Polyhedral number representing a tetrahedron From Wikipedia, the free encyclopedia
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number, Ten, is the sum of the first ntriangular numbers, that is,
A pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers.
Tetrahedral and triangular numbers are related through the recursive formulas
The equation becomes
Substituting for in equation
Thus, the th tetrahedral number satisfies the following recursive equation
The pattern found for triangular numbers and for tetrahedral numbers can be generalized. This leads to the formula:[1]
Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number (Te5 = 35) can be modelled with 35 billiard balls 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-n tetrahedra built from Ten spheres are used as a unit, it can be shown that a space tiling with such units can achieve a densest sphere packing as long as n ≤ 4.[2][dubious–discuss]
By analogy with the cube root of x, one can define the (real) tetrahedral root of x as the number n such that Ten = x:
which follows from Cardano's formula. Equivalently, if the real tetrahedral root n of x is an integer, x is the nth tetrahedral number.
The only tetrahedral number that is also a square pyramidal number is 1 (Beukers, 1988), and the only tetrahedral number that is also a perfect cube is 1.
The parity of tetrahedral numbers follows the repeating pattern odd-even-even-even.
An observation of tetrahedral numbers:
Te5 = Te4 + Te3 + Te2 + Te1
Numbers that are both triangular and tetrahedral must satisfy the binomial coefficient equation:
The third tetrahedral number equals the fourth triangular number as the nth k-simplex number equals the kth n-simplex number due to the symmetry of Pascal's triangle, and its diagonals being simplex numbers; similarly, the fifth tetrahedral number (35) equals the fourth pentatope number, and so forth
The only numbers that are both tetrahedral and triangular numbers are (sequence A027568 in the OEIS):
Te1 = T1 = 1
Te3 = T4 = 10
Te8 = T15 = 120
Te20 = T55 = 1540
Te34 = T119 = 7140
Ten is the sum of all products p × q where (p, q) are ordered pairs and p + q = n + 1
Ten is the number of (n + 2)-bit numbers that contain two runs of 1's in their binary expansion.
The largest tetrahedral number of the form for some integers and is 8436.
Number of gifts of each type and number received each day and their relationship to figurate numbers
Te12 = 364 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".[3] The cumulative total number of gifts after each verse is also Ten for verse n.
The number of possible KeyForge three-house combinations is also a tetrahedral number, Ten−2 where n is the number of houses.