Skip to main content
MathOverflow

Return to Answer

added 33 characters in body
Source Link
Christopher King
Christopher King
  • 6.4k
  • 1
  • 32
  • 60

For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points?

Probably

The paper "Fracture and Damage Behaviors of Concrete in the Fractal Space" compares the fracture toughness of concrete when you take fractals into account v.s. when you do not take them into account. Their experiments suggest that the fractal analysis is more accurate, and that the non-fractal approach often overestimates the strength of concrete (which is obviously bad for bridges).

FractalsFunctions whose graphs are fractals just so happen to be examples of continuous functions that are no where differentiable. In fact, the first known example of such a function washad a fractal graph. Therefore, it is plausible that a bridge engineer, assuming no such functions exist, overestimated the strength of concrete, which concrete was the cause of the bridge's failure. However, it is unlikely that he thought to apply fractals in determining the cause of failure as well, so we may have trouble finding specific examples.

In general, fractals occur very commonly in nature. In many contexts, fractals are the only thing that occur (try finding a natural coast line that is differentiable). Therefore, I suspect that concrete is not the only time that this assumption misled us gravely.

Easier to prove, however, is the application that fractals, and the modern theory of functions, has given us. And it all started with a question: How weirdly can a function act? Being blinded by "intuition" can have us miss the forest for the well-behaved trees.

For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points?

Probably

The paper "Fracture and Damage Behaviors of Concrete in the Fractal Space" compares the fracture toughness of concrete when you take fractals into account v.s. when you do not take them into account. Their experiments suggest that the fractal analysis is more accurate, and that the non-fractal approach often overestimates the strength of concrete (which is obviously bad for bridges).

Fractals just so happen to be examples of continuous functions that are no where differentiable. In fact, the first known example of such a function was a fractal. Therefore, it is plausible that a bridge engineer, assuming no such functions exist, overestimated the strength of concrete, which concrete was the cause of the bridge's failure. However, it is unlikely that he thought to apply fractals in determining the cause of failure as well, so we may have trouble finding specific examples.

In general, fractals occur very commonly in nature. In many contexts, fractals are the only thing that occur (try finding a natural coast line that is differentiable). Therefore, I suspect that concrete is not the only time that this assumption misled us gravely.

Easier to prove, however, is the application that fractals, and the modern theory of functions, has given us. And it all started with a question: How weirdly can a function act? Being blinded by "intuition" can have us miss the forest for the well-behaved trees.

For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points?

Probably

The paper "Fracture and Damage Behaviors of Concrete in the Fractal Space" compares the fracture toughness of concrete when you take fractals into account v.s. when you do not take them into account. Their experiments suggest that the fractal analysis is more accurate, and that the non-fractal approach often overestimates the strength of concrete (which is obviously bad for bridges).

Functions whose graphs are fractals just so happen to be examples of continuous functions that are no where differentiable. In fact, the first known example of such a function had a fractal graph. Therefore, it is plausible that a bridge engineer, assuming no such functions exist, overestimated the strength of concrete, which concrete was the cause of the bridge's failure. However, it is unlikely that he thought to apply fractals in determining the cause of failure as well, so we may have trouble finding specific examples.

In general, fractals occur very commonly in nature. In many contexts, fractals are the only thing that occur (try finding a natural coast line that is differentiable). Therefore, I suspect that concrete is not the only time that this assumption misled us gravely.

Easier to prove, however, is the application that fractals, and the modern theory of functions, has given us. And it all started with a question: How weirdly can a function act? Being blinded by "intuition" can have us miss the forest for the well-behaved trees.

Source Link
Christopher King
Christopher King
  • 6.4k
  • 1
  • 32
  • 60

For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points?

Probably

The paper "Fracture and Damage Behaviors of Concrete in the Fractal Space" compares the fracture toughness of concrete when you take fractals into account v.s. when you do not take them into account. Their experiments suggest that the fractal analysis is more accurate, and that the non-fractal approach often overestimates the strength of concrete (which is obviously bad for bridges).

Fractals just so happen to be examples of continuous functions that are no where differentiable. In fact, the first known example of such a function was a fractal. Therefore, it is plausible that a bridge engineer, assuming no such functions exist, overestimated the strength of concrete, which concrete was the cause of the bridge's failure. However, it is unlikely that he thought to apply fractals in determining the cause of failure as well, so we may have trouble finding specific examples.

In general, fractals occur very commonly in nature. In many contexts, fractals are the only thing that occur (try finding a natural coast line that is differentiable). Therefore, I suspect that concrete is not the only time that this assumption misled us gravely.

Easier to prove, however, is the application that fractals, and the modern theory of functions, has given us. And it all started with a question: How weirdly can a function act? Being blinded by "intuition" can have us miss the forest for the well-behaved trees.

Post Made Community Wiki by Christopher King