Triangular numbers represent the number of dots or points in an equilateral triangle, with each additional row adding the next positive integer of dots. The n-th triangular number is calculated by adding all the integers from 1 to n, representing the total number of handshakes if n people greet each other or nodes connected in a fully meshed network. Triangular numbers model connectivity in situations that can be arranged in an triangular format.

2.  A triangular number is the number of dots in an equilateral triangle with dots uniformly filled.
 For example, the three dots may form a triangle and hence 3 is a triangular number. The n-th
triangular number is the number of dots of a triangle on the side of the n points.
 Equivalently, a triangular number is the addition of the first n positive integers from 1 to n.
[2]

3. [4] [5]

4. [5]
The handshake problem:
“If a group of n people greet each other, the total number of handsheaks will be triangular number”.
To make a fully connected network of PC. The nodes of the network are connected to more than one
other node in the network with a point-to-point link
This is the triangular number of the internet nodes.

5.  [1] http://en.wikipedia.org/wiki/Triangular_number
 [2] http://www.shyamsundergupta.com/triangle.htm
 [3] https://mathblag.wordpress.com/2012/06/23/a-product-rule-for-triangular-numbers/
 [4] http://unurthed.com/2007/11/21/polygonal-numbers-and-a-slice-of-pie/
 [5] http://en.wikipedia.org/wiki/Network_topology#Mesh