A tetrahedral number, or triangular pyramidal number, is a
figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean
* polyg ...
that represents a
pyramid
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
with a triangular base and three sides, called a
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
. The th tetrahedral number, , is the sum of the first
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
s, that is,
:
The tetrahedral numbers are:
:
1,
4,
10,
20,
35,
56,
84,
120 120 may refer to:
*120 (number), the number
* AD 120, a year in the 2nd century AD
*120 BC, a year in the 2nd century BC
*120 film, a film format for still photography
* ''120'' (film), a 2008 film
* 120 (MBTA bus)
* 120 (New Jersey bus)
* 120 (Ken ...
,
165,
220, ...
Formula
The formula for the th tetrahedral number is represented by the 3rd
rising factorial
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
:\begin
(x)_n = x^\underline &= \overbrace^ \\
&= \prod_^n(x-k+1) = \prod_^(x-k) \,.
\e ...
of divided by the
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \ ...
of 3:
:
The tetrahedral numbers can also be represented as
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s:
:
Tetrahedral numbers can therefore be found in the fourth position either from left or right in
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
.
Proofs of formula
This proof uses the fact that the th triangular number is given by
:
It proceeds by
induction.
;Base case
:
;Inductive step
:
The formula can also be proved by
Gosper's algorithm.
Generalization
The pattern found for triangular numbers
and for tetrahedral numbers
can be generalized. This leads to the formula:
Geometric interpretation
Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number () can be modelled with 35
billiard ball
A billiard ball is a small, hard ball used in cue sports, such as carom billiards, pool, and snooker. The number, type, diameter, color, and pattern of the balls differ depending upon the specific game being played. Various particular ball ...
s and the standard triangular billiards ball frame that holds 15 balls in place. Then 10 more balls are stacked on top of those, then another 6, then another three and one ball at the top completes the tetrahedron.
When order- tetrahedra built from spheres are used as a unit, it can be shown that a space tiling with such units can achieve a densest
sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three- dimensional Euclidean space. However, sphere pack ...
as long as .
Tetrahedral roots and tests for tetrahedral numbers
By analogy with the
cube root
In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. F ...
of , one can define the (real) tetrahedral root of as the number such that :
which follows from
Cardano's formula. Equivalently, if the real tetrahedral root of is an integer, is the th tetrahedral number.
Properties
*:, the
square pyramidal number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts 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 bro ...
s.
*:, sum of odd squares.
*:, sum of even squares.
*
A. J. Meyl proved in 1878 that only three tetrahedral numbers are also
perfect squares, namely:
*:
*:
*:.
*
Sir Frederick Pollock conjectured that every number is the sum of at most 5 tetrahedral numbers: see
Pollock tetrahedral numbers conjecture.
* The only tetrahedral number that is also a
square pyramidal number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts 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 bro ...
is 1 (Beukers, 1988), and the only tetrahedral number that is also a
perfect cube
In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together.
The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or .
...
is 1.
* The
infinite sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
of tetrahedral numbers' reciprocals is , which can be derived using
telescoping series:
*:
* The
parity
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
of tetrahedral numbers follows the repeating pattern odd-even-even-even.
*An observation of tetrahedral numbers:
*:
*Numbers that are both triangular and tetrahedral must satisfy the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
equation:
*:
: The only numbers that are both tetrahedral and triangular numbers are :
::
::
::
::
::
* is the sum of all products ''p'' × ''q'' where (''p'', ''q'') are ordered pairs and ''p'' + ''q'' = ''n'' + 1
* is the number of (''n'' + 2)-bit numbers that contain two runs of 1's in their binary expansion.
Popular culture
is the total number of gifts "my true love sent to me" during the course of all 12 verses of the carol, "
The Twelve Days of Christmas".
The cumulative total number of gifts after each verse is also for verse ''n''.
The number of possible
KeyForge three-house combinations is also a tetrahedral number, where is the number of houses.
See also
*
Centered triangular number
A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers.
The followin ...
References
External links
*
Geometric Proof of the Tetrahedral Number Formulaby Jim Delany,
The Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
{{DEFAULTSORT:Tetrahedral Number
Figurate numbers
Simplex numbers