I don't know where to begin, so let me just get a few ideas on the table, to see if a whole will emerge.
A degree of freedom is something like movement in the x-direction. Move over, move back. This degree of freedom is big, because we can see the movement if it is large enough.
So we might call this idea of a freedom of movement in the x-direction a dimension.
A line of this movement, or a filamentary curve is something we can slide a bead along. We assume that we can label subsequent positions of the bead as we move over and move back, and if these labels are consecutive numbers, we can use them to say where we were, where we are, and where we will be.
This filament of possible movement is so large we will call it a large dimension - a large degree of freedom.
But now take the bead and twirl it. The bead can also have another degree of freedom, a rotation.
But assume for a moment that we allow the bead to shrink, ever smaller and smaller, till it is just the size of the filamentary curve itself. We can talk about the rotational station of this bead, but the degree of freedom itself is curled up, too small to see, so we might call this a small dimension - a small degree of freedom.
We can also think about labeling how curled up the bead is by naming the turns or parts of a turn the bead has made. If these names are consecutive numbers we can use them to say where we were, where we are, and where we will be.
Now we are ready to talk about the first idea.
If we have something that is moving in a large dimension along a curve or line, we can resolve this movement as movements along a set of complementary axes. Then we can say that movement along our arbitrary filamentary curve has this much movement in the x-direction, this much movement in the y-direction, and we can use these pairs of labels as perfectly adequate alternative names for the position of the bead along the arbitrary filament.
If you have ever spun a bicycle wheel and held onto the axle in each hand, you will know that the wheel doesn't like to be tilted. It resists this tilting with an inertial force called the gyroscopic force. The gyroscopic force wants to keep the wheel spinning in its original direction and complains by resisting if the wheel is tilted to spin in some other plane of rotation.
All the points on the wheel except the very center, can be represented as translations through space, they aren't rotating at all! But the very center of the wheel (we pretend the axle is rotating too) has an axle, an axis of rotation, and there is an infinitesimally-wide dimension where the movement is pure rotation with no translation. This is a small dimension, because it can never be seen. A more apt name for it might be an invisible dimension.
If the bicycle wheel became like the bead, and became infinitely small, it would have no gyroscopic force. So it could be spinning in one direction, and then that whole assembly could be spun in a second direction and we could resolve spins in one direction into components of spins in multiple other directions as we did with translation above.
Now for the second idea.
If we count up the directions that we live in there are three, and for each of these directions there are three rotational degrees of freedom. Then there is the passage of time. So we really live in a six dimensional sort of space, if we count the invisible dimensions of filamentary rotation, seven if we consider time to be a dimension, but it doesn't have the same freedom as the others. We are stuck in it.
The third idea.
I should stop here, but I am afraid I might lose an important idea, so I will just add something that I find interesting. We rarely talk about the position of a photon. We can talk about where it originated, or where it might be after a time, but the native state of a photon is not really its position, but rather its velocity, which in a vacuum is just c - the speed of light.
But this is a translational velocity and I want to know if there is also some kind of native rotational velocity of a particle, say a photon, or even some other kind of a fundamental building block, perhaps an electron. The spin of an electron is one of its four pieces of state information. My question is, how fast is it spinning?
So putting the last two ideas together we see that it isn't just where something is that is its native state, but rather how fast it is going that characterizes something important about it. How fast something goes is a degree of freedom - a dimension also.
If we add up where things are, and how fast they are going in translation and rotation we come up with 12 degrees of freedom, or 12 dimensions. That plus time makes 13, which just happens to be my lucky number...
