Fold the piece of paper in two halves by taking one corner and lining it up on top of the corner directly across from it. Crease the fold to create a right triangle.
Unfold the paper. Repeat Step 1 by folding the corner at one end of the crease over to align with the corner at the opposite end. Crease it.
Unfold the new fold so the paper is divided by the two creases that intersect at the center of the square, dividing it into four equal triangles.
Fold one corner into the square so the point touches the spot where the two creases meet. Repeat this step with the other three corners to create a smaller square divided into eight small triangles.
Lay the square down so the four flaps face up. Choose two of the four folded in flaps next to one another. Mark them as the "A" and mark the other two as "B."
Mark a point 1/2 inch from the outside corner on the line between the A's. Make a similar mark 1/2 inch from the outside corner on the line between the B's.
Draw a line across each flap from the point on one of its side to the point on its other side.
Fold the flaps back from the center of the square and crease them on the drawn lines. The flap points should hang over the square, creating a shape with eight points.
Fold the middle point of three upward points in toward the center and crease before the other two points bend. Similarly, fold the center point of the lower three points up and crease them.
Fold the top of the layer back onto itself so the three points extend 1 inch over the flat top. Flip the entire structure on its back.
Fold up the three downward points up so they align with the upward points.
Fold the top three points, which you folded up to align with the other lower three in the previous step, back onto itself so the two flat spaces between the three points line up with the flat base of the shape.
Flip the structure on its back. Fold the the three upward points down to align with the three downward points folded in the previous step.
Unfold the crease that makes the flat top surface of the shape. Each set of three points should extend from a trapezoid that faces up with the upward points and down with the downward points.
Find the center crease. Follow it from the tip of the large leftward point. There is 1 1/4 inch of paper before the bases of the trapezoids begin on the upper and lower layer. Fold the point of this extra inch back onto itself to make a triangle. Do this before the trapezoid begins on the left and right for a total of four folds.
Turn the structure so it is taller than it is wide. Fold the bottom third of the structure up and crease. Fold the flap back onto itself about 1/4 inch from the new base and crease.
Fold the bottom section of the structure up so the crease is directly behind the two smallest outward-pointing triangles. Fold the flap back onto itself. Crease 1/4 inch up from the new base.
Fold the bottom section up so the crease is directly in the middle of the highest outward-facing points. Fold the flap back onto itself. Crease it 1/4 inch up from the new flat base.
Fold the structure in half along the crease along the entire vertical length. The structure can stand on the table on six little points and features four segments. The largest of the end segment serves as the caterpillar's head, the middle segments are its body and the smaller end segment is the tail.