The Silver Ratio - Numberphile

of approximately 17 degrees. You can solve it by comparing these two formally. And just by comparing the two that you put here in the golden

ratio

, you can extract what the tone is. It turns out about 17 degrees. For the corresponding

silver

, the step is about 29 degrees. For bronze it is 37 degrees. For n equal to 4 we get 42 degrees, for n equal to 5 we get 46 degrees, n is equal to 6 - 49 degrees. Remember that Japanese proportion we said was root 2? It has a pitch of approximately 12 degrees. People always say, "Oh, the golden

ratio

, it appears everywhere in nature." It appears in some places in nature.

But when they talk about spirals, it is actually this family of spirals that appears. For example, you'll sometimes hear it said that the Milky Way sees the golden ratio, well, through its spiral. Is not true. In fact, a logarithmic spiral is seen, with a medium tone. It's about 12 degrees. So the Milky Way is not a golden spiral; In reality, it looks much more like a Japanese spiral. There is another one that I really like, they are the Brady peregrine falcons. I love peregrine falcons, they are the fastest creature on the planet, you even get them in England, right?

So there's something cool, an animal that we actually found in England. He also wins at Top Trumps, you know, the predatory Top Trumps my kids have. The peregrine falcon is one of the winners. That's why I love the peregrine falcon. The peregrine falcon also knows these proportions, right? Imagine you are a peregrine falcon, you are flying. Here is your prey, it is sitting here in the center. Now you want to keep your eyes on the prey. But you can't move your eyes, you don't want to start moving your head, right? So you... that remains in a fixed position.

Therefore, you must maintain a fixed angle with respect to the prey. That necessarily means that you have to follow one of these logarithmic spirals. The angle you should be looking at is actually 40 degrees. Which means the spiral pitch is about 50 degrees, which is almost one of our metal spirals; the... I'm going to call it n equals 6 one. So I mean I don't know what this is, right? So do we have gold,

silver

, bronze, copper, nickel, aluminum? Anyway, it's there, it's practically there; I mean that's a good approximation, right? So people talk about seashells, that they see the Golden Spiral.

They don't really see it, they see all the possible images, they are seeing these logarithmic spirals. And there's a good reason why, for example, seashells see these log spirals. They have a very nice property, everyone, everyone; the gold, the silver, all of them. They have what is called a self-similar property. So you go around and then you go around again and when you go around again you're basically looking at a scaled down version of the exact same thing. And you keep seeing that as you go around and around and around. And when you think about growth in a biological system, that allows you to grow very efficiently if you follow that repeating pattern; repetition, repetition, repetition.

These trunk spirals are also seen a lot in architecture and art. Again, there is probably a good reason for this and it is related to this similar property. Basically, when you look at something, your eye makes visualizations at many different scales and then correlates them. When you look at this and you look at it on many different scales, you're seeing the same thing. And so maybe in some sense that's a good thing because we correlate a lot of the same thing on a lot of different scales. - (Brady: Are we reducing cognitive load?) Exactly and in some ways we like that.

That's why these spirals appear. - Did you know you can get Numberphile t-shirts? There is a whole range of them; most recently, these very special, limited edition, gold stamped ones. They look even better in real life than in the pictures. And as you can see, they're based on the videos we posted this week on Numberphile. If you want to see all the shirts, including the gold ones, I'll put a link in the video description. The gold ones will actually only be available for a very limited time because we have to make them to order. But anyway, thanks for watching, see you in the next video.

Repetition, repetition, repetition. Repetition, repetition, repetition. Repetition, repetition, repetition. Repetition, repetition, repetition. Repetition, repetition, repetition. Wow, great deal, what's so good about that?