Infiniti Q45

[jinydu]

One can also regard the unit circle as the set of all points in [0, 2pi], provided that we identify 0 and 2pi as the same point.

[m_f_h]

Originally Posted by robert44444uk

If you type in infinity into Google you get the result:
Results 1 - 10 of about 43,600,000 for infinity
This is a disappointing reality of the finite nature of humankind, in my humble opinion.

Well, maybe merely the finite nature of computer storage media connected to internet ?
Also, note that you have to end the question in "=" in order to force mathematical evaluation in case of ambiguity.

However, I can't be bothered to click through to the last of the 43,600,000 references, or maybe there is a surprise at the end of this bounded universe?

The "last" just refers to some humanly arbitrarily chosen partial(!) preorder relation.
Also, surprises are usually hidden somewhere (of course not exactly) in the middle and not necessarily at one of the extremities : even if it's the farest away, it would be easier to find, thus contradicting to several basic principles of statistical thermodynamics and computer science (e.g. Murphy's law).

[m_f_h]

Originally Posted by jinydu

One can also regard the unit circle as the set of all points in [0, 2pi], provided that we identify 0 and 2pi as the same point.

[0,2pi] is the same than [0,1] etc. Anyway all of this refers to the same manifold.

Originally Posted by xilman

In one dimension, circle is finite but unbounded.

I think you mix up "unbounded" and boundary-less.

More mathematically: a circle is the boundary of an open ball in 2-D. (i.e. an Open disc). The open ball has finite volume (area), hence its closure is also finite. As you can see from the subtext under my username, the boundary (that's the outer of the 2 deltas) of a boundary (that's the "delta omega") is the empty set.

A bound is not at all the same than a boundary.
The first has to do with POsets, the second with topology.

Finally, I'd rather suggest that the boundedness of the universe remains an open question so far.

[xilman]

Originally Posted by m_f_h

I think you mix up "unbounded" and boundary-less.

I don't think so. I was using the terms in the sense used earlier in the thread for continuity and to avoid introducing another possible cause of confusion.

Further, the word "unbounded" is used in precisely this same sense in cosmology, which leads me on to:

Originally Posted by m_f_h

Finally, I'd rather suggest that the boundedness of the universe remains an open question so far.

As I said, the best available observational evidence suggests a boundary at the big bang and no boundary in the future. Surely, that's (a) an accurate summary of present knowledge and (b) acknowledges that the definitive answer is not yet known.


Paul

[mfgoode]

Originally Posted by Wacky

You must be reading a different book from the one that I do.

For a sphere, I have with the equality replaced by < or ≤, as needed, depending on whether you are speaking of the shell or the volume enclosed, etc.

It is not the exponent, but the number of degrees of freeedom that relates to the dimentionality.

Thank you both Richard and Paul for giving me a keen insight into dimensions and exponents.

I was thinking of the areas, such as length (0 area but length one dimension)

Area of circle *pi*r^2. Area of sphere 4*pi*r^2. Area of square a^2

And volume of cube a^3, Volume of ball 4/3*pi*r^3.

There seemed to be some relation between exponents and dimensions at least where areas are concerned

The next logical question : what would be the volume of a 4D cube (tesseract). Would it include the exponent 4 in its formula? From the net I get it is of R^4.

Why I ask is that in my opinion after 3D solids we go onto 4 dimensions. Here our quest branches in to different geometries not necessarily Euclidean. This is a lacuna I would like to plug.

In browsing the net I find that ewmayer has also explained it well and used the correct terminology.

["Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1996, p. 21). As a result, geometers call the surface of the usual sphere the 3-sphere, while topologists refer to it as the 2-sphere and denote it ".

"Regardless of the choice of convention for indexing the number of dimensions of a sphere, the term "sphere" refers to the surface only, so the usual sphere is a two-dimensional surface. The colloquial practice of using the term "sphere" to refer to the interior of a sphere is therefore discouraged, with the interior of the sphere (i.e., the "solid sphere") being more properly termed a "ball." ]

Regards the equation Richard gave of the sphere this is where the < or = comes in for the surface and the ball. The same is for circles which is the difference between the circle itself and the disc.

Milli Gratia, Ernst, Paul and Richard.

Mally

[m_f_h]

well, maybe it's a matter of definition.
It's surprising how even (relatively reasonable, I'd say) mathematicians can disagree about such basic questions:

Originally Posted by xilman

In one dimension, circle is finite but unbounded.

To me a circle is the set of points having a given distance from a chosen origin, which is (as I see it) usually infinite, but bounded.

So it's exactly the opposite of your statement...

[ewmayer]

I think Paul meant "finite but unbounded" in the sense of "having finite extent but no boundary", i.e. finite length but no endpoint in the case of a circle.

[m_f_h]

Originally Posted by ewmayer

I think Paul meant "finite but unbounded" in the sense of "having finite extent but no boundary", i.e. finite length but no endpoint in the case of a circle.

d-d-did he ?

PS: hey, since any of my profound philosophical discoveries are immediately flooded by trivialities, I'll simply re-edit it hereafter. --
It's surprising how even (relatively reasonable, I'd say) mathematicians can make seemingly contradictory statements about things that one would never suspect to have an ambiguous definition.

Originally Posted by xilman

In one dimension, circle is finite but unbounded.

Originally Posted by m_f_h

To me a circle is the set of points having a given distance from a chosen origin, which is (as I see it) usually infinite, but bounded.

These statements seem to be manifestly contradictory, while both of them can be considered to be true...
(Reminds me of some other "fuzzy truth" thread...)

[mfgoode]

Originally Posted by m_f_h

d-d-did he ?

PS: hey, since any of my profound philosophical discoveries are immediately flooded by trivialities, I'll simply re-edit it hereafter. --
It's surprising how even (relatively reasonable, I'd say) mathematicians can make seemingly contradictory statements about things that one would never suspect to have an ambiguous definition.


These statements seem to be manifestly contradictory, while both of them can be considered to be true...
(Reminds me of some other "fuzzy truth" thread...)

Naturally m_f_h !

As I see it linearity is the dimension of a straight line i.e. it only has length but no area. Lets call it one dimensional. Its equation is X=0 or Y=0 in one dimension; y=mx + c in two D as it lies in between the X and the Y axis but is still linear!

One cannot visualise a circle in one dimension. At most we can study its projection from its two dimensions on to the visualised straight line and that will be a straight line.

This straight line can be finite with two end points (bounded) or one point, and an unreachable second end and we call it unbounded, which means it cannot be measured along the visualised line. Also it if it has no end points no matter how far you go it is also unbounded in the one dimension.

A bounded straight line is the length of part of a circle with an infinite radius
if you want to bring a circle into the definition. We cannot visualise an infinite circle only parts of it and these are straight lines.

To draw a str. line you have one direction of freedom say the X axis. For a circle you require two axis both the X and Y axis and so on as it has area pi.r^2

A finite circle has finite area but no limits to its end as one simply goes 'round and round the mullberry bush' in the words of an old poem

If you can visualise a ball in the third dimension, as ewmayer puts it then it(the circle) is the circumference formed by the intersection of the ball or sphere by a plane This is a true circumference and not the disk which is inside the ball.

Please note that a ball is the full content whereas a sphere is only the sur face.

No fuzzy logic there!

m_f_h you said something of boundary and boundedness ? Kindly clarify the difference if you care.

Mally

[ewmayer]

Originally Posted by m_f_h

To me a circle is the set of points having a given distance from a chosen origin, which is (as I see it) usually infinite, but bounded.

So it's exactly the opposite of your statement...

The set of points is infinite, but the resulting curve is not.

[robert44444uk]

Just taking a view on where we are... the direction of the thread is a little random, and that is a good thing.... but..... I am particularly interested in the concept that, we as humans, cannot deal with infinity except in a conceptual (mathematical) fashion, and that relatively small numbers, 4.36*10^7 for example, are big, big numbers for everyone.

Take Bill Gates' wealth.... quoted in cents....even he got bored with this small number and created a useful foundation for the rest of the world to benefit from. (And I am not a big fan of the Gatester, but I would hope to do the same thing as he, should such luck befall me)

Out here in Bangladesh, they have the lovely numbering system, lacs and crores. A lac is 10^5 and a crore 10^7, but nothing above is commonly used, as it is beyond everyday occurrence. Steal a crore taka from the people and you will make the front page headlines here, as many politicians and businessmen have found to their surprise recently. But a taka is a small amount, and a rickshaw driver will laugh if you offer him only 12 taka for a ride. (don't worry, I pay more!!!) But have a look at http://en.wikipedia.org/wiki/Indian_numbering_system for those inflationary days still to come.

As a human, I am always amazed at infinity. I remember I had conceptual problems not very long ago in realising that the set of factors of the integers 2^n-1, n from 1 to infinity, (an infinitessimally small group of integers) contained all of the prime numbers. Hilbert's hotel and all of that.

In any case there seems to me to be a smaller infinity, that defined by human endeavour, and that is quite a small quantity in the scale of things. So we will pride ourselves on finding small primes of the order 2^n-1 where n is 30 million or so, and think it has taken all of the last 4.5 billion years to find this. How the infinity god (small g) must be laughing at our puny efforts!!!!