Spiral Explorations
From Math Images
Alternative Fibonacci Spiral |
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Alternative Fibonacci Spiral
- The picture on the right demonstrates a way of constructing the Fibonacci Spiral (explained below) using arcs. If you are not familiar with the fibonacci sequence, you can click here. Instead of using the squares to determine arcs, this uses circles to determine the arcs (their position and size) that make up the spiral. The traditional squares are overlaid to demonstrate the pattern: the ratio between circle sizes matches the ratio between square sizes.
Contents |
Basic Description
The basic definition of a spiral is a curve that winds in a gradually widening pattern. In this page, we are going more in depth on what a spiral is—what exactly causes a spiral to be defined as a spiral—and how to construct spirals. We are also exploring the relationships between different types of spirals. Each spiral starts with a circular pattern: one circle is tangent to a second slightly larger circle, then the second circle is tangent to a third circle, and so on. The spiral is made up of parts of each circle; each circle contains one arc that contributes to the spiral. One arc is the part of the circle from one tangent point to another tangent point. For example, one part of the spiral would be the arc from the point where the first circle is tangent with the second circle, to the point where the second circle is tangent with the third circle., and so on. Each arc is bigger than the previous arc so any given point is farther away from the center than the previous point. Together, the arcs form the spiral.Shown above is a Fibonacci Spiral, in green, constructed from arcs on circles. The squares are included to show how parts of circles are used in spirals. The Fibonacci Spiral is created using arcs which are 1/4 of their respective circles. The radii of the circles fit the Fibonacci Sequence with the smallest circle having a radius of 1, the next largest being 2, then 3, then 5, and so on. Each circle has a radius along the length of one of the squares. The centers of the circles are rotated 90 degrees from each other at a distance that also fits the Fibonacci Sequence. That is, the center of the smallest circle is rotated 90 degrees and translated 1 unit away. Then, the next center is rotated 90 degrees and translated 2 units away. This pattern continues. This way, the circles are tangent to each other, and arcs become visible:
Shown above, on the left, the centers (shown in red) are rotated counterclockwise 90 degrees at a distance determined by the Fibonacci Sequence (1, 1, 2, 3, 5, 8, etc. units).
In the middle, circles were constructed around each of the points with radii starting at the second number in the Fibonacci Sequence (1, 3, 3, 5, 8, 13, etc. units).
On the right, the Fibonacci Spiral is constructed, shown in thick green. It is important to determine which direction the figure should spiral. The spiral shown above spirals counterclockwise away from the center. Each arc on the circles is 1/4 of the circle's circumference except the smallest arc, which is 1/2. The smallest arc is 1/2 of the circumference to clearly mark the beginning of the spiral. Any spiral where the radii are based on a predetermined set of numbers that start at one specific, unchangeable number and that get progressively bigger have beginnings. For example, the Fibonacci Spiral and the Archimedean Spiral start at a predetermined number—1 and respectively—and get progressively bigger. However, spirals where the radii of the circles follow a pattern like the spiral shown below do not have a constant beginning. These spirals are constructed of arcs on circles where the ratio between the radii of the first (smallest) circle and the second circle is 1:2. As the spiral continues, the pattern continues; the radius of one circle is doubled to get the radius of the next circle. These spirals have no constant beginnings because the measurement of the smallest radius can simply be halved, resulting in a smaller arc.
The 1:2 spiral may look similar to the Fibonacci Spiral, but as both spirals get bigger, the differences between the radii increase, causing the spirals' shapes and sizes to differ. From the tenth to the seventeenth iteration, the Fibonacci Spiral follows the pattern 55; 89; 144; 233; 377; 610; 987; and 1,597. The spiral with the 1:2 ratio, with the first iteration of 1 unit, follows the pattern 512; 1,024; 2,048; 4,096; 8,192; 16,384; 32768; and 65,536. Obviously, the spiral with the 1:2 ratio is much larger than the Fibonacci Spiral. But what does this mean? How can it be measured?
It is possible to compare spirals based on the areas of their arc sectors and the lengths of the arcs that make up the spiral.
Spiral Centers
A More Mathematical Explanation
Exploring 180° Rotation
Both the Fibonacci Spiral and the 1:2 ratio spiral have centers that are [...]Exploring 180° Rotation
Both the Fibonacci Spiral and the 1:2 ratio spiral have centers that are rotated 90° and then translated. We explored what would happen if we rotated the centers by 180 degrees (the centers would be on a straight line) but still translated using the previous pattern (the Fibonacci Sequence and the 1:2 pattern). The outcome is the following picture:
This spiral looks "tighter" than the Fibonacci Spiral; the spiral is more "controlled." The arcs that create the spiral now are 1/2 of the individual circle instead of 1/4 because the circles are tangent to each other in different places. When the centers are rotated 180°, they all lie upon a line. This line connects all of the centers and all of the tangent points of the circles. Because this line intersects each circle twice, there are only two possible places for two circles to be tangent to each other. In a spiral, three circles cannot be tangent in the same place, so the tangent points must alternate. When the line that connects the centers and tangent points is vertical, the alternation can be described like this: image the spiral on a coordinate grid. If a circle is tangent to "the bottom" (a point lower on the y-axis) of a smaller circle, then it must be tangent to a bigger circle at a point higher on the y-axis. Because of this alternation, the arcs make up 1/2 of their individual circles.
Next, instead of using the Fibonacci Sequence, we tried a 1:2 ratio. To summarize: the circle's centers are rotated 180 degrees from each other, or in other words, they are on a straight line. The radii of the circles are dependent on a 1:2 ratio. The circles are tangent to each other, so the distance between the centers is doubled each time another circle is created. Each bold arc is half of its individual circle instead of 1/4:
Spiral Lengths and Areas
Spirals can be compared in terms of size and shape, but they can also be compared using "spiral area" and "spiral length."
Spiral Length
Technically, it is impossible to find the length of a spiral because spirals go on forever, so the length would therefore be infinite. However, it is possible to find the length for iterations of the spiral. We came up with a formula to estimate the length of a spiral to a certain iteration. Each iteration would be added to find the desired spiral length after iteration n.
Note: These formulas serve as approximations because we are adding arcs at discrete (non-moving) intervals. Realistically, the radii increase gradually as the spiral progresses, not jumping from integer to integer, as is assumed here.
This formula works with all spirals that follow an iterative pattern; in other words, all spirals in which the increase from one circle to the next can be described using a ratio (so it does not work with the Fibonacci Sequence or the Archimedean Spiral, for example).
The formula for the spiral length estimate is as follows:
Spiral Length=x*[(absolute value of (180-a))/180]*π*[1+r+(r^2)+(r^3)...+(r^(n-1)]
x=radius of the first (smallest) circle
a=angle/degree in which the centers of the circles are rotated
r=ratio of increasing circles (written as an improper fraction, with the larger number in the numerator)
n=number of iterations
How we found this formula/why it works:
Spiral Area
Just like with spiral length, there is technically no spiral area. However, we wanted to come up with a way to estimate the area of the arc sectors that make up the spiral. The sum of the arc sectors that make up the spiral, as shown in the image below, is what approximate what we call "spiral area".
Note: Again, this is an approximation for similar reasons provided in "Spiral Length"
The formula for approximating spiral area is as follows:
Spiral Area=x*[(absolute value of (180-a))/360]*π*[1+(r^2)+(r^4)+ (r^6)...+(r^(n-1))^2]
(variables used in this formula are the same as those used in the formula for spiral length)
How we found this formula/why it works:
Examples
We will approximate the spiral area and the spiral length of certain spirals.
1:2 Ratio Rotated 180°
This is a 1:2 ratio spiral with a 1st iteration circle with a radius of 1 cm.
Following the formula, the length of this spiral is
(1)*[180/180]*π*[1+(2)+(4)+(8)+(16)]
The absolute value of 180-180 is 0, but, following the rule for spirals with a 180° rotation, we will substitute 180° in. π(1+2+4+8+16) = π(31) ≈ 97.39 cm
The area of this spiral is
1*[(180/360]*π*[1+(2^2)+(2^4)+ (2^6)+(2^8)] (1/2)π*[1+4+16+64+256] ≈ 535.64 cm^2
1:2 Ratio Rotated 90°
Notice the effect on the length and area of the 1:2 ratio spiral when the centers are rotated 90° instead of 180°:
This is a 1:2 ratio spiral (where the smallest circle has a radius of 1) with the centers rotated 90°. Substituting the radii and the angle of rotation into the length equation results in the following equation: 1*[(absolute value of (180-90))/180]*π*[1+2+(2^2)+(2^3)+(2^4)] = (1/2)π*(31) ≈ 48.69 cm.
Area: 1*[(90/360]*π*[1+(2^2)+(2^4)+ (2^6)+(2^8)] = (1/4)π*(1+4+16+64+256) ≈ 267.82 cm^2
1:2 Ratio Rotated 45°
Now, instead of rotating the 1:2 ratio spiral by 90°, the spiral is rotated by 45°:
Substituting the radii and the angle of rotation results in this equation: 1*[(absolute value of (180-45))/180]*π*[1+2+(2^2)+(2^3)+(2^4)] = (135/180)π*(31) ≈ 73.04 cm
Area: 1*[(180-45)/360]*π*[1+(2^2)+(2^4)+ (2^6)+(2^8)] = (135/360)π*[341] ≈ 401.73 cm^2
3:4 Ratio Spiral Rotated 180°
This is a 3:4 ratio spiral with a 1st iteration circle with a radius of 1, and a rotation of 180°.
Following the formula, the length of this spiral is 1*[180/180]*π*[1+(4/3)+(4/3)^2+(4/3)^3+(4/3)^4] π*[1+(4/3)+(16/9)+(64/27)+(256/81)] ≈ 30.29 cm
Area: 1*[(180/360]*π*[1+((4/3)^2)+((4/3)^4)+ ((4/3)^6)+((4/3)^8)] = 32.27 cm^2
Spiral Requirements
There are certain properties common to all spirals. They include:
1) All spirals are made of connected arcs belonging to tangent circles. If the circles are not tangent, then there is no place for the arcs to connect to each other. The result looks like a broken spiral, where the "pieces" are the unconnected arcs. The tangent points follow a pattern. For example, spirals where the centers of the circles are rotated 180° from each other have tangent points on alternating sides of the circle. This means that if the smallest circle is tangent to the second-smallest circle on "top" (if the circle was on a grid, the highest point on the y-axis), then the second circle will be tangent to the third circle on the "bottom" (the lowest point on the y-axis).
Spirals that are made of circles rotated 90° also follow a pattern. If the centers of the circles are rotated counterclockwise and the first circle is tangent to the second circle on "top," then the second circle will be tangent to the third circle on the left, the third will be tangent to the fourth on the bottom, and the fourth will be tangent to the fifth on the right. These patterns repeat as the spiral gets bigger.
2) Spirals can go on forever; they circle around a central point, and they get progressively farther away from a different central point. Each point of the spiral is farther away from this center of the spiral than the previous point. All spirals are never-ending because the patterns that the radii of the circles they are constructed of never end. The Fibonacci Sequence is infinite, for example. Also, spirals that are based off of ratios never end because those patterns can continue indefinitely, too. If the points do not get progressively farther away from a center, then the spiral appears to circle back, like in the pictures below:
This shape circles back to one point and it is obviously not a spiral.
This one seems like a spiral, since it is made of tangent circles, but it does not grow progressively bigger. The distance of L2 is shorter than L1, which means that the shape grew closer to the center than farther. This violates the personality of a spiral because it grew closer to the center at one point rather than growing farther. This may be why the shape seems more elliptical than round; it is lopsided and irregular. Also, it does not appear to have a rotational center because of this lopsidedness.
3) The proportion between every two circles that make up a spiral must not be 0 or 1. The ratio between one circle to the next circle determines how tight or wide the spiral is. If the ratio between the circles approaches 1, the spiral will be tighter (the arcs would be closer to the same size because the circles would be more similar in size ).
If the ratio between the circles approaches 0, then the spiral will be wider (one arc would be much bigger than the previous arc because the circles would be very different in size).
If the proportion is 0, then only one circle exists, so there is no tangent point and no place for the spiral to continue.
If the proportion is 1, then the circles are the same size. The circles must be tangent to each other to continue the spiral, but when a circle is tangent to a circle of the same size, then they are tangent at all points on the circles. The circles would be on top of each other. There would be no place for the spiral to continue here as well because there would be no visible pace for another arc.
4) Two subsequent circles cannot intersect If the subsequent circles intersect, then they are not tangent; circles that intersect cannot be tangent to each other.
Why It's Interesting
These spirals are interesting because they occur in nature! Pages like Romanesco Broccoli and Fibonacci Numbers explore these phenomenon.Spirals in Plants
Spirals not only appear in nature but they play a big part in a plant's survival.
Phyllotaxis is the study of a plant's ordered position of leaves on its stem, and research shows that leaves are arranged in a spiral pattern that maximizes the space for each leaf, or how much light they receive.
The following picture is an example of a plant exhibiting a spiral pattern, the clearest representation is the angle on the lower left side:
Sunflowers themeselves are arranged in a spiral. The spiral arrangement gives them a biological advantage because it maximizes the quantity of seeds that can be stored in seed head.
Hence, we see that spirals help maximize a plant's chance of survival!
Teaching Materials
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References
Images were created in the Sketchpad by group members Natalia, Jane and Hannah
Future Directions for this Page
Further explorations include making actual formulas rather than approximations.
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.