Math Is Fun Forum

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

2-D or 1-D jumble

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
-1,1  1
0,1   2
1,1   3
-1,0  4
0,0   5
1,0   6
-1,-1  7
0,-1   8
1,-1   9
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.
-10,10   10
-9.9,10  11
-9.8,10  12
-9.7,10  13
-9.6,10  14
-9.5,10  15
-9.4,10  16
-9.3,10  17
-9.2,10  18
-9.1,10  19
-9.0,10  20
-8.9,10  21
-8.8,10  22
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.
-100,100  40420
-99.99,100 40421
-99.98,100 40422
-99.97,100 40423
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.

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igloo myrtilles fourmis

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: 2-D or 1-D jumble

Not possible.
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.

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X'(y-Xβ)=0

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: 2-D or 1-D jumble

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

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: 2-D or 1-D jumble

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?
Ha!

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igloo myrtilles fourmis