In mathematics, a polygonal number is a number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers.
Polygonal numbers were first studied during the 6th century BC by the Ancient Greeks, who investigated and discussed properties of oblong, triangular, and square numbers.
Definition and examples
The number 10 for example, can be arranged as a triangle (see triangular number):
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But 10 cannot be arranged as a square. The number 9, on the other hand, can be (see square number):
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Some numbers, like 36, can be arranged both as a square and as a triangle (see square triangular number):
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By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as in red.
Triangular numbers
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The triangular number sequence is the representation of the numbers in the form of equilateral triangle arranged in a series or sequence. These numbers are in a sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45, and so on.
Square numbers
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Polygons with higher numbers of sides, such as pentagons and hexagons, can also be constructed according to this rule, although the dots will no longer form a perfectly regular lattice like above.
Pentagonal numbers
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Hexagonal numbers
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Formula
Polygonal numbers differ mainly by the difference between each consecutive number. The difference between each triangular number increases by 1 each iteration. For square numbers the difference increases by 2. For pentagonal numbers it increases by 3.
The size of the difference between each N starts at 1 for polygonal numbers (and for centered polygonal numbers the size of the difference starts at 0). For example: the first triangular is 1 and the "difference" starts at 1. To get the next triangular, you increment the difference by 1 and add it to the current one. The 2nd triangular number is 3, the difference is 2. Increment the difference again, it's 3. 3+3 is 6, the 3rd triangular. The difference between the 3rd and the 2nd is 3. Given that triangular numbers are the sum of the first N numbers this is unsurprising, however the same method holds for all other polygonal numbers as well.
With squares for the Nth and diff we start at the first square 1 and a diff (or offset) of 1. For the next square the difference is first incremented by 2: 1+2=3. We then add the difference to the current square: 1+3=4. The 2nd square number. Continuing the pattern for Nth+diff: 4+5=9, 9+7=16, 16+9=25. In other words, for the Nth polygonal number the difference between it and the next is X * N + 1. Where X is the number of sides (beginning at 1 for triangular).
thumb|An s-gonal number greater than 1 can be decomposed into s−2 triangular numbers and a natural number.If is the number of sides in a polygon, the formula for the th -gonal number is
:<math>P(s,n) = \frac{(s-2)n^2-(s-4)n}{2}</math>
The th -gonal number is also related to the triangular numbers as follows:
:<math>P(s,n) = (s-2)T_{n-1} + n = (s-3)T_{n-1} + T_n\, .</math>
Thus:
:<math>\begin{align}
P(s,n+1)-P(s,n) &= (s-2)n + 1\, ,\\
P(s+1,n) - P(s,n) &= T_{n-1} = \frac{n(n-1)}{2}\, ,\\
P(s+k,n) - P(s,n) &= k T_{n-1} = k\frac{n(n-1)}{2}\, .
\end{align}</math>
For a given -gonal number , one can find by
:<math>n = \frac{\sqrt{8(s-2)x+{(s-4)}^2}+(s-4)}{2(s-2)}</math>
and one can find by
:<math>s = 2+\frac{2}{n}\cdot\frac{x-n}{n-1}</math>.
Every hexagonal number is also a triangular number
Applying the formula above:
:<math>P(s,n) = (s-2)T_{n-1} + n </math>
to the case of 6 sides gives:
:<math>P(6,n) = 4T_{n-1} + n </math>
but since:
:<math>T_{n-1} = \frac{n(n-1)}{2} </math>
it follows that:
:<math>P(6,n) = \frac{4n(n-1)}{2} + n = \frac{2n(2n-1)}{2} = T_{2n-1}</math>
This shows that the th hexagonal number is also the th triangular number . We can find every hexagonal number by simply taking the odd-numbered triangular numbers:
{| class="wikitable"
|-
! rowspan="2"|
! rowspan="2"|Name
! rowspan="2"|Formula
! colspan="11"|
! rowspan="2" align="right" | Sum of reciprocals
! rowspan="2" align="center" | OEIS number
|-
! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !!
|-
| align="right" | 2
| Natural (line segment)
| align="center" |
| align="right" | 1
| align="right" | 2
| align="right" | 3
| align="right" | 4
| align="right" | 5
| align="right" | 6
| align="right" | 7
| align="right" | 8
| align="right" | 9
| align="right" | 10
| align="right" |
| align="center" | ∞ (diverges)
|
|-
| align="right" | 3
| Triangular
| align="center" |
| align="right" | 1
| align="right" | 3
| align="right" | 6
| align="right" | 10
| align="right" | 15
| align="right" | 21
| align="right" | 28
| align="right" | 36
| align="right" | 45
| align="right" | 55
| align="right" |
| align="center" | 2 proved that if three different integers , , and are all at least 3 and not equal to 6, then only finitely many numbers are simultaneously -gonal, -gonal, and -gonal.
Katayama, Furuya, and Nishioka proved that if the integer is such that <math>s=5</math> or <math>7\le s\le 12</math>, then the only -gonal square triangular number is 1. For example, that paper gave the following proof for the case where <math>s=5</math>. Suppose that <math>P(3,n)=P(4,p)=P(5,q)</math> for some positive integers , , and . A calculation shows that the point <math>(x,y)</math> defined by <math>(x,y)=(48p^{2}+3,24p(2n+1)(6q-1))</math> is on the curve <math>Y^{2}=X^{3}-X^{2}-9X+9</math>. That fact forces <math>(x,y)=(51,360)</math> (as an elliptic curve database confirms), so <math>p=1</math> and the result follows.
The number 1225 is hecatonicositetragonal (), hexacontagonal (), icosienneagonal (), hexagonal, square, and triangular.
See also
- Centered polygonal number
- Polyhedral number
- Fermat polygonal number theorem
Notes
References
Bibliography
- The Penguin Dictionary of Curious and Interesting Numbers, David Wells (Penguin Books, 1997) [].
External links
- Polygonal Numbers: Every s-polygonal number between 1 and 1000 clickable for 2<=s<=337
- Polygonal Number Counting Function