I have heard about some amazing uses for 3-D printing, not the least impressive being artificial organs. As I haven’t needed a transplant so far, I had no interest in a 3D printer until now. Check this Fibonacci sculpture project out. I copied and pasted the page info below the video as it does not travel with the embed link. I am not sure why the other code travel instead of the links, but I can’t remove it.

<p><a href=”

http://vimeo.com/116582567″>Fibonacci Zoetrope Sculptures</a> from <a href=”

http://vimeo.com/pier9″>Pier 9</a> on <a href=”

https://vimeo.com”>Vimeo</a>.</p>

These are 3-D printed sculptures designed to animate when spun under a strobe light. The placement of the appendages is determined by the same method nature uses in pinecones and sunflowers. The rotation speed is synchronized to the strobe so that one flash occurs every time the sculpture turns 137.5º—the golden angle. If you count the number of spirals on any of these sculptures you will find that they are always Fibonacci numbers.

For this video, rather than using a strobe, the camera was set to a very short shutter speed (1/4000 sec) in order to freeze the spinning sculpture.

John Edmark is an inventor/designer/artist. He teaches design at Stanford University.

Visit John’s website here: web.stanford.edu/~edmark/

and Vimeo site: vimeo.com/johnedmark/videos

Learn how he made these sculptures here: instructables.com/id/Blooming-Zoetrope-Sculptures/

And more about the Pier 9 Artist in Residence program here: autodesk.com/air

Music – “Plateau” by Lee Rosevere – freemusicarchive.org/music/Lee_Rosevere/Farrago_Zabriskie/Lee_Rosevere_-_Farrago_Zabriskie_-_03_-_Plateau

Cinematography and editing by Charlie Nordstrom

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## About Steve Gingold

I am a Nature Photographer with interests in all things related. Water, flowers, insects and fungi are my main interests but I am happy to photograph wildlife and landscapes and all other of Nature's subjects.

That was great. Thanks for passing it along.

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Now that I look at this again I’m still awestruck and wondering, however, why the mathematician among us hasn’t chimed in about the wonders of numbers.

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Steve dropped in below in Andrew’s comment. But I am interested in what he has to say, especially as they figure into several of hisimages of sunflowers etc.

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Someone shared it on FB and I immediately thought some of the folks here would enjoy seeing it.

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This is sooooo neat. Thank you for sharing it, Steve!

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I am happy you enjoyed it, Melissa.

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I could play with those for a long time. In physics class, we stretched a string from a loud speaker and the other end over a pulley with a weight on it. Play a tone into the speaker to make the string vibrate. Shine a strobe light on it to freeze the waves or make them move slowly forward or backward.

Remember the stage coach wagon wheels that appeared to rotate backward in some movies?

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I think I may have seen that in a science class at some point too, Jim. When you started your comment, I thought you were going to mention the paper drinking cups with strings attached for communications. My friends and I did that a lot when we were kids. Obviously, they were close enough that we could have just spoken a little loudly to each other, but the fun trumped the logic.

I do remember the coach wheels and that was another experiment we did in science class.

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I don’t understand the science – hey, I’m a languages grad – but I do like Fibonacci. Usually carbonara. 🙂

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I’m a languages grad too, but that doesn’t mean we can’t coexist in the world of math-science as well. As you point out, Fibonacci is an excellent connecter of the realms.

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“Can’t we all just get along?”

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I don’t really understand it either, Andrew-hey, I’m a college dropout. I like most anything carbonara….or marinara for that matter.

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Now if someone would just invent a dish named Fibonacci marinara we could all spiral bites of it into our mouths. A restaurant could sell 1 portion to 1 person for $23.58, with a preparation time of just 13 minutes and 21 seconds.

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I still don’t understand it…but now I’m hungry.

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I’ll feed your hunger for knowledge by explaining that the Fibonacci sequence is generated by letting the first two terms be 1 and 1. From then on, each new term is created by adding the two previous ones. 1 + 1 = 2, so the third term is 2. Then 1 + 2 = 3, so the fourth term is 3. As a result, the Fibonacci sequence is:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

(You can now see that I incorporated the first eight terms in my previous comment.)

The Fibonacci numbers come into play in certain composite flower heads, where the number of spirals of seeds in one direction is a Fibonacci number, and the number of interwoven spirals of seeds in the other direction is the next Fibonacci number.

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That answers two questions. First of all mine and I understand completely now. Second, I took one of those ridiculous online IQ tests and there were a few questions with series of numbers. Most were just straightforward, but a few were a little more challenging with three or four digit numbers in varying sequences. Probably a piece of cake for you. I think I figured them all out but had no idea the theory behind them. Now I have a better idea.Thanks.

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All of those figure-out-the-next number quizzes have an inherent flaw: there are infinitely many formulas that will generate the terms you’re presented with, but will then generate

any desired next term. For example, if you’re presented with 1, 2, 3, you might reasonably assume the next term is 4. That’s certainly a possibility, and clearly the most likely one, but I can find a formula that will generate 1, 2, 3, as the first three terms but then will generate 5 as the next term.LikeLike

1+2=3+2=5+2=7 etc?

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If I understand what you’re proposing, your example is different from what I had in mind because you started with one rule (add the two previous terms to get the next) but then you switched to another rule (keep adding 2 each time regardless of what the previous terms were). I could tell you what I had in mind but I don’t want to bog down your comments any more than I already have.

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I guess it would depend on whether the first rule had been stated as adding the first two numbers. I looked at it as adding 2 to 1, so I then added 2 to three etc.

Bog away. I am interested in what you had in mind.

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The way a formula for a sequence works is that to get the value of term number n, you replace every occurrence of n in the formula with the desired term number, carry out the calculations (following the standard algebraic order of operations), and out pops the value of term number n.

With a very simple formula like just plain n, when you replace n with 1, the formula says the value of the first term is 1. When you replace n with 2, the formula says the value of the second term is 2. When you replace n with 3, the formula says the value of the third term is 3. This simple formula will produce a fourth term whose value is 4.

Now take a more complicated formula (where the symbol ^ means ‘raise to a power’):

[n^3 – 6n^2 + 17n – 6] / 6

If you replace every n in that formula with 1 and carry out the calculations, you get that the value of the first term is 1. If you replace every n in that formula with 2 and carry out the calculations, you get that the value of the second term is 2. If you replace every n in that formula with 3 and carry out the calculations, you get that the value of the third term is 3. In other words, like the very simple formula above, this formula also generates the sequence 1, 2, 3. However, when you replace every n in this more-complicated formula with 4 and carry out the calculations, you get that the first term is 5.

In a similar way, I could give you a formula that generated the sequence 1, 2, 3, 6 or another formula that generated the sequence 1, 2, 3, 2015. As a result, if someone tells you that the first three terms of a sequence are 1, 2, 3, there’s no way to know what the “right” fourth term is.

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Well, I asked for it. Way over my head. But then, most math is. Beyond my fingers and toes gets risky.

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It’s all scifi to me but I reckon it’s a great invention. Wonder what will come up next to benefit mankind? Veeery interesting.

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All this stuff does seem to have mostly benefits, Yvonne. Such as someone figuring out how to scar the inner wall of a beating heart to control its beat. Isn’t that just amazing? But I often wonder whether all these advances are really good for us and the planet. It seems that the more advanced we become, at least those of us fortunate enough to live in lands that can put them to use, the farther removed we become from nature…both in reality and figuratively.

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Yes you’re fight about the good things that result from all the miraculous inventions. And also that we’re becoming more removed from nature- at least for some of mankind. I can not imagine life without all the wonderful and beautiful things the wild world has to offer.

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Fascinating…Watched the video and read your post. It was very interesting.

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The video was mesmerizing for sure.

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That’s a delicious combination you linked to, Fibonacci and 3D photography. At John Edmark’s site I clicked Para(llel) below his 3D pictures and after zooming out to make the stereo pairs small enough was able to free-view them and see the three dimensions.

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I went to his site, but I don’t see any 3-D pictures…just a link to images and no controls there. Where did you see that?

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I found his stereo sets here:

http://web.stanford.edu/~edmark/StereoPhotos/StereoPhotos.htm

After you click on any of the five links for sets of 3D pictures, the controls will appear right under the stereo images.

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As soon as I hit Post I figured it out. I thought about editing my comment, but if you are like me…and if you are you have my full sympathies…you read the reply in the email notification and edits don’t catch up. I changed the title slightly of my post this afternoon, but all the email notifications will only have “Frosty” instead of the full title.

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Steve, I read your comment up above (” A restaurant could sell 1 portion to 1 person for $23.58, with a preparation time of just 13 minutes and 21 seconds.) and “got it” immediately. Very clever way to present it!

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It’s great that the spirit of Fibonacci came upon you and opened your eyes.

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When I first looked at the image, I thought it was another view of your Coral Slime (Ceratiomyxa fruticulosa). It’s amazing how they resemble each other.

I’ve been fascinated by the Fibonacci numbers since I first learned about them. There’s a good bit here I don’t understand, but I just keep trucking along. Every time I read something about them, I learn a little more.

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You are a few jumps ahead of me, Linda. Even after Steve’s example I still don’t understand. I may need Fibonacci for Dummies.

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