Square pyramidal number

Geometric representation of the square pyramidal number 1 + 4 + 9 + 16 = 30.

In mathematics, a pyramid number, or square pyramidal number, represents the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes.

As well as counting spheres in a pyramid, these numbers can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an octahedral number.

Formula

Six copies of a square pyramid with n steps can fit in a cuboid of size n(n + 1)(2n + 1)

If spheres are packed into square pyramids whose number of layers is 1, 2, 3, etc., then the square pyramidal numbers giving the numbers of spheres in each pyramid are:

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, ... .

These numbers can be calculated algebraically, as follows. If a pyramid of spheres is decomposed into its square layers with a square number of spheres in each, then the total number $P_{n}$ of spheres can be counted as the sum of the number of spheres in each square, $$P_{n}=\sum _{k=1}^{n}k^{2}=1+4+9+\cdots +n^{2},$$ and this summation can be solved to give a cubic polynomial, which can be written in several equivalent ways: $$P_{n}={\frac {n(n+1)(2n+1)}{6}}={\frac {2n^{3}+3n^{2}+n}{6}}={\frac {n^{3}}{3}}+{\frac {n^{2}}{2}}+{\frac {n}{6}}.$$ This equation for a sum of squares is a special case of Faulhaber's formula for sums of powers, and may be proved by mathematical induction. Equivalent formulas for sums of consecutive squares are given by Archimedes, who used this sum as a lemma as part of a study of the volume of a cone, and by Fibonacci, as part of a more general solution to the problem of finding formulas for sums of progressions of squares.

More generally, figurate numbers count the numbers of geometric points arranged in regular patterns within certain shapes. The centers of the spheres in a pyramid of spheres form one of these patterns, but for many other types of figurate numbers it does not make sense to think of the points as being centers of spheres. In modern mathematics, related problems of counting points in integer polyhedra are formalized by the Ehrhart polynomials. These differ from figurate numbers in that, for Ehrhart polynomials, the points are always arranged in an integer lattice rather than having an arrangement that is more carefully fitted to the shape in question, and the shape they fit into is a polyhedron with lattice points as its vertices. Specifically, the Ehrhart polynomial L(P,t) of an integer polyhedron P is a polynomial that counts the number of integer points in a copy of P that is expanded by multiplying all its coordinates by the number t. The usual symmetric form of a square pyramid, with a unit square as its base, is not an integer polyhedron, because the topmost point of the pyramid, its apex, is not an integer point. Instead, the Ehrhart polynomial can be applied to an asymmetric square pyramid P with a unit square base and an apex that can be any integer point one unit above the base plane. For this choice of P, the Ehrhart polynomial of a pyramid is ⁠(t + 1)(t + 2)(2t + 3)/6⁠ = P_{t + 1}.

Geometric enumeration

All 30 squares in a 4×4 grid

As well as counting spheres in a pyramid, these numbers can be used to solve several other counting problems. For example, a common mathematical puzzle involves finding the number of squares in a large n by n square grid. This number can be derived as follows:

• The number of 1 × 1 squares found in the grid is n².
• The number of 2 × 2 squares found in the grid is (n − 1)². These can be counted by counting all of the possible upper-left corners of 2 × 2 squares.
• The number of k × k squares (1 ≤ k ≤ n) found in the grid is (n − k + 1)². These can be counted by counting all of the possible upper-left corners of k × k squares.

It follows that the number of squares in an n × n square grid is: $$n^{2}+(n-1)^{2}+(n-2)^{2}+(n-3)^{2}+\ldots ={\frac {n(n+1)(2n+1)}{6}}.$$ That is, the solution to the puzzle is given by the nth square pyramidal number. The number of rectangles in a square grid is given by the squared triangular numbers.

The square pyramidal number $P_{n}$ also counts the number of acute triangles formed from the vertices of a $(2n+1)$-sided regular polygon. For instance, an equilateral triangle contains only one acute triangle (itself), a regular pentagon has five acute golden triangles within it, a regular heptagon has 14 acute triangles of two shapes, etc.

Relations to other figurate numbers

A square pyramid of cannonballs at Rye Castle in England

4900 balls arranged as a square pyramid of side 24, and a square of side 70

The cannonball problem asks for the sizes of pyramids of cannonballs that can also be spread out to form a square array, or equivalently, which numbers are both square and square pyramidal. Besides 1, there is only one other number that has this property: 4900, which is both the 70th square number and the 24th square pyramidal number. After previous incomplete proofs by Claude-Séraphin Moret-Blanc and Édouard Lucas, the first complete proof of this fact was given by G. N. Watson in 1918.

The square pyramidal numbers can be expressed as sums of binomial coefficients: $$P_{n}={\binom {n+2}{3}}+{\binom {n+1}{3}}.$$ The binomial coefficients occurring in this representation are tetrahedral numbers, and this formula expresses a square pyramidal number as the sum of two tetrahedral numbers in the same way as square numbers are the sums of two consecutive triangular numbers. If a tetrahedron is reflected across one of its faces, the two copies form a triangular bipyramid. The square pyramidal numbers are also the figurate numbers of the triangular bipyramids, and this formula can be interpreted as an equality between the square pyramidal numbers and the triangular bipyramidal numbers. Analogously, reflecting a square pyramid across its base produces an octahedron, from which it follows that each octahedral number is the sum of two consecutive square pyramidal numbers.

Square pyramidal numbers are also related to tetrahedral numbers in a different way: the points from four copies of the same square pyramid can be rearranged to form a single tetrahedron of slightly more than twice the edge length. That is, $$4P_{n}={\binom {2n+2}{3}}.$$

Other properties

The alternating series of unit fractions with the square pyramidal numbers as denominators is closely related to the Leibniz formula for π, although it converges more quickly. It is: $${\begin{aligned}\sum _{i=1}^{\infty }&(-1)^{i-1}{\frac {1}{P_{i}}}\\&=1-{\frac {1}{5}}+{\frac {1}{14}}-{\frac {1}{30}}+{\frac {1}{55}}-{\frac {1}{91}}+{\frac {1}{140}}-{\frac {1}{204}}+\cdots \\&=6(\pi -3)\\&\approx 0.849556.\\\end{aligned}}$$

External links

• Weisstein, Eric W., "Square Pyramidal Number", MathWorld