Polygonal
Numbers
- We all know and love
the triangular numbers:
- and the square
numbers:
- but you might not be
aware of the developing pattern:
|
1 + 1 + 1 + 1 + 1 +
...
|
1 , 2, 3, 4, 5
...
|
Counting numbers
|
|
1 + 2 + 3 + 4 + 5 +
...
|
1, 3, 6, 10, 15
...
|
Triangular
numbers
|
|
1 + 3 + 5 + 7 + 9 +
...
|
1, 4, 9, 16, 25
...
|
Square numbers
|
|
1 + 4 + 7 + 10 + 13 +
...
|
1, 5, 12, 22, 35
...
|
Pentagonal
numbers
|
|
1 + 5 + 9 + 13 + 17 +
...
|
1, 6, 15, 28, 45
..
|
Hexagonal
numbers
|
|
???
|
???
|
Heptagonal
numbers
|
You might like to calculate
the 5th Heptagonal number. How is it related to the 5th
Hexagonal number and the 4th Triangular number? `Is this
just coincidence? Investigate further! Here is a diagram
showing the amazing Pentagonal numbers:
- Can you see the 1 + 4 +
7 + 10 pattern?
- Multiplying each
pentagonal number by 3 gives: 3, 15, 36, 66... Compare
these with the triangular numbers (you'll need to
calculate a few more) and be amazed!
Alex Bogomolny has created
a wonderful Java Applet which shows diagrams of Polygonal
Numbers. You will find it two-thirds of the way through the
article From
L Carroll To Archimedes.
|
No
57
