gopiv wrote:
How is a degree of freedom different from the dimension?
"Dimension" is derived from the Latin "dimensio", which means "to measure."
"Degree of Freedom" comes from the concept of the "degree", derived from the Latin "densus" (density), which means "frequency of occurence" (how close things are together). From that, we derive the mathematical concept of "degree", which are small graduations of measurement.
I've never really examined the origins of these words before, but do find it amusing that it comes down to that basic concept put forth in the
Law of One material of "density" versus "dimension." In essence, "dimension" and DOF are inverses of each other, though commonly confused to be the same thing.
"Dimensioning" is counting the number of degrees of measure. If you have no "degrees of freedom" to count, then you can't have a dimension. Therefore, the "dimension" is the "shadow" of the degrees of freedom.
I consider the "degrees of freedom" to be a more generic term than "dimension", like "ratio" compared to "speed." When I use DOF, then I am just talking about an independent, algebraic variable, with no other assumptions. When I use Dimension, there is an implied one-dimensional, axial assumption with orthagonal relationships to other dimensions.
Thus, three "scalar dimensions" can be viewed as the common 3-axis set of measure, connected at a point so they are an aggregate of relationships, but otherwise act independently.
Three "degrees of freedom", however, can mean anything. When I use a paint program to pick a color, I have 3 DOF in my selection, "hue", "saturation" and "lightness", or "red", "green" and "blue". They are independent variables, but are not considered "dimensions", even though they measure steps, because the color wheel does not have an orthogonal relationships between its degrees of freedom.
gopiv wrote:
Don't the three dimensions of speed create the degrees of freedom in the first place?
When it comes to projective geometry, I've noticed that they use DOF when talking about the transformation, itself (the cells in the projective matrix), with the result of the applied transform as having x,y,z "dimensions."
That makes the "dimension" the shadow of the projection of "degrees of freedom". What you call it, depends on which strata you place the screen to cast the shadow on.
So, the initial 15 degrees of freedom create Larson's scalar dimensions of speed.
I'll have to re-evaluate this a bit, because there ARE 6 degress of freedom regarding scale in the affine stratum. Larson apparently splits the linear and polar geometries, interpreting linear in Euclidean (fixing scale at unity) and polar in Affine (atomic rotations).
Every dogma has its day...