If we want to find a specific periodicity, we can rearrange the equation to In general, the periodicity P = 360 / A, where is the angle around the circle. The next iteration takes the point to 810 degrees, which is the same angle as 720 + 90, or 2 and 1/4 times around the circle. This is the same as 360 + 180, so it's halfway around the circle. Then the second iteration adds 270 degrees, which is 540 degrees. The first iteration takes the point 3/4 of the way around the circle or 270 degrees. We can see why this works by examining the progression of the angles. There is another angle that will generate period-4 patterns:Ģ70 degrees. Then it returns to 360, which is the same as 0 degrees, and the pattern begins again. The progression of the points is just 0, 90, 180, 270 degrees. The periodicity can be determinedīy dividing the angle of a full circle, 360 degrees, by the rotation angle. If you set the angle to be 90 degrees, The dots will grow in a square pattern, that is, with a period of 4. If you set the angle to 180 degrees, the point will rotate to the other side, and then back again at the next iteration, and so on, oscillating with a Click on "Connect Dots" to make the connections easier to see. It can become difficult to determine the order of the dots. But as you increase the angle, all sorts of interesting patternsĮmerge. Applet courtesy of Andrei Buium.Īt very low angles, the applet creates a simple spiral. The distance of each point from the center keeps increasing at the same rate.Īdjust the angle with the slider, or use the arrow keys to amke small adjustments. What is the ratio of an 8.This Spiralizer generates dots at a given angle. What is the error between this value and φ What is the ratio of a Hi-Def TV that has a resolution of 1920x1080 pixels? (Use 4 digits of precision) It is difficult to really know how common the appearance of φ is, or how it might have appeared in some of the places it is claimed.ThereĪre lots of proportions in nature and art that are close to φ, but the question is, how close? In other words, what is the error It is commonly stated that the Golden Ratio φappears in art, architecture and design, for instance in the proportions ofįamous buildings such as the Parthenon in Greece. The ratio of the Red to the Green line is the same as the ratio of the Green line to the Blue Line, which is the same as the ratio of the Blue line to the Purple line.Īll of these ratios are φ, the Golden Ratio. It has the beautiful property that you can subdivide it by scaling and rotating the same shape to fit inside itself perfectly forever as shown below. The Golden Rectangle is a rectangle whose long side is 1.61803399 times longer than its short side. Now let's look at the Golden Ratio in geometry. The error equals the absolute value of 1.61803399 - 1.66666667 = 0.048632680Īfter only a dozen iterations the sequence has converged quite close to the value of φ. For instance, to find the error for F 6 / F 5, first see that 8 / 5 = 1.66666667, so The difference you get, but without a plus or minus sign. Only interested in the absolute value, that is the size of the difference, not whether it's bigger or smaller than φ, so just enter Do your calculations with 8 decimals of precision to match the numbers above. For this exercise, calculate the ratio of consecutive numbersĪnd find the difference between your answer and φ. How quickly does the value of the ratio of Fibonacci numbers converge to the number φ? Let's measure the error, or differenceīetween various values of the ratio of numbers in the sequence and φ. What is the ratio of F 11 / F 10: (Use 8 decimals of precision for your answers.) The value it settles down to as n approaches infinity is called by the greek letter Phi or φ, and this number, called the Golden Ratio, The ratio of the successive Fibonacci Numbers gets closer and closer to a certain value as n gets larger and larger.
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