Have you ever pondered the geometric structure of K'Nex?
Probably not.
But in fact it is quite interesting.
Beyond the simple angle-connectors in increments of one-fourth pi (they could have done one-sixth pi perhaps, but you'll see why one-fourth is better in a moment), the designers of those colorful construction sets made a marvelous choice in selecting the lengths of their rods:
Each differs by a factor of (what I call) the Platinum Ratio, the ineffable square root of two.
Yes, Platinum is better than Golden, a lot better. The Parthenon may be made as 1.618, but doesn't it look wide and strange? Your television set is most likely 1.333. My "wide-screen" computer is 1.6; film letterbox is 1.777. Index cards look wide, don't they? 1.666. Hardcover books? Only 1.285. Letter-size paper? A mere 1.294. A PDA? About 1.4.
And the clincher: The human torso? 1.4.
These pleasing ratios don't cluster around the famous 1.6180335... They cluster around 1.414213..., or maybe simply 1.5.
But why might this be? The Platinum Ratio has a power much more useful in nature than the Golden Ratio ever could: Cut it in half, and you get two of them. Double it instead, and you get another.
Which is to say that they are easily nested, and easily made by--you guessed it--an angle of one-fourth pi.
Make a square of green rods, the diagonal is a white rod (= a^1).
Make a square of those, the diagonal is a blue rod (= a^2 = 2). (Same as two green rods.)
Make a square of those, the diagonal is a yellow rod (=a^3). (Same as two white rods.)
Make a square? The diagonal is a red rod (=a^4). (Same as two blue rods.)
Make a square? The diagonal is a gray rod (=a^5). (Same as two yellow rods.)
Make a square of those, and the diagonal is two red rods (=2a^3 =(a^2)(a^4) = a^6).
Make a square again, and the diagonal is two gray rods (=2a^4 = (a^2)(a^5) = a^7)...
And so on, and so on. There's no limit to how far you can extend this.
You couldn't do that with angles of pi/6 or lengths of the Golden Ratio.
With pi/6, the lengths would match up only in sums of three (and you'd have to deal with a, 2a, 3a = a^2 , 4a, 5a, 6a = 2a^2, 7a = a^2 + 4a, 8a = 2a^2 +a, 9a = 3a^2 = a^4), and with the Golden Ratio, they'd never match!
By using the Platinum Ratio and its associated angles, the makers of K'Nex make pleasing sizes and complex shapes without confusing the children.
I hereby salute them.
But in fact it is quite interesting.
Beyond the simple angle-connectors in increments of one-fourth pi (they could have done one-sixth pi perhaps, but you'll see why one-fourth is better in a moment), the designers of those colorful construction sets made a marvelous choice in selecting the lengths of their rods:
Each differs by a factor of (what I call) the Platinum Ratio, the ineffable square root of two.
Yes, Platinum is better than Golden, a lot better. The Parthenon may be made as 1.618, but doesn't it look wide and strange? Your television set is most likely 1.333. My "wide-screen" computer is 1.6; film letterbox is 1.777. Index cards look wide, don't they? 1.666. Hardcover books? Only 1.285. Letter-size paper? A mere 1.294. A PDA? About 1.4.
And the clincher: The human torso? 1.4.
These pleasing ratios don't cluster around the famous 1.6180335... They cluster around 1.414213..., or maybe simply 1.5.
But why might this be? The Platinum Ratio has a power much more useful in nature than the Golden Ratio ever could: Cut it in half, and you get two of them. Double it instead, and you get another.
Which is to say that they are easily nested, and easily made by--you guessed it--an angle of one-fourth pi.
Make a square of green rods, the diagonal is a white rod (= a^1).
Make a square of those, the diagonal is a blue rod (= a^2 = 2). (Same as two green rods.)
Make a square of those, the diagonal is a yellow rod (=a^3). (Same as two white rods.)
Make a square? The diagonal is a red rod (=a^4). (Same as two blue rods.)
Make a square? The diagonal is a gray rod (=a^5). (Same as two yellow rods.)
Make a square of those, and the diagonal is two red rods (=2a^3 =(a^2)(a^4) = a^6).
Make a square again, and the diagonal is two gray rods (=2a^4 = (a^2)(a^5) = a^7)...
And so on, and so on. There's no limit to how far you can extend this.
You couldn't do that with angles of pi/6 or lengths of the Golden Ratio.
With pi/6, the lengths would match up only in sums of three (and you'd have to deal with a, 2a, 3a = a^2 , 4a, 5a, 6a = 2a^2, 7a = a^2 + 4a, 8a = 2a^2 +a, 9a = 3a^2 = a^4), and with the Golden Ratio, they'd never match!
By using the Platinum Ratio and its associated angles, the makers of K'Nex make pleasing sizes and complex shapes without confusing the children.
I hereby salute them.
posted by PNRJ at 18:37
1 Comments:
Hey I agree that the square root of 2 is way more useful. But they are linked.. there is a pattern you can make, if you can figure it out ^^.. where the square root of 2 +1 is the 2nd number and the golden ratio is the first.
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