Dimensions contra degrees of freedom. Damned if I know how to define this. We live in a four dimensional SpaceTime, right. We use parameters defining where you are, relative some agreed on grid. The grid consist of the degrees of freedom something can have. Physically (not strictly mathematically now) a test particle should be defined by three degrees of freedom spatially and one temporal. Meaning your position in the room as well as the time. To that there might be other parameters you can add, as 'energy' 'spin' etc, but for a point particle that you want to place the first four should be enough.

Then we have Phasespace.

"In mathematics and physics, a phase space is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables....

In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system, or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (a phase space trajectory for the system) through the high-dimensional space. The phase space trajectory represents the set of states compatible with starting from one particular initial condition, located in the full phase space that represents the set of states compatible with starting from any initial condition.

As a whole, the phase diagram represents all that the system can be, and its shape can easily elucidate qualities of the system that might not be obvious otherwise. A phase space may contain a great many dimensions. For instance, a gas containing many molecules may require a separate dimension for each particle's x, y and z positions and momenta as well as any number of other properties." from

http://en.wikipedia.org/wiki/Phase_space Mathematically it becomes even weirder. A circle can mathematically be defined as having three degrees of freedom, its radius being one, and two center coordinates. A angle has four degrees of freedom, two coordinates of its vertex and the slopes of its rays.

So what is it? What is a dimension, and what is a degree of freedom? I like degrees of freedom, and naively i would define those as the possible ways something can move in a dimensional system, from an idea of physics. We better take a look at dimensions too.

"In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it.

Thus a line has a dimension of one because only one coordinate is needed to specify a point on it (for example, the point at 5 on a number line).

A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and its longitude).

The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces."

http://en.wikipedia.org/wiki/Dimension_%28mathematics_and_physics%29One might find dimensions easier to comprehend, as a description of the universe you can 'touch'. But I find degrees of freedom better. And why is just because I'm questioning what it means.