Graphics Reference
In-Depth Information
This replacement of a triangle by two smaller triangles is repeated
indefinitely, so that at each stage of replacement there are 1, 2, 4, 8, . . .,
2
, . . . congruent triangles with hypotenuses filling the line segment
l
,
and at the
n
th stage, the combined perimeter of these triangles is
p
and
their combined area is
A
.
(a) What is the relation between
p
and
p
?
(b) What is the relation between
A
?
(c) What can you say about the two sequences (
p
and
A
) and (
A
)?
82 (
Koch
, 1904) An equilateral triangle is given with perimeter
p
and
area
A
. This is stage 1. Now an equivalent triangle is added to
each side on the outside of the original figure, at its points of
trisection
Stage1
Stage2
The result is a polygon with 12 sides, perimeter
p
and area
A
.
This is stage2.
The same kind of extension is now made on each of the 12 sides of
this polygon, resulting in a further polygon with 48 sides, perimeter
p
and area
A
. This is stage3.
