How many sides does a pentagon have?

Five, and unlike the triangle, square and hexagon it will not tile a flat surface - regular pentagons leave gaps however you arrange them, because 360 does not divide by the 108-degree interior angle. That single fact makes it the most interesting shape in the family.

It is bound up with the golden ratio. Draw the diagonals of a regular pentagon and they cross to form a pentagram, and every one of those lines is divided in the golden ratio - about 1.618. The Pythagoreans used the pentagram as their emblem, and the tradition holds that discovering the ratio could not be expressed as a fraction was a serious crisis for a school built on the idea that everything is number.

Because pentagons cannot tile periodically, they became central to a modern question: can a set of shapes tile a plane only without ever repeating? Roger Penrose found a two-tile solution in the 1970s using pentagonal symmetry, which was assumed to be a mathematical curiosity - until Dan Shechtman found real crystals with exactly that forbidden fivefold symmetry in 1982. He was told to go and read a textbook, was asked to leave his research group, and was awarded the Nobel Prize in Chemistry for it in 2011.