Patterns in Shapes

Shape Sequences

Just as we have patterns in numbers, we have patterns in shapes. This is the foundation of Geometry.

1. Regular Polygons

A regular polygon is a shape with equal-length sides and equal angles. The sequence starts with 3 sides.

The number of sides follows the Counting Numbers sequence starting at 3 ($3, 4, 5, 6, \dots$). The number of corners (vertices) follows the exact same sequence.

2. Complete Graphs

A Complete Graph is a shape where every corner is connected to every other corner by a line.

• K2 (2 points): 1 line.
• K3 (3 points): 3 lines (Triangle).
• K4 (4 points): 6 lines (Square with X inside).
• K5 (5 points): 10 lines (Pentagon with star inside).

The sequence of the number of lines in Complete Graphs is: 1, 3, 6, 10, 15…

3. Koch Snowflake

The Koch Snowflake is a fractal pattern. You start with a triangle and repeatedly add smaller triangles to the middle of every side.

• The boundary gets more complex with each step.
• The number of line segments grows rapidly: 3, 12, 48, 192…
• This sequence is $3 \times$ Powers of 4.