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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

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.

**igloo** **myrtilles** **fourmis**

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**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,320

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.

**X'(y-Xβ)=0**

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,685

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

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

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!

**igloo** **myrtilles** **fourmis**

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