Choosing the Strobe Rate

The videos shown at the beginning of the of this instructable were shot with a video camera running at 24 frames/second. If you animate a bloom using a strobe running at 24 flashes/second, you will find that the sense of flashing is quite pronounced, and detracts from the effectiveness of the animation. I suggest running the strobe at a minimum of 33 flashes/second to get a satisfyingly coherent illusion. Keep in mind that this will also mean spinning the turntable faster in order to maintain the strobe and turntable in synch. At 33 flashes/second, your turntable will need to be spinning 756 RPM (or 12.6 revs/sec).

An Inconvenient Truth

Now I have a confession to make. In order not to confuse the casual reader, I told a little white lie at the beginning of this instructable, when describing what was going on in the videos. The vast majority of people reading the introduction to this instructable will never try to recreate the effect, and thus will not be impacted by the slight inaccuracy of my description. But you, having read this far, are far more likely to be actually trying to animate a bloom. I certainly do not want to create unnecessary confusion for you in your pursuit, so, I am now going to give you the unvarnished truth.

To refresh your memory, here is what I wrote in the introduction:

What you are viewing in each of the above videos is a bloom spinning at 550 RPMs while being videotaped at 24 frames-per-second with a very fast shutter speed (1/4000 sec). The rotation speed is carefully synchronized to the camera's frame rate so that one frame of video is captured every time the bloom turns ~137.5º—the golden angle...If you follow what appears to be a single petal as it works its way out and down the bloom, what you are actually seeing is all the petals on the bloom in the order of their respective distances from the top-center.

And now, here is a version with the white lies removed:

What you are viewing in each of the above videos is a bloom spinning at 340 RPMs while being videotaped at 24 frames-per-second with a very fast shutter speed (1/4000 sec). The rotation speed is carefully synchronized to the camera's frame rate so that one frame of video is captured every time the bloom turns ~85º—not the golden angle...If you follow what appears to be a single petal as it works its way out and down the bloom, what you are actually seeing is every second petal on the bloom in the order of their respective distances from the top-center.

"Huh???"

Ok, let's see what's going on here. The reason for the slightly more complicated story (and thus the white lie) is that even at a very fast shutter speed, the video camera we used (a Canon EOS 5D Mark III) does not record every pixel of the image at exactly the same moment, and this results in distortions in the image. If you look carefully at the videos you can see evidence of this distortion in certain parts of the image. When we tried to record the bloom spinning at the speed required to achieve 137.5º rotation every frame, the distortion became very pronounced, so we slowed things and spun it at ~61.8% of the "proper" speed (550/340 = .618--golden ratio, anyone?).

Why this particular speed, you ask? Well, if running it at the "proper" speed causes the camera to capture every petal of the bloom in sequential order, the next slower speed we could run it at while still achieving a consistent animation would be one that led to the camera capturing every second petal in sequential order. We know that it takes a rotation of 137.5º to get from any given petal to the next one in the sequence, so to skip a petal we need to rotate 2 * 137.5º, or 275º. Now, that may initially seem counterproductive, since we've doubled amount we have to rotate for each frame. But because we're dealing with a circle, 275º is the same as -85º. So if we just reverse the direction the turntable is spinning, we can slow it down to 85º per frame and get an animation that moves twice as fast, because it only sequences half as many petals.

(It's interesting to note the rather counter-intuitive fact that by slowing down the turntable to 61.8%, we double the speed of the animation.)

Of course, capturing every second petal is not the only speed change we can achieve. We could also choose to capture every third, every fourth, every fifth petal, and so on. Let's look at how much we'd need to rotate the turntable for the first eight of these:

A few observations from the chart above:

1) As the frequency increases, those that are Fibonacci numbers always lead to a smaller resulting angle.

2) The sign of the resulting angle for the Fibonacci numbers alternates between positive and negative.

3) The % of full speed for subsequent Fibonacci numbers decreases by the golden ratio.

The chart above only shows the first eight frequencies, with the Fibonacci frequencies in bold. I encourage you to continue the process for higher frequencies to convince yourself that the above observations remain true for all values.

Finally, I'd like to note that in actual practice I run the strobe at a higher frequency than 24fps. Because the light is on for such a tiny part of each cycle, it looks quite "flashy" at that rate. So instead I run it at 36fps (50% faster) and rotate the turntable at 825rpm, which yields a more constant-appearing illumination.