Since it's now lantern season, I thought I would re-post this tutorial I had put up some time ago...
During last lantern season I was looking everywhere for instructions to make these beautiful paper lanterns and couldn't find them anywhere. I ended up purchasing a lantern that was already made and since I've been struggling with some difficult folds and designs of other paper crafts, I decided to unfold the seemingly simple lantern I bought and figure the thing out for myself. Like all traditional origami, no glue or cutting (once the basic outside shape of the paper is made) are required. I feared that once I unfolded my lovely store-bought version I wouldn't be able to get it back together but thankfully it wasn't nearly as difficult as I assumed.
Since in my own quest for instructions I crossed paths with many other people looking for the same thing, I thought I'd do my best to share what I learned. There may be better ways to do this and certainly my origami terminology will lack a little something but this is how I worked it out:
(Feb 2012 edit note: A video tutorial of this project can now be found here.)
Begin with a 12" square of paper to end up with a lantern that is about 7" across when complete. (Those beautiful wet-on-wet water colored paintings our children are turning out each season make for the lovliest lanterns. Also, 12x12 is the standard size of the amazing array of scrapbook paper found at local craft stores.)
With what will ultimately be the outside of your lantern (the painted side) face up, fold the paper in half horizontally and vertically and then again on each diagonal effectively dividing the page into eight pie wedges.
For crisper creases and to help fold thicker card stocks, try using a bone folder. The more exact the folds, the nicer the finished project will be.
(The dashed and dotted lines shown here are guides for the next step.)
To further divide the paper into 16 wedges, turn the paper over (painted side down so this second set of 8 folds are "valleys" to the first 8 fold's "hills") and make another fold between each of the eight sections by lining up the existing folds and the center point.
To help illustrate I marked the paper with dotted and dashed lines. The dotted lines (marked on both the front and back of the page) fold to align directly on top of the dashed line. (For more exact alignment, check the line at 90° and make sure it also aligns exactly with the corner fold underneath it.)
At the same time these folds are being made, lightly mark with a pencil the triangle piece that is sticking out at each of the four corners.
The only scissor work of this project takes place here.
Cut off the four marked corners of the paper.
With all of these folds complete and the corners removed, the octagonal "pie" will now have 16 wedges.
With the painted side down, using the existing lines between the eight points as a guide (the dotted lines), fold down the paper edge (on the dash-dot line that connects the end points of the dotted lines).
For extra visual assistance, note that the solid lines lay one atop the other.
Repeat this fold eight times.
It might be easiest to understand this by first folding in the top, bottom and two sides, creating a square, then unfolding the paper, rotating it 45 degrees and repeating the process of creating a second square.
In the end there will be an eight pointed star formed by the folds (or two squares - one overlaying the other).
Fold in the edges of one of those squares.
Again, if the paper has a painted side, it is face down for this step.
Flip over the paper (so the outside is up) and fold in the already creased corners.
This is where it gets tricky and the instructions are best understood with paper in hand just trying to work it out I think...
Between each folded-in corner, reach around to the other side of the paper and gently grab the center point (where the arrow is pointing) from the back side.
It will, by nature of the existing folds, turn into a triangle as it is pulled.
Flip that triangle over the edge until it too points to the center of the side of the page that is currently face up. (For illustration purposes I drew the arrow on both sides of the paper.)
The center of the paper will start to crinkle and this is totally fine.
Continue doing this all the way around. By carefully cupping the paper and helping pop the folds into their proper place when necessary, the star pattern will begin to take shape.
I have to say that when it all just fell into place I was quite surprised.
Press the project down onto a flat surface to flatten the bottom into a level cup shape.
The lantern I purchased was coated with oil which increases the translucency (and sort of magical quality) of the paper. To do this simply rub with any vegetable or olive oil prior to folding and let it dry overnight. (I would suggest perhaps attempting your first version without. )
Of course, these beautiful pieces are made of paper and so are quite flammable. I would suggest placing a glass votive around an open flame prior to putting it in the lantern and, as with all burning objects, caution should be used and candles should never be left unattended.
My mum likes to use those little battery powered tea lights and they are, of course, quite safe and a great option for very small children.