Complex Baloney Space

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Dale10101

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I am currently thinking of "complex space" as baloney on a stick, that is, a cylinder of baloney speared longitudinally down the center. The stick (z axis) is marked in radians, the baloney is sliced thin enough to accommodate a slice for every real number along the stick and the slices have a radius of one.

A complex number z(theta) is found at theta along the stick plus cos(theta) + i sin(theta) on the corresponding slice (i.e the corresponding orthogonal complex plane).

The evolution of z( f(theta) ) is then plotted in 3-D using cylindrical coordinates along the stick.

Am I full of baloney, is this obvious (no, not me, the idea!). I do not find any such representations on the internet or in tutorials, just the 2-D complex plane.

PS. Is the complex plane considered 1-D ?, would Baloney space be considered 2-D ?, identifying dimensionality with the number of coordinates rather then the dimensions of the representation.


 
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Dale10101 said:
I am currently thinking of "complex space" as baloney on a stick, that is, a cylinder of baloney speared longitudinally down the center. The stick (z axis) is marked in radians, the baloney is sliced thin enough to accommodate a slice for every real number along the stick and the slices have a radius of one.

A complex number z(theta) is found at theta along the stick plus cos(theta) + i sin(theta) on the corresponding slice (i.e the corresponding orthogonal complex plane).
Click to expand...


You are limiting yourself greatly here. These are only complex numbers having a norm of 1. For example 1+i is not of that form.

The evolution of z( f(theta) ) is then plotted in 3-D using cylindrical coordinates along the stick.


Am I full of baloney, is this obvious (no, not me, the idea!). I do not find any such representations on the internet or in tutorials, just the 2-D complex plane.

PS. Is the complex plane considered 1-D ?, would Baloney space be considered 2-D ?, identifying dimensionality with the number of coordinates rather then the dimensions of the representation.
Click to expand...

As a function, \(\displaystyle \theta \to e^{i\theta}\) is a set map \(\displaystyle \mathbb{R}\) to \(\displaystyle \mathbb{C} = \mathbb{R}^2\). The result is a one-dimensional curve as a subset of \(\displaystyle \mathbb{R}^3\) (think helix) along the \(\displaystyle \theta-\) axis. I doubt this is what you had in mind.

The complex plane is one dimensional, as a vector space, over itself. But we usually think of it as \(\displaystyle \mathbb{R}^2\) with a special multiplication, as a 2-dimensional vector space over the reals. Dimension is relative.
 
OK

daon2 said:
You are limiting yourself greatly here. These are only complex numbers having a norm of 1. For example 1+i is not of that form.



As a function, \(\displaystyle \theta \to e^{i\theta}\) is a set map \(\displaystyle \mathbb{R}\) to \(\displaystyle \mathbb{C} = \mathbb{R}^2\). The result is a one-dimensional curve as a subset of \(\displaystyle \mathbb{R}^3\) (think helix) along the \(\displaystyle \theta-\) axis. I doubt this is what you had in mind.

The complex plane is one dimensional, as a vector space, over itself. But we usually think of it as \(\displaystyle \mathbb{R}^2\) with a special multiplication, as a 2-dimensional vector space over the reals. Dimension is relative.
Click to expand...

OK, I understand the helix. Baloney space is of course just an imaginative attempt to use cylindrical coordinates to express the evolution of a complex function, with respect to the independent variable Theta, in three spatial dimensions. In short, can one map such an evolution using the cylindrical coordinates (r,Theta,z) where z = Theta. I keep thinking of reasons why you can, and cannot. The reason for doing this would possibly be only pedagogic, illustrative.
 
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Dale10101 said:
OK, I understand the helix. Baloney space is of course just an imaginative attempt to use cylindrical coordinates to express the evolution of a complex function, with respect to the independent variable Theta, in three spatial dimensions. In short, can one map such an evolution using the cylindrical coordinates (r,Theta,z) where z = Theta. I keep thinking of reasons why you can, and cannot. The reason for doing this would possibly be only pedagogic, illustrative.
Click to expand...

Well, again a complex number has two components. Whether you want to use r and theta or x and y, using a single parameter is just not going to work (there are interesting "space filling curves" but they cannot be used as a normal coordinate system). The cylinder you are imagining is the domain and range of a function from a 1-dimensional space to a two-dimensional space over the real numbers. Neither the function, its image nor the graph can represent the complex plane.

Polar coordinates do work. The tuple \(\displaystyle (r, \theta) \) is the point \(\displaystyle (r\cos\theta, r\sin\theta)\) in the cartesian plane. The set of all such points with theta considered modulo 2*pi can model the complex numbers in the obvious way.

A complex function from \(\displaystyle \mathbb{C}\to\mathbb{C}\) has a four-dimensional euclidean graph. The function you describe is from \(\displaystyle \mathbb{R}\to\{x+iy\in \mathbb{C};\,\, x^2+y^2=1\} = \{z\in \mathbb{C};\,\, |z|=1\}\)
 
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Thanks

daon2 said:
Well, again a complex number has two components. Whether you want to use r and theta or x and y, using a single parameter is just not going to work (there are interesting "space filling curves" but they cannot be used as a normal coordinate system). The cylinder you are imagining is the domain and range of a function from a 1-dimensional space to a two-dimensional space over the real numbers. Neither the function, its image nor the graph can represent the complex plane.

Polar coordinates do work. The tuple \(\displaystyle (r, \theta) \) is the point \(\displaystyle (r\cos\theta, r\sin\theta)\) in the cartesian plane. The set of all such points with theta considered modulo 2*pi can model the complex numbers in the obvious way.

A complex function from \(\displaystyle \mathbb{C}\to\mathbb{C}\) has a four-dimensional euclidean graph. The function you describe is from \(\displaystyle \mathbb{R}\to\{x+iy\in \mathbb{C};\,\, x^2+y^2=1\} = \{z\in \mathbb{C};\,\, |z|=1\}\)
Click to expand...

The original coordinates that I wanted to use were z(theta) = ( g(theta)cos(theta), h(theta)sin(theta), theta )

BUT ... I hear you when you say that this will not work.

Thanks for saving me from a long trip around the block.
 
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At last

OK, now I see it. In retrospect the obvious appears obvious. Yes, complex numbers like rational numbers are specified by two independent parts. You cannot keep track of both parts with a reference to only one of them unless you specify a further condition and that of course narrows the range of what can be described and therefore defeats the intent of the invention. Alas, Baloney space, perpetual motion machines, fountains of youths ... back to the cutting board, but ... a more general perspective emerges, so ... OK :)
 
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