XIII D : Role of Mathematics behind a flowing river.

XIII D:Role of Mathematics behind a flowing river.
            [Contd. A Journey to the Wonderland of Math.by Ajay Kumar Chaudhuri.]

           "The river moves, but it follows a path. When it tires of one journey,it rubs through some rock to forge a new way.Hard work,but that`s its nature.------Kekla Magoon.

All of us must agree that rivers are indispensable to us and have many important roles to play in our lives in many ways, Rivers are cradle of civilization. All major civilizations such as the Mesopotamian, Indus valley, the Egyptian and Chinese civilizations have developed on the bank of rivers. Rivers are used to provide fresh water to billions of people, to sustain lives of all living things including flora and fauna, on this Earth, to build our healthy economy like inland waterways for transportation and producing hydroelectricity, and in many other innumerable purposes. Some rivers regarded as sacred, such as the Yangtze in China, the Ganges in India, the Nile in Egypt and in parts of north- eastern Africa.

Most rivers have their source in lakes, springs, wetlands and glaciers. They flow towards the sea where they empty their water. However ,there are some inland rivers, often called virgin rivers. They have their source and month inland. The Amu Darya and the Syr Darya, life lines of central Asia, are examples of inland rivers.

There are about 165 major rivers in the world. There are thousands of small and large rivers in the world. Their number is difficult to determine.

Amongst the major rivers of the world, the Amazon, the Nile and Yangtze rivers are worthy to mention. The Amazon is the largest river by volume of water, and has length of 6,516 km, the Nile at 6,695 km. long and longest river in the world. The Mississippi-Missouri river system in United States is more than 5,969 km. long.

In this ongoing chapter, we are mainly interested in the role of mathematics behind the patterns in nature around us. Is there any role of math related to rivers? Yes, of course. The flow and meanders are the patterns which rivers follow. These patterns follow some sort of mathematical rules and also have some geometrical aspects. But what is a Meander? A meander in general, is a bend in a Sinuous (having many curves and turns) water course or river. A meander is formed when moving water in a stream erodes the outer banks and widens its valley, and the inner part of the river has less energy and deposits silt.

Our common experience about rivers tells that some rivers are relatively straight while other rivers twist and turn across landscape like scribble of someone checking if his pen is dry.

We can measure how bending a river is by measuring its total length and dividing by straight route from its source to mouth; this measure is called “Sinuosity”. So, a totally straight river would have sinuosity of 1, while very bendy rivers can have very high sinuosity, with no limit to how high it can go. Yet it is claimed that the average sinuosity of rivers around the world is pi* (π) which is a very important and famous ratio, like golden ratio (denoted by phi, Φ). The value of π is approximately 3.14159, a little more than 3.

[* The history of pi, the constant ratio of the circumference to the diameter of any circle, is as old as man’s desire to measure. In 1700, a little known mathematics teacher William Jones from west of U.K. first used a symbol for the concept of this ratio, which in numerical terms can be approached, but never reached. It is widely believed that the Swiss born great mathematician Leonhard Euler (1707-83) introduced the symbol Greek alphabet pi (π) for this ratio. It is one of the most extremely useful and fundamental quantities we know of. The length of the circumference and area of a circle can be determined using this ratio. If r be the radius of a circle then its length of the circumstances will be 2πr or in other words a little more than three times the diameter or six times the radius of the circle, since the value of π is little more than 3. The area of a circle is found to be πr² or a bit more than three times the Square on the radius of the circle.

Some interesting facts about pi are : It is an irrational number which means its value cannot be expressed as ratio of two numbers. It is also a transcendental number meaning there by no sequence of algebraic operation using integers (such as powers, roots, sums etc.) can be equal to its value. Its actual value cannot be calculated, it is a non-repeating and non-recurring decimal. Interestingly, the Guinness - recognised record for remembered digits of π is 67,890 digits held by Lu Chao, a 24 year- old graduate student from China. It took him 24 hours and 4 minutes to recite to the 67,890^th decimal place of π without an error. Pi is used for different tasks in different professions and  to assist in different aspects of design

work, particularly for round objects. When pi is used in home project and often it is approximated at 3. It appears in all sorts of calculations in physics, engineering, electrical systems, probability, statistics etc.]

   Interestingly, pi is in our body as the double helix our DNA revolves around pi. It is in the pupil of our eye when we see a rainbow and when a rain drop fells into water pi emerges in spreading rings.  It appears in colours and in music. It also appears when calculating number of deaths in a population, in sinuosity of a river, as has already been mentioned, which was also shown by Einstein himself.

Look at the aerial view of a river in Pic. No. 17. It appears a lazily snake back and forth through a gentle sloping field. They change shape over time by continually eroding their outer bank of each curve and depositing silt on the inner bank. Occasionally two bends in the river come together, changing the topology of the river, and ultimately leavings behind an oxbow lake.

                  Pic. No. 17

 

                                                   Meandering of a river.

There is a very interesting mathematical fact: It turns out that the size of a river cannot be determined by its shape on a map, say for example, if we look at an aerial snapshot of a meandering river, we would not be able to tell whether it is the Amazon or a small any other stream.

The stunning fact is that it appears nature forces some geometric relations* on the features of the rivers.

   [ *The Pic. No. 18 is the sketch that of a meandering river. The curves in the meander is roughly a circle. Let λ (small lambda, a Greek alphabet) denote the length of a meander, may be termed wavelength, as in a wave motion, ω (another Greek letter,small omega) be the width of the river. Let r denote the radius of the curvature of the bend. It is found that these variable quantities λ, ω and r are almost related in the following way  λ=II ω and  r»2.3ω.

What a stunning fact!]

      Pic.No18.

Reference : Internet
                   Pic.No17--Aerial view of a river : Attribution https:(//en.wikipedia.org/wiki/Meander#/media/File:Mean)
             
 Pic.No 18 Geometry of meandering :Image credit:Dave Richeson (https://divisbyzero.com/2009/11/26/the-geometry-of-meandering-rivers/ )
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