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Can the golden ratio be expanded into more dimensions?
- Thread starter Braznor
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I highly doubt it. In 2D, an aspect ratio (length/width) defines a whole group of rectangles. They all look the same, just differing levels of 'zoom'. That's b/c rectangles have only 2 dimensions: length & width. So if I give you their ratio, I've fixed the shape.
In 3D, a rectangular prism would have 3 dimensions. So one ratio is not sufficient to fully specify anything; you need 2 ratios. My guess is that the best you can do is to have an object whose faces are composed of 2D features that make use of the golden ratio (lots of Greek architecture come to mind as examples).
One possible generalization from 2D to 3D would be to use the ratio of circumcircle to incricle radii to define the golden ratio. In 3D, a corresponding ratio would be circumsphere to insphere, but again the issue is that now would be rectangular prsims that satisfy the specified ratio, but they do not "look the same" when you zoom in/out. But perhaps all of these would have some appealing properties? Dunno.
In 3D, a rectangular prism would have 3 dimensions. So one ratio is not sufficient to fully specify anything; you need 2 ratios. My guess is that the best you can do is to have an object whose faces are composed of 2D features that make use of the golden ratio (lots of Greek architecture come to mind as examples).
One possible generalization from 2D to 3D would be to use the ratio of circumcircle to incricle radii to define the golden ratio. In 3D, a corresponding ratio would be circumsphere to insphere, but again the issue is that now would be rectangular prsims that satisfy the specified ratio, but they do not "look the same" when you zoom in/out. But perhaps all of these would have some appealing properties? Dunno.
DrPizza
Administrator Elite Member Goat Whisperer
The golden ratio is a 1 dimensional ratio. You're jumping from one dimension to three dimensions.
In one dimension, two amounts are in the golden ratio if the ratio of the smaller to the larger is the same as the larger to the sum.
Or rather, I would be better to reverse that - the larger to the smaller and the sum to the larger. Either statement holds, but the value of the golden ratio is the larger to the smaller, not the reciprocal.
Now, as several people have noticed, this ONE dimensional concept can be applied to 2 dimensional objects. i.e. examples of the golden ratio in rectangles where the length divided by the width is the golden ratio. I see no problem extending it to the 3rd or greater dimensions, although at first thought, you don't have such simple examples as "take a square out of a golden rectangle, you're left with another golden rectangle."
However, consider the regular pentagon & how the golden ratio can be represented by the intersections of line segments within. Thus, while all regular pentagons are the same "fixed shape", the golden ratio is present by looking at a ratio of line segments within. So, I suppose that a pentagonal prism could be created such that the height of the prism was the shorter length of the intersection of a pair of segments inside the pentagon. Or, hurting my brain a little more, if three golden rectangles are intersected in three dimensions, one along each plane, centered at the origin, the 12 corners form the centers of a dodecahedron. <brain hurting> there's gotta be something you can do with a dodecahedron to create a smaller dodecahedron in 3-D that demonstrates the golden ratio. (i.e. remove a cube from the center & collapse it in on itself? Something along those lines.)
In one dimension, two amounts are in the golden ratio if the ratio of the smaller to the larger is the same as the larger to the sum.
Or rather, I would be better to reverse that - the larger to the smaller and the sum to the larger. Either statement holds, but the value of the golden ratio is the larger to the smaller, not the reciprocal.
Now, as several people have noticed, this ONE dimensional concept can be applied to 2 dimensional objects. i.e. examples of the golden ratio in rectangles where the length divided by the width is the golden ratio. I see no problem extending it to the 3rd or greater dimensions, although at first thought, you don't have such simple examples as "take a square out of a golden rectangle, you're left with another golden rectangle."
However, consider the regular pentagon & how the golden ratio can be represented by the intersections of line segments within. Thus, while all regular pentagons are the same "fixed shape", the golden ratio is present by looking at a ratio of line segments within. So, I suppose that a pentagonal prism could be created such that the height of the prism was the shorter length of the intersection of a pair of segments inside the pentagon. Or, hurting my brain a little more, if three golden rectangles are intersected in three dimensions, one along each plane, centered at the origin, the 12 corners form the centers of a dodecahedron. <brain hurting> there's gotta be something you can do with a dodecahedron to create a smaller dodecahedron in 3-D that demonstrates the golden ratio. (i.e. remove a cube from the center & collapse it in on itself? Something along those lines.)
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