One and a half dimensions

Before I start, how many dimensions does this figure have?



I've noticed that there were alot of threads about the 4th dimensions and such, so I decided to add to the discussion by introducing fractional dimensions.

How is it possible for a figure to have 1 and a half dimensions? Not that complex actually, by being fractal. If you don't know about fractals already and their unique properties, then you should really find out. Trust me, I thought it was a bunch of silly blots and colors, but I found truely how facinating they were. Seriously, check it out!

Here's a really good link about how fracitonal dimensions are possible. hterea re a whole other bunch of stuff on that site too.

math.rice.edu...

Now you can see that other theories like time is the 4th dimension don't really make sense, but I may be wrong.

You make a good point in that it is true that there is always exceptions to any rule. We often ignore this and expect everything to fit into an easy black or white, yes or no kind of ranking.

Oh, the number of dimensions that the Sierpinski Triangle has is defined by : log2 3

The 2 is the base of the logorithm, so it's subscripted. See? That's what logorithms are used for!

I forgot to add, a 4th dimensional object is defined by a figure, so that if you double it's dimension (width, height, depth, and whatever you call the "4th dimension"), it produces a object that's 16 times bigger than the original figure, but exactly the same except for the scale.

2^4= 16

And note that the Sierpinski Triangle is infinate in terms of the iterations of triangles.!!!

Actually, dimensionality is not so easy to pin down in the way you and the website describe. The point of having the next dimension means that their is an infinite integration through a new dimensional axis into the next "dimension." In layman terms, any area that appears as being perpendicular to a line is the sign of the second dimension (the plane dimension). It doesn't matter if the fractal is a triangle or if it is partly empty. The problem is that the site is comparing apples and oranges. Squares, cubes, and hypercubes all have defined special characterists to their structures. A triangle, triangular pyramid, or Sierpinski Triangle do not have the same properties as the figures listed above. The log formula used to determine the dimensions on the website is dependent upon a uniform grid structure that does not apply to all figures. This is why the fractal appears to have a different dimension, when it is still 2 dimensional. To give the site credit, it does define what a dimension is in a new way. However, I don't believe I would use the word "dimension" to describe the difference between fractals and standard geometric shapes.

The problem with knowning this information is that you need to know many of the underlying developmental steps in mathematics, think math history here, where a dimension is defined by specific characterists that have taken millenia to develop. If you want to know more, you'll have to read about how calculus developed.

This link should explain the idea of infinite pieces defining a new dimension

Sorry if this seems complex, but it is an field of mathematics and geometry that I've been studying for months now.

The different dimentions that exist are very interesting. Go to www.pbs.com and find the String Theory. It predicts the existance of 11 dimentions. 7 of which we are unable to persieve. I found it very ammusing.

Since the figure is displayed on a 2-dimensional suface of my monitor screen, I'd have to say that the *figure* is also 2-dimensional...The perception of anything else is mere optical illusion.


[Edited on 10-11-2003 by MidnightDStroyer]

That triangle shaped thing has two dimensions. Length and width as far as I can see. Dont look into things too much. Take them for what they are.

Originally posted by MidnightDStroyer
Since the figure is displayed on a 2-dimensional suface of my monitor screen, I'd have to say that the *figure* is also 2-dimensional...The perception of anything else is mere optical illusion.

[Edited on 10-11-2003 by MidnightDStroyer]

Ah, but you see, the format an object is portrayed as doesn't necessarily reflect it's state. If you saw a picture of a 3D object on your computer screen, would you think that's it's 2D?

I know that's not a very good example, but what I'm trying to say is that things shouldn't always be evaluated graphically. Assumptions can't be made.

If you do a search on fracitonal dimensions, you can see that there's alot more sites supporting the theory. It's a difficult concept to grasp, but it's the truth.

math.bu.edu...
crca.ucsd.edu...
www.math.umass.edu...
www.math.umass.edu...
www.math.umass.edu...


I know it's the truth, alot of mathematicians know it's the truth, Mandelbrot knows it's the truth!

[Edited on 11-10-03 by Saucerat]

[Edited on 11-10-03 by Saucerat]

The only problem I see with this is that it is trying to explain complexity in terms of dimensionality. In other words, if you look at a fractal that uses chaos theory to constantly modify the shape of a structure as you get closer and closer, the shape will continue to morph given the range of complexity of that object. However, I do not associate the complexity of Chaos Theory and fractals with geometric dimensions. The sites claim that dimensions are simply the number of pieces that result mathematically from the next dimension. However, I believe this is far from true geometric formation. The concept of how dimensions form is much more complex, in fact, we still don't understand exactly how we ended up as three/four dimensional.