## What 3.5 dimensions means to me

What is the basis for reason? And mathematics?

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### What 3.5 dimensions means to me

Before explaining, Google magic cube and look at Wikipedia's description of a magic cube.

What was done there is Wiki took a two-dimensional object and projected it into a three-dimensional object.
Clearly though what you're looking at is neither two-dimensional nor three-dimensional and I would describe the object as in-between, 2.5 dimensional.

Now I'm ready to describe a 3.5 dimensional object. Immerse yourself in VR (virtual reality). Take a three-dimensional object and project it into a four-dimensional object. Since what results won't be completely three- nor four-dimensional, then for me, this object is 3.5 dimensional.

Math can be that strange (btw if I had the right tools, I could do a better projection of a two-dimensional object
into a three-dimensional object).

PhilX

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### Re: What 3.5 dimensions means to me

There's a well-developed theory of fractal dimension.

The Wiki writeup is pretty clear. Imagine a line or a curve traveling through the plane. It has topological dimension 2. The plane itself has dimension 2. I hope we all agree on that.

You've heard of those space filling curves that are lines that snake around so deviously that they fill up the plane.

The fractal dimension of a curve is a number between 1 and 2 that says how close the curve is to being like a plain old well-behaved curve of dimension one, and a crazy space-filling curve of dimension 2. You can have curves that take on any value between 1 and 2 inclusive.

Likewise there are fractal dimensions between 2 and 3 to measure how close a 2-D figure in three space is like a regular old 2-D surface and how much it's like a 3-space filling surface.

Fractal dimension 3.5 would be measure of how some 3-space fills up 4-space in the analogous way.

This notion of fractal dimension is due to Felix Hausdorff, a brilliant German mathematician of the 1930's who worked in topology, set theory, and functional analysis. In 1942 Hausdorff, his wife, and his wife's sister were ordered to a concentration camp. Rather than go, the three of them committed suicide.

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