clifford_a-_pickover_surfing_through_hyperspacebookfi-org

174 appendix Q
Hypertetrahedral Numbers
Tetrahedral numbers form the sequence: 1, 4, 10, 20, 35, 56, 84, 120 . . . with a
generating formula (\IG)n(n + !)(» + 2). This can be best visualized using cannonballs
in a pyramid-shaped pile with a triangular base. Starting from the top of
the pile, the number of balls in each layer is 1, 3, 6, 10, 15, . . . , which forms a
sequence of triangular numbers because each level is shaped like a triangle. Tetrahedral
numbers can be thought of as sums of the triangular numbers. We can
extend this idea into higher dimensions and into hyperspace. In 4-D space, the
piles of tetrahedral numbers can themselves be piled up into 4-D, hypertetrahedral
numbers: 1, 5, 15, 35, 70. . . . We can form these numbers from the general formula:
(l/24)n(n + 1)(« + 2)(n + 3). Can you impress your friends by generating
hyper-hypertetrahedral numbers?