You're missing the point. Within the set of points along the curved line, I can define any point by a single coordinate: how far along the line that point is. That makes the space defined by that set of points one-dimensional. The fact that the line itself is defined by multiple coordinates within the larger space it is embedded in, is irrelevant.
I understand what you are trying to say, but I don't think it is entirely accurate to say that being able to define any point by a single coordinate is what makes it "one-dimensional."
The reason is that R^n has the same cardinality as R. You could create a bijection between the set of 2-dimensional real coordinates to the set of 1-dimensional reals by using every even positioned digit to represent the X value and odd positioned digits for the Y.
The issue with this is that the intuitive metric you were using in the space breaks down and you need a totally unintuitive one to treat this 1-dimensional version as a metric space. It wouldn't be fun to play a game in.
> I understand what you are trying to say, but I don't think it is entirely accurate to say that being able to define any point by a single coordinate is what makes it "one-dimensional."
You're missing the fact that several different definitions of dimension exist. If we're talking about topological dimension, then yes, the fact that you need exactly one parameter to define a point is what makes it one-dimensional.
I played around with this idea. First transforming the visual representation of the chess board and then seeing how I would describe the moves of the pieces and whether it was more / less intuitive.
It is easy to view the board as 64 consecutive spaces (left-to-right through the ranks, then snaking through the ranks, and then a circular representation. The circle was cool because the moves became rotations (a rook can stay in its octant or move an exact multiple of 45 degrees...). I did some 2D transformations with a one space skew on each rank (this made the bishop act simultaneously like a rook and knight which was interesting).
So it is easy to preserve the rules of chess and create a different visual representation of the board. Of course, every one I tried just made the game more difficult to understand. I wonder if there is a transform that could aid in understanding chess.
Using that bijection, you are still describing two sets of coordinates, you have just encoded them into a single number by interleaving them. Also, you won't be able to define points in certain quadrants of the cartesian coordinate space using that system because you have no way to make just the x coordinates or just the y coordinates negative. Depending on how you define the meaning of a negative interleaved coordinate, you could have access to any pair of two quadrants (+x/+y and -x/+y or +x/+y and +x/-y, etc.).
Unless you just use the first two digits as sign bits - if num would encode to (x,y) then instead you have 00num = (x,y), 01num = (x,-y), 10num = (-x,y), 11num = (-x, -y)
Any line, really, but let's go with a spiral. The natural coordinate is its arc - there is no rule saying only cartesian coordinates are allowed. Indeed, geographers use latitude and longitude instead, saving a full coordinate.
...which is one less than the three it would take to represent points on a sphere in cartesian coordinates.
If you want me to describe the curve itself, you are essentially asking for an embedding - there are infinitely many of those, and they will depend on the dimension and coordinate system of the superspace.
Of course it's impossible to explicitly describe an embedding without reference to the superspace, but that only goes for one level - you can ignore said superspace when describing the interior of the subspace.
