A dimension between 1 and 2, is that possible?
What is a dimension?
A line has 1 dimension, a plane has 2 dimensions
and our space has 3 dimensions.
What do we mean by that?
More or less everybody knows what we mean by that; with a line
one can talk about "length";
with a plane one can talk about "length" and "width" and
in our 3-dimensional space one can talk about "length",
"width" and "height".
Mathematicians have a more precise definition for a dimension.
To define one point in space they prefer to use a system of coordinates.
On a line one can manage with 1 coordinate,
namely the distance to a chosen point of origin.
In a plane you need 2 coordinates, usually called x- and y-coordinates.
In our three-dimensional space one needs three coordinates,
namely x-, y- en z-coordinates.
Another definition for the idea of a "dimension"
Now suppose that you have to measure the length of a line segment.
How would you do that?
You could, for instance, take a piece of wood
that was precisely 1 dm long, and measure the number of times
you could fit the piece of wood alongside the line segment.
If you could do this 13 times,
you would say that the line segment was 13 dm long, wouldn't you?
What will happen if you do the same thing with a different piece of wood
that is 10 times as small?
If the new piece is 10 times as small,
then it will fit 10 times as often along the line and you will get 13*10=130 cm.
I prefer to write 13*101 = 130 cm.
if you measure the length of a line segment with a unit that is 10 times as accurate,
then the answer is 101 as large.
Exactly the same thing happens if you measure 3 more times accurately;
the answer becomes 31 times as large, etc...
Because I have written 101 and 31, I can say:
the dimension of the line segment is 1.
What does this definition say about the plane?
Now imagine that you have to measure the surface area of a rectangle.
How would you do that?
You saw a square piece of wood that is precisely 1 dm by 1 dm large.
And you measure how often it fits into the rectangle.
If you can manage this 65 times, you
can say that the surface area is 65 dm2.
What happens if you take a piece of wood
that is 1 cm by 1 cm? How many times would this fit?
Yes, you guessed right, 65*100=6500 times.
The surface area is therefore 6500 cm2,
or rather 65*102 cm2,
Conclusion: if you measure the surface area with a unit
that is 10 times as accurate,
the answer becomes 10 times as large.
If you measure 3 times as accurate, the
answer becomes 32 times as large.
I have written 102 en 32,
I can say the dimension of the plane is 2.
What if the line is twisting and the rectangle is not a rectangle,
but a part of one? Is this whole story still true?
Good questions, but the answer is: if the figures aren't too twisty,
then this story really is true.
But what do I mean by "too twisty"? Is that "Mathematician-speak"?
What now, if the line does appear to be very twisty?
Now it is the time to look at Koch's curve.
You can see it below!
The curve is generated by dividing each line segment into three further,
The central segment is removed and two segments of the same length are added.
In that way, the length of each line becomes 3 times as small,
but the total length of the "curve" becomes 4/3 times as large.
This may be a little difficult to understand,
but (try having a look at the first pictures of Koch's curve ) if you
measure 3 times as accurately, your answer becomes 4 times as large.
What is its dimension then?
If you measure 3 times as accurately,
and your answer becomes 31 times as large, then its dimension is 1.
If your answer becomes 32 times as large,
then its dimension is 2.
But what if your answer becomes 4 times as large?
Now 4 ~ 31,261859507, so the conclusion seems easy:
the dimension of Koch's curve is ~ 1.261859507.
The curve is so twisty that actually it is not a line anymore, but
something that lies between a line and a plane.
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