If we represent numbers using pebbles, there are special numbers that can form “polygons.” Numbers that can form polygons are called polygonal numbers. For example, the square numbers 1, 4, 9, 16 and so on form a “square.” As shown below, it is easy to see that the 10th square number is a square with 10 rows and 10 columns. This means that the 10th square number is equal to 100 (has 100 dots).
Like square numbers, triangular numbers form a shape. You guessed it right, the shape is a triangle. Finding the 15th triangular number seems hard, but we have already learned that triangular numbers is connected to the sum of the first m positive integers. The mth triangular number has
Notice that each “polygonal number” has its own formula. For example, we have not discussed pentagonal numbers yet, so it may take quite sometime to get the 10th pentagonal number. The question now is, can we find a formula to get the the number of dots given the “size” and “shape” of a polygonal number? For example, what what is the 8th heptagonal number?
To answer this question, consider the next figure. In the figure, we see the geometric representations of square, pentagonal, and hexagonal numbers. Study the grouping of the dots before scrolling down and see if you can find some patterns.
First, we can observe that the yellow dots are triangular numbers. For brevity, let us call the yellow dots, triangles. Observe that the square numbers, pentagonal numbers, and hexagonal numbers have 2, 3 and 4 triangles respectively.
Second, we can observe the group of blue dots. Their rectangular border makes them look like a stick so we call each group a stick . Observe that the square numbers, pentagonal numbers, and hexagonal numbers have 3, 4, and 5 sticks respectively.
Third, notice that we have isolated the red dot.
The table above shows the relationship among the number sides, the number of triangles, and the number of sticks in a polygon. We use these relationships to derive the general formula for polygonal numbers.
In finding the number of dots in a polygon, two quantities vary. The first varying quantity is the number of sides of the polygon which we have denoted above as n. The second quantity is the size of the polygon which we denote as k. The size k is the number of dots in each side.
Total Number of Dots on Sticks
The number total number of dots on sticks is the product of the number of sticks in a polygon with n sides and the number of dots in each stick. From the table above, a polygon with n sides has (n – 1) sticks and the number of dots on each stick is (k – 1); that is, the number of dots on each side minus the red dot (see third figure). So, the total number of dots on sticks is .
Total Number Number of Dots on Triangles
The total number of dots on triangles is equal to the number of triangles times the number of dots on each triangle. Again, from the table above, a polygon with n sides has (n-2) triangles.
Now, the number of dots in each triangle is the sum of 1 + 2 + 3 + … + (k – 2) as shown above. Recall that that the sum of 1 + 2 + 3 + … + (m – 1) + m = 1/2 m (m + 1). If we let m = k + 2, then the total number of dots in each triangle is 1/2(k-2)(k-1).
Therefore, the total number of dots on all the triangles is equal to
The General Formula
To get the general formula, we add 1 (the red dot) to the total number of dots on the sticks and the total number of dots on the triangles. Therefore, the kth polygonal number (P) is described by the formula
We now answer the question above: “What is the 8th heptagonal number?”
A heptagon has 7 sides, so n = 7. The 8th heptagonal number has 8 dots on each side so k = 8. Substituting the values to the formula, we have
Therefore, the 8th heptagonal number is 253.
Proof by Dave Logothetti, Proofs without Words: Exercises in Visual Thinking