Page 27 - Ganit Kaushal

Step 4: Repeat the process: For each side of the new shape, repeat the same procedure of dividing each
side into three parts and adding a smaller equilateral triangle in the middle. Remove the base of the
new triangle each time, leaving the bump.
This process continues indefinitely, and with each iteration, the snowflake’s perimeter increases. As one does
this more and more times, the changes become thinner and thinner with tiny line segments.

1.8 RELATION OF SHAPE SEQUENCES WITH NUMBER SEQUENCES

Sometimes, patterns in shapes are connected to patterns in numbers in surprising ways. These connections can
help us understand both the shapes and the numbers in a better way. The following examples help us understand
this relation.

Ex 1. Which number sequence does the number of sides in the shape sequence of regular polygons follow?
Give its significance.

Sol. The number of sides in the shape sequence of regular polygons follows the counting sequence starting
from 3, that is, 3, 4, 5, 6, 7, 8, and so on.

So, a regular triangle has three sides, a square (or quadrilateral) has four, a pentagon has five, a hexagon
has six, and so on. These shapes are referred to as regular triangles, squares, pentagons, hexagons, and
so on, depending on the number of sides they have.
The term “regular” refers to a shape with all sides of the same length and angles of the same measure.

Ex 2. Count the number of edges in each shape of the shape sequence of complete graphs. Which number
sequence do you get?

Sol. The number of edges in each shape of the sequence of complete graphs is as follows: 1, 3, 6, 10, 15,
21, and so on.

Clearly, this is also the sequence of triangular numbers.
\ The number of edges in each shape of the shape sequence of complete graphs gives the sequence
of triangular numbers. Amazing connection! isn’t it?
Ex 3. How many little triangles are there in each shape of the sequence of stacked triangles? What number
sequence does this give? Can you explain why? (NCERT)
Sol. The number of little triangles in each shape of the sequence of stacked triangles is 1, 4, 9, 16, 25, …
and so on. This is a sequence of squares.
Let us explain it by counting the number of small triangles in each row of the shapes in the sequence
of stacked triangles as shown in the figure below.
1
1 3
1 3 5
1 3 5 7
1 3 5 7 9
This gives the sequence –

1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, 1 + 3 + 5 + 7 + 9, and so on.
This is the sequence which we get by adding up the odd numbers. And we know that this gives the
sequence of squares (by example 1 of section 1.6).
Other shape patterns also have interesting connections with number patterns.

Readers are advised to explore and explain them.

Patterns in Mathematics 25