The Math and Art in Origami - How to Make Geometric Wireframes

I was never truly intrigued by origami until I ran into modular origami. I have folded for a few years now, and, while I am still no master of this art, I want to share at least a part of it with others. Origami is a lovely example of the too-often-ignored (although frequent) intersections between science and art. Modular origami leans more toward the mathematical side, but the potential for a practically infinite variety of shapes that are (both visually and theoretically) beautiful is also rather artistic. The majority of the models shown above are compounds of multiple intersecting shapes, which is where the variety really opens up.

In this instructable, I'll talk about only one variety of modular origami: wireframes made from Ow units. A wireframe is a 2-d or 3-d shape where only the edges are solidified, leaving open faces. The units (the individual modules in a model) that I will explain were more or less originally conceptualized by Francis Ow.

To equip you to come up with new compounds and modular constructions, I will cover a bit of theory on polyhedra and the polygons that make them up, explain how to fold Ow units and how to adjust the design to a variety of angles for making different polygons, and then leave you with some pictures, links, and ideas for going further.

When you're done, please give me ideas for improvement and addition.

That aside, let's get started! All you need is:

-Paper: origami paper is better, but printer paper usually works fine

-Papercutter or scissors (unless you want to crease and tear, which isn't quite as pretty)

-Time. More is better

Step 1: Theory: Polyhedra

Polyhedron: from poly (many) and hedron (face). Polyhedra are 3D geometric shapes. They are built from polygons (gon - angle/corner), which are 2D geometric shapes. Polyhedra are named and classified based on the number and type of faces, and their symmetry, as well as how they can be constructed.

Possibly the most fundamental group is the Platonic Solids (all consist of identical, regular polygons, shown in a render above). They are the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. Another common group is the Archimedean Solids, which all consist of identical vertices but a variety of faces (two or more types of regular polygons). There are 13 of these (depending on the exact definition - for more, see the going further page), shown in the second render. There are several other groups, but those groupings are less commonly used.

The aforementioned groupings are only for polyhedra with regular-polygonal faces; of course, a theoretical infinity of irregular-polygonal polyhedra, as well as irregular-verticed, regular polygonal polyhedra are possible, but only the simpler ones are practical to construct with Ow's methods, due to their size and interlocking limitations imposed by the geometry of the given model.

Step 2: Theory: Polygons

Many models are comprised of polyhedra made from regular polygons; these are probably the easiest ones to understand. This is because the inside angles of a regular polygon are, by definition, equal. Thus, these angles are easy to determine, using the formula

θ = (180(n - 2)) / n

when θ is one inside angle, and n is the number of side/angles.

Irregular polygons' angles can be calculated only if all sides, or a sufficient mixture of sides and angles, are known. Once the given polygon is decomposed into triangles, the individual angles can be calculated using a variety of trigonometric techniques. All the angles will still sum to 180(n-2), but this can only be used to discover, say, the one remaining angle if all others are known. To determine the length of paper needed to fold the units in compound models, the side lengths of the polygons making up each polyhedron must be known, and have some length added to them, as some is used up in the locking mechanism. The method for calculating this amount will be discussed later.

Step 3: Folding: Ow Units

An Ow unit is mostly folded on the end of a strip of paper. Thus, the length of the unit (and, therefore, the length of the resultant side of the polygon) is independent of the size of the joining mechanism, which depends on the width of the strip (and, therefore, the "thickness" of the "wire", or edge, in the wireframe model).

Thus, I am only demonstrating on one end of a unit, as opposed to showing the whole thing:

Start with piece of paper, eventually hidden side up
Fold in half lengthwise
Unfold
Fold each side lengthwise to centre line
Unfold
Fold angle for pocket. For an equilateral triangle (as seen in images above), fold top right corner from centre line until corner point is on line 1/4 from left side of paper. See images
Reverse-fold flap along right-most 1/4 line. This will create crease lines as seen in unfolded view, and will draw rightmost 1/4 of paper back to the centre line
Repeat steps on the opposite end of the paper, resulting in the first end being closed as seen in the third-last image
Fold unit in half so that the lengthwise open slit is on the inside. This will crease across the part of the pocket that sticks over the centre line

Step 4: Folding: Assembly of Units

To lock two separate units together:

Hold units with the flap/unfolded corner of one unit pointing to the pocket on another
Open the pocket slightly, slide the first unit in all the way (so that the centre lines of the units coincide)
Crimp the second unit along the centre fold, locking the units together.

The jump from angle to closed polygon to polyhedron is not difficult once the units and their attachment is understood. Keep in mind that, on a completed model, each vertex formed at the juncture of units should be surrounded by a complete ring of units, with no extra flaps or pockets (see image). Depending on the angles in the corners of the polygons that are meeting, there may be anywhere from two to perhaps five units meeting at a single vertex.

Step 5: Theory: Generalization of Concept

How can these angles be varied to produce any polygon?

Analyzing how the internal 60° angle was produced allows the folding of any angle up to about 140° before the units stop locking together. The reverse fold made for the pocket used the reference points of the centre line and the left-most quarter line. These reference lines were used to fold a triangle with proportions of base 1 and hypotenuse 2.

θ = sine^-1(1/2) = 30°

This is shown in the first image.

Next, follow my blue thought-process arrows. The 30° angle can be geometrically carried over to the angle between the centre line and the edge of the flap. Since this is an angle in a right triangle, we can find the angle just clockwise from this because we know two of the three angles as well as the sum of the angles:

180° - 90° - 30° = 60°

This can be carried over to the angle between the centre line and the reverse fold, which forms the bottom of the pocket.

In the third picture, I have the two units we assemble flipped over from the previous step, such that the pocket is still on the right side of the unit. It can be seen that the right unit's edge lies along the reverse-fold in the pocket of the left unit. Thus, the red angle, which we already found, can be carried over to the angle between the units. Therefore, the units have an angle of 60°, or the inside angle of an equilateral triangle's corner, between them.

Similar reasoning, done backwards, can be used to find sufficiently accurate reference folds for other angles, as desired. Understanding that it is the pocket angle that determines the angle between the units is necessary for folding units which are the shared edge between two differently shaped polygons, as exist, for example, in the Archimedean solids. A bit of trigonometry, geometric manipulation, somewhere to write, and practice will allow you to easily produce a variety of polygons, regular and irregular, to make polyhedra from.

When making compounds of polyhedra, precise lengths of units are important to make the final model fit snugly. When calculating the dimensions of the rectangle of paper needed to produce a given length of edge, keep in mind the following artifact of the locking mechanism. Any paper past where the angled fold crosses the centre line (as shown in the last picture) will not contribute to the edge's length (this is intuitively obvious after you do some folding). Therefore, for precision edge lengths, it is necessary to find how much will be truncated on the units' ends, and to add this onto your original paper dimensions.