More on Polygonal and Figurate Numbers

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 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 You Do The Maths... icon means there is a You Do The Maths... section of questions to start your own investigations.
The calculator 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:
size1234
Square
number of
matchsticks
4122440
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?

4, 12, 24, 40, ... A046092
2 n (n+1)

Matchstick Squares with Diagonals

If we squash our squares a little, we can make a rhombus (a 4-sided shape with all sides the same length) and have a matchstick as a diagonal.
=
These give rise to many other patterns and sequences.
size1234
Square
with
Diagonal
Rhombic
number of
matchsticks
5163356
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?

5, 16, 33, 56, ... A045944
n (3n + 2) = 4 p3(n) + p4(n) = 4 T(n) + n2
Surprisingly, these numbers are the octagonal numbers with negative size! The number of matchsticks in a square of side r is p8(–r).
You can also investigate what happens to pn(–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:
size1234
Triangle
number391830
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 nth 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:
tiers1234
House
of
Cards
HoC1 HoC1 HoC1 HoC1
number of
cards
271526
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 nth is the sum of the next n numbers after n:
1st2 = 2
2nd7 = 3 + 4
3rd15 = 4 + 5 + 6
4th26 = 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 nth 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:
size1234
Hexagon
number of
matchsticks
124290156
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?

12, 42, 90, 156, ... A045945
3 n (3 n + 1)

Centred Polygonal Shapes

Centred shapes

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
Rank12345
Shapes tri tri tri tri tri
Counts14101931

Centred Squares
Rank12345
Shapes tri tri tri tri tri
Counts15132541

Centred Pentagons
Rank12345
Shapes tri tri tri tri tri
Counts16163151

Centred Hexagons
Rank12345
Shapes tri tri tri tri tri
Counts17193761

/ You Do The Maths...

  1. 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?
  2. Find a formula for the Centred Triangle Numbers c3(r), Centred Square Numbers c4(r) and Centred Pentagonal Numbers c5(r).
  3. Find a formula for the centred n-gonal numbers cn(r)
  4. Can you find the formula for cn(r) in terms of Triangle numbers T(n-1) from this pattern which uses shapes of side 6?
  5. 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
  6. Express this pattern using polygonal numbers:
    22 + 12 32 + 22 42 + 32 52 + 42
  7. Here is an alternative pattern for the even-shaped centred polygonal numbers. Can you use it express C2n(r) algebraically?

    2n = 4, r=6
    Square Centred

    2n = 6, r=6
    Hexagonal Centred

    2n = 8, r=6
    Octagonal Centred

Centred polygon formulas

Centred Polygonal Numbers cn(r)
r=1r=2r=3r=4r=5r=6r *OEIS *
Triangularn=31 4 10 19 31 46½ ( 3r2 – 3r + 2) A005448
Squaren=41 5 13 25 41 61½ ( 4r2 + 4r + 2)
= 2r2 – 2r + 1
A001844
Pentagonaln=51 6 16 31 51 76½ (5r2 – 5r + 2) A005891
Hexagonaln=61 7 19 37 61 91½ (6r2 – 6r + 2)
=3r2 – 3r + 1
A003215
n-gon n11+n1+3n1+6n1+10n1+15n ½(n r2 – n r + 2)
= n p3(r – 1) + 1
 
* 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 pn(r) formula.
Centred Polygonal Numbers cn(r)
nr123456789101112131415
314101931466485109136166199235274316
4151325416185113145181221265313365421
51616315176106141181226276331391456526
61719376191127169217271331397469547631
718224371106148197253316386463547638736
819254981121169225289361441529625729841
9110285591136190253325406496595703820946
1011131611011512112813614515516617819111051
11112346711116623230939749660672785910021156
12113377312118125333743354166179393710931261
131144079131196274365469586716859101511841366
141154385141211295393505631771925109312751471
151164691151226316421541676826991117113661576
1611749971612413374495777218811057124914571681
17118521031712563584776137669361123132715481786
18119551091812713795056498119911189140516391891
191205811519128640053368585610461255148317301996
201216112120130142156172190111011321156118212101

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
Numbers common to several Centred shapes

simultaneously: -shaped
in the range

up to (optional)
Which polygonal shapes?

all Centred-polygon shapes for cn(r) =
exclude rank 2?
Runsums

for centred -gonal

R E S U L T S

calculator: Centred Plain&Centred

The Star Centred Figurates

The game of Chinese Checkers in played on a star-shaped board and gives us another nice series of shapes variously called

sq+2tri=hex sq+2tri=hex sq+2tri=hex sq+2tri=hex sq+2tri=hex
11337 73121
sq+2tri=hex sq+2tri=hex sq+2tri=hex sq+2tri=hex sq+2tri=hex
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 pb(r) and the centred polygonal numbers pb(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:
pn(r) = r + (n – 2) r(r–1)
2
p4(6)
= 2 T(5) + 6
p5(6)
= 3 T(5) + 6
p6(6)
= 4 T(5) + 6
p7(6)
= 5 T(5) + 6
p8(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:
cn(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:
c3(6)
= 3 T(5) + 1
c4(6)
= 4 T(5) + 1
c5(6)
= 5 T(5) + 1
c6(6)
= 6 T(5) + 1
c7(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
Numbers common to several shapes

-shaped and
centre-shaped
show their ranks too?
in the range 
up to

R E S U L T S

calculator: Centred Plain&Centred

Pyramids: Number Shapes as Solids in 3 dimensions

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 viewExpanded view of layersTop view
and here is a Pentagonal-based Pyramid of height 5:
side view top view
The layers are just the plane centred pentagonal numbers we saw earlier:
side12345
shape tri
number of
dots
15122235
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 Pn(r) using a capital P for our 3D shapes and the small p is used pn(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:
Pn(r) = pn(1) + pn(2) + ... + pn(r-1) + pn(r)

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 Pn(r) form.
oranges plums apples tins 1 tins 2
4=P3(2) 5=P4(2) 6=P5(2) 14=P4(3) 20=P3(4)

Higher dimensions and Pascal's Triangle

The Triangular shapes of 2D and 3D are part of a more general pattern:
r:123456
p3(r):136101521
P3(r):1410203556
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:
"Triangular" numbers in D dimensions
Dimensionr=1r=2r=3r=4r=5r=6OEIS
0:1111111...
1:r123456...A000027 the natural numbers
2: p3(r)136101521...A000217 the plane triangle numbers
3:P3(r)1410203556...A000292 the triangular-based pyramid numbers
4:15153570126...A000332
5:162156126252...A000389
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)
rDD(D-1)(D-2)...3×2×1

/ You Do The Maths...

  1. 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?
  2. 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
  3. Find a formula for the Pyramid Number Pn(r).
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