Pascal's Pyramid - Overview of The Tetrahedron

Overview of The Tetrahedron

• There is three-way symmetry of the numbers in each layer.
• The number of terms in the nth Layer is the nth triangular number: (n + 1) × (n + 2) / 2.
• The sum of the values of the numbers in the nth Layer is 3n.
• Each number in any layer is the sum of the three adjacent numbers in the layer above.
• Each number in any layer is a simple whole number ratio of the adjacent numbers in the same layer.
• Each number in any layer is a coefficient of the Trinomial Distribution and the trinomial expansion. This non-linear arrangement makes it easier to:
  • display the trinomial expansion in a coherent way;
  • compute the coefficients of the Trinomial Distribution;
  • calculate the numbers of any Tetrahedron layer.
• The numbers along the three edges of the nth Layer are the numbers of the nth Line of Pascal's triangle. And almost all the properties listed above have parallels with Pascal's triangle and Multinomial Coefficients.