and 5th di'ension, thinking that if I can not 'ove back in ti'e (which I can not), then there are no alternative ti'elines, so this all is just theory, 'ade up, like a cute fairytale. unless I a' 'issing so'ething.

One can avoid a need to move freely within the 4th dimension by considering only possibilities from the present: for instance, you can choose to remain sitting or stand up or stand up and immediately sit back down again, etc. Each such option represents a different path in their 5th dimension of probability.

In physics, when we're dealing with coordinate systems/dimensions, we're interested in how they describe a set of points (an n-dimensional space has every point within it being described by an n-dimensional vector or n-vector). As such, it is convenient to consider "motion" through said n-space because we have ready analogs for that idea in normal physics.
Noli me tangere! Nescio ubi fuisti!
Don't touch me! I don't know where you've been!

Marquess of Pembroke
Duke of Saxony in Her Majesty's Court
Knight of the Swan for Her Imperial Highness

...resistance was obviously useless against a family that could invent italics.

"Let the locative live."

http://my.ddo.com/referral/Delling87