Page 26 - Ganit Kaushal

• Connectivity: Complete graphs are fully connected, meaning there are no isolated vertices, and
the graph is as connected as possible.

Examples:
• K : A graph with 2 vertices and 1 edge.
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• K : A triangle, 3 vertices with 3 edges, where each vertex is connected to every other vertex.
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• K : A tetrahedron, 4 vertices with 6 edges, and so on.
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A few complete graphs are shown in the following figure:

(Complete Graphs)

K 2 K 3 K 4 K 5 K 6
3. Stacked Squares: Stacked squares generally refer to a series of squares that grow horizontally as well as
vertically as shown in the given picture. The following picture demonstrates the growing shape sequence
of stacked squares.

(Stacked Squares)

4. Stacked Triangles: Stacked triangles refer to a series of triangles where rows of triangles are arranged
one on top of the other, and the growing row at the base corresponds to the sequence 1, 3, 5, 7, ... . These
triangles are identical in size, creating a progression or pattern. The following picture demonstrates the
growing shape sequence of stacked triangles.

(Stacked Triangles)

*
5. Koch Snowflakes: The Koch Snowflake is a famous fractal curve and one of the earliest fractals to be
described. It is constructed by repeatedly adding a speed bump ‘ ’ to the sides of an initial equilateral
triangle. Over each iteration, the snowflake becomes more complex, creating an infinitely detailed and
non-smooth boundary. The following picture demonstrates the shape sequence of Koch snowflakes.

(Koch Snowflake)

Construction of the Koch Snowflake: Follow the given steps–
Step 1: Start with an equilateral triangle.

Step 2: Divide each side into three equal parts.
Step 3: Construct an equilateral triangle on the middle segment of each side. The base of this new triangle
is removed, leaving a “speed bump” of type ‘ ’ in the middle of each side.

* Fractal curves are a fascinating class of geometric shapes, that displays a repeating pattern at all points.

24 Mathematics-6