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Spirals and Quilt Design
Jessica Korth
In partial fulfillment of the requirements for the Master of Arts in Teaching with a
Specialization in the Teaching of Middle Level Mathematics in the Department of
Mathematics.
Dr. Fowler, Advisor
July 2010
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Have you ever looked at a quilt and thought about the math behind it? Have you
ever sat in class and drawn a spiral to consider the mathematics of it? We will look at a
variety of spirals, focusing mainly on two, and how to incorporate them into
mathematical quilts. In simple terms, “a spiral is a curve traced by a point that moves
around a fixed point, from which it moves further and further away” (Mathematical
Quilts, 1999).
The first spiral is a logarithmic spiral. Descartes first studied the logarithmic
spiral in 1638. Later Jakob Bernoulli studied the spiral and became so fascinated with it
he had it engraved on his tombstone with the words “eadem mutata resurgo” (“Though
changed I rise unchanged”). The curve, however, was not quite drawn correctly on his
tombstone and looked more like an Archimedean spiral. The logarithmic spiral is also
known as the equiangular spiral, Bernoulli spiral, and Fibonacci spiral. A logarithmic
spiral is a spiral in which the radius of the spiral grows exponentially with the angle.
There is a geometric progression in the distances of the radius from the point of origin to
where it meets the curve.
The second spiral is the Archimedean spiral, named after the Greek
mathematician Archimedes who explored the curve for the first time in 225 BC. He
wrote about his work on spirals. The Archimedean spiral is a plane curve generated by a
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point moving away from a point at a constant rate, which gives a fixed constant for the
distance between the successive arms.
The difference between the logarithmic and Archimedean spiral is found in the
distance between each turn in the spiral. The logarithmic spiral increases based upon a
geometric sequence instead of an arithmetic sequence as in the Archimedean spiral. In
simpler terms, the logarithmic spiral describes a growth related to size, whereas the
Archimedean spiral shows a growth that is constant. There are two polar equations that
define the two spirals. A polar equation is an algebraic curve expressed in polar
coordinates on a two-dimensional plane. In particular, the logarithmic spiral has a polar
equation of r=aebθ and, in contrast, the Archimedean spiral has a polar equation of r=aθ,
where the polar coordinates are (r,θ), e is the base of natural logarithms, and a and b are
positive real constants. You can see the differences in these equations in the following
figure.
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Logarithmic Spiral
Archimedes Spiral
The book Mathematical Quilts, published by Key Curriculum Press, is our guide
for the next sections of this paper.
Exploring the Wheel of Theodorus
This spiral was first studied by mathematics seeking the meaning of irrational
numbers. The Greek mathematician Theodorus (400 BC) studied irrational numbers and
the use of the root spiral to explain these numbers. The root spiral, or the Wheel of
Theodorus, is made by joining a sequence of right triangles as seen below and has some
interesting properties. We construct a table for each specific triangle numbered below.
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Triangle Number
Exact length of the
short leg
Exact length of the
long leg
1
2
3
4
5
6
7
8
9
1
1
1
1
1
1
1
1
1
1
Exact length of the
hypotenuse
Ratio of successive
hypotenuse lengths
1.225
1.155
1.118
1.095
1.080
1.069
1.061
1.054
You can see many patterns in this table. For example, notice that the exact length of the
long leg in triangle n is the square root of n. Also, observe that the exact length of the
hypotenuse is the length of the long leg of the next triangle in succession. If we were to
calculate the ratios of hypotenuse lengths for the 99th and 98th triangles it would be
, and for the 100th and 99th triangles it would be
. The
ratios appear to be approaching a limit of one. The Wheel of Theodorus, however, is
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not a logarithmic spiral because the ratios of successive hypotenuse lengths are not
constant. When you look at the measures of each acute angle at the common vertices
you see that the angle measure gets smaller as the numbers of triangles increase.
Theodorus is cited in Plato’s Theaetetus as having discussed the irrationality
of square roots from 2 to 17. Theodorus stopped at 17 because the 17th triangle
begins to overlap the previous triangles. The picture below shows how 16 triangles
fit in the Wheel of Theodorus, but the 17th triangle would overlap triangle one.
Side Lengths and Areas of Squares
We now consider patterns of configurations of successive squares. Start with a
square whose side length is 1 unit. Use the exact value of the diagonal of this square as
the side length of the next square. Using diagonals, continue computing the exact value of
the side lengths of each successive square and configure as demonstrated below.
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Square
Exact length of each
side
1
2
3
4
5
6
7
8
9
1
Exact length of the
diagonal
Area of the square
1
2
4
8
16
32
64
128
256
We see some interesting patterns in the above table. In particular, the areas of the squares
are 20, 21, 22, 23,..., 28. Also, the exact length of the diagonal is the same as the exact
length of each side in the successive triangle.
Rational Number Patterns
You can determine the decimal approximations of rational numbers very quickly
when rounding to two or three decimal places, unless you are dividing by sevenths. The
digits following the decimal point in a rational number will terminate or follow a pattern.
There is no pattern for the digits following the decimal point in an irrational number.
Below are the decimal equivalents in each “family” of rational numbers.
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There are many patterns in the “family” of rational numbers. In fractions with a
denominator of 2, 4, or 8 all of the decimals terminate because 2 is a prime factor of 10.
In fractions with a denominator of 5 all of the decimals terminate because 5 is a prime
factor of 10. In fractions with a denominator of 3 or 9 the decimals all have a single-digit
repeating number. Fractions with a denominator of 6 sometimes terminate and sometimes
repeat because 2 and 3 are factors of 6 and you can apply the previous observations of 2
and 3. The fractions with a denominator of 7 are the most interesting since the decimals
all have a six-digit repeating pattern.
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Spiraling Squares
Spiraling Squares is a pattern using multiple circles of squares inset on each other.
Using geometry we can find the measures of every angle in such a picture. We will not
use any measuring tools to find the angles. By calculating some angles below, one can
see that the others are thus also calculated by applying basis facts such as all three angles
in a triangle add to 180°, a straight line is equal to 180°, supplementary and
complementary angle rules.
4
Using the single piece of spiraling squares below, some of the polygons one can
find 18 total squares; there are 12 squares in red, 3 squares in blue, and 3 squares in
green. One can also find 12 hidden equilateral triangles.
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0
One problem when drawing the spiraling square design is deciding on an
appropriate radius for the starting circle. You can use the Law of Sines to find the
measure of unknown angles or sides of a triangle. Recall that the sine of an angle in a
right triangle is the ratio of the length of the side opposite the angel to the length of the
hypotenuse. The Law of Sines states that for any triangle ABC,
. The
triangle ABC has angles labeled A, B, C and the sides opposite those angles are labeled a,
b, c respectively. As an example, we will find the radius of the starting circle if we want
our smallest-size square to have a side length of 1 unit.
A
75°
b
B
75° (There are 180° in a
triangle, angle A and B are the
same; 180-30=150/2=75)
a
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1
In
order to find the length of a, we use the Law of Sines:
30° (There were 12
triangles in the circle:
360/12=30)
So the radius of the starting circle would be approximately 2.7 units if you wanted the
smallest-size square to have a side length of 1 unit.
Indiana Puzzle Spirals
The Indiana Puzzle spirals are logarithmic spirals because the ratios between the
successive radii of the spirals remain constant. In the picture below we measure the radii
in dark and find the successive ratios to show that the spiral is logarithmic.
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Radius 1-5 from shortest to
longest
Length of radius (inches)
(Note: measurements are from the
Activity in the Mathematical
Quilts book, not the
representation above)
Ratio of successive radius lengths
1
2
1.444
3
1.461538
4
1.42105
5
1.407407
We can call this spiral approximately logarithmic, because the ratio is approximately 1.4
units.
The Indiana Puzzle quilt results from the square design above. All of the squares
are similar nesting inside of one another. Using the initial inside square you find the
length of the diagonal. This is the length of the next size square in the pattern. You
continue to repeat this process until you have a size quilt square you like. “This quilt is
based upon Baravalle spirals which nest inscribed similar squares to form a logarithmic
spiral. The synergistic pattern formed by the spirals capitalizes on the structure of the
sixteen squares and the rotating four-patch in the center of each square. The resulting
self-similarity provides opportunities to begin to talk about simple fractal patterns
although this pattern is not a true fractal since the shape is not repeated only the steps are
repeated.” (http://www.mi.sanu.ac.rs/vismath/meel2008/index.html)
Mathematical Quilts
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Quilts have been used as visual introductions to mathematical concepts, such as
patterns, spirals, and geometric shapes. Here are a few examples of mathematical quilts
and hands-on activities to complete them.
The Wheel of Theodorus spiral is formed by constructing a series of right
triangles on the hypotenuse of the previous triangle. The outer edge of the curve may be
smoothed to look more like a spiral. Most Wheel of Theodorus quilts are constructed
using 36 right triangles where any triangle over the 16th one is placed on top of the
previous spiral. The simplest construction method is to use a note card to draw the square
root spiral by forming the right angles in the triangles and mark the lengths of the outside
edges of the spiral. The following steps describe this procedure step-by-step.
1. Choose one of the corners of the note card and mark equal lengths on the
adjacent sides.
2. Starting at the center of the paper, use the unit markings to trace the first right
angle and to mark the lengths of the adjacent sides of the angle.
3. Use a ruler to draw the hypotenuse of the first right triangle.
4. Place the index card so that one side of the right angle lies on the hypotenuse
of the first triangle, with the vertex of the right angle on the index card exactly
aligned with the vertex of the hypotenuse and leg of the right triangle. Trace
the segment for the unit leg of the second triangle.
SPIRALS AND QUILT DESIGN 1
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5. Use the ruler to draw the hypotenuse of the second triangle.
6. Repeat steps 4 and 5, drawing as many triangles as you have room for on the
paper. When you get to the 17th triangle, the triangles will begin to overlap.
Draw the overlap radii lightly in pencil so you can erase them when done.
The Spiraling Squares Quilt can be quite difficult when using a compass and ruler
so the Mathematical Quilts book offers an easier construction. First decide what size
squares you want to start with. Cut a template for this square out of tag board, and
determine the radius needed for the beginning of the circle. (Refer to Spiraling Squares
section on Laws of Sines and Logarithmic Spirals.) Use a compass to draw the circle and
then divide the circle into 12 equal arcs using either a compass or protractor. Trace the
square into the appropriate positions. Next, make a pattern for the next size square by
using the diagonal length of the previous square as the side length for the new square.
Continue adding larger and larger squares until your design is complete.
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The Indiana Puzzle quilt is also known as the Snail’s Trail pattern. The pattern
consists of 16 squares, each of which contains four spirals. When the squares are
combined, a far more complex pattern of spirals emerges. Follow these steps to create the
spiral design for the Indiana Puzzle quilt. (This quilt was previously mentioned in
Logarithmic Spirals, pg. 11.)
1. Draw a square on graph paper. Because you will be bisecting the sides of the
square, it works best if the dimensions of the sides are a power of 2.
2. Mark the midpoint of each side of the square.
3. Connect the midpoints to form a new square inscribed in the original square.
4. Repeat steps 2 and 3 until you have the desired number of squares.
5. To complete the design, shade the similar right isosceles triangles to form a spiral.
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Conclusion
It is natural to wonder where the logarithmic and Archimedean spirals are found
in nature. Archimedean spirals can be found in pinecones, a Hawaiian fern, and the Great
Mosque of Samarra.
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A particularly beautiful logarithmic spiral is found in the chambered nautilus, a
sea creature that lives in South Pacific. As it grows, the nautilus shell curves and
resembles this spiral. In the nautilus shell each chamber is 6.3% larger than the one
before. Other examples of logarithmic spirals are draining water, low-pressure systems,
Romanesco broccoli, and patterns in sunflowers.
Mathematical quilts can produce visual images of spirals and patterns found in the
real world. While exploring these quilts students can begin to make connections with the
Logarithmic and Archimedean spiral and the mathematics that support these spirals.
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References
http://www.2dcurves.com/spiral/spirallo.html.
http://www.mi.sanu.ac.rs/vismath/meel2008/index.html.
Venters, Diana & Ellison, Elaine Krajenke. Mathematical Quilts, published by Key
Curriculum Press, 1999.
Weisstein, Eric W. “Logarithmic Spiral.” From MathWorld-A Wolfram Web Resource.
http://mathworld.wolfram.com/LogarithmicSpiral.html.
Weisstein, Eric W. “Archimedes’ Spiral.” From MathWorld-A Wolfram Web Resource.
http://mathworld.wolfram.com/ArchimedesSpiral.html.