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# Tag Archives: Geometry

## The Area of a Disk

[This post is under review.]

If you are aware of elementary facts of geometry, then you might know that the area of a disk with radius $R$ is $\pi R^2$.

The radius is actually the measure(length) of a line joining the center of disk and any point on the circumference of the disk or any other circular lamina. Radius for a disk is always same, irrespective of the location of point at circumference to which you are joining the center of disk. The area of disk is defined as the ‘measure of surface‘ surrounded by the round edge (circumference) of the disk.

The area of a disk can be derived by breaking it into a number of identical parts of disk as units — calculating their areas and summing them up till disk is reformed. There are many ways to imagine a unit of disk. We can imagine the disk to be made up of several concentric very thin rings increasing in radius from zero to the radius of disc. In this method we can take an arbitrary ring, calculate its area and then in similar manner, induce areas of other rings -sum them till whole disk is obtained. (more…)

## Triangle Inequality

Triangle inequality has its name on a geometrical fact that the length of one side of a triangle can never be greater than the sum of the lengths of other two sides of the triangle. If $a$, $b$ and $c$ be the three sides of a triangle, then neither $a$ can be greater than $b+c$, nor$b$ can be greater than $c+a$, nor $c$ can be than $a+b$.

Triangle

Consider the triangle in the image, side $a$ shall be equal to the sum of other two sides $b$ and $c$, only if the triangle behaves like a straight line. Thinking practically, one can say that one side is formed by joining the end points of two other sides.
In modulus form, $|x+y|$ represents the side $a$ if $|x|$ represents side $b$ and $|y|$ represents side $c$. A modulus is nothing, but the distance of a point on the number line from point zero.

Visual representation of Triangle inequality

For example, the distance of $5$ and $-5$ from $0$ on the initial line is $5$. So we may write that $|5|=|-5|=5$.

Triangle inequalities are not only valid for real numbers but also for complex numbers, vectors and in Euclidean spaces. In this article, I shall discuss them separately. (more…)

## Tesseract

Static View of Hypercube Via Wikipedia

## Image Information

Image URL: http://wpgaurav.files.wordpress.com/2011/06/wdir.gif
Image Dimension: 320 X 240
Copyright: None

About Tesseract:

The Tesseract is a four-dimensional object in which each of the “faces” is actually a cube (much the same way that in a three- dimensional cube each face is a square, and in a two-dimensional square each face is a line). There are a plethora of mathematical features of the hypercube: for example, a line has 2 vertices, a square has 4, a cube has 8, and the hypercube has 16, illustrating the fact that an n-dimensional “square” has $2^n$. vertices.

## Understanding Poincaré Conjecture

Image via Wikipedia

## Introduction & Statement of Poincaré Conjecture

In 1904, the french Mathematician Henri Poincaré posed an epoch making question in one of his papers, which asked:

If a three-dimensional shape is simply connected, is it homeomorphic to the three dimensional sphere?