Edited by Jane Rosemarin

Before folding the blintz, a single corner has to be folded ... and for that we need landmarks. In this article, we make a systematic review of the landmarks of the corner fold.

## The Corner Needs Landmarks

Folding a single corner to the center of a square seems simple. However, there are a large number of ways even this simple fold can be made.

Using a machine, one would press a score line and then fold the corner over that crease, maybe bending the material over a hard edge.

In origami, we need to construct some landmarks to which the corner can be aligned. The figure below shows the possible reference points for the fold. These may be the points where the crease touches the edge, the lines along which the edges of the folded corner lie or the center point of the square.

Luckily, all these references may be obtained via simple side-to-side or corner-to-corner folds,1 either by making entire creases or by pinching marks at the reference points. In addition to not using any marks and guesstimating the position, the following marks are possible:

## Systematic Matrix of Corner Landmarks

There is one case with no marks and 10 cases with one mark. Then there are 100 cases with two marks, as depicted in the figure below. Altogether, there are 111 cases with at most two marks. In special cases, one might imagine the use of more than two marks, but for our purposes, we will stop the analysis with two.

However, not all two-mark cases are meaningful.

• The diagonal of the table consists of cases where the first and the second marks are the same, and it corresponds to the 10 single-mark cases. Thus, AA is the same as A, BB as B, and so forth.
• The upper and lower triangles of the table are the same, in mirror positions. It might give a different or even better flow to do the marks in opposite order, but the resulting marks will be the same. Hence, we will consider only the upper half in addition to the diagonal.
• In some cases, one mark is a pinch of the full crease of the other mark, which is not meaningful. These cases are colored red.
• Some cases are a mirror of others. For example, case BG mirrors AH. Again, there might be reasons, such as lefthandedness or mountain/valley considerations of the final model, why one might prefer one or the other flow, but essentially the result is the same. Such mirror cases are yellow.
• The two marks may be at a weak angle with each other. This leads to greater uncertainty in determining the correct point. Hence we disregard those cases where the marks do not meet at a 90-degree angle. Painted gray.

The remaining 21 cases may be divided into underspecified (white, four cases), specified (light blue, 12 cases) and overspecified blue, 5 cases). The underspecified cases will seldom be used, but the other cases may have their use according to which crease one wishes to include or avoid in the final model. In particular, the medians (case AB) or the diagonals (case CD) are the most common, and almost always the ones diagrammed. However, experts will often use one of the other cases.

A special case is DD which is underspecified, but nevertheless used by some authors when folding the corner (not the blintz), for simple models where precision is not paramount, and instead the simplicity of the folding takes priority.

Another special case is ABCD, not in the table, in which both the diagonals and the medians are creased. These four folds appear in many models.

Let us review selected gray cases more carefully.

• Why EG and not EH? When folding the corner anchored to one edge mark, it will swing over the paper in a circular curve. In case EG, the corner will bump into the median mark, whereas in case EH, the corner will glide along the mark, the optimal position of which is much more ill-defined. Essentially, EG is the same as A, just with cleaner paper.
• Why AC and AD, but not CG? Yes, in AC the diagonal crosses the median at the same weak angle, but it is the full median that includes the edge mark and the line along the corner edge. Remember, the median alone is enough to define the position of the corner fold. Basically AC and AD are overspecified and treating them like case A, we just ignore the diagonal and its weak angle.

Finally, note that if one corner has been positioned, the other corners may use the first as a sufficient landmark. For instance, the minimal EG is enough to position all corners in a blintz, even if this landmark is insufficient to fold the two lower corners first.

## Endnote

1. Kazuo Haga, Origamics — Mathematical Explorations Through Paper Folding (Singapore: World Scientific, 2008). ISBN 981-283-489-3.