Relation to Number Sequences
Often, shape sequences are related to number sequences in surprising ways. Such relationships can be helpful in studying and understanding both the shape sequence and the related number sequence.
Example: The number of sides in the shape sequence of Regular Polygons is given by the counting numbers starting at 3, i.e., 3, 4, 5, 6, 7, 8, 9, 10, .... That is why these shapes are called, respectively, regular triangle, quadrilateral (i.e., square), pentagon, hexagon, heptagon, octagon, nonagon, decagon, etc., respectively.
The word ‘regular’ refers to the fact that these shapes have equal-length sides and also equal ‘angles’ (i.e., the sides look the same and the corners also look the same). We will discuss angles in more depth in the next chapter.
The other shape sequences in Table 3 also have beautiful relationships with number sequences.
Figure it Out
1. Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?
| Polygon | Equilateral Triangle | Square | Pentagon | Hexagon | Heptagon | Octagon |
|---|---|---|---|---|---|---|
| No. of sides | ||||||
| No. of vertices(corners) |
2. Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?
| Graph | |||||
|---|---|---|---|---|---|
| No. of lines |
3. How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?
| No. of small squares per side | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Total no. of small squares |
4. How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?)
| No. of rows of small triangles | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Total no. of small triangles |
5. To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment ‘—’ by a ‘speed bump’ 
.
As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48, ..., i.e., 3 times Powers of 4; this sequence is not shown in Table 1.)
| Koch snowflakes | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| No. of lines |