Part 1:
- With your partner, discuss
and draft a definition for a spiral. Share your definition with
the class, and formulate a class definition.
A spiral is a curve
traced by a point that moves around a fixed point, called the pole,
from which the point continually moves towards or away. There are many
different types of spirals. We are going to create a specific type of
spirals called
Baravelle
spirals. They are created by constructed
nested regular polygons and shaded triangles in a clockwise manner.
- The first Baravelle spiral we will construct is formed by a group
of nested equilateral triangles. Let's begin by constructing a large
equilateral triangle, T0 , in your sketch window. Construct the midpoints
of each of the three sides of the triangle and connect them by segments.
The figure you have constructed separates your equilateral triangle
into smaller triangles. How many smaller triangles are there? What kind
of triangles are they? What other conjectures can you make about the
four triangles?
- Construct the polygon interior
of one of the corner triangles and label it T1. Formally prove that
T1 is congruent to the three other inner triangles. The area
of T1 is what fraction of T0? Verify your conjecture by a formal proof.
To
label a figure in a sketch window:
-
Select
the Text tool in the tool box.
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Drag
a text box in your sketch window.
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When
a flashing cursor appears in the text box, you may type in the
box.
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To
edit your text box once you’ve clicked outside of it, select
the Text Tool and click inside the text box. A
flashing cursor should appear, allowing you to edit.
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- We would like to continue
the construction of the Baravelle spiral by finding and connecting the
midpoints of each of the sides of the center triangle. The figure you
have constructed separates the center equilateral triangle into smaller
triangles. How many smaller triangles are there? What kind of triangles
are they? What else can you say about the four smaller triangles?
- Moving in a clockwise manner, construct the polygon interior of the
corner inner triangle, which is adjacent to T1, and label
it T2 . Be sure to keep the color of your polygon interiors
consistent. The area of T2 is what fraction of T0?
- We have created two levels
of this construction. Create two or three additional shaded triangles
using the same process as above. If this construction was continued
indefinitely, the resulting shaded region is called a Baravelle spiral.
Part 2:
- Looking at your constructed Baravelle spiral, visually estimate the
spiral’s area as a fraction of the area of T0. Justify your
answer. How could we find the actual area of the spiral?
- In your own words, what is a series? How does a series differ from
a sequence? Can an infinite series sum to a finite number? If so, how
can we find that number?
- Record the ratios of the
areas of the shaded triangles, TN , in comparison to the
area of the original equilateral triangle, T0. Fill in the
chart below, computing the partial sums of the areas of each of the
levels of triangles.
|
Level of Triangle
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Ratio of Area of TN:
The area of T0
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Computed Partial Sums of the
areas T1 through TN
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T0
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1
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T1
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T2
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T3
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T4
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T5
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T6
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- What did you notice about
consecutive ratios of the areas of the triangles? Furthermore, what
did you notice about the column of computed partial sums? As N increased,
what number did the column of partial sums approach? Describe your observations
in your own words. Use your knowledge of infinite series to verify this
sum. See the spreadsheet activity Exploring and Analyzing Sequences.
- Express the area of the
Baravelle spiral as an infinite series. The spiral's area is what fraction
of the area of the original triangle?
- Shade (each in a different
color) two other Baravelle spirals using your original equilateral triangle.
Part
3:
- Another infinite series
is similarly found by considering the length of the outer edge of the
spiral. The outer edge of a spiral is the sum of the lengths
of one side of each of the shaded triangles. Record the length of each
side of the triangle, TN , in comparison to the length of
each side of the original equilateral triangle, T0. Fill
in the chart below, computing the partial sums of the length of the
outer edge for each of the levels of triangles.
|
Level of Triangle
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Ratio of the length of a side
of TN:
the length of a side of T0
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Computed Partial Sums of the
lengths T1 through TN
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T0
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1
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T1
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T2
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T3
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T4
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T5
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T6
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- What did you notice about
consecutive ratios of the lengths of a side of the triangles? Furthermore,
what did you notice about the column of computed partial sums? As N
increased, what number did the column of partial sums approach? Describe
your observations in your own words. Use your knowledge of infinite
series to verify this sum.
- Express the outer edge
of the spiral as an infinite series.
Part
4:
- As you can see from the previous construction, the process to construct
a Baravelle spiral is a recursive process. In this context, discuss
with your neighbor what it means to be a recursive process? Share your
ideas with the class.
- To further explore infinite series and recursion, we will direct focus
on a Baravelle spiral generated by a square. To begin, construct a square
in your sketch window.
The basic construction for a Baravelle spiral is to find and connect
the midpoints of each of the sides of the center square multiple
times. We would like to create a script to do this construction for us.
Sketchpad allows us to create a recursive script, using a script command
called Loop. Loop allows the script to be played on an object
or group of objects previously created in the script. Remember, when you
use Loop, the objects you select to match the script's Givens
must be of the same number and type. Follow the steps in the gray box
below to create a recursive script for the construction of a Baravelle
spiral generated by a square.
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To record the construction of a Baravelle spiral
generated by a square recursively:
- Under the File command, choose New Script.
- In the Script window, click on the Rec. (Record)
button.
- Construct the midpoints of the sides of the previously constructed
square.
- Connect consecutive midpoints with segments.
- Direct your attention to the Given objects listed in
your script window. In your sketch window, select these objects
for the next iteration. Click the Loop button in the Script
window.
- End the script by clicking in the Script window and then
click on the Stop button.
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- Construct a new square in a new sketch window. What do you think
will happen when you play back your recursive script, entering 1 as
the Depth of Recursion? Play back your script, entering 1 as
the Depth of Recursion, and describe what happened in your sketch
window.
To play back the recursive construction of a Baravelle spiral
generated by a square:
- Select the Given objects listed in the Script
window.
- Click on either the Play button.
- In the Recursion window, enter a reasonable Depth
of Recursion, taking under consideration the number of objects
produced.
- Click on the Play button.
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- The figure you have constructed separates the square into a smaller
squares and triangles. Within the first level of the construction, how
many smaller triangles are there? What kind of triangles are they? What
other conjectures can you state about the four corner triangles?
- Construct the polygon interior of one of the corner triangles, label
it T1. The area of the shaded triangle is what fraction of
the large square?
- Play back your recursive script, using the sides of the center square
as the necessary given objects. Enter a reasonable Depth of Recursion,
taking under consideration the number of objects produced. Finish shading
in one of the Baravelle spirals.
- Record the ratios of the areas of the shaded triangles, TN
, in comparison to the area of the original square. Fill in the chart
below, computing the partial sums of the areas of each of the levels
of triangles.
|
Level of Triangle
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Ratio of the area of TN:
the area of the square
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Computed Partial Sums of the
areas T1 through TN
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T0
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1
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T1
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T2
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T3
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T4
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T5
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T6
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- What did you notice about
consecutive ratios of the areas of the triangles? Furthermore, what
did you notice about the column of computed partial sums? As N increased,
what did the column of partial sums approach?
- Express the area of the
spiral as an infinite series. The spiral’s area is what fraction of
the area of the original square?
- Construct the three other
Baravelle
spirals using your original square, each in a different color.
Extensions:
- Explore a similar investigation using a hexagon as the original starting
figure. By constructing the midpoints of each of the six sides of the
hexagon and connecting them by segments, continue a similar procedure
as the triangle and square. The shaded spiral's area is approximately
what fraction of the original hexagon? Use your knowledge about infinite
series to verify this sum?
- Rewrite your recursive script so that it not only constructs the Baravelle
spiral generated by a square, it also correctly shades in one of the
Baravelle spirals. Hint: When constructing your script, be cautious
of the order in which you select the Given objects.
- In each of the Baravelle spirals constructed we constructed the midpoints
of each of the sides of our polygon and connected them by segments.
A different type of spiral could be constructed by choosing points other
than the midpoints of each of the sides of the polygon. Recursively
construct a spiral that does not use the midpoints of the sides of the
polygon. Write a report describing your construction and showing examples
of your spirals.
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