We start with a 1×1 square and draw an arc, center C, through 2 corners such that the sides of the square are tangent to the arc (that is, they touch once only). We can obtain a spiral that looks quite similar to the Golden Spiral by using arcs of circles that increase in size by the Golden Ratio, as follows. You can see on the graph above each spiral arm makes an angle of 73° with the x-axis (and y-axis, or any line from the center).Īpproximating the Golden Spiral using arcs of a circle This is the angle our spiral arms make with a line from the center of the spiral. If we keep going, we'll get a spiral as follows (this is 2 complete revolutions, or 4π = 720°):ī = arccot 0.30635 = 1.274 radians or around 73° We can see these values are correct on our spiral graph above. units along the positive y-axis, that is (6.854, 0). One more quarter turn will bring us to φ 4 = 6.85410. So the spiral will cut the y-axis at (0, -4.236). We could work out our next position, along the negative y-axis, by just multiplying this last value by φ = 1.6180, giving us: We are now 2.618 units away from the origin along the negative x-axis, or φ = 1.6180 times the distance from the origin we were at the last quarter turn. That is, φ = 1.6180 times the distance we started from.Īnother rotation of a quarter turn brings us to t = π, where: Note that we are now 1.618 units from the origin up the y-axis. You can see the starting point in the following graph of the spiral. So it means we start 1 unit from the origin along the positive x-axis. Starting at t = 0, we get the starting point of the curve: Let's substitute a few important values to see what this expression means. To convert the polar form (which we've got) to rectangular form (which we need for the graph) in Geogebra, we need to set up and graph the following function: Note that r increases at an ever-increasing rate (it gets steeper) as t increases.īut to see a spiral, we need to graph the curve using polar coordinates. Now if we graph our function on ordinary rectangular coordinate axes in GeoGebra, we get the following exponential curve. Setting up the Golden Spiral using GeoGebra
#Fibonacci spiral free#
We'll use the excellent free graphing tool GeoGebra from here on. Using this value, and taking the simple case where a = 1, our function becomes: The Golden Spiral has the special property such that for every 1/4 turn (90° or π/2 in radians), the distance from the center of the spiral increases by the golden ratio φ = 1.6180.įor this to occur, cot b must take the value (which comes from solving our function): Note: Normally, we use θ for the independent variable, but we often use t as we can think of the spiral being traced out over time. We can write the general logarithmic spiral as a function in polar coordinates using t as follows: The Golden Spiral is a special case of the logarithmic spiral. Choosing the start point for the fern is not an exact science!) Golden Spiral (In an actual logarithmic spiral, they are exactly the same. In the fern case above, b ≈ 1.4 radians (≈ 80°).Īs a consequence of the way we defined the logarithmic spiral, the ratio of the distances from the center to each spiral arm of an adjacent pair is constant.ĭistance to the first arm: distance to second armĭistance to the second arm: distance to third arm So the coordinates of a point on the curve in polar coordinates is given by ( r, θ).ī is the angle (in radians - the "equal" angle) that the line from the center of the spiral makes with the tangent to the spiral. Θ is the angle (in radians) from the horizontal axis. R is the distance from the origin (or "pole") The formula for a logarithmic spiral using polar coordinates is: Otherwise, if we use ordinary rectangular coordinates, the formulas become very complex. We normally use functions in Polar Coordinates when describing spirals. To see what this means, the 3 acute angles marked in the following fern image are approximately 80° This is why they are also known as "equi-angular" spirals. Logarithmic spirals grow such that the angle of a line from the center of the spiral to the tangent to the curve at that point is constant. The Golden Spiral that Pehr is asking about is a special case of the logarithmic spiral.
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