Difference between revisions of "Spiral Explorations"
Line 146: | Line 146: | ||
Following the formula, the length of this spiral is | Following the formula, the length of this spiral is | ||
− | 1*[ | + | 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: | Area: | ||
− | 1*[(180/360]*π*[1+((4/3)^2)+(( | + | 1*[(180/360]*π*[1+((4/3)^2)+((4/3)^4)+ ((4/3)^6)+((4/3)^8)] = 32.27 cm^2 |
Revision as of 16:49, 16 June 2013
Alternative Fibonacci Spiral |
---|
Alternative Fibonacci Spiral
- This is a guide to constructing arcs. The picture on the right demonstrates an alternative way of constructing the Fibonacci Spiral. 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. On 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 circle pattern: one circle that is tangent to a second slightly larger circle; then the second circle is tangent to a third circle, with the third larger than the second; and the pattern continues with tangent circles inside of each other. 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. The next part would be the arc from the point where the second circle is tangent with the third circle, to the point where the third circle is tangent to the fourth 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.
The bolded spiral is the well-known Archimedean Spiral, or the beginning of it. The pattern of the spiral is determined by corresponding points on right triangles that have a relationship to each other in the lengths of their hypotenuses, which is: , , , ...etc. All of the right triangles meet at a point: the center of the spiral. The points on the spiral continuously get farther away from this point.
This is the first image we started with. 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 1/4 sections of circles (arcs). 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.
These spirals have no constant beginnings, but it is important to realize that they still have a center: a point that the spiral continuously travels away from. Similarly to the spirals with constant beginnings, the center of this type of spiral is located at the first point on the spiral: the beginning--which is not constant for all spirals, but it present on all spirals. For example, think of spiral with a ratio of the radii of the circles of 1:2. This first iteration of this spiral could have the smallest radius measuring 1 unit; however, it could also have a smallest radius of 1/2 units, or 1/4, or 1/8, etc. The location of the center is different in each of those cases, but the rule for finding the center of a spiral is as follows: it is the first point on the first arc of the spiral.
The Fibonacci Spiral is on the left, and a spiral with a 1:2 ratio is on the right. The orange segments connect the centers (in bright green) to random points on the spiral. The orange segments get progressively longer, demonstrating that spirals originate from this exact point. However, this one point is only one example of a center of a spiral. All spirals can have more that one center from which all points get father away. The easiest example of this is the Archimedean Spiral: all of the triangles meet at one vertex. Every point on the spiral gets progressively farther away from this point. Shown below, a 1:2 ratio spiral demonstrates the fact that all spirals have more than one center:
In the image, point G is a center. The requirement for all centers of spirals are as follows: the center must be closer to the first point on the first arc of the spiral, point D, than any other point on the spiral, for example, points E and I. To ensure that a center is closer to point D than any other point, one can draw a triangle connecting the three points. As long as the measure of angle GDE (or GDI) is greater than the measure of angle DEG (or DIG). The measure of segment GD must be greater than the measure of segment GE (or GI).
This is one type of center that all spirals have. The other type of center is the point around which the spiral revolves, which we will refer to as a rotational center. This point is the center of the circle used in creating the smallest arc on the spiral. The spiral as a whole appears to 'spiral' away from this one point.
The 1:2 spiral may look similar to the Fibonacci Spiral, but as they 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.
A More Mathematical Explanation
Both the Fibonacci Spiral and the 1:2 ratio spiral have centers that are rotated 90° and then transl [...]
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:
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 calculate the length of a spiral to a certain iteration. This formula works with all spirals that follow a repetitive 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).
Here is the formula for 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:
Basically, the spiral length is equal to the sum of the lengths of the arcs that make up that spiral. An arc is simply a fraction of the circumference of a circle. Therefore, the length of an arc is the circumference (2π*radius) multiplied by (number of degrees of the arc/360). If you look at the spirals on our page, you will notice that each arc takes up the following fraction of the entire circle: (the absolute value of (180-a)/360, with "a" equaling the angle in which the centers of the circles are rotated. Assuming the radius of the first circle is 1, the length of the first arc would be [(absolute value of (180-a))/360]*2π*1, which simplifies to [(absolute value of (180-a))/180]*π. The second arc would be the length of the first arc multiplied by the ratio in which the circles increase: [(absolute value of (180-a))/180]*π*r. The third arc would be the length of the second arc multiplied again by the ratio: [(absolute value of (180-a))/180]*π*r*r, or [(absolute value of (180-a))/180]*π*r^2. The pattern continues like this until the last arc: [(absolute value of (180-a))/180]*π*r^(n-1). Add all these together, and you get: {[(absolute value of (180-a))/180]*π} + {[(absolute value of (180-a))/180]*π*r} + {[(absolute value of (180-a))/180]*π*r^2}... + {[(absolute value of (180-a))/180]*π*r^(n-1)}. Because of the distributive property, you can simplify this to the following: [(absolute value of (180-a))/180]*π*[1+(r^2)+(r^2)+(r^(n-1))]. However, this equation only works if the radius of the first/smallest circle is equal to 1. If the radius of the first circle is any number other than one, this changes the length of the spiral. So we added "x" into the equation ("x" equals the radius of the first/smallest circle). If the radius of the first circle is any number other than one, you simply multiply the entire equation by that number. Therefore, the formula for spiral length is: x*[(absolute value of (180-a))/180]*π*[1+r+(r^2)+(r^3)...+(r^(n-1)]
Note: the formula explained above works with almost all spirals that follow a repetitive pattern (where the increase from one circle to the next can be explained using a ratio), but there is one exception. When the centers of the circles that form the spiral are rotated by 180 degrees, the formula is changed slightly. This is because [absolute value of (180-180)] is equal to 0, so the formula above would calculate the length of the spiral as zero, which is obviously not the length. For spirals in which the centers of the circles are rotated by 180, the formula has [absolute value of (180)] instead of [absolute value of (180-a)]. The entire formula would be [(absolute value of 180)/180]*π*[1+(r^2)+(r^2)+(r^(n-1))]. Since (absolute value 180)/180=1, it can be simplified to simply x*π*[1+(r^2)+(r^2)+(r^(n-1))]
Spiral Area
Just like with spiral length, there is technically no spiral area. However, we wanted to come up with a way to calculate the area of the arc sectors that make up the spiral. The sum of the arc sectors that make up the spiral is what we call "spiral area".
Here is the formula for 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:
We came up with this formula in basically the same way we came up with the formula for spiral length: we found the area of each of the arc sectors, added them together, then simplified. Similar to how arc length is degrees multiplied by circumference, area of an arc sector is degrees multiplied by area of the circle, or degrees*π*radius^2. Assuming the radius of the first circle is 1, the area of the first arc sector would be [(absolute value of (180-a))/360)]*π*1^2. The radius of the second arc sector is the radius of the first multiplied by r (ratio in which the circles increase). So the area of the second arc sector would be [(absolute value of (180-a))/360)]*π*r^2. The radius of the third arc sector is the radius of the second multiplied again by r (radius of the third arc sector would be r^2). So the area of the third arc sector would be [(absolute value of (180-a))/360)]*π*((r^2)^2). Following this pattern, the area of the fourth would be [(absolute value of (180-a))/360)]*π*((r^3)^2) This continues until the last arc sector: [(absolute value of (180-a))/360)]*π*(n^(n-1))^2. If you add them together, then simplify in the same way that we simplified the spiral length formula, you get: [(absolute value of (180-a))/360)]*π*[1 + r +(r^2)^2 + (r^3)^2 + (n^(n-1))^2]. But if the radius of the first circle is any number other than 1, you have to multiply the entire equation by that number (shown as "x" in the formula), just like with the formula for spiral length. Therefore, the final formula that we came up with for spiral area is: x*[(absolute value of (180-a))/360]*π*[1+(r^2)+(r^4)+ (r^6)...+(r^(n-1))^2]
The length of a spiral is infinite, but it is possible to calculate the length for iterations of the spiral. The spiral length, in this case, is defined as the arc length of each arc on each circle for a certain number of circles. Below, we calculated the arc lengths of certain spirals to the 5th iteration.
1:2 ratio rotated 180 degrees
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
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
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
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
In summary, spirals have a bunch of things in common. Here's a list:
1) All spirals are make 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. Then, the only things that remain are arcs that do not connect. It ends up looking 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. Meaning, 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 is closer to 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 is closer to 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.
Teaching Materials
- There are currently no teaching materials for this page. Add teaching materials.
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.
[[Category:]]