Counting in shape sequences

Counting the parts of shape sequences gives number sequences we already know.

Complete graphs. K2 has 2 dots, K3 has 3 dots, up to K6 with 6 dots, and every pair of dots is joined by a straight line.

• K2: a single line (1)
• K3: a triangle (3)
• K4: a square with both diagonals (6)
• K5: a pentagon with a five-pointed star inside (10)
• K6: 15 lines

1, 3, 6, 10, 15: the triangular numbers! Each new dot is joined to all the old ones: K5 has K4's 6 lines plus 4 new ones (6 + 4 = 10), and K6 has 10 + 5 = 15.

Stacked squares are big squares made of little squares: 1 row of 1, 2 rows of 2, up to 5 rows of 5. The little squares number 1, 4, 9, 16, 25: the square numbers.

Stacked triangles are big triangles cut into little triangles, some pointing up and some down. Count by rows: the top row has 1, the next 3, then 5, then 7. So the shapes have 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16, then 25 little triangles: the square numbers again, because adding odd numbers from 1 gives squares.

The Koch Snowflake starts as a triangle with 3 line segments. For the next shape, every segment is replaced by a 'speed bump' of 4 smaller segments. So each shape has 4 times as many segments as the one before: 3, 12, 48, … Your book calls this 3 times the powers of 4.