Beaconator: Did you actually read my post #9, above? Or did you perhaps read it and not understand? I'll break it down for you. Start by considering the following equation: $x=1$. Is there a "variable" in that equation? Yes, it's $x$. But $x$ has a fixed value. Its value is dependent, in this case on the number to the right of the equals sign. Next consider the following description of a line: $y=x$ How many independent variables are there here? The answer is one. If you choose $x$ as the independent variable, then whatever value $x$ has immediately determines the value of $y$, making $y$ dependent on $x$. You are free to choose $x$ to have whatever value you like, but once you've made that choice you are not free to choose $y$. Now consider this equation: $x + y + z = 0$ How may independent variables are there here? The answer is two. Suppose you say that $x$ and $y$ are independent. Then the value of $z$ is completely determined by the values of $x$ and $y$, so $z$ is a dependent variable. What does that last equation describe, geometrically? It describes a two-dimensional flat plane. It is two dimensional because precisely two values need to be specified in order to locate a unique point on the plane. For instance, if we specify the values of the coordinates $x$ and $z$, then the third coordinate, $y$, is completely determined by those values and the equation itself. So, look what I just did there. I gave you an example of a two-dimensional object specified using three variables - something you asserted was impossible for me to do. The important takeaway message, that you missed first time around, is that it is not the number of variables that matters, but how many independent variables there are. Do you understand now?
Oh I read it, and I understand it. Still I believe the discussion is semantic and arbitrary. Never said it was impossible. Just seems to me to be scatological to say a sphere has two dimensions or assert a different plane as being a new dimension. By your logic I could easily argue a sphere has an infinite number of dimensions because I can ascribe an infinite number of tangent planes to its outside surface. I mean to say a circle can be drawn on one plane seems far more logical than saying it is one dimensional.
One way to think of this is, if you convert to polar coordinates, how many do you need to specify a point? If you are on the surface of a sphere, or on a circle, the radius is fixed as part of the definition of the object, so it does not form part of a coordinate system needed. (I did not think of this at first, I admit.) Consequently on a circle you just need one angle relative to a given starting point, and on a surface of a sphere you need two angles (i.e. a latitude and a longitude). But if you are dealing with a disc, or a solid sphere, the radius needs to be defined as well, and so the dimensionality increases accordingly.
That settles it then in my head, a sphere is a 1D object, it's made up of many, many hollow circles. Let's be brutally honest, it doesn't really matter what I think about Spheres does it. lol
Your logic is faulty. If it is made up of many circles then to specify a location you have to say which circle and then where on that circle, i.e. 2 dimensions.
If you think it's semantic and arbitrary, then I'd say you haven't really understood. Scatological? I don't think that word means what you think it means. But here's the thing: a circle can be drawn on a plane, certainly. But circles can also be embedded in three-dimensional space, or in any space of a higher dimension. Therefore, it isn't very helpful to say that the dimensions of the circle are defined with reference to the dimensions of any particular embedding space.
Did you read my post above? How many independent parameters are needed to specify a point on a sphere? The answer is two for a hollow sphere, and three for a solid sphere. It's never one. Certainly it's irrelevant if you can't make a valid argument to support your position.
