Infiniti Q45

[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.

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?

And how annoying that Google thinks I want to buy a luxury car!!!!

[ewmayer]

Originally Posted by robert44444uk

And how annoying that Google thinks I want to buy a luxury car!!!!

Googling "Infiniti" returns 26,100,000 hits. I'm not sure if that should make us better about humankind or not. If not, perhaps we could sue the manufacturer for false advertising.

[mfgoode]

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.

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?

And how annoying that Google thinks I want to buy a luxury car!!!!

I dont see why you should guage the infinitude of human kind to a finite quantity of references of infinity. It just does not click logically.

Of course you can question my statement as I believe in eternal life- that it is eternal. I cant prove it scientifically though. Its a belief in Faith of the Word.

The universe is not bounded. Would you call a sphere bounded ? Where is the boundary ? And thats only in 3 dimensions. 4 dimensions will be even more interesting I can assure you.

Mally

[axn]

Originally Posted by robert44444uk

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?

Just as well. Google will show you only the first 1000 anyway

[xilman]

Originally Posted by mfgoode

The universe is not bounded. Would you call a sphere bounded ? Where is the boundary ?

Boundedness and finiteness are independent quantities. Objects can exist which can have either, both or neither of these characteristics.

In one dimension, circle is finite but unbounded. A line open at both ends is infinite and unbounded. A line segment with one end point is infinite and bounded. A line segment with two end points is finite and bounded.

As you point out, a sphere is unbounded. It is, however, finite.

Observational evidence suggests that the universe is probably bounded and infinite. The boundary for which the best evidence exists is the Big Bang, where *all* spacetime forms a singularity analogous to the end of a line segment. Recent measurements of the expansion of spacetime and theoretical models of black holes suggests that the future is unbounded with no long-lasting singularities.


Paul

[mfgoode]

Originally Posted by xilman

In one dimension, circle is finite but unbounded.
Paul

Could you please explain and clarify why this is so?

Mally

[ewmayer]

Originally Posted by mfgoode

Could you please explain and clarify why this is so?

You find yourself on a circular walkway. Staying on the walkway, start walking. Stop when you reach the end.

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.

(At least in general; the topological caveats on this are beyond the scope of this discussion, but are out there on the web, for the interested or merely mathematically masochistic.)

[xilman]

Originally Posted by mfgoode

Could you please explain and clarify why this is so?

Mally

Ernst has given a reasonably rigorous explanation. Here's a more picturesque explanation.

Pick a circle large enough to walk around, then step on to it with a pot of paint and a paint brush in your hand. Walk around the circle, backwards by preference or you'll get your feet messy, and paint the circle where you've been. After a little while the entire circle is painted, using a finite amount of paint. The circle is therefore finite (to be more precise, it has finite length) but you have not yet come across a boundary, nor will you ever do so because after going around once without finding a boundary you are in exactly the same situation as when you started (except, possibly, by having paint on your feet).

The analogous situation in two dimensions would have you paint the entire surface of a sphere with a finite amount of paint without ever finding an edge to the sphere.


Paul

[mfgoode]

Thank you both Ernst and Paul for giving me a down to earth clarification on my query.

Whereas I understand that a circle is boundless, I am confused in which least dimension it can exist. I was always under the impression that a line requires just one dimension (linear) with one direction of freedom for it to be drawn. The second direction of freedom would put it into the second dimension.

I have checked my 4 dictionaries of maths and surprisingly find there is no strict geometrical definitions for lines and circles (and discs). They say there are only intuitive definitions, for the straight line at least.

Then again a circle can be traced with one direction of freedom but its a planar curve and so can only be generated where there is a plane and that's 2dimensions A sphere can only exist in 3 minimum dimensions I consider only 3dimensions.

If we consider it algebraically in The equations for a straight we have the exponent as 1, (y=mx + c). I take that to be one dimension. The circle eqn is x^2 + y^2 = r^2 and so I take that to be 2 Dim.

Similarly the eqn of the sphere has an exponent of 3, hence it is in 3 dim. The area of the surface of the sphere is in 2 dim. though I would say to generate it 3 dim is required. If we take a semi circle (2 dim) and rotate it we require the 3rd dim to form a sphere.

In both your posts I find that you have gone down one dim less. i.e 'circle' 1 dim.. 'open ball' 2 dim. This is what I find very confusing so please clarify and let me know where I have gone wrong, unless 0 is a dimension in which case the dim.1 will be the 2nd. dim and so on.

A point was made about open and closed discs. We have adequately thrashed this out in a former thread and I wont to labour on it here.

For the record an open disc is the set of points such that x^2 + y^2 < r. That is the points are all inside except on the circumference. For the closed disc the points are also on the circumference i.e. x^2 +y^2 <= r^2.

Thank you once again for bearing up with me,

Mally

[Wacky]

Originally Posted by mfgoode

Similarly the eqn of the sphere has an exponent of 3, hence it is in 3 dim.

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.

[xilman]

Originally Posted by mfgoode

:If we consider it algebraically in The equations for a straight we have the exponent as 1, (y=mx + c). I take that to be one dimension. The circle eqn is x^2 + y^2 = r^2 and so I take that to be 2 Dim.

As I thought, you are confusing dimensionaility with exponents in the govenrning equation in Cartesian coordinates.

A circle in radial coordinates with origin at the center of the circle is r=constant. A linear equation and an extremely simple one. However this may also be misleading as, once again, you may be tempted to confuse dimensionality with an exponent in a particular coordinate system.

When a mathematician speaks of the dimensionality of a geometric object (at least for relatively simple ones, we'll ignore pathological cases such as fractals and disconnected sets of points for the moment), what is meant is the minimum number of quantities required to specify any point on the object. In the case of a circle, you need only one number --- the distance from a specific point (the place where you started painting) to the point in question. A circle is thus one-dimensional. That's not to say you can't use more numbers if you wish --- your x and y coordinates for instance --- but the important point is that you don't need to.

To specify a location on a sphere you need two numbers ---- one is not sufficient. Conventionally the numbers are the latitude and longitude, but there are many other reasonable choices --- the (x,y) coordinates of the location on a Mercator projection, for instance.


Paul