Pascal's Pyramids

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!