Mathematical concepts began early in the history of man. From
earliest times, crude understanding of mathematical relationships
were observed, at least quantitatively, in the physical world.
The great breakthroughs in mathematics come when these concepts
are generalized. In the following discussion, we will examine
how specific situations involving proportions can be useful in
leading to generalizations in which valid rules can be stated.
We will begin with a specific instance of a constant geometric
ratio and from this we will try to arrive at a general case.
Suppose we construct the midpoints of a square of length 1 and then connect the midpoints to form another square. In repeating this process, a sequence of concentric squares are generated connected at the midpoint of the outer square. Consider the spiral formed by the sequence of segments starting at a corner and moving inward as each midpoint is reached. The spiral is highlighted in blue in the following sketch.
What is the length of the spiral as this process is continued?
If we look at the first few lengths of each segment, a pattern
emerges. These terms are
From this we obtain that the lengths of each segment form a
geometric series and that the length of the spiral is
At this point, we have
Additionally, we can look at the polygonal area formed between
two of these spirals. This area is highlighted in blue in the
From the sketch, it is easy to see that the area formed by
the two spirals is going to be the sum of the areas of the corner
triangles in each of the concentric squares. Using the Pythagorean
Theorem, we obtain the following ratio and corresponding sum
For each of the above sums, we can use spreadsheets to explore
the rate of convergence and find both the length and area to the
desired degree of accuracy.
From the spreadsheet, we can observe that the sum of the lengths
of the first 30 segments is approximately 1.7071 which coincides
with our earlier results. Similarly, we have that the polygonal
area after 30 iterations is one-fourth. As you examine the sketch
of the polygonal region, this coincides with what we observe looking
at the triangular regions with respect to the entire square.
Can these results be generalized for any point on the edge of the square or are they specific to the midpoint? Consider the following sketch.
In the above sketch, the first segment of the spiral is labeled
as having length "a" making the remainder of the side
"1-a". Using the Pythagorean Theorem, we can generalize
the lengths of the first few segments of the spiral as
The sum of the segments of the spiral can be expressed as the
following geometric series
For the area enclosed by two spirals, we obtain the following
Again, we wish to establish a general rule for the area of
the polygonal region which can then be used on a spreadsheet for
any value of "a". We can use the Pythgorean Theorem
again to establish the area of the polygonal region as
Once we have the general rules established, we are able to
use spreadsheets to find the length and area for different values
It is also important to take into account the rate of convergence for different values of a. For each of the following values of "a", the length is given with 3 decimal place accuracy.
1200 .99 99.5019194 .25 40 .75 3.58090464 0.249999999 35 .25 1.1933932 0.249999982 30 .50 1.707054684 .25 1200 .05 1.02646013 0.249999929 1000 .01 1.00503081 0.249999999
As the general case is obtained, we are able to find the lengths
and areas between spirals with relative ease using spreadsheets.
From this, we are able to make several conjectures concerning
both the length of the spirals and the area between spirals. The
length of the spirals seem to converge towards 1 as a 0 (the
length of the side of the original square) and the polygonal area
formed by the two spirals is constant.
As we have explored the problem, we have been able to find a number of relationships that exist, however, there are several extensions which could be explored further. Does the length of the spiral converge to the length of the side for any regular polygon as a 0? Is the area between two spirals a fixed number perhaps equal to 1 over the number of sides?