Thirty-six is the sum of the numbers from 1 to 8, which makes it triangular, and it is six times six, which makes it square. Numbers that are both are rare: the sequence begins 1, 36, 1225, 41616, and grows very fast. Thirty-six is the first non-trivial member, and that is a genuine and checkable distinction rather than a mystical one.
It is also the sum of the first three cubes, 1 plus 8 plus 27, which is an instance of a general result: the sum of the first n cubes always equals the square of the sum of the first n numbers. And it is highly composite, with nine divisors, more than any smaller number.
The magic square of order six has thirty-six cells, and it is the smallest magic square whose construction resists the simple methods that work for odd orders. Its magic constant is 111, and the sum of all thirty-six cells is 666. That relationship is arithmetic, the numbers 1 to 36 sum to 666 whatever arrangement they are put in, and it is the origin of the association between the six-by-six square, the planet Sun in Renaissance planetary magic, and the number of the beast. Cornelius Agrippa set out the seven planetary squares in the Three Books of Occult Philosophy of 1533, assigning the six-by-six square to the Sun, and that is where the pairing enters Western occultism.
None of this requires interpretation to be interesting, and the reader who knows the mathematics will notice immediately whether an account of thirty-six understands what it is claiming. Both properties are provable in a line.