andersonracing.org.uk - 3 is a Magic Number. 

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3 is:
a rough approximation of π (3.1415...) and a very rough approximation of e (2.71828..) when doing quick estimates.

• the number of non-collinear points needed to determine a plane and a circle.
• the first odd prime number and the second smallest prime.
• the first Fermat prime (2²n + 1).
• the first Mersenne prime (2ⁿ − 1).
• the second Sophie Germain prime.
• the second Mersenne prime exponent.
• the second factorial prime (2! + 1).
• the second Lucas prime.
• the second triangular number. It is the only prime triangular number.
• the fourth Fibonacci number.
• the smallest number of sides that a simple (non-self-intersecting) polygon can have.

Three is the only prime which is one less than a perfect square. Any other number which is n² − 1 for some integer n is not prime, since it is (n − 1)(n + 1). This is true for 3 as well (with n = 2), but in this case the smaller factor is 1. If n is greater than 2, both n − 1 and n + 1 are greater than 1 so their product is not prime.

A natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three (3 times 7) and the sum of its digits is 2 + 1 = 3. Because of this, the reverse of any number that is divisible by three (or indeed, any permutation of its digits) is also divisible by three. For instance, 1368 and its reverse 8631 are both divisible by three (and so are 1386, 3168, 3186, 3618, etc.). See also Divisibility rule. This works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one (bases 4, 7, 10, etc.).

Three of the five Platonic solids have triangular faces – the tetrahedron, the octahedron, and the icosahedron. Also, three of the five Platonic solids have vertices where three faces meet – the tetrahedron, the hexahedron (cube), and the dodecahedron. Furthermore, only three different types of polygons comprise the faces of the five Platonic solids – the triangle, the square, and the pentagon.

There are only three distinct 4×4 panmagic squares.

According to Pythagoras and the Pythagorean school, the number 3, which they called triad, is the noblest of all digits, as it is the only number to equal the sum of all the terms below it, and the only number whose sum with those below equals the product of them and itself.[2]

The trisection of the angle was one of the three famous problems of antiquity.

Gauss proved that every integer is the sum of at most 3 triangular numbers.