The Number Spiral

The number spiral is all natural numbers arranged in a spiral.


If we remove all the numbers in this spiral that are not prime and leave only the prime numbers, we see that a pattern forms.

The prime numbers seem to line up along diagonal lines. Expanded, diagonal lines are clearly visible.


This type of spiral is known as the Ulam spiral, as it was devised by mathematician Stanislaw Ulam.

After many tests, the prime numbers are found to line up along diagonals with the function:

The number spiral has another interesting quality.

The numbers in the black boxes are square numbers. The even square numbers move in a diagonal to the top-left. The odd square numbers move in a diagonal to the bottom-right.

Trigonometric Proofs

Fundamental Law of Trigonometry

Let us assume that α > β > 0.

Consider a unit circle with centre O.

Let the terminal side of angles α and β cut the unit circle at A and B respectively. 

Evidently AOB = α - β. 

Take a point C on the unit circle so that XOC = AOB = α - β.

Join A with B and C with D.

The coordinates of A are ( cos α , sin α ). 
The coordinates of B are ( cos β , sin β ). 
The coordinates of C are [ cos(α - β) , sin(α - β) ].
The coordinates of D are ( 1 , 0 ) Now AOB and COD are congruent. 

How would you work out this question using the rule above?

You could break this down into cos(60 - 45), and put these numbers into the formula.