I was thinking about the 90° angle between the x, y, and z-axis.
It is all so perfect. So I said you need two independent real
numbers (cooridinates) to describe a point in a plane. Then I
thought about putting the x and y axis at a 10° angle instead of
90°, and noticed you could still get to all the points on the plane.
Next I attempted to find a way to describe all the points on a plane
with only 1 number. Is it possible? So at first I played around with
the idea of a spiral that was so close together at each pass around that
it would take infinite times to make up some area, but this idea soon
I decided was dumb and incomprehensible. Then I came up with a
plausible idea, if you don't mind a jumbled mess in place of two nice
x and y axis. So this is only a first try at this and I hope to come up
with refined examples that use negative numbers as well as decimals
later, but for now I am only using whole numbers. I started at (-1,1)
and called this 1. Then (0,1) would be 2. Here is a chart.
(x,y) 1-D number
At this point we expand the grid larger by factor of ten and make the intervals
smaller by a factor of ten. So we continue. Some duplicate points will exist
as we go over the previous ones.
This continues to the right until we hit 10,10 and
then the rows continue down to form a square grid.
Then we will be at approximately the number 201 * 201 + 9.
Next we again expand the grid ten times larger and ten times more intricate.
etc. etc. You get the picture.
Does anybody think it is interesting??
Do you think by going to infinity, you will pass through both a large surface area
and work toward getting more intricate as well?
Obviously it is a jumbled mess, but if you ignore the mess, it is pretty cool,
hence in the "This is Cool" category.
igloo myrtilles fourmis
2 lines may determine a plane, whether being verticle or not.
But a plane can never be determined by 1 line. Because it's more complex by one dimension.
Not a plane in the usual sense, but interesting nonetheless. Let's call it a "pseudoplane"
I imagine you could have any number of methods for this "pseudoplane-filling" line - spirals, bisection, etc. You could even decide that the first so many digits would be the x-coord and the rest the y-coord.
Properties? I don't think the "pseudoplane" would be continuous. (You could drive a car over it, but ants might slip through)
It also has the flavor of Fractals to me, see http://en.wikipedia.org/wiki/Space-filling_curve
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
The fractals are lines, not points, usually, I take it by seeing the pictures.
Or can some fractals be points not touching?
And the coordinate (3 1/3, 6 2/3) could never be attained unless you went to infinity, and which infinity would you pick?
igloo myrtilles fourmis