Friday, May 06, 2011

'Rhombicized' Classic Origami

This article is written for folders that are familiar with the basics of origami and the classic models such as the Crane and the Masu box -- and especially for folders that like to experiment.


I have recently experimented with some simple modifications of classic origami bases. First, I was developing a modular Water Lily design using bird bases, and I wanted more slender petals. So I modified the classic bird base to make what I call the "skinny bird base" (photo below).

Later, I was contemplating the classic Masu box, wondering if there was a way to make it rectangular. So I envisioned a rectangular Masu box, unfolded it in my mind, and found that I got a rhombus rather than a rectangle. I then verified the mental exercise with a physical experiment: I started with a rhombus and did all the same folds as I would use on a square to get a Masu box. The result was a rectangular tray, with a few surprises that I did not anticipate. The photo below shows two trays and a rhombus of the size and shape used for the trays.

This encouraged me to try folding other classic bases and designs using a rhombus instead of a square:

preliminary and water-bomb bases
blintz fold
fish base
bird base
classic crane
classic flapping bird
petal-topped container

I'm calling these "Rhombicized Classic Origami", and I am reporting my findings here. But first, let's return to the "skinny bird base".

The Skinny Bird Base

When making a bird base, one starts with a preliminary base and folds each 'wing' angularly in half. Two wings are shown folded in the photo at the right.

For the skinny version, each 'wing' is angularly folded in thirds. The first step is to fold a pair of wings over each other so that they divide the 90 degrees of the bottom corner in thirds, making three 30-degree angles, as shown in the photo at the left.

Then each wing is folded again at the center line, so that the 45 degrees of each wing is divided in thirds, making three 15-degree angles (stacked) for each wing, as shown in the photo below right.

As for the classic bird base, these folds are repeated behind, and all folds are converted to reverse folds, to obtain the skinny bird base (below left).

Notice that there are two corners instead of one on either side of each wing. The 'top' corner is easily seen on the outer layers, but the 'bottom' corner is below it on the inner layers. If a wing needs to be hinged upward, it cannot fold along a horizontal line joining the two outer top corners without tearing the paper. Instead the fold line must be between the two inner bottom corners. We will call this the 'hinge line'.

If a sink fold is needed at the top of the bird base (sometimes leading to a twist fold), the preparation for this sink fold should be to fold the top corner (central point) down to the center of the 'hinge line' as shown in the photo at the right. (The wing in front is hinged toward the viewer, and the wing behind is hinged to the right.)

Making a Rhombus

Like a square, a rhombus has four equal sides; but instead of having four corners with equal (90 degree) angles, two opposite corners have equal angles less than 90 degrees, and the other two corners have equal angles greater than 90 degrees. We will call these the 'sharp' corners and the 'blunt' corners.

To make a rhombus from a rectangle such as an 8.5 by 11 inch sheet of letter paper, fold the rectangle in half by bringing two opposite corners of the rectangle together, as shown in the photo at right. Then cut off the two triangular areas that are only one layer thick. Unfold the remaining two-layer-thick area, and you have a rhombus, already creased on the diagonal between its blunt corners. You can crease the other diagonal by bringing together the two blunt corners and bisecting the angles of the two sharp corners.

Folding the Rhombus

There are two different ways of adding two more creases after creasing the two diagonals:

On the left of the photo, the angles between the diagonals are bisected by bringing two half-diagonals together and creasing the paper between them. This folding method is appropriate when making a rhombicized bird base.

On the right of the above photo, each new crease is made by bringing one side of the rhombus over to align with the opposite side as shown in the photo at left. (See how the corners don't meet.) Each of these creases is parallel to two sides of the rhombus, and intersects the mid-points of the other two sides. This folding method is appropriate when making a rhombicized blintz fold, and also for dividing a rhombus into four similar rhombuses.

To make the blintz fold, fold each short edge of the top layer over to the folded edge, as shown in the photo at right, being careful not to go past the folded edge.

Turn over and repeat, then unfold the first (longest) fold.

Some Results

Here are the rhombicized bird base, blintz fold, and windmill base:

Notice that two wings of the rhombicized bird base are longer than the other two wings. Notice that the rhombicized blintz fold is rectangular. Notice that the rhombicized windmill base is shaped like a rhombus.

Here on the left are two fish bases, both folded from identical rhombuses.

The difference between these two fish bases is which diagonal of the rhombus is used as the central axis of the fish base. The 'tail' angle of one version is exactly half the 'head' angle of the other version.

Here on the right is the classic Masu box (left) and the rhombicized Masu box (right).

When making the rhombicized Masu box, the short sides must be closed last, because they are taller. The result of using existing creases to close the box is that the long sides lean outward.

Here on the left are two Cranes, both folded from identical rhombicized bird bases.

Recall that two of the wings of the rhombicized bird base are longer than the other two wings. The difference between the two cranes is which pair of wings of the bird base were used for the head and tail of the crane.

Here on the right is a rhombicized Flapping Bird.

The Petal-Topped Container can be made from any regular polygon. An easy polygon to use is an octagon, made by modifying a square. For the rhombicized version, we modify a rhombus in a similar way, getting an octagon that appears to inscribe an ellipse. The Petal-Topped Container is then folded from this 'elliptical' octagon. In the photo on the left, the rhombus for one container was made from an 8.5 x 11 inch rectangle, and the other from an 8.5 x 10 inch rectangle.

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