# ReferenceFinder

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When one is designing an origami model using mathematical tools such as TreeMaker, the crease pattern is usually defined mathematically, rather than by folding. Important points in the pattern — junctions of several creases, which are called reference points of the crease pattern — are commonly specified entirely numerically, rather than as a result of a series of folds. To fold such patterns, one can record the coordinates of the important points and then compute, measure and mark their location on the paper to be folded. But making marks on the paper is awkward and inelegant; it is inconvenient to require a ruler and calculator to fold origami! There is an aesthetic benefit to being able to start with an unmarked square and proceed to the completed model entirely by folding. To do this, we need a way of finding reference points by folding alone when all we have is an algebraic or numerical description of the point. This challenge of finding folding sequences for the location of a reference point given its

mathematical definition is a problem of both mathematical and practical interest, and it has seen considerable progress in recent years. Finding reference points is closely related to the concept of geometric construction — a field familiar to anyone who has manipulated compass and

straightedge in high-school geometry. The field of origami constructions is considerably richer than compass-and-straightedge constructions: a partial listing of some of the accomplishments and discoveries in this field includes:

— Construction of any rational fractional distance (expressed as a fraction of the side of the unit square) with algorithms by Husimi, Fujimoto, Noma, Haga, Mosely, and me;

— Construction of any binary rational fraction (a fraction whose denominator is a perfect power of 2);

— How to trisect an arbitrary angle, with constructions by Justin and Abe;

— Construction of cube roots, including a marvelous construction of the cube root of 2 by Peter Messer;

— Solutions of general quadratic and cubic equations and construction of perfect N-gons for broad classes of N by Robert Geretschläger (including all N up to 20 except N=11).

Using these and related techniques, it is mathematically possible to construct by folding alone any point whose coordinates in the unit square are given by any combination of integers, simple arithmetic (+, –, *, /), and square and cube roots.

However, these constructions are not always useful in a practical sense. The mathematical origamist looks at the problem of origami geometric construction as an intellectual exercise where mathematical exactness is an absolute requirement and the construction of extraneous or leftover creases is of little or no consideration. To the practical origami designer, however, many of the mathematical constructions described above leave the paper littered with extraneous creases even for the construction of a single point. Furthermore, these constructions can be very difficult to do precisely, involving the transfer of distances from one location on the paper to another and/or locating points by finding the intersection of two creases at very shallow angles.

Paradoxically, a mathematically exact folding sequence can lead to a very imprecise result when one actually applies hands to paper.

Furthermore, mathematical exactitude is not required for most practical purposes, and is, in fact, utterly unachievable, since paper is an imperfect medium and there is uncertainty inherent in the process of folding itself. There is no point in using a mathematical folding sequence that has

theoretically infinite accuracy if either the needed or the attainable accuracy is only some finite value.

And in fact, I have found empirically that an accuracy of one part in 1000 is sufficient to fold any model and it is difficult, if not impossible, to do better than this when folding. For a standard 25 cm square, 1 part in 1000 is a quarter-millimeter — about three hair diameters! In many cases, 1

part in 100 accuracy will suffice.

It is also very important to the origami designer that unnecessary creases be minimized or eliminated and those that remain should be small and unobtrusive. If the folder must make 100 sequential creases to locate a single reference point, or the paper must be covered with creases

that serve as intermediate reference points — the folding sequence, the finished model, or both, will be aesthetically unpleasing.

And so, a very practical problem faced by the origami designer can be stated quite succintly, albeit imprecisely: how can you locate an arbitrary point on a unit square (1) by folding alone, (2) to reasonable accuracy, (3) with as few folds as possible, (4) leaving as few creases on the paper as possible?

This statement is, to a mathematician, an ill-defined problem. How accurate is “reasonable accuracy?” How few is “as few as possible?” There are some tradeoffs. In general, higher accuracy requires more creases. We can, however, place some quantitative limits on these tradeoffs. I have previously looked at the problem of finding a reference point along the edge of the paper. The binary folding method† lets you construct an approximation to any point along an edge with a well-defined accuracy; a sequence of N folds can be found with a maximum error of 2–(N+1). Thus, for example, an accuracy of 1 part in 256 — a maximum error of .004 — can be obtained with no more than 7 creases for any location on the edge of the square. (Of course, some distances can be found with fewer creases, but the worst-case scenario is 7 creases.)

A nice feature of the binary folding sequence is that all intermediate marks take the form of small pinches which can be made arbitrarily short. There are no extended creases that run across the square that potentially mar the surface of the finished model, and so it is reasonable when searching for alternate algorithms to add the restriction that the only creases allowed are short pinches.

The binary folding sequence utilizes pinches to form intermediate marks along a single edge of the square — the same edge that the desired reference point is located upon. More recently, I’ve looked at the problem of locating a point along an edge of the square where the intermediate

marks can be made along any of the four edges. I did this by simply looking at all possible marks that can be made recursively by bringing one mark on any edge to another mark on any edge, starting with the four corners of the square as the original four “marks.” Remarkably, it turns out

that it is possible to locate any point to an accuracy of better than 1 part in 100 with no more than 4 creases — whereas the binary sequence would require up to 6 creases for the same level of accuracy.

The problem changes when you start to consider that there are usually multiple reference points that need to be found, as the amount of precreasing required can grow significantly. Even with 4 creases per reference point, if there are 10 reference points to be found, there could be as many as 40 folds required to find those 10 points. That’s a lot of precreasing! The number of folds can be reduced significantly if several of the reference points can share earlier creases, or better yet, if all reference points can be derived from a small set of starting marks.

One set of starting marks that is fairly easy to find is the set of marks spaced 1/8 of the side of the square around the outside of the square (see figure). Each of these marks individually can be made with no more than 3 creases, and the entire set of 28 crease marks can be constructed with

14 creases (making pinches at each end of each crease). With the addition of the original 4 corners of the square as marks, the set of 1/8 marks gives 32 points around the outside of the square that can be used to find other points along the edge.

Last updated on

**February 9th, 2008**