3.14159..... Where does it exist in the 4 -D universe?
It seems to me that circles can only exist as abstractions in a make-believe 2-D universe (?) In the reality of a 4-D universe circles are 3-D spirals embedded in time, If Pi was a real relationship, would it not be a real number?
Pi is a real number, defined as the ratio of a circle's circumferenceC to its diameterd. Since the real numbers are a field, this ratio is also a real number.
Pi can also be defined in terms of limits (http://functions.wolfram.com/Constants/Pi/09/). By a property of real numbers from analysis (http://en.wikipedia.org/wiki/Real_number#.22The_complete_ordered_field.22) real numbers are a complete ordered field, so these limits converge to a real number, which is pi. Hence pi exists as a number. Looking at the definition of abstraction, "something that exists only as an idea", you can call pi an abstraction if you aren't a Platonist.
Circles are sometimes called a "2 dimensional ball"; the set of all points in R^2 a set distance from a center point. You also have 4 dimensional balls, which is just the set of all points in R^4 a set distance from a center point. I'm not sure how spirals and circles are related in your thought, but that is the closest thing I could come up with to having a 4 dimensional circle.
Looking up the definition of embedding, In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup, I have trouble seeing how you can embed a 3-d spiral in time, which is 1-dimensional.
I'm assuming your spiral looks a bit like this. We could smash that down into two dimensions, I suppose.
I might not just imaginative enough. Can you put in more rigorous terms what you're talking about? Maybe define some terms, write out your embedding, etc.
If analysis can be formulated with paraconsistent logic, I don't see how it could make a difference to his question's mathematics. I'm not exactly sure what he is asking though.
If analysis can be formulated with paraconsistent logic, I don't see how it could make a difference to his question's mathematics. I'm not exactly sure what he is asking though.
There is no speaking of any inconsistency, so I do not see why one should or would go into paraconsistency? Either way, it should, tho one might require more strict premises, lead to the same conclusions assuming we never exploited any inconsistencies into explosion to begin with. (Seems a rather fair assumption in this case...)
To the direct topic : Simplest definition of Pi is in 2D, yes. However, firstly, there's the generalisation in practically any dimension, including possibilities for definitions of Pi in these dimensions. http://en.wikipedia.org/wiki/N-sphere
Then, there's the fact that Pi shows up much more frequently than simple circles. For example, the gamma function, being the analytical complex extention of the factorial (!) function has value (root(Pi)) at 1/2. Please, tell me where the circle comes in? http://en.wikipedia.org/wiki/Gamma_function
Define exponential functions over the complex numbers through series in any dimension, including matrix-exponentials. They'll lead to Pi, yet I don't see anyone mentioning any circles. (Yes, one can relate circles and polar coordinates in this case. In any, general dimension, however, including 4D.) http://en.wikipedia.org/wiki/Exponential_function.
Definition is no more than a series, it leads right to Pi, tho you do need complex numbers and can relate this to circles, this makes no matter in the dimensional discussion. Exponentials can be related to Pi in any case which is more than 2D (over R, 1D over C)
NO WONDER SOME OF YOU HAVE TROUBLE SEEING THE BEAUTY IN MATHS. Pi is far more common than you'd think. it's not just circle-related. Open your eyes.
@DeadonDreaming Jumping Geometry, Batman! You're an example of math education gone right. Have a philosopher's cookie: baked in the fires on the other side of the sun, cooled behind Mars, and delivered by an unseeable light-red fantastical quadruped that only seems conscious, it is truly a treat that can certainly be mentally conceived of, but for which no compelling evidence can be found.
There is no speaking of any inconsistency, so I do not see why one should or would go into paraconsistency? Either way, it should, tho one might require more strict premises, lead to the same conclusions assuming we never exploited any inconsistencies into explosion to begin with. (Seems a rather fair assumption in this case...)
To the direct topic : Simplest definition of Pi is in 2D, yes. However, firstly, there's the generalisation in practically any dimension, including possibilities for definitions of Pi in these dimensions. http://en.wikipedia.org/wiki/N-sphere
Then, there's the fact that Pi shows up much more frequently than simple circles. For example, the gamma function, being the analytical complex extention of the factorial (!) function has value (root(Pi)) at 1/2. Please, tell me where the circle comes in? http://en.wikipedia.org/wiki/Gamma_function
Define exponential functions over the complex numbers through series in any dimension, including matrix-exponentials. They'll lead to Pi, yet I don't see anyone mentioning any circles. (Yes, one can relate circles and polar coordinates in this case. In any, general dimension, however, including 4D.) http://en.wikipedia.org/wiki/Exponential_function.
Definition is no more than a series, it leads right to Pi, tho you do need complex numbers and can relate this to circles, this makes no matter in the dimensional discussion. Exponentials can be related to Pi in any case which is more than 2D (over R, 1D over C)
NO WONDER SOME OF YOU HAVE TROUBLE SEEING THE BEAUTY IN MATHS. Pi is far more common than you'd think. it's not just circle-related. Open your eyes.
I think this gives him more than enough information on Pi to pretty much satisfy his appetite (har har). Did you answer his question? I'm still not sure what he is asking.
@DeadonDreaming Jumping Geometry, Batman! You're an example of math education gone right. Have a philosopher's cookie: baked in the fires on the other side of the sun, cooled behind Mars, and delivered by an unseeable light-red fantastical quadruped that only seems conscious, it is truly a treat that can certainly be mentally conceived of, but for which no compelling evidence can be found.
I think this gives him more than enough information on Pi to pretty much satisfy his appetite (har har). Did you answer his question? I'm still not sure what he is asking.
Whelp. "Where does it exist in the 4 -D universe?"
-Answered for any dimension higher than dimension 2. I'm actually not 100% sure on dimension one talking in terms of practical use. The series-definition works, ofcourse, but one could argue that series don't quite flow forth naturally and show the same ammount of elegance without it's higher dimension applications. A function's representation requires at least 2 dimensions, after all.
"In the reality of a 4-D universe circles are 3-D spirals embedded in time"
Implying linear 4D universe, but acceptable, however not a question. "If Pi was a real relationship, would it not be a real number?"
Pi is a real number... Question does not pose itself. Perhaps he meant rational, in which case I'd just answer 'No'
"If Pi was a real relationship, would it not be a real number?"
Pi is a real number... Question does not pose itself. Perhaps he meant rational, in which case I'd just answer 'No'
Whelp. "Where does it exist in the 4 -D universe?"
-Answered for any dimension higher than dimension 2. I'm actually not 100% sure on dimension one talking in terms of practical use. The series-definition works, ofcourse, but one could argue that series don't quite flow forth naturally and show the same ammount of elegance without it's higher dimension applications. A function's representation requires at least 2 dimensions, after all.
"In the reality of a 4-D universe circles are 3-D spirals embedded in time"
Implying linear 4D universe, but acceptable, however not a question. "If Pi was a real relationship, would it not be a real number?"
Pi is a real number... Question does not pose itself. Perhaps he meant rational, in which case I'd just answer 'No'
Ah. I don't think he'll be happy by what you consider the universe though. I'm not exactly sure what he means by that, though I think the way its been explained in this thread is as close as you'll get to showing where it "exists in the 4-d universe". You're showing it exists in a 4-dimensional object, but I don't think its the object he wants, which is why I think he needs to respond with more information.
While I enjoy the pi = C/d relationship, I find many others even more enjoyable. And, fortunately for a discussion like this, many of those other relationships are geometry independent. My own understanding of pi is more along the lines of this...
Those sorts of things make far more sense to me, when trying to understand pi, than geometry examples (though I've come to have a deeper appreciation for the geometry cases the older I've gotten and the longer I've had to consider all these things).
3.14159..... Where does it exist in the 4 -D universe?
It seems to me that circles can only exist as abstractions in a make-believe 2-D universe (?) In the reality of a 4-D universe circles are 3-D spirals embedded in time, If Pi was a real relationship, would it not be a real number?
yes, the difficulty is in the comprehension (i avoid the word definition) of what illusion is. pi may be an illusion, but it exists and functions within the world of math, which is the world of cognition, which must also be illusion as a whole, if pi is illusion. so relatively speaking it only makes sense, that people who are attached to math-cognition will deny the notion that pi, of all things, might be an illusion (as in "standing out as something weird"). integrating math with perception is apparently something that humanity hasn't achieved yet.
"The decimal expansion of an irrational number never repeats or terminates, unlike a rational number."
You can prove by contradiction, as Rudin does at the end of Chapter 1 of his real analysis book, that this is true.
Since Pi can be defined in a number of different ways, including a limit definition (which through another proof in Rudin, since the real numbers are complete, we know that it must be a real number) or a ratio definition (since the real numbers are a field, it must again be a real number), it is a real number with an infinite decimal expansion.
You're going to have to define what you mean by reality. Most Platonists would state that mathematics is "more real" than things we perceive in the every day world.
yes, the difficulty is in the comprehension (i avoid the word definition) of what illusion is. pi may be an illusion, but it exists and functions within the world of math, which is the world of cognition, which must also be illusion as a whole, if pi is illusion. so relatively speaking it only makes sense, that people who are attached to math-cognition will deny the notion that pi, of all things, might be an illusion (as in "standing out as something weird"). integrating math with perception is apparently something that humanity hasn't achieved yet.
I think it matters what you mean by real. All numbers are imaginary in a sense; you can't go into a museum and touch a number. Platonists believe we live in a world of shadows, and that numbers inhabit a realm that is more real than the one we live in. They're eternal and pure.
How 'real' is any relationship? Can relationships be sensed in any manner? Human relationships have been described as the "Space between us", so what kind of 'space' do relationships, such as Pi exists - certainly not traditional time/space - for there they would be directly observable (?)
It does seem to be something of a paradox, for we have ordered the chaos of the universe using this cognitive heuristic we have named, relativity.
However, instead of "every thing is relative" What if "No thing is relative"?
Relativity may be a mere human attribution rather than a deterministic causality.
Is not the perspective that encompasses relativity and relationships, necessarily, a transcendental one?
Is it not true that the 2-D relationship of Pi actually does not exist in two dimensions, but only in three (or More)?
It seems to me that attributing Pi is very much like seeing around a corner...
@DeadonDreaming I was actually thinking of spirals as viewed from the top, at right angles to the illustrations one provided, along the axis.
I would contend that it would be difficult to prove that Time exists independently as a single dimension, so that the concept of 'being embedded within time" is not without merit.
I started this thread because I was thinking about Fourier transforms and wondering if Pi, as the relationship between amplitude (diameter) and frequency (circumference), could be said to exist as a constant. My initial conclusion was that it could not, because all 2 - D representations are figments of imagination, mere abstractions of reality.
Fourier transforms, on the other hand, seem to be able to represent reality in four dimensions... (?)
but 4D is a figment of imagination as well. the differentiations of dimensions are made up by us. no concept is capable of representing reality. ever. [.....] we just need to understand how our theoretical 'spaces' relate to each other.
How 'real' is any relationship? Can relationships be sensed in any manner? Human relationships have been described as the "Space between us", so what kind of 'space' do relationships, such as Pi exists - certainly not traditional time/space - for there they would be directly observable (?)
I think you're confusing relativity with relativity; Einstein's theory of relativity with the every day or mathematical notion of relativity. I'm not sure really.
I would contend that it would be difficult to prove that Time exists independently as a single dimension, so that the concept of 'being embedded within time" is not without merit.
I can prove it. Let time be a dimension in Minkowski space, as it usually is. Since Monkowski space is a real vector space, time exists independently as a single dimension. Q.E.D.
I think that is how it goes. I'm not familiar with very much physics, but looking at the wikipedia for Minkowski space leads me to believe that time is simply a dimension within Mikowski space. " An orthonormal basis for Minkowski space necessarily consists of one timelike and three spacelike unit vectors."
You might accuse me of mistaking the map for the territory, but then I don't really think anyone can answer your question, since any talk of time will have to be about an abstraction of it they've formed in their own mind.
The rest of your post didn't really make sense to me, because I'm not very familiar with physics.
3.14159..... Where does it exist in the 4 -D universe?
It seems to me that circles can only exist as abstractions in
a make-believe 2-D universe (?) In the reality of a 4-D universe circles
are 3-D spirals embedded in time,
If Pi was a real relationship, would it not be a real number?
=. What is 4 -D universe? It is 2-D Pseudo-Euclid's universe. What is the 2-D Pseudo-Euclid's universe ? It is Nothingness, Vacuum, Zero- Gravity continuum: T=0K. And there are laws of physics that say: at T=0K the particle must have flat - circle geometrical form: pi = 3.14159..... Pi = 3.14159..... is the real number in the universe. =.
socratus
Once again, I believe that is an illusion. Pi seems to be the relationship between a circumference and a diameter in 2-D, but it is actually the relationship of two identical circumferences at a 90 degree orientation in three dimensions.
Pi is a measure of abstraction, a reflection of the mental gymnastics necessary to convert a plane into a volume and the converse, a volume into a plane.
Divide the 'universe' by Pi and what model results - that of Einstein's?
I started this thread because I was thinking about Fourier transforms and wondering if Pi, as the relationship between amplitude (diameter) and frequency (circumference), could be said to exist as a constant. My initial conclusion was that it could not, because all 2 - D representations are figments of imagination, mere abstractions of reality.
Fourier transforms, on the other hand, seem to be able to represent reality in four dimensions... (?)
What you're saying makes some sense. Tho i'd argue formal mathematical fourier transformations go from one complex plane to the next, so they're not inherently 4D. However, again, 2D ideas are easily expanded to higher dimensions, 2D is part of 3D, after all. (and so on.) (Atleast when we remember that the dimensions we're talking about are vectorspaces.)
Once again, I believe that is an illusion. Pi seems to be the relationship between a circumference and a diameter in 2-D, but it is actually the relationship of two identical circumferences at a 90 degree orientation in three dimensions.
Pi is a measure of abstraction, a reflection of the mental gymnastics necessary to convert a plane into a volume and the converse, a volume into a plane.
Divide the 'universe' by Pi and what model results - that of Einstein's?
Aaaaanndd i've lost you entirely. Pi is more than just 'the relationship between a circumference and a diameter'. It comes back in a LOT of places, because they're mathematically connected. See the beauty in it, not the illusion. In the end, pi is nothing but converging series.
Also, divide the universe by pi? What? Eh. What? Devide every number by pi? You've rescaled everything, which means you've effectively done ... nothing? Honestly, that last sentence makes no sense to me whatsoever.
This feels like someone getting crazy ideas because the definition of Pi which you learn at age 8 and the definition of Pi which you learn at age ... 18? are hard to relate to one another...
Once again, I believe that is an illusion. Pi seems to be the relationship between a circumference and a diameter in 2-D, but it is actually the relationship of two identical circumferences at a 90 degree orientation in three dimensions.
Pi is a measure of abstraction, a reflection of the mental gymnastics necessary to convert a plane into a volume and the converse, a volume into a plane.
This doesn't make any sense. Divide the universe by pi? Division has a very rigorous definition from abstract algebra, and since the 'universe' isn't a number, what do you mean by this? Can you show me how you're dividing it, or are you just saying weird stuff?
Reading the wikipedia article on division, it seems there isn't any special division that allows you to divide objects in the real world by real numbers. All division seems to be just as it is defined in a basic abstract algebra course, which leads me to believe you're trolling.
I don't think you have any idea what you're talking about, or you're using words that have specific meanings in ways they aren't meant to be used.
Right now, give me the exact transformation you're using to convert a volume into a plane. First off, any object with volume can not be mapped into a plane, so that is out of the picture. Spheres are not developable surfaces under any metric as they cannot be unrolled onto a plane. Converting a volume ( a number ) to a plane just doesn't make any sense.
I can't imagine why you see me as an illusion. I am as real as you are. Are you ratio-nalizing? Go figure.
You may think you can't get my number but you can count on me even though I'm not always presently round about. I am around now.
To be more exact, my wife and I were cruising around the Adriatic Sea arcing part of the globe but am back now full circle. Any illusions about this are rationalizing.
Okay, to clarify. There really are spheres moving through space in 4-D reality. However there are not any 3-D spheres frozen in time, in status or any 2-D artifacts/circles orbiting a gravity well in the reality of the objective universe.
The reduction of real phenomena to dimensions less than four (or five?) is a popular means of analysis via reductionism. These figments of imagination, models of less than four dimensions exist only in a subjective universe.
If one was walking in the woods and encountered what seemed to be a circle with a diameter just floating in space - what 4-D object could it possibly be?
The diameter would almost have to be a plane viewed edge on. Of the possible finite planar shapes of a diameter is that of a circle. So one could be looking at a sphere, as two circles, a circle 'within' a circle (or a Wheel within a Wheel) as 3-D object in status.
The relationship of the circles is one to one, so where does Pi come from but from the unique perspective where a plane is temporarily seen as a diameter? Is not Pi a relationship that indicates a reduction of reality has occurred via a dimensional transformation of a model?
As far as the phrase 'divide the universe by Pi" I was attempting to share the idea of folding dimensions/unfolding dimensions...
Think of a generalization of dimension and call it, "variable." There are many variables. When we move in space-time, we experience heat, light, pain, oxygen in the air, etc. All of these are variables and we are not aware of them all. We are especially not aware if they remain constant. If they remain constant, for all practical purposes, they don't exist. But let pain increase, or the lights go out, or it gets to 105 degrees F, and we notice. If we have never experienced a variable at such an altered state, we don't know how to deal with it.
Reasoning to a special case, those living in a two-dimensional world who unexpectedly experience a reality of three-dimensions are not going to know how to deal with it. The illusion (non-existence) of three-dimensions has turned into a reality.
Idae, that just "proves" Pi<4, since you haven't proven it will converge to the circumference of the circle. Method of exhaustion; you're missing a portion of the proof.
Idae, that just "proves" Pi<4, since you haven't proven it will converge to the circumference of the circle. Method of exhaustion; you're missing a portion of the proof.
Oh what about 'pi', the ratio of the circumference of a circle to its diameter. And you can never come to the end of pi. There is no creature in the universe, no matter how smart, who can calculate pi to the last digit_because there is no last digit, only an infinite number of digits. Mathematicians have made the effort. Let's say it is in the ten-to-the-twentieth place_something happens. The randomly varying digits disappear and for an unbelievably long time there's nothing but ones and zeros. Suppose the zeros and ones suddenly stop and you get back to the random sequence of digits. What one could say is maybe there is a message from some advanced civilization coded in there, using prime numbers. Kind of a universe communication using numbers/mathematics. A point, pi has the same value all over the whole universe. And it's all built into the fabric of the universe. It is all about zeros and ones.
And the first beings that discovered pi, have been waiting billions of years for us earthlings, for 10 fingered mathematicians and fast computers to come along.
Maybe this message was addressed to us earthlings.
"Do we, holding that the gods exist,
deceive ourselves with insubstantial dreams
and lies, while random careless chance and
change alone control the world?
_Euripides
Hecuba
Pi, let’s see the diameter of a circle exists, the circumference also exists. How could the ratio be irrational? Same problem with a right angle triangle with two side of length one, the hypothesis also exists and yet it is irrational.
How is length measured? You compare it to a known standardized length. If I were to measure theabsolutely smallest length measurable you would be unable to do so, as you would have nothing to compare to but itself. At its smallest length is relative so Pi is not so irrational after all.
Pi, let’s see the diameter of a circle exists, the circumference also exists. How could the ratio be irrational? Same problem with a right angle triangle with two side of length one, the hypothesis also exists and yet it is irrational.
How is length measured? You compare it to a known standardized length. If I were to measure theabsolutely smallest length measurable you would be unable to do so, as you would have nothing to compare to but itself. At its smallest length is relative so Pi is not so irrational after all.
You're comparing measurement errors to theoretical (and thus idealistic) maths. I'm sorry to dissapoint you, but what you have proven is that whether or not Pi is irrational does not matter in reality. You have not proven that it isn't irrational (which would be wrong, it has been proven that it is, same for sqrt(2)).
Since, in maths, both lenghts could be -defined- as being 1, measurement errors do not occur. Even more, while figures and easy definitions certainly help your understanding of the matter, we do not find further decimals of Pi or sqrt(2) through measurement... (Series, integrals, converging rows are all far more appropriate to come to an approach of an irrational number.)
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) as 3-D object in status. 
