$Fibonacci fractals in ultra fractal$

$fibonacci fractals in ultra fractal$

FIBONACCI FRACTALS IN ULTRA FRACTAL FULL

When the angle of rotation is not simple like 1/2, 1/3, or 1/4, but instead is an irrational fraction of the circle, then the angle never quite repeats itself, and This is a relatively small proportion, and is a poor use of space. In a period-4 pattern, and the percentage of the space filled by dots is 11.436%. For instance, when the angle of rotation is 90 degrees, the dots fall You can see the percentage of space filled by dots at the bottom of the Spiralizer. This is a simple pattern, but it is NOT the most efficient way to pack a lot of seeds The whole way around the circle, the dots (or seeds) line up in radial arms. How does the angle affect the outcome of the pattern of seeds? As we saw with the spiralizer, when you set the angle to be a simple fraction of It keeps rotating by the same angle and adding seeds, which all keep growing in scale and distance from the center. Then it rotates again by the same angle andДЇorms a third seed.

$fibonacci fractals in ultra fractal$

It creates a seed at the origin, and then it rotates by a certain angle and creates another seed. Shyamal and Jonathan Wolfe.ДЄ sunflower pattern can be created by a simple repetitive process similar to how the spiralizer forms its patterns. The Fibonacci Sequence appears in many plants, and can be seen in the distribution of the seeds in the sunflower below. The more complex details in the next section, when we relate this behavior to the periodicities of the Mandelbrot Set. There are many other higher-order periodicities that emerge in between the simple angles that create triangles, square, pentagons and hexagons, and we will explore You can nudge the angle 0.1 degreesДЄt a time with the arrow keys to see the effect of perturbing the angle just slightly.ДЇind an angle that generates a perfect triangle pattern (period 3): ДЇind another angle that generates a perfect triangle pattern: ДЇind an angle that generates a perfect pentagon pattern (period 5): ДЇind another angle that generates a perfect pentagon pattern: ДЇind an angle that generates a perfect hexagon pattern (period 6): ДЇind another angle that generates a perfect hexagon pattern: One interesting thing to observe is that when the angle is set exactly to a value that repeats perfectly, such as 90 degrees, the points line up in straight lines.Д«ut if you adjust the angle just a tiny bit larger or smaller than one of these perfect values, then the dots start to twist into spirals. 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:Дў70 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.

FIBONACCI FRACTALS IN ULTRA FRACTAL FULL

The periodicity can be determinedД«y 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.

$fibonacci fractals in ultra fractal$

It can become difficult to determine the order of the dots. But as you increase the angle, all sorts of interesting patternsД®merge. Applet courtesy of Andrei Buium.ДЄt very low angles, the applet creates a simple spiral. The distance of each point from the center keeps increasing at the same rate.ДЄdjust the angle with the slider, or use the arrow keys to amke small adjustments.

$fibonacci fractals in ultra fractal$

This Spiralizer generates dots at a given angle.

$Fibonacci fractals in ultra fractal$