GEOMETRY VS TOPOLOGY

x = V - E + F
Euler characteristic = Vertices - Edges + Faces.

Draw triangle on a sheet of paper. It has 3 vertices, 3 edges and 1 face.
Hence the
Euler char. = 3 -3 + 1 = 1
For a square
Euler char. = 4 - 4 + 1 = 1
For a circle, imagine two points or vertices on the circle.
So, Euler char. 2 - 2 + 1 = 1.
The circle, square and triangle all have same Euler characteristic 'one'. They are geometrically different, but topologically same. That means, all the shapes can be converted into one another, if they are made of rubber sheet.

"Topology is concerned with the properties of space that are preserved under continuous deformations; such as stretching, crumpling and bending, but not tearing or gluing" (Wikipedia).

Now we get the idea of topology. Let us see one more example.
Take a 'tetrahedron' solid (prism). It has 4 vertices, 6 edges, 4 faces.
Euler char. = 4 - 6 +4 = 2
for a cube
Euler char. = 8 - 12 + 6 = 2
A sphere has also Euler char. 2.
Hence prism, sphere, cube, all belongs to same topological family. But have different geometrical properties.

The shape of coffee cup and Doughnut have zero as Euler characteristic. They are one and the same. Euler's formula helps us to find the topologically invariant shapes. ( you might have noticed one point in the above cases, that is, all the vertices meet with same number of edges).

DNA has a special shape called double helix (a twisted staircase). This Euler formula helped us to understand the DNA even better.

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THE EARTH, A SUPER ORGANISM

JOIN MY COURSE: "Become a programmer in a day with python" A man called 'love lock' (what a name) proposed a theory called Gaia theory, named after Greek Goddess. It says, "Earth is a self-regulating organism like a human being. The organic life in it interacts with in-organic matter and maintains atmosphere, temperature and environment". Hence the earth is still suitable for the life to thrive. Imagine, in a particular place, there are lot of flowers. Some flowers are white and some are darkly coloured. We know, white reflects light and heat while dark absorbs the same. White flowers can thrive in hot climate. But dark flowers requires cold climate. The absorption and reflection balances and the environment reaches average, warm temperature at which both the flowers can co-exist. This is the essence of "Gaia" theory. On our earth, the oxygen constitute 20% of the atmosphere. The oxygen level is always mai

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

DISORDER IS THE "ORDER OF THE DAY"

Imagine a balloon full of air. The air molecules are moving randomly inside the balloon. Let us pierce the balloon with a pin. The air rushes out. Why should not the air molecules stay inside the balloon safely and ignore the little hole? That is not the way the world works. The molecules always "want to occupy as many states as possible". Hence the air goes out in the open to occupy more volume. The things always goes into disorder (entropy) and the disorder increases with time. The above statement is what we call "second law of thermodynamics". Consider a cup of coffee on the table. Suppose the heat from entire room flows to your cup of coffee, the coffee will boil and the rest of the room will freeze. Freezing means bringing things to order and arrangement. It violates the second law. Hence it will never happen. Hence heat must flow from high temperature to low temperature and not the other way. The air molecules in y

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