As mentioned
in a previous blog, Pascal’s Triangle is a famous triangular array of numbers that
has applications for the expansion of binomials, powers of two, sequences of
numbers, statistics, and properties of simplexes. Each line of the triangle starts and ends
with the number 1, and the numbers in between can be obtained by finding the
sum of the two numbers above it. The
triangle can then be continued indefinitely.
A similar
method can be extended in three dimensions to make an array of numbers in the
shape of a pyramid with either a triangular base or a square base. Both of these pyramids also have amazing
properties.
Triangular-Based Pascal’s Pyramid
The
triangular-based Pascal’s Pyramid can be formed by layers of expanding
equilateral triangles. This means the
first layer has one number (a triangle with one number per side), the second
layer has three numbers (a triangle with two numbers per side), the third layer
has six numbers (a triangle with three numbers per side), and so on.
Expanding a Pyramid by Adding Equilateral Triangle Layers
The numbers
in this pyramid can be obtained by finding the sum of the three numbers in the
layer above it. For example, the center
number 6 in the fourth layer was obtained by adding the three 2’s in the third
layer. The pyramid can also be continued
indefinitely.
Steps for Finding a New Triangular Layer
Layers in a Triangular-Based Pascal’s Pyramid
The
triangular-based Pascal’s Pyramid has several interesting properties. The
numbers along the edges are the same numbers as Pascal’s Triangle, and
therefore all of the interesting properties of Pascal’s Triangle also apply to
Pascal’s Pyramid. In addition to this, the coefficients for the expanded form of the
trinomial (x + y + z)n match up with the numbers in the triangular
layers of Pascal’s Pyramid. For
example, (x + y + z)2 = x2 + y2 + z2
+ 2xy + 2xz + 2yz (1, 1, 1, 2, 2, 2), and (x + y + z)3 = x3
+ y3 + z3 + 3x2y + 3x2z + 3xy2
+ 3xz2 + 3y2z + 3yz2 + 6xyz (1, 1, 1, 3, 3, 3,
3, 3, 3, 6). (Recall that the
coefficients for the expanded form of the binomial (x + y)n match up
with the numbers in Pascal’s Triangle.) Finally,
the numbers of each triangular layer of
Pascal’s Pyramid add up to a power of 3.
For example, the numbers in the third layer (1, 2, 1, 2, 2, 1) add up to
9 which is 32, and the numbers in the fourth layer (1, 3, 3, 1, 3,
6, 3, 3, 3, 1) add up to 27 which is 33. (Recall that the numbers of
each row of Pascal’s Triangle adds up to a power of 2.)
Square-Based Pascal’s Pyramid
The
square-based Pascal’s Pyramid can be formed by layers of expanding squares
instead of triangles. This means the
first layer has one number (a square with one number per side), the second
layer has four numbers (a square with two numbers per side), the third layer
has nine numbers (a square with three numbers per side), and so on.
Expanding a Pyramid by Adding Square Layers
The numbers
in this pyramid can be obtained by finding the sum of the four numbers in the
layer above it. For example, one of the
9’s in the fourth layer was obtained by adding the 1, 2, 2, and 4 in the third
layer. This pyramid can also be
continued indefinitely.
Steps for Finding a New Square Layer
Layers in a Square-Based Pascal’s Pyramid
The square-based
Pascal’s Pyramid also has several interesting properties. Like the triangular-based Pascal’s Triangle, the numbers along the edges of the
square-based Pascal’s Pyramid are the same numbers as Pascal’s Triangle,
and therefore all of the interesting properties of Pascal’s Triangle also apply. In addition, just as the numbers of each row
of Pascal’s Triangle adds up to a power of 2, and just as the numbers of each
triangular layer of the triangular-based Pascal’s Pyramid add up to a power of
3, the numbers of each square layer of
Pascal’s Pyramid add up to a power of 4.
For example, the numbers in the third layer (1, 2, 1, 2, 4, 2, 1, 2, 1)
add up to 16 which is 42, and the numbers in the fourth layer (1, 3,
3, 1, 3, 9, 9, 3, 3, 9, 9, 3, 1, 3, 3, 1) add up to 64 which is 43. Finally,
each square layer of Pascal’s Pyramid
represents a multiplication table, in which any number in the square is the
product of the row header and column header.
For example, any of the 9’s in the fourth layer is can be found by
multiplying the row header of 3 by the column header of 3. Because the numbers in each square layer are
symmetrical, all numbers in the diagonals
are square numbers.
Variations of Pascal’s Pyramid
There are a
few other variations of Pascal’s Pyramid.
In the triangular-based pyramid, the next layer can be obtained by
adding a triangle of numbers that is upside-down from the usual triangle of
numbers. The result is a new hexagonal
layer that still holds the property that the numbers along the edges are the
same as the numbers in Pascal’s Triangle, and also the property that the sum of
the numbers in each layer is a power of 3.
Usual Way of Obtaining a New Layer
Alternate Way of Obtaining a New Layer
A
hexagonal-based pyramid can also be made by numbers in a honeycomb
pattern. Each new number can be obtained
by finding the sum of the three numbers in the layer above it. Once again, this pyramid still holds the
property that the numbers along the edges are the same as the numbers in
Pascal’s Triangle, but also has the additional property that the sum of the
numbers in each layer is a power of 6.
Steps for Finding a New Hexagonal Layer
Layers in a Hexagonal-Based Pascal’s Pyramid
Conclusion
Extending
Pascal’s Triangle into three dimensions reveals even more unique
properties. The edges of each pyramid
have the same numbers as Pascal’s Triangle, and the sum of the numbers in each
layer can be represented as a power between two integers. Each layer in a triangular-based Pascal’s
Pyramid has application in expanding trinomials, and each layer in a
square-based Pascal’s Pyramid is a multiplication table. All of these properties are truly amazing!
