How can I fold paper into 3 x 4 grid? Or prove that it can't be done? [duplicate]

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I am trying to fold paper so that it looks like 3 x 4 grid of 12 rectangles of equal size.

Like this

Its easy to get 4 rectangles. Just fold twice. But how to get 3?

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3 Answers

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Let the paper be $[0,a]\times[0,b]$ By halving you find $(\frac a2,0)$, make a crease to find the line through $(\frac a2,0)$ and $(0,b)$. This intersects the diagonal at $(\frac a3,\frac b3)$.

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If your piece of paper is a square ABCD of side length 1, then from the diagram, if AP=1/2, then QC=1/3.

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(Stealing from Hagen von Eitzen's answer and Ross Millikan diagram at How can a piece of A4 paper be folded in exactly three equal parts?)

1. Fold the paper into four equal horizontal strips with two folds

2. Fold a diagonal across the top three horizontal strips

3. Fold vertical strips at the two points where the diagonal crosses the horizontal strips.

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