I have a concept that I have been pondering for quite some time now. Suppose you have a square. Say, 10 in. x 10 in. We know that the area of the square is 100 in. and so it is a finite quanity. However, what if you were to place smaller squares into the 100 in square. You would be able to have an infinite amount of smaller squares, yet the total area of the 100 in square is finite. I have represented this concept by an equation.


---E = epsilon
---s = sum of all n (intergral sign)
---n = area for 1 smaller square
---x = infinite


= n E s
= n-->0
= s-->x


Please leave me some comments. Thanks

Originally posted by S. Dalal
I have represented this concept by an equation.
The concept of an area integral is due to Leibniz and Newton, hundreds of years ago.

The general form is the integral of the differential (infinitesimal) area element dA over the area of the square yields A, the total area -- in this case, 100.

A = integral(dA, all area) = 100

Or, if you prefer, you can represent the calculation as a double integral over dx and dy, the linear elements of width and height, such that dx dy = dA, and

integral from x=0 to x=10 of (integral from y=0 to y=10 of (dx dy)) = 100.

It's basic calculus.

- Warren

An old concept would be the fractal geometry of space...

for example, the idea that an island can have a finite area but have an infinite (due to the tiny gaps in the grains of sand, etc) (perimeter) shoreline, has alway fascinated me..

another idea is that the space gets exponentially smaller as you near the center...

But how does this all relate to the expanding4 dimensional hyper-balloon universe?

perhaps gravity is a product of acceleration of the skin of the balloon?

I'm always fascinated with ideas such as that. They tease the mind. You speak of a 3-dimensional object. The Sierpinski carpet(it's a square) is a 2-dimensional series with the opposite idea as your cube. You start with a 1x1 square and remove the center 1/9th of it. There is now a square hole in the square. Mentally divide the holed square into 8 squares and take the center out of those, divide and repeat to infinity. The remove peices all add up to give an area of 1, thus it implies that the Sierpinski carpet has an area of 0. The Cantor set is a 1-dimensional series with the same idea as the carpet. The Cantor set works like this: On the closed interval [0,1] you remove the open inverval (1/3, 2/3) *the middle third*. This leaves you [0,1/3] and [2/3,1]. Remove the middle thirds of each of those and repeat this process to infinity. The Cantor set is the set of numbers that remains in [0,1]. Even though all of these points add to 1, and yet there is still an infinite number of numbers.

If you want more info on the carpet, search on the net, there is bound to be something out there.

For the Cantor set there is this website. It told me much more about the Cantor set than I ever knew. It's awesome.
http://www.mathacademy.com/pr/prime/articles/cantset/index.asp

Math is beautiful and amazing. Gabriel's horn with the infinite surface area, but finite volume. Crazy stuff.

*edit* btw- those sets have to do with fractals and such.

There was a maths lecture last year at uni (I missed it) about some mathematical method for cutting up a orange's surface and putting it back together so that it has a surface area as large as the Sun. Anyone know much about this?

Interesing statement there, Adam. I'll do some research on it and see what I come up with.

It's the Banach-Tarski theorem:

http://www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml

- Warren

Yep, that's the one. Very interesting. At my uni the maths department runs a series of lectures in the lunch hour, they call it Lunchmaths, each Tuesday. It's not part of a course, they just do it for those who are interested. Each lecture is some odd thing like that.