為什麼一條自然河流長度與起始距離的比值是pi？

01-11

https://www.youtube.com/watch?v=Y4TaH40uy3o

找到一篇英文報到的原文，翻譯了一下，有小部分概念沒有翻譯，直接替換成了通俗易懂的中文。

在上世紀90年代中期，哥倫比亞大學的Hans-Henrik Stolum教授發現河流真正的長度和其直線距離（源頭與盡頭距離）的比值--彎度通常約為3。對於那些發展漫長、路線曲折的古老的河流而言，彎度通常接近pi的值3.14。而眾所周知pi的值是圓周與直徑的比值。

之所以此數值與pi有關係很大一部分原因歸結為混沌理論（ chaos theory）。混沌理論的主導思想是,宇宙本身處於混沌狀態,在其中某一部分中似乎並無關聯的事件間的衝突,會給宇宙的另一部分造成不可預測的後果。例如河流只要不被窄的峽谷限制，就是以類似的方式延伸。

在河流延伸問題上，愛因斯坦提出了一種理論。（我們熟知的愛因斯坦的兒子Hans Albert Einstein感謝知友指正）在研究了巴爾定律後，愛因斯坦的論文中指出，在水流沖刷河岸的過程中，河岸受到一股向內的推力，相當於水流對其向外的推力，這就沿著河岸建立起一個壓力梯度。沿著河流底部流淌的水體受到摩擦作用速度減慢，導致向外的推力減小，因而壓力梯度會從側面向中間侵蝕河床。愛因斯坦認為，這就是河流一側被侵蝕，以及河床形成的機制。但是，Stolum指出：河流的這個過程很快會重新開始，並且河流的彎度會趨向pi。

研究發現在真正的河流中，Stolum提出的這個數值更像是個極限值。延伸的河流的彎度不會總是趨向pi，比如狹窄的山谷和茂密的植被在河的一邊，可以影響河流的延伸。

英文原文：

| Is there a relationship between the length of a river and its "as the crow flies" distance from source to mouth?

ANSWER: During the mid-1990s, Cambridge University researcher Professor Hans-Henrik Stolum found that the ratio between the actual length of a river and the straight-line distance from its source to its mouth—known as its sinuosity—is typically about three. For older rivers, which will have had a chance to develop lengthy, meandering courses, the ratio often approaches 3.14, similar to the value of pi, the irrational number that links the radius of a circle to its circumference and area.

The reason why pi plays a big part in the relationship comes down to chaos theory. Chaos theory is based on the idea that tiny changes can lead to large-scale effects over time and that apparently random fluctuations can still lead to surprisingly similar shapes, such as the way that rivers tend to meander in the same way as long as they aren』t constrained by narrow valleys.
In the case of rivers, it was Albert Einstein who came up with a reason why rivers meander. When a river forms, it will have little kinks and bends in it. Einstein noticed that the water that flows around the outside of a bend moves faster than the water flowing around the inside. This erodes the outer bank more quickly than the inner bank and the river moves outward, creating a larger bend. Eventually, the bends become so sharp that they meet and the river forms a short cut through them, straightening it out and possibly forming a cut-off oxbow lake. But, Stolum noted, the process soon starts again and the ratio of the river』s actual to straight-line length wanders back towards pi.
Research looking at real rivers has found that Stolum』s number is pretty much a maximum. Meandering rivers don』t go much past pi and any constraints, such as narrow valleys and dense vegetation at the side of the river, can keep it from meandering too far.
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有翻譯錯誤希望指正，謝謝。

以不同的精度去量河流和海岸線的長度，得到的結果是不同的，和首尾距離湊出Pi來，是容易做到的。

黃河

5464/2100 = 2.601905

長江

6280/2893.7 = 2.170232

珠江

2320/1060 = 2.188679

以上是百度與騰訊地圖上測的大概數據~

算了一下，長江、黃河與遼河的此比值分別為2.2，2.6和5。沒看Science上http://raaf.org/pdfs/meandering_river.pdf等相關文章，估計是有一定的適用條件

微分河流長度視作圓