In an article entitled River Meandering as a Self-Organization Process published in Science in 1996, Hans-Henrik Stølum proposes a model for calculating the meandering of a river and concludes that the ratio of the river length to the distance between its source and mouth, called the river sinuosity, tends to be equal to Pi1.
The sinuosity of a river can in principle be as low as 1 if the river flows in a straight line from its source to its mouth (or confluence point), or much higher if the river flows through a hilly landscape.
In 2014, James Grime2, a mathematician at the Cambridge University, published a video on this topic in a youtube channel called Numberphile. In this video, which is just about 6 min 28 s long – is this a coincidence – , he asks people to figure out the sinuosity of their local river. Following this call, a viewer called Laurence Roberts created a participative website, PiMeARiver.com (no longer online), where readers could enter the data about their local rivers.
From 258 rivers, the average sinuosity was found to be 1.94, significantly smaller than Pi3.
A quick look at the sinuosity of French rivers
Unfortunately, I don’t know if the data collected by PiMeARiver.com is available. So I decided to perform my own check with some French rivers, asking Chatgpt to gather data. It created the following table:
| River | Length L (km) | Source (lat, lon) | Mouth (lat, lon) | d (km) | L / d |
|---|---|---|---|---|---|
| Loire | 1006 | 44.83000, 4.22222 | 47.26917, -2.18583 | 563.77 | 1.784 |
| Seine | 777 | 47.5007, 4.6967 | 49.43389, 0.20667 | 394.61 | 1.969 |
| Rhône | 813 | 46.5770, 8.3889 | 43.33083, 4.84556 | 456.00 | 1.783 |
| Garonne | 529 | 42.607295, 0.965424 | 45.04139, -0.60667 | 298.58 | 1.772 |
| Dordogne | 483 | 45.52833, 2.81417 | 45.04139, -0.60667 | 273.03 | 1.769 |
| Lot | 485 | 44.51493, 3.78462 | 44.31667, 0.33500 | 274.85 | 1.765 |
| Saône | 473 | 48.094444, 6.181944 | 45.72750, 4.81778 | 282.84 | 1.672 |
| Allier | 421 | 44.58972, 3.80111 | 46.95944, 3.07889 | 269.38 | 1.563 |
| Doubs | 453 | 46.70472, 6.20944 | 46.90083, 5.02417 | 92.81 | 4.881 |
| Tarn | 380 | 44.41472, 3.81500 | 44.086111, 1.04250 | 223.82 | 1.698 |
| Marne | 514 | 47.3000, 4.6333 | 48.8417, 2.3833 | 262.70 | 1.956 |
| Meuse | 950 | 47.9800, 6.3600 | 51.9717, 4.1131 | 461.88 | 2.057 |
| Charente | 381 | 45.6833, 0.9833 | 45.6167, -1.0167 | 172.60 | 2.208 |
| Adour | 308 | 42.8617, 0.1083 | 43.5083, -1.5450 | 143.70 | 2.144 |
| Vienne | 372 | 45.7500, 2.1833 | 46.1875, 0.6061 | 197.44 | 1.885 |
| Oise | 341 | 49.6544, 4.3847 | 49.0167, 2.5000 | 176.56 | 1.932 |
| Dordogne + Garonne (Gironde system) | 647 | 44.8711, 0.1597 | 45.5500, -1.0667 | 350.22 | 1.847 |
| Ariège | 163 | 42.7000, 1.4833 | 43.3500, 1.3333 | 86.21 | 1.891 |
| Aude | 224 | 42.6200, 2.1500 | 43.2000, 3.2000 | 118.35 | 1.892 |
| Somme | 245 | 49.9500, 2.4500 | 50.1833, 1.5833 | 93.87 | 2.611 |
I have not checked yet all the values found by Chatgpt. For the Loire river, the values given for its length and for the locations of the source and the mouth are the same as the values found on Wikipedia: 1006 km, 44°50’N, 4°12’E for the source, 44°16’N, 2°11′ for the mouth.
If the values computed by Chatgpt are correct, the average sinuosity of these 20 rivers is 2.14, definitely lower the Pi (rather equal to Pi – 1!), with the highest value, 4.881, for river Doubs (its mountainous course and pronounced meanders explain this high ratio) and the lowest value, 1.563, for the river Allier.
Conclusion
The average river sinuosity doesn’t seem to be equal to Pi, and something in the justification given by Hans-Henrik Stølum must be wrong. Does anyone know if a more recent work has been done to calculate or simulate the sinuosity of rivers ?
A last word: there is a river called the Pai river in Thailand4. Its length L is 180 km, and its value of d is 110 km. Even this river doesn’t have a L/d ratio equal to Pi!
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