ACK!!
Math doesn''t allow PI to exists, it is a universal constant.
And anyway, 3 / 1 is not equal to .33333333. It is equal to .333333333 (with final digit having a bar over it.) Thus, when multiplied back by 3, you get 1.
Tim
Is pi truely random?
Here''s my 2 cents.
Pi isn''t a random number. It is an irrational number. If we can find this number by some mechanical means then it is deterministic. The digits of Pi can be found by Taylor Series (although it may take a long while
) So Pi is a deterministic, irrational number.
There is chaos theory however. One finds sometimes in differential equations (for instance) with certain parameters, equations that are sensitive to initial conditions. It is completely deterministic (since we have our equations), however, if we measure our initial conditions inaccurately, trajectories in state space diverge (long term prediction is hard). For instance, if we had a weather system described by a chaotic system (e.g. group of differential equations), and measured temperatures with inaccurate (all sensors are somewhat inaccurate) and other variables for our initial conditions, we might be able to predict the weather for short term, but not very well for long term.
So, I wonder if the question really was not of random (nondeterministic) but of the predictability.
What the heck did I just write? I''m not going to check it over.
"... you act as if stupidity were a virtue."
-- Flight of the Phoenix
Pi isn''t a random number. It is an irrational number. If we can find this number by some mechanical means then it is deterministic. The digits of Pi can be found by Taylor Series (although it may take a long while
) So Pi is a deterministic, irrational number.There is chaos theory however. One finds sometimes in differential equations (for instance) with certain parameters, equations that are sensitive to initial conditions. It is completely deterministic (since we have our equations), however, if we measure our initial conditions inaccurately, trajectories in state space diverge (long term prediction is hard). For instance, if we had a weather system described by a chaotic system (e.g. group of differential equations), and measured temperatures with inaccurate (all sensors are somewhat inaccurate) and other variables for our initial conditions, we might be able to predict the weather for short term, but not very well for long term.
So, I wonder if the question really was not of random (nondeterministic) but of the predictability.
What the heck did I just write?
I''m not going to check it over."... you act as if stupidity were a virtue."
-- Flight of the Phoenix
Oh, I forgot to mention. I did think of trying to perform compression using chaotic systems and initial conditions. However, it turns out (from asking a professor if it was feasible), all the information would wind up in the initial condition (grumble). No compression is the result. I still haven''t spent much time convincing myself of this, but it seems reasonable.
"... you act as if stupidity were a virtue."
-- Flight of the Phoenix
"... you act as if stupidity were a virtue."
-- Flight of the Phoenix
Sorry that I have nothing to contribute to this conversation. I just wanted to say: cool.
Dark Lord Pi
Dark Lord Pi
whoa, you could use it for compression. Not sure if it would be worth it, but imagine everybody had like a gig worth of pi on their drives. To send data you just say where it starts and where it stops. For things not found in the stored pi you just assemble them from pieces, so you wouldn''t need so much either. Hmm maybe it could be like stored on a special pi card which your modem connects to. So you need to send like a full motion video at 1024x768? Each frame or part of a frame just becomes a lookup to part of pi. Hey why even use pi at all? Why not have it be the most optimal sequence of numbers, oh but that might be pi. You know I remember writing this in a dream like ten years ago when I was a little kid and didn''t know anything about programming. That is odd isn''t it?
Hmm if you have enough pi though.. the lookup to the section might be as long as the information itself. There''s probably someway you could optimize it, or perhaps you could do some crazy resolution thing, where it skips over detail bits depending on how precise the index is. So like you could download in whatever detail the line is currently supporting... hey that''s cool. So like if there is an internet tv show and everyone is watching the quality would degrade slightly, perhaps dropping frames or decreaseing resolution or color depth but everyone would still be able to watch fine. Grr got to go write a stupid english report.
Hmm if you have enough pi though.. the lookup to the section might be as long as the information itself. There''s probably someway you could optimize it, or perhaps you could do some crazy resolution thing, where it skips over detail bits depending on how precise the index is. So like you could download in whatever detail the line is currently supporting... hey that''s cool. So like if there is an internet tv show and everyone is watching the quality would degrade slightly, perhaps dropping frames or decreaseing resolution or color depth but everyone would still be able to watch fine. Grr got to go write a stupid english report.
While Pi is transcendental (a slightly stronger statement than Pi is irrational) it hasn''t been proven that every possible finite sequence of digits is a subsequence of Pi. In fact, there is a large class of transcendental numbers where it is intuitively obvious that they don''t contain every possible finite sequence.
Even given that Pi contains every possible finite sequence, in some random distribution the probability of matching a given sequence with random digits after n digits is approximately a poisson distribution with mean equal to the cardinality of the set of digits to the power of the length of the sequence. So for the case where we represent Pi by bytes, and trying to match a single screen shot of 1024x768x8bpp the distribution matches a poisson distribution of mean 2^6291456. This is about 10^1893916, for those of you who don''t think in binary. So the number that would index that would have to be about around 6291456 bits long. Which is 786432 bytes. Which is the same size as a screenshot of a 1024x768x8bpp if you just used a bitmap. Of course, a poisson distribution with a mean of 2^6291456 has a standard deviation of only 2^3145728, so it''s reasonable to assume that only two or three extra bits will be needed to resolve the index of the outliers on the far side, heck we might as well add a full 8 bits, so we''re back up to an integral number of bytes. (This would be the point where we''ve show that "compression" in a chaotic system requires all the information to be stored in the initial conditions.)
Of course in order to determine the index number of the screen-shot, you need to examine about 2^6291456 * 2^20 bytes. (More precisely 2^6291456 bytes 2^20 times.) Even if we assume a google processors (10^100 or about 2^332) each running at 100 Terahertz (10^14 cycles per second) it would still take about 2^6291130 seconds to find this index. In comparison a billion years is about 2^55 seconds.
If we assume DNA computing instead of traditional computers, it would take about 6,300,000 PCR cycles in order to finish the computation. The fastest I''ve heard of a PCR cycle done is 10 minutes, so it''d only take about 1200 years in order to complete. Of course we''d need a DNA chain of at least 1600 kilobases (dead minimum, realistically probably closer to 160,000 kilobases). I believe that''s about 1 nanogram of material. Let''s be conservative and call it 10^-24 grams. (If each base actually massed about as much as a hydrogen atom.) So after PCR the sample necessary would mass about 6.3 gigagrams. So about the same mass as a mountain, and that''s being conservative. I don''t even want to think of the size of the gel that you would need to do the separations.
So compression in this manner is will either take more time than the universe has existed, or requires you to do about a thousand years of lab work in a facility that generates it''s own gravitational fields. Even then, if it does come up with an answer, it probably won''t compress the data, and it still isn''t guaranteed to even find a "compressed" form.
Even given that Pi contains every possible finite sequence, in some random distribution the probability of matching a given sequence with random digits after n digits is approximately a poisson distribution with mean equal to the cardinality of the set of digits to the power of the length of the sequence. So for the case where we represent Pi by bytes, and trying to match a single screen shot of 1024x768x8bpp the distribution matches a poisson distribution of mean 2^6291456. This is about 10^1893916, for those of you who don''t think in binary. So the number that would index that would have to be about around 6291456 bits long. Which is 786432 bytes. Which is the same size as a screenshot of a 1024x768x8bpp if you just used a bitmap. Of course, a poisson distribution with a mean of 2^6291456 has a standard deviation of only 2^3145728, so it''s reasonable to assume that only two or three extra bits will be needed to resolve the index of the outliers on the far side, heck we might as well add a full 8 bits, so we''re back up to an integral number of bytes. (This would be the point where we''ve show that "compression" in a chaotic system requires all the information to be stored in the initial conditions.)
Of course in order to determine the index number of the screen-shot, you need to examine about 2^6291456 * 2^20 bytes. (More precisely 2^6291456 bytes 2^20 times.) Even if we assume a google processors (10^100 or about 2^332) each running at 100 Terahertz (10^14 cycles per second) it would still take about 2^6291130 seconds to find this index. In comparison a billion years is about 2^55 seconds.
If we assume DNA computing instead of traditional computers, it would take about 6,300,000 PCR cycles in order to finish the computation. The fastest I''ve heard of a PCR cycle done is 10 minutes, so it''d only take about 1200 years in order to complete. Of course we''d need a DNA chain of at least 1600 kilobases (dead minimum, realistically probably closer to 160,000 kilobases). I believe that''s about 1 nanogram of material. Let''s be conservative and call it 10^-24 grams. (If each base actually massed about as much as a hydrogen atom.) So after PCR the sample necessary would mass about 6.3 gigagrams. So about the same mass as a mountain, and that''s being conservative. I don''t even want to think of the size of the gel that you would need to do the separations.
So compression in this manner is will either take more time than the universe has existed, or requires you to do about a thousand years of lab work in a facility that generates it''s own gravitational fields. Even then, if it does come up with an answer, it probably won''t compress the data, and it still isn''t guaranteed to even find a "compressed" form.
Word!
Proof positive that programmers really do have nothing better to do.
Dare To Think Outside The Box
_____________________________
/____________________________/
http://www.inversestudios.com
Edited by - INVERSED on 5/1/00 7:36:02 AM
Proof positive that programmers really do have nothing better to do.
Dare To Think Outside The Box
_____________________________
/____________________________/
http://www.inversestudios.com
Edited by - INVERSED on 5/1/00 7:36:02 AM
I know our lab generates some kind of gravitational field. I just hope I don''t get sucked in for a 1000 years. 
"... you act as if stupidity were a virtue."
-- Flight of the Phoenix
"... you act as if stupidity were a virtue."
-- Flight of the Phoenix
Holy crap, I just got out of a lecture with one of the mathematicians that is considered an authority on pi. In any case, he clearly explained that it''s not random. It''s the same all the time. I guess there is some Japanese guy who''s memorized the first 40,000 digits of it (he holds the record), but I guess he has an advantage since the Japanese language makes each digit faster to speak (takes 6 hours for him to do it). Anyway, our guest lecturer did point out that there is a point at which we cannot compute any more digits of pi, and that''s hecka less than 10^51 digits. 10^51 is a pretty good estimate of how many elementary particles there are in the universe, and it is simply impossible to calculate and store that many digits with only so much matter.
Anyway, that''s the food for thought from me.
Pythius
Anyway, that''s the food for thought from me.
Pythius
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