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I do not want to know if some phenomena in nature have normal distribution, but whether we can somewhere see shape of normal curve as we can see it for example in Galton box. See this figure from Wikipedia.

enter image description here

Note that many mathematical shapes or curves are directly seen in nature, for example golden mean and logarithmic spiral can be found in snails.

First naive answer is whether nonskewed hills would often "fit" normal distribution :-).

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    $\begingroup$ This example is a favorite of mine. $\endgroup$
    – cardinal
    Commented Oct 10, 2012 at 12:23
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    $\begingroup$ @cardinal That's an intriguing example, but wear on steps is highly unlikely to be at all Normal. In fact, it would be a puzzle if it were. The CLT might possibly be invoked to describe horizontal variation in where people walk, but that will not lead to a Gaussian shape in the wear on the step. $\endgroup$
    – whuber ♦
    Commented Oct 10, 2012 at 14:40
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    $\begingroup$ Many years ago, the East Wing of the National Gallery of Art in Washington DC had a beautiful (and unintentional) example of a normal distribution showing on an exterior wall where two exterior walls met at a 45-degree angle instead of the usual 90-degree angle. People presumably had touched the edge to see if it felt sharp, and the smudges from their fingers left a stain on the wall which showed as a bell curve (rotated 90 degrees clockwise) at about chest height. On a more recent visit, I found that the exterior walls had been cleaned and the smudges had disappeared. $\endgroup$
    – Dilip Sarwate
    Commented Oct 10, 2012 at 15:02
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    $\begingroup$ I have not seen wear like that, @cardinal. In your model the wear would be much flatter near the peak than a Gaussian and--of course--would be confined within natural limits, unlike any Gaussian. (The latter is not such a big deal because we cannot demand that the Gaussian fit perfectly.) The wear on steps I have examined is far flatter than a Gaussian. (Often it is bimodal, too, because most long-lived stairs have been used in both the up and down directions.) A better model would be a convolution of a fairly narrow Gaussian with a uniform distribution. $\endgroup$
    – whuber ♦
    Commented Oct 10, 2012 at 15:16
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    $\begingroup$ This blog post shows the example that @Dilip mentions as well as one example of wearing patterns on stone steps (with links to other pictures of wear patterns). Some might find it interesting. $\endgroup$
    – cardinal
    Commented Oct 10, 2012 at 17:20

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I wouldn't think any pattern of erosion or deposition on Earth would fit because skewing factors including gravity and Coriolis are always involved (rivers meander more as they age, for example, and valley floors are sort of the average of rivers). Maybe the cross section of a stalagmite, assuming the drip remained in one fairly exact central location? I would think the drips would deposit the most precipitate right where they are moving slowest, which would be at the point of impact.

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I thought a lot about my question and probably I found something. U-shape of many valleys imitates "reversed" normal curve. Are there any reasons why this should not be gaussian (note that water makes the valleys smooth)?

Here is an example.

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    $\begingroup$ There seems to be a strong tendency for people to hope than any unimodal curve is normal. I see no reason why such a valley would be closely approximated by an inverted normal curve, and many factors such as the erosion from water which may be unimodal, but where any accurate physical model predicts something other than a normal curve. $\endgroup$
    – Douglas Zare
    Commented Oct 11, 2012 at 22:26
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    $\begingroup$ This is an interesting empirical question - how closely the shape can be approximated as normal would depend on the age of the various features. A valley probably begins more poisson shaped, becomes normal-ish, and as the tops of the hills wear heads back in a poisson direction. $\endgroup$
    – N Brouwer
    Commented Oct 12, 2012 at 13:46

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