The golden ratio is only the first in an infinite sequence of fellow "metallic ratios."

Study of the golden ratio, which is very cool, has obscured the rest of the metallic ratios.

All the metallic ratios share unifying mathematical ideas, and the metallic ratios form a set.

Two math students in Singapore are drawing the world’s attention back to the metallic means, which include the golden ratio as well as all roots of the quadratic equation x² - nx = 1 for positive values of n. Progressively larger values are named silver ratio, bronze ratio, and after that, we can just use the many-color system for swim meet ribbons.

The real award is that you competed. QuickTrophy.com

The golden ratio is famous, but maybe more so as an image of a nautilus shell with a rectangle drawn around it than as the actual idea of the ratio it embodies: “A straight line is cut in accordance with the golden ratio when the ratio of the whole line to the longer segment is the same as the ratio of the longer segment to the shorter segment,” the students write, paraphrasing ancient Greek mathematician Euclid. In other words, each progressive cut makes a new golden ratio. It's golden ratios all the way down. In their article, the students offer an interactive graph where a slider shows each progressive spiral drawn for increasing metallic ratios.

An illustration of metallic graphs.

Each pair of adjacent integers has its own metallic mean, which is the collective name for the full set of roots that includes the golden ratio. The golden ratio is the metallic mean between 1 and 2, the silver ratio represents the metallic mean between 2 and 3, and so forth, all the way to the fermium ratio (between 100 and 101—I just used the matching chemical element) and beyond.

The golden ratio is closely associated with aesthetic balance and beauty, used by everyone from interior decorators to cinematographers. And the golden ratio is related to the famous Fibonacci sequence (1, 1, 2, 3, ... ) because the ratio of progressive Fibonacci terms get closer to the golden ratio. The students point out that the silver ratio has its own version, called the Pell sequence . (Silver is definitely the Jan Brady to golden’s Marcia in this setup.) In fact, each subsequent metallic mean has its own sequence as well, with mathematical derivations to match.



"Artistic" rendition of the golden ratio. Caroline Delbert

The famous rectangle and spiral associated with the golden ratio have counterparts for the other metals as well. For the golden ratio, you draw the rectangle according to the ratio, then block off a square with sides as long as the short leg of the rectangle. What’s left, when you rotate it 90 degrees, is a new golden ratio rectangle to divide, and so on, and so on. If this sounds like the beginning of a fractal, you’re not wrong—fractals don’t rely on the golden ratio, but the golden ratio spiral has elements in common with fractals and can be drawn as a beautiful one .

"Artistic" rendition of the silver ratio. Caroline Delbert

"Artistic" rendition of the bronze ratio. Caroline Delbert

The silver ratio and other metallic ratios have rectangles of their own. In each, you block off as many squares as you can, which corresponds with the integer floor of the metallic mean. So in a silver ratio rectangle, you block off two squares and are left with a new, smaller silver ratio rectangle.

Yes, the golden ratio is the famous one, but the fact that the related phenomena scale across all the metallic ratios could be even more interesting, especially with the right PR. Maybe we could name them after yearly anniversary gifts ? The flowers ratio, the linen ratio, the wine ratio ...

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