Mathematical Essay :: Investigating Figurate Numbers With Technology
By: Kate Hobgood and Clay Kitchings
What are Figurate Numbers? Perhaps it might be helpful to consider various examples that are models for specific figurate numbers.
For a detailed method for finding a closed formula for the nth triangular number, use the link below. The method relies on the use of a TI83/84 calculator (or a similar calculator with matrix multiplication capabilities):
Pythagoras is credited to have initially explored the mathematical relationship within figurate numbers. Pythagoras was a Greek mathematician and was even titled as “the father of numbers”. He is most well known for the Pythagorean Theorem, and he “believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns or cycles”. Mathematicians would work with the numbers using pebbles or seeds and arranging them into figures. Working in this manner, they were able to work with regular polygon numbers noticing a sequence to the growing figures. Depending on the number of sides of the polygon and the number of pebbles along a given side, the figurate numbers were discovered.
Figurate numbers are numbers that can be represented by a regular geometrical arrangement or sequence of evenly spaced points. Figurate numbers are most commonly expressed in the form of regular triangles, squares, pentagons, hexagons, etc. For this reason, figurate numbers are also known as the polygon numbers. For instance, a triangle created with three pebbles along a given side results in a total of six pebbles. Adding one more pebble to each side of the triangle results in a total of ten pebbles. Therefore, we can find the triangular sequence of numbers as 1, 3, 6, 10, 15, 21.... *The formulas for figurate numbers will be discussed later on.* The triangular numbers are shown in the following figure.
Figure from Wolfram Mathworld.
The next set of polygonal figurate number that is frequently discussed is called the square numbers. Similar to the triangular numbers, each square number is found by the addition of one unit to two adjacent arms of the square. Refer to the following figure.
Figure from Wolfram Mathworld.
The figurate numbers correspond to the number of sides of the polygon. Lengthening two adjacent sides by one point enlarges the polygon. With each increase, an additional “layer” is added to the figure. This “layer” is also referred to as the gnomon. The gnomon is the piece of the figure that needs to be added to a given figurate numbers in order to get the next greater figurate number. Gnomons also refer to an architectural template that is used to mark off similar forms or shapes. The Greek translation of gnomon is “carpenter square”. For instance, the formation below represents the sixth square number for there are six units along a given side. The gnomon can be visualized as the L-shaped formation of additional units (the figure contains five gnomons).
The gnomon is also related to Pythagoras’ finding of the commonly known Pythagorean Theorem. Consider the square created below:
Note that the backwards L-shaped gnomon or “carpenter’s square” contains nine black dots, which can be thought of as three groups of three black dots. Therefore, after the total number of dots in the gnomon, we find that the gnomon can be represented by dots. Leaving us with a smaller red square with a width of four dots and a length of four dots. So the smaller red square can be represented by dots. Finally, we know that the entire square figure has a width of five dots and length of 5 dots. So here we now have a proof of the Pythagorean triple of and thus supports the Pythagorean Theorem for a triangle with two legs of length 3 and 4 and a hypotenuse of 5 units.
Notice that the gnomon of the figure is represented by an odd number . The smaller imagined gnomon of the smaller red square (not drawn in the figure) contains seven dots. The even smaller gnomon of a square has five dots. Recall the figure:
If one were to continue adding units to the square figure, then the next gnomon of a square would have 11 dots, a would have 13 dots, and so on. Surprisingly, the sequence of gnomons can be represented by the odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17…. Therefore, with helpful use of the gnomon, one can find support for the conjecture that the sum of the first n odd numbers is .
(We think he is a distant cousin, twice removed).
One can explore the possible formulas for finding the gnomon of the nth figurate number. Explore some of the polygons briefly mentioned above.
The pentagonal figurate numbers can be expressed as the figure below. The gnomons are shown by the additional “rings” linking the red dots.
Notice that with each consecutive pentagonal number, one red dot is added to only two adjacent sides. Since there are two fixed adjacent sides that do not contribute to the gnomon structure, we have a three-sided gnomon shape. Then, working with the three sides of the gnomon figure, we can begin summing the gnomon number from the dot placed on one of the fixed sides. Extend this dot to create a side length of n dots. We now have to consider the two sides remaining to complete the gnomon. The next side will contain n-1 dots because we do not want to count the dot used on the previous side. The final third side of the gnomon will also have n-1 dots. Hence, the formula for the gnomon of a regular pentagonal figure is .
The gnomon number for the hexagonal number follows in a similar manner.
In this case, the gnomon figure is created by four sides. Working in the same way, take the dot on one adjacent side and create a side length of n dots. This accounts for one side of the gnomon, now we must create the remaining three sides. The next three sides will each contain dots. The formula for the gnomon of a regular hexagonal figure is .
Notice that the number of sides of the gnomon is found by (the number of sides of the polygon) – (two adjacent sides) = (n – 2) where n represents the number of sides of the polygon.
For the square, the gnomon had only two sides. Note that (the number of sides of a square) – (two adjacent sides) = (4 – 2) = 2.
The gnomon for the pentagon had 5 – 2 = 3 sides.
The gnomon for the hexagon had 6 – 2 = 4 sides.
Since we now know that the gnomon shape has n – 2 sides, and we also know that one side of the gnomon has exactly n dots, the remaining dots are found on (n – 3) sides. Each of these remaining sides has (n – 1) dots.
A more general formula can thus be found for the number of sides of a gnomon shape for a particular polygon. Letting n be the number of sides of the polygon, we have:
n + (n-3)(n-1)
Extension: Two-dimensional polygons are not the only figures used to represent the figurate numbers. They can also be represented by other shapes such as L-shaped and three dimensional figures. Expanding the formations to L-shaped and three-dimensional formations gives rise to a new sequential relationship. Consider the pyramidal numbers where each layer of the three-dimensional pyramid has one less unit along a side length as the layer below it. It may be helpful to picture the pyramidal numbers as a layering or stacking of the polygonal numbers. This way of thinking is useful when considering any three-dimensional figure created by regular congruent polygonal faces.