Spring 2001

Essay 2

Polygonal Numbers

(Review the definition, if you like.)

Let Kn = the nth K-gonal number. What is the general form for any K-gonal number? We will proceed with a geometric derivation.

Recall that all polygons can be "triangulated." That is, all polygons can be divided into triangles by choosing a vertex and drawing a segment to every non-adjacent vertex. A square can be divided into 2 triangles; a pentagon into 3 triangles; a hexagon into 4 triangles. In general, a K-gon can be divided into (K-2) triangles. Consider the following pictures of polygonal numbers that have been triangulated with red segments. :

Notice the blue dot located at the bottom left corner of each figure. Let this represent the first number of that K-gon sequence. The second number is represented by the number of dots used to make the smallest version of the figure. The next "level" of the figure (including all the dots within it) represents the next K-gonal number. In other words, to represent the nth K-gonal number

begin at the blue dot (Count the blue dot as the 1st number)

follow an outer edge to the nth dot

trace out the figure at that level

The total number of dots around the edges and inside that level is the nth k-gonal number.

Observe that the triangles within each polygon follow that pattern as well. For example, the fourth pentagonal number (54) contains three triangles that each represent the fourth triangular number. And the third hexagonal number (63) contains four triangles that each represent the third triangular number.

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Since every K-gon can be triangulated into (K-2) triangles, every K-gonal number can be expressed in terms of a triangular number. recall that the nth triangular number is given by

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For example, 54 contains three sets of T4. Does it follow that 54 = 3*T4 ??? If we count the dots, we find that 54 = 22. Since T4 = 10, then 3*T4 = 3*(10) = 30. What happened???

We must be careful to realize that the inner triangles share some dots. We CANNOT simply state that the number of dots in the nth K-gon is equal to (# of triangles)*Tn = (K-2)*Tn. We must consider the dots that are "overcounted."

The dots that are overcounted lie along the red diagonals. There are (K-3) diagonals in a K-gon. There are n dots on each diagonal. Two triangles share a diagonal as a common side. For every diagonal, each dot on the diagonal is counted twice. That's like having twice as many diagonals!!! In other words, we must subtract the number of dots on the diagonals that are counted twice:

Kn = (# of triangles)*Tn - (# of diagonals)*(number of dots on each diagonal)

Kn = (K-2)*Tn - (K-3)*n

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Whew, wasn't that a task? Let's put that it all together. So, the general form for Kn (the nth k-gonal number) is:

.

So,

.

And,

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You can use the general form to get the formulas for each polygonal number.

Triangular numbers

Square numbers

Pentagonal numbers

Hexagonal numbers

OTHER ESSAYS