4-space anyone?



#15

I initiated the "0" and the "inf." threads. Wow, the "inf." finally halted at 100 responses!

There was one thing that never came up. Some mathematicians went into "special analysis" with this particular view: the entity "d" or "iota", d = 0.000...001. In other words infinitesimally small. And 0.999... + d = 1. Now, if someone insists that d = 0, well it would still be OK, since one could say that 0.999... + 0 = 1. Or?

I know one retired mathematician who doesn't disagree with 0.999... = 1, but he says that 0.999... + d = 1 is more correct, since it eliminates certain doubts that some have about it.

Anyway, since I first read "Flatland" many years ago, I've been fascinated with the concept of 4-space. I'm quite familiar with the tesseract, a similar figure for a tetraeder, etc. I have been wondering about a 4-space sphere. Can it be two spheres which coincide? Anybody know? I've been searching Wikipedia for an answer.


#16

Quote:
And 0.999... + d = 1. Now, if someone insists that d = 0, well it would still be OK, since one could say that 0.999... + 0 = 1. Or?

I know one retired mathematician who doesn't disagree with 0.999... = 1, but he says that 0.999... + d = 1 is more correct, since it eliminates certain doubts that some have about it.


You may find this interesting:
http://news.slashdot.org/story/10/10/14/135219/Proving-0999-Is-Equal-To-1

#17

Or this:

WE ALREADY KNOW ABOUT THE .999 --> 1 THING (and 0/0 etc)

#18

May it be that I'm too naive (mathematically speaking) but I've always found 0.99999...=1, so whatever (different from 0) you add to 0.99999... will make it also different from 1.

1/9 = 0.1111111....

n·1/9 = 0.nnnnn.....

9·1/9 = 0.99999.... = 9/9 = 1

Without any advanced math background, and just by commom sense, concepts like infinity, zero, i (sqrt(-1)), d (infinitesimal), and the like, are certainly unvaluable tools which allow mankind to deal with several aspects of the physical universe and its behaviour. However it is my opinion that many such abstract concepts haven't got a realworld counterpart.

Mathematics are the most powerful tool set to help us in understandig that real world by showing us a "simplified" sight. The same way a tomography can halp us to "see" the internals of our bodies.

Likewise a tomography "is not" a body, a mathematical model "is not" the real world.

Universe is what "really" exists... (regardless we are or not aware of any -or most- of its existance). Math is a rational artifact (built by us humans) in an attempt to make some parts of the universe to fit into the little (I really mean *little*) amount of "intelligence" inside our brains.

As stated above, may I'm just too naive... :-)

Cheers.


#19

Agree.

#20

Quote:
I have been wondering about a 4-space sphere. Can it be two spheres which coincide? Anybody know?

Nobody knows, at least no human body, since we lack a sense for the fourth dimension like flatlanders lack a sense for the third. It certainly cannot, however, be "two spheres which coincide" since these are 3D bodies definitively.

#21

I'm not so sure about the last argument. A tesseract is a 3D representation (projection) of a 4D cube. Yet it consists of eight 3D cubes.


#22

Looks like the good old game of incomplete induction. We used to play this some 40 years ago in school (honestly, it wasn't what the teacher wanted to tell us in that class then ;)

Anyway, eventually you find that a 4D analogon of a tetraeder shall have 5 vertices, 10 edges, and 5 tetraedric "face volumes" around its 4D "hypervolume". And a 4D analogon of a cube (call it like you want) shall have 16 vertices and 8 cubic "face volumes" around its 4D "hypervolume". I won't dare to guess a hypersphere - too little vertices and edges ;)

FWIW

Edited for correct wording (would have been easier in my mother tongue).

Edited: 17 Oct 2010, 9:22 a.m.

#23

You can build up a cube of 6 squares but you can't build up a
sphere of circles in an analogous way. However you can use parallel
circles of lattitude to build a sphere. This works also for the cube:
lots of parallel squares make up the cube.
Now extrapolate this idea to the 4th dimension.

Best regards

Thomas

Edited: 17 Oct 2010, 9:30 a.m.

#24

Quote:
I've been searching Wikipedia for an answer.

Here's a picture of a 3-sphere in 3-space:

n-sphere

IMHO it's rather confusing though. What helped
me is the following approach from 3D to 4D:
Consider a cube and its dual the octahedron.
Now project the grid of the octahedron to the
enclosing 2-sphere. You get 3 meridians.
Chose one axis and rotate the meridian into the
3rd dimension: you get the 2-sphere.

Now do the same with the tesseract (4-cube) and
the 4-octahedron (4-cross polytope).
Search WolframAlpha if you need a picture of these.
If you replace the squares (or rather parallelograms)
by circles (or rather ellipses) you get a nice picture.
Choose one of these 2-spheres and rotate it into
the 4th dimension. (The 2-sphere you choose
defines 3 dimensions. Choose one of them as the
axis of rotation. Now choose one of the others and
move this to the 4th axis.)

A picture would certainly help but unfortunately
currently I don't have any. Still hope this helps.

Best regards

Thomas


Edited: 17 Oct 2010, 8:51 a.m.


#25

Quote:
A picture would certainly help

This video of a
Hypersphere
comes closest to what I tried to explain.


#26

In case the folks here haven't seen it this series of short videos are superbly done: http://www.dimensions-math.org/Dim_E.htm

I liked them so much I bought the DVD and my children enjoyed watching it & I think understood some of it.

- Pauli




Edited: 18 Oct 2010, 6:30 a.m.


#27

Thanks for the link! I just did a short tour d:-)


#28

I've downloaded the films to watch them on my AppleTV. :)


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