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 SPIRALS AND QUILT DESIGN 2 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 SPIRALS AND QUILT DESIGN 3 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. SPIRALS AND QUILT DESIGN 4 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. SPIRALS AND QUILT DESIGN 5 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 SPIRALS AND QUILT DESIGN 6 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. SPIRALS AND QUILT DESIGN 7 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. SPIRALS AND QUILT DESIGN 8 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. SPIRALS AND QUILT DESIGN 9 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. SPIRALS AND QUILT DESIGN 1 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 SPIRALS AND QUILT DESIGN 1 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. SPIRALS AND QUILT DESIGN 1 2 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 SPIRALS AND QUILT DESIGN 1 3 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 4 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. SPIRALS AND QUILT DESIGN 1 5 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. SPIRALS AND QUILT DESIGN 1 6 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. SPIRALS AND QUILT DESIGN 1 7 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. SPIRALS AND QUILT DESIGN 1 8 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.
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