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Numberphile goldenratio4/27/2023 ![]() I hope you get the idea by now and I don't know how close the analogy really is. Then if you choose half a turn, so pi*R which is -1, it alternates between -1 and 1, left and right., just like in this video. or right, right, right., like in this video. Starting to the right of what is essentially a trigonometric cirlce (so 1) but that you draw as a X, it gives you 1, 1, 1. Since what you did there in the mandelbrot video is basically taking a C (which is a fraction) within this circle and squaring it iteratively in the 2D plane, this is exactly the same thing the imaginary flower of the example is doing: choosing a fraction of a turn (a C) and square it iterativelly, which is equal to doing the same fraction of a turn over and over because we are in the 2D plane and multiplying two complex numbers is like turning and stretching until their points met. In Ben's mandelbrot video, we see paterns like that forms when you move the value of the constant C within the "non exploding area", between -1, 1, i and -i.īecause of that radius 1 centered on 0, this means this circle contains all complex numbers containing only rational parts. The Golden Ratio ( why it is so irrational), Youtube.I just saw the other video Ben Sparks did on the mandelbrot set and watching both of them made me realize something about complex numbers.ģBlue1Brown once explained and showed multiplying a number by another on the complex plane was like turnning and stretching a transparant of this plane such as to make the point of the first number correspond to the resulting point/number. The Algorithmic Beauty of Plants, Springer-Verlag. Przemyslaw Pruskinkiewicz and Aristid Lindenmayer (1990). ![]() We discovered that when the number of seeds emplaced is equal to the golden ratio, the resulting pattern resembles that found in a sunflower □! References The Golden Ratio (why it is so irrational) - Numberphile Numberphile 3.5M. By varying the number of emplaced seeds per 360 degree rotation about the flower’s centre, we can generate a whole range of different seed patterns, from straight spokes to space filling patterns. mathematics in nature golden ratio Activity 1 Math in Nature - Name: John. In this article, we explored how to model the placement of seeds within a flower. It is because of this irrationality that results in the space filling pattern when we emplace φ seeds per 360 degree rotation - as φ cannot be well approximated by any rational number, no well defined, repeating number of spokes appear unlike when π, 10 or 2 seeds are emplaced per 360 degree rotation. Hence, the golden ratio is also sometimes known as the most irrational number! This is in contrast with other rational numbers such as π which can be well approximated by rational numbers such as 3 or 22/7. The continued fraction for φ continues in this recursive relation forever, meaning that φ is not well approximated by any rational number. The Golden ratio expanded as a continuous ratio. For example, if we use nseeds = 10, we get a flower where the seeds are arranged neatly in 10 spokes! flower_seeds(10) Intuitively, we would expect the seeds to form some sort of neat repeating pattern, as the angular position of the seeds repeats for each full rotation. if count >= nseeds: count = 0 L = L + D = np.dot(L * np.eye(2), np.array() / np.sqrt(X**2 + Y**2)) x.append(X) y.append(Y) plt.plot(x, y, 'o') plt.axis('equal') plt.axis('off') plt.show() Placing an Integer Number of Seedsįirst let us see what happens if we place an integer number of seeds per 360 degree rotation. = np.dot(R, np.array(, y])) count = count + 1 # If a full rotation is made around the centre, # increase the seed position outwards. for i in range(NTRIES): # Rotate and place a new seed. # Seeds cannot lie on top of each other! L = D count = 0 # To-do: optimize the code below. x.append(D) y.append(0) # L is a scaling factor, set equal to D initially. # Rotation matrix: R = np.array(, ]) # Empty lists to hold the x and y seed locations. theta = theta * np.pi / 180.0 # Convert to radians. theta = 360.0 / nseeds # Rotation angle in degrees. # NTRIES = number of total iterations to run the algorithm. Pentagons and the Golden Ratio David: QM Emergence of Numbers and Geometry from probability-> dominance-distribution properties, observable phenomena of. # D = offset distance of the seed location. import numpy as np import matplotlib.pyplot as plt phi = (1 + np.sqrt(5)) / 2.0 # golden ratio def flower_seeds(nseeds = np.pi, D = 1, NTRIES = 200): # nturns = number of seeds to place in 360 degrees. Instead of specifying the angle of rotation, we instead specify how many seeds we want to place in a full rotation of 360 degrees in the argument nseeds. ![]() ![]() This algorithm above can be coded in python quite easily as follows. We mainly post videos about mathematics and just numbers in general. Algorithm for seed placement in a flower, with a rotation angle of 90 degrees (or 4 seeds per 360 degrees).
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