A triangular number is a number that can be arranged in the shape of an equilateral triangle (by convention, the first triangular number is 1):
1:
+ x3:
x x + + x x6:
x x x x x x + + + x x x10:
x x x x x x x x x x x x + + + + x x x x15:
x x x x x x x x x x x x x x x x x x x x + + + + + x x x x x21:
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + + + + + + x x x x x xSince each row is one unit longer than the previous row it can be seen that a triangular number is the sum of consecutive integers.
The formula for the nth triangular number is ½n(n+1) or (1+2+3+...+ n-2 + n-1 + n).
It is the binomial coefficient
will accurately show the number of that simplex. For example, a tetrahedron with sides of length 2 has a number of , or 4. (Note: A tetrahedron can be created by taking a number, getting the triangle of that number, and then adding to it all the triangles of the numbers before it, so a tetrahedron of 2 would have 2 triangled=3 plus 1 triangled=1 =4.)
One of the most famous triangular numbers is 666, also known as the Number of the Beast. Every perfect number is triangular.
The sum of two consecutive triangular numbers is a square number. This can be shown mathematically thus: the sum of the nth and (n-1)th triangular numbers is {½n(n+1)} + {½(n-1)n}. This simplifies to (½n2+½n) + (½n2-½n), and thus to n2. Alternatively, it can be demonstrated diagrammatically, thus:
x + + +
x x + +
x x x +
x x x x
x + + + +
x x + + +
x x x + +
x x x x +
x x x x x
In each of the above examples, a square is formed from two interlocking triangles.
See also: square number, polygonal number, triangular square number.