I've been making origami cranes and flapping birds since I was in elementary school, and even have a collection of cranes I've been making since undergrad with an eventual goal of hitting 1000. Clearly, I haven't been working non-stop on it, but I'm probably about 75% the way there. ;o)
Recently, my friend wanted to make a bunch and she noted how much longer it was taking her than she thought it would. After realizing that she was folding the square base the "traditional" way (the way most books tell you to fold it), I tried to find online directions for how I fold the square base, and discovered that it was basically impossible to find a way much different from the typical method. So, here's how I fold the square base in origami:
Start with your origami square:
Step 1: Fold your paper in half.
Optional: Step 1.5: Unfold and then fold your paper in half the other direction. This will give your paper an extra guideline and fold line to follow, if it is needed. I typically skip this direction.
Step 2: With the paper oriented as above, fold the upper right corner down to begin folding a triangle.
Step 2.5: Flip your piece over.
Step 3: Repeat Step 2, and fold the upper right corner down on the opposite side.
This next picture is just to demonstrate how the triangle folds are oriented. They are on opposite sides of each other, not the same.
Step 4: The next series of pictures are all to explain step 4. You need to open up the middle section, and pinch the outer two corners (circled in red below) together. As that happens, you will notice it folding into the square base.
And, ta-da! Square base! Replay of the steps:
(1) Fold in half, (optional, fold in half the other direction), (2) fold corner down, flip, (3) fold other corner down, (4) reach into the middle and fold outer corners together. Done! :oD Please let me know if any step is confusing, I can certainly try to explain in a different manner or take better pictures.
Enjoy!