http://chestofbooks.com/crafts/children/Paper-Folding/V-The-Hexagon.html#.Um999RAzJAY

Found this image while looking at hexagonal gazebos...

Could come in handy, although I'm not fully sure how it works...will have to give it a go first!

They do seem confident though...

"AHGBFE is a regular hexagon.

It is unnecessary to give the proof."

"An example of ornamental folding into equilateral triangles and hexagons."