Boston - Mar 13, 2002Fractal Antenna Systems, Inc. today disclosed that it has
filed for patent protection on a new class of antenna arrays
that use close-packed arrangements of fractal elements to
get superior performance characteristics.
7. The Koch snowflake is six of these put together to form . . .
. . . well, a snowflake.
8. Notice that the perimeter of the Koch snowflake is infinite . . .
. . . but that the area it bounds is finite (indeed, it is
contained in the white square).
9. The Koch snowflake has even been used in technology:
Boston - Mar 13, 2002
Fractal Antenna Systems, Inc. today disclosed that it has
filed for patent protection on a new class of antenna arrays
that use close-packed arrangements of fractal elements to
get superior performance characteristics.
Fractal Tiling Arrays -- Firm Reports
Breakthrough in Array Antennas
11. Each of the six sides of the Koch snowflake is
self-similar: If you take a small copy of it . . .
. . . then dilate by a factor of 3 . . .
. . . you get four copies of the original.
12. But self-similarity is not what makes the Koch snowflake
a fractal! (Contrary to a common misconception.)
After all, many common geometric objects exhibit
self-similarity. Consider, for example, the humble
square.
13. If you take a small square . . .
. . . and dilate by a factor of 2 . . .
. . . then you get 4 copies of the original.
A square is self-similar, but it most certainly is not a fractal.
14. If you take a small square . . .
. . . and dilate by a factor of 3 . . .
. . . then you get 9 copies of the original.
15. Let k be the scale factor.
Let N be the number of copies of the original that you get.
Note that for the square, we have that:
2
log
N
k
N
k
2
Or in other words, we have:
22. That’s right: N
k
log tells us the dimension of the shape.
(Note that for this to make sense, the shape has to be
self-similar.)
So for a self-similar shape, we can take N
k
log
to be the definition of its dimension.
(It turns out that this definition coincides with a much more
general definition of dimension called the fractal dimension.)
23. Now let’s recall what k and N were for one side of the
Koch snowflake:
k = scale factor = 3
N = number of copies of original = 4
...
261
.
1
4
log
log 3
N
k
24. So each side of the Koch snowflake is approximately
1.261-dimensional.
That’s what makes the Koch snowflake a fractal – the fact that
its dimension is not an integer.
Even shapes which are not self-similar can be fractals. The
most famous of these is the Mandelbrot set.
26. Start with a line segment of length 1.
Now cut away the middle third.
Then cut away the middle third of each remaining piece.
]
1
,
0
[
1
C
Iterate.
]
1
,
3
2
[
]
3
1
,
0
[
2
C
. . . . . .
]
1
,
9
8
[
]
9
7
,
3
2
[
]
3
1
,
9
2
[
]
9
1
,
0
[
3
C
]
1
,
3
1
3
[
]
3
2
3
,
3
2
[
...
]
3
1
,
3
2
[
]
3
1
,
0
[ 1
2
1
2
2
2
1
1 n
n
n
n
n
n
n
n
n
C
27. The Cantor set is what’s left after you’re finished cutting.
In other words:
1
n
n
C
C = Cantor set
28. We can ask several questions about the Cantor set, such as:
• What is its cardinality?
• What is its length? Indeed, does the concept of length
apply to it?
• Between any two points in the Cantor set, can you
find another point in the Cantor set?
• Is it complete?
• What is its fractal dimension?
29. If you take the Cantor set . . .
. . . and dilate by a factor of k = 3 . . .
. . . then you get N = 2 copies of the original.
...
63
.
0
2
log
log 3
N
k
30. So the Cantor set is approximately 0.63-dimensional.
33. Start with a square of side length 3, with a square of side
length 1 removed from its center.
perimeter = 4(3) + 4(1)
area = 1
3
2
2
34. Think of this shape as consisting of eight small squares, each
of side length 1.
area =
3
1
8
1
3
2
2
2
From each small square, remove its central square.
perimeter =
3
1
4
8
1
4
3
4
40. Like the Cantor set, the Sierpinski carpet is what’s left after
you’re finished removing everything.
In other words, it’s the intersection of all the previous sets.