The previous page on Polygonal Numbers introduced flat (plane) shapes made of dots that have fascinated mathematicians
since the times of the Ancient Greek mathematicians such as Pythagoras (around 500BC)
and Diophantus (around 250AD).
I recommend that you look at it first.
Here we extend the idea of counting the dots in dot patterns to
other ways of arranging the dots on a plane: the Central Polygonal Numbers
flat shapes made with sticks (matchsticks) instead of dots
shapes in 3D and even higher dimensions
The last two are called figurate numbers because they are not regular flat polygons in shape.
If the shape is not planar or is not a simple regular polygon, the number of dots (spheres etc)
is called a figurate number.
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Contents of this page
The icon means there is a
You Do The Maths... section of questions to start your own investigations.
The calculator icon
indicates that there is a live interactive calculator in that section.
Matchstick Numbers
Instead of using dots let's use matchsticks to make patterns.
Matchstick Squares
Here, for instance, are the Matchstick square numbers:
size
1
2
3
4
Square
number of matchsticks
4
12
24
40
Can you find the pattern in this sequence of numbers?
What is the next number?
What is a formula for the number of matchsticks in a square of side n?
Surprisingly, these numbers are the octagonal numbers with negative size! The number of
matchsticks in a square of side r is p_{8}(–r).
You can also investigate what happens to p_{n}(–r) for the other
polygons with negative rank. Can we find an interpretation for these numbers in terms of diagrams?
Matchstick Triangles
Let's see what we get if we form equilateral triangles with the matchsticks:
size
1
2
3
4
Triangle
number
3
9
18
30
Can you find the pattern in this sequence of numbers?
What is the next number?
What is a formula for the number of matchsticks in a triangle of
side n?
3, 9, 18, 30, ... A045943
3 n (n+1) / 2 which we see are also just 3 T(n).
These numbers also have a special overlapping runsum property:
the n^{th} is the smallest number
that is a sum of both n-1 and of n consecutive numbers:
9
= 2 + 3 + 4
= 4 + 5
18
= 3 + 4 + 5 + 6
= 5 + 6 + 7
30
= 4 + 5 + 6 + 7 + 8
= 6 + 7 + 8 + 9
...
House of Cards
Once we have the formula for Matchstick Triangles, it is easy to find the formula for
a House of Cards:
tiers
1
2
3
4
House of Cards
number of cards
2
7
15
26
What is the next number?
What is a formula for the number of cards in a House of Cards of
n tiers (n levels high)?
2, 7, 15, 26, 40, 57, 77, ... A045943
are the Matchstick Triangle numbers
n (3n + 1) / 2
These numbers also have a runsum property: The n^{th} is the sum of the next n numbers after n:
1^{st}
2
= 2
2^{nd}
7
= 3 + 4
3^{rd}
15
= 4 + 5 + 6
4^{th}
26
= 5 + 6 + 7 + 8
...
Can you also write each of these House of Cards numbers as a pronic number + a triangle number?
Take n from the n^{th} House of Cards number. You should find a series that we have met before - but which is it?
n(3n+1)/2 = n(n+1) + n(n–1)/2
2−1=1, 7−2=5, 15−3=12, 26−4=22, ... and 1,5,12,22,...
are the Pentagonal Numbers
Matchstick Hexagons
We can extend our matchstick triangle patterns to hexagons:
size
1
2
3
4
Hexagon
number of matchsticks
12
42
90
156
Can you find the pattern in this sequence of numbers?
What is the next number?
What is a formula for the number of matchsticks in a hexagon of
side n?
Up to now, our dot shapes have grown outward from one corner. We have found many useful series and patterns that are used extensively
throughout mathematics. But we can also make polygonal shapes that grow
from the centre by adding a new layer all around the outside.
Here is a table:
Centred Triangles
Rank
1
2
3
4
5
Shapes
Counts
1
4
10
19
31
Centred Squares
Rank
1
2
3
4
5
Shapes
Counts
1
5
13
25
41
Centred Pentagons
Rank
1
2
3
4
5
Shapes
Counts
1
6
16
31
51
Centred Hexagons
Rank
1
2
3
4
5
Shapes
Counts
1
7
19
37
61
You Do The Maths...
Find the next number in each of the series in the tables above by looking at what we add to one
number to get the next. Can you see why this is so from the diagrams?
Find a formula for the Centred Triangle Numbers c_{3}(r),
Centred Square Numbers c_{4}(r)
and Centred Pentagonal Numbers c_{5}(r).
Find a formula for the centred n-gonal numbers c_{n}(r)
Can you find the formula for c_{n}(r) in terms of
Triangle numbers T(n-1) from this pattern which uses shapes of side 6?
Which polygonal shape corresponds to the following series of sums of odd numbers:
1
1+3+1
1+3+5+3+1
1+3+5+7+5+3+1
Express this pattern using polygonal numbers:
2^{2} + 1^{2}
3^{2} + 2^{2}
4^{2} + 3^{2}
5^{2} + 4^{2}
Here is an alternative pattern for the even-shaped centred polygonal numbers. Can you use it
express C_{2n}(r) algebraically?
* Note that the formulas in this table are not the same as those in the OEIS;
here we use r as the number of dots on an outside edge of the polygon
so as to be be consistent with the planar non-centred p_{n}(r) formula.
Centred Polygonal Numbers c_{n}(r)
_{↓n}^{r→}
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
3
1
4
10
19
31
46
64
85
109
136
166
199
235
274
316
4
1
5
13
25
41
61
85
113
145
181
221
265
313
365
421
5
1
6
16
31
51
76
106
141
181
226
276
331
391
456
526
6
1
7
19
37
61
91
127
169
217
271
331
397
469
547
631
7
1
8
22
43
71
106
148
197
253
316
386
463
547
638
736
8
1
9
25
49
81
121
169
225
289
361
441
529
625
729
841
9
1
10
28
55
91
136
190
253
325
406
496
595
703
820
946
10
1
11
31
61
101
151
211
281
361
451
551
661
781
911
1051
11
1
12
34
67
111
166
232
309
397
496
606
727
859
1002
1156
12
1
13
37
73
121
181
253
337
433
541
661
793
937
1093
1261
13
1
14
40
79
131
196
274
365
469
586
716
859
1015
1184
1366
14
1
15
43
85
141
211
295
393
505
631
771
925
1093
1275
1471
15
1
16
46
91
151
226
316
421
541
676
826
991
1171
1366
1576
16
1
17
49
97
161
241
337
449
577
721
881
1057
1249
1457
1681
17
1
18
52
103
171
256
358
477
613
766
936
1123
1327
1548
1786
18
1
19
55
109
181
271
379
505
649
811
991
1189
1405
1639
1891
19
1
20
58
115
191
286
400
533
685
856
1046
1255
1483
1730
1996
20
1
21
61
121
201
301
421
561
721
901
1101
1321
1561
1821
2101
The Centred Polygonal Number Calculator
C A L C U L A T O R : C e n t r e d P o l y g o n a l N u m b e r s
The game of Chinese Checkers in played on a star-shaped board
and gives us
another nice series of shapes variously called
the hexagonal star
the USA Sheriff's badge but some have 5 points and others 7
the Star of David
1
13
37
73
121
1, 13, 37, 73, 121, 181, 253 ... A003154
These numbers are also the Centred 12-gonal numbers.
Each 12-gon
is made up of 12 triangles plus an extra dot and can therefore be
can be transformed into the corresponding star shape: for example...
Centred Polygonal Numbers versus Corner Polygonal Numbers
There are several important differences between the ordinary or corner polygonal numbers
p_{b}(r) and the centred polygonal numbers
p_{b}(r).
Integer Representations
On the first Polygonal Numbers page we saw that Fermat's theorem states that
we can represent any integer as a sum of those polygonal numbers in many ways,
as a sum of 3 triangular numbers or a sum of up to 4 square numbers, or 5 pentagonal etc.
This is not true for
centred polygonal numbers as the numbers are too far apart.
For instance, with centred triangular numbers:
1, 4, 10, 19, 31, 46, ...
7 needs at least 4 centred triangle numbers: 7 = 4 + 1 + 1 + 1, and
17 needs at least 5 centred triangle numbers: 17 = 10 + 4 + 1 + 1 + 1.
For centred square numbers:
1, 5, 13, 25, 41, 61
9 needs at least 5 centred square numbers: 9 = 5 + 1 + 1 + 1 + 1, and
22 needs at least 6 centred square numbers: 22 = 13 + 5 + 1 + 1 + 1 + 1.
As sums of triangular numbers
The cornered n-gonal polygonal numbers are made up of n-2 (cornered) triangular numbers of rank
r-1 plus (a line of) r:
p_{n}(r) = r + (n – 2)
r(r–1)
2
p_{4}(6) = 2 T(5) + 6
p_{5}(6) = 3 T(5) + 6
p_{6}(6) = 4 T(5) + 6
p_{7}(6) = 5 T(5) + 6
p_{8}(6) = 6 T(5) + 6
whereas a centred n-gonal polygon is made up of n (cornered) triangular numbers of rank
r-1 plus an extra 1:
c_{n}(r) = 1 + n
r(r–1)
2
which is an easy way to remember their formulas.
The centred polygonal numbers formula is illustrated by these dot-diagrams from the
You Do The Maths... question above:
c_{3}(6) = 3 T(5) + 1
c_{4}(6) = 4 T(5) + 1
c_{5}(6) = 5 T(5) + 1
c_{6}(6) = 6 T(5) + 1
c_{7}(6) = 7 T(5) + 1
The Plain and Centred Polygonal Number Calculator
C A L C U L A T O R : P l a i n & C e n t r e d P o l y g o n a l N u m b e r s
Let's move into three dimensions and start off with at Pyramid Numbers since these are related to the
Polygonal Numbers where our 2-dimensional dots now become 3-dimensional balls!
As these 3D solid shapes are not planar, the dots become balls
and the numbers are called figurate numbers.
Here are three views of a stack of 5 squares making a Square-based Pyramid:
Square based Pyramid of 5 layers
Side view
Expanded view of layers
Top view
and here is a Pentagonal-based Pyramid of height 5:
The layers are just the plane centred pentagonal numbers we saw earlier:
side
1
2
3
4
5
shape
number of dots
1
5
12
22
35
You can see that each layer is one of our polygonal images.
A Pyramid of r layers has each of
the polygonal numbers with ranks from 1 to r.
We will denote these as P_{n}(r)
using a capital P for our 3D shapes and the small p
is used p_{n}(r) for the 2D polygonal numbers
we looked at earlier.
Since we stack r layers to make a pyramid of height (or rank)
r, we have the definition:
For the Polygonal numbers, we summed arithmetic series, accumulating sums from the beginning,
1, 2, 3, 4, 5, ... to get the series 1, 3, 6, 10, 15, ... The Triangular numbers 1, 3, 4, 7, 9, ... to get the series 1, 4, 9, 16, 25, ... The Square numbers 1, 4, 7, 10, 13, ... to get the series 1, 5, 12, 22, 35, ... The Pentagonal numbers
and so on
If we now do the same thing but sum the polygonal number series, we get the Pyramid Numbers.
If we stack Triangles on top of each other, we make a pyramid which has a Triangle on each layer, and so on.
1, 3, 6, 10, 15, ... to get the series 1, 4, 10, 20, 35, ... The Triangular Pyramid numbers 1, 4, 9, 16, 25, ... to get the series 1, 5, 14, 30, 55, ... The Square Pyramid numbers 1, 5, 12, 22, 35, ... to get the series 1, 6, 18, 40, 75, ... The Pentagonal Pyramid numbers
and so on
Here are some simple examples of Pyramid Numbers. How many objects in each picture?
Write each in P_{n}(r) form.
4=P_{3}(2)
5=P_{4}(2)
6=P_{5}(2)
14=P_{4}(3)
20=P_{3}(4)
Higher dimensions and Pascal's Triangle
The Triangular shapes of 2D and 3D are part of a more general pattern:
r:
1
2
3
4
5
6
p_{3}(r):
1
3
6
10
15
21
P_{3}(r):
1
4
10
20
35
56
We could even invent a 2D version since each row is formed by summing the items on the previous row from the beginning up to
that column, and continue with a Fourth Dimension and Fifth too, although it would be difficult to see how we
could represent these in a (2D) diagram:
If we look at the upwards diagonals in this table starting each in the leftmost column you might find that this new
triangle of numbers is one you have met before:
Pascal's Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
...
So the formula for the entries in the Trianglular Numbers in D dimensions table above is
(
D + r
)
=
(
D + r
)
=
(D+r)(D−1+r)...(r)
r
D
D(D-1)(D-2)...3×2×1
You Do The Maths...
You are working in a greengrocers shop and have just arranged the new batch of oranges
into a polygonal pyramid of 3 layers. The shopowner says he now wants them
rearranging into a square-based pyramid.
You manage to arrange them all as he requires with none left over. How many oranges could you have had?
What was the polygon shape that you used in your original pile?
Why is it much easier to arrange round fruit in layers which are Hexagonal polygon numbers?
Make a list of all the Hexagonal Pyramid Numbers up to 100.
Check your answer with
A002412