I wanted to design an origami puppet with a moving mouth. Like a flapping bird, but with a flapping mouth. :)
The interesting thing I discovered about this one, is that the deep inset eyes always look like they are staring at you. It looks something like a cross between a bat, pig or a bird:
To make, start with a standard sheet of letter paper, and fold in half (in the long direction):
Fold up the corners
Fold bottom tip to the center edge:
Fold tip to the bottom
Fold the top tip to meet the bottom tip
Fold up along edge meeting the bottom tip. This is the base of the mouth.
Lightly fold half way to the center, for the hinge of the mouth: Repeat on both sides, top and bottom. This will extend to a moving mouth.
Now fold seven guide lines about a 1/4" spaced, starting around the center line. You can just fold in half (to the center) repeatedly. These will be used for the eyes. It doesn't have to be perfectly sharp ... it can be more of a rounder, organic shape.
Collapse paper along guides (side view)
Lightly pinch the eye corner and create diagonal folds for socket.
Repeat for each corner:
If you want a self-standing sculpture that stares menacingly at you, flip it around and interweave the ends:
Or if you want ears, flip over and fold the outside edge to inside edge. Pull up the compressed folds for an ear. Also, if there's a space between the lips, you can easily pull up the bottom jaw and refold. I also reshaped the bottom jaw to make it pointier (like a bat):
And that's it. Fold along the center crease (or hold the corners of his lips and pull outwards)... and his mouth should move. Here the mouth is closed.
With a little planning, and a longer sheet of paper, you could make a much longer beak (by double-folding sharper top and bottom tips).