Origami part 1 : Introduction

Introduction

I found about modular origamis originally with a video on making a Menger’s sponge out of business cards [5]. Quickly, I become interested in this method to build things out of paper cubes but my cubic realizations are for a future post. More or less at the same time, I found out about some videos on Micmaths channel [2] on modular origamis which includes the business card technique [1] as well as some others [3]. The technique that particularly interested me uses identical paper modules that are interlocked to form an augmented variant of regular polyhedra.

I like modular origami because it does not necessitate the use of any paper cutting (apart from making the correct ratio for the starting piece of paper) nor glue (nor tape). So, in theory, models can be mounted and unmounted indefinitely. I also like it because I could easily construct the 3D shapes of regular polyhedra though in their augmented version.

About augmentation

The augmented variant of a polyhedron is obtained by replacing each face of the base polyhedron with a pyramid, forming a 3D starred shape, [4]. This operation works well when the faces of the polyhedron are regular convex polygons with the same edge length. The faces of the pyramids are all the same isosceles triangles and the center of the pyramid is above the barycenter of the face at a height that defines the ratio between the edge and base of the triangular faces. When the ratio is 1, all faces are equilateral triangles producing a non-convex deltahedron (a polyhedron with only equilateral triangles as faces). Such a polyhedron, both a deltahedron and the augmentation of another polyhedron, seems really interesting and is beautiful. However, when the base polyhedron has some faces with 6 or more edges, using a ratio of 1 would not work for obvious reasons. Technically, it would work with a regular hexagon as the base face but would produce the exact same flat face and not the pointy thing one would expect.

As a matter of fact, one can compute the minimum edge-to-base ratio for any regular polygons. This minimum is attained when the vertex of the pyramid is in the plane of the face and matches its center. The limit angle α, see Figure 1, depends on n, the number of edges of the regular polygon:

(1)

The minimum edge-to-base ratio M is then :

(2)

Values from 3-gons (triangles) to 20-gons (dodecagons) are displayed in Table 1.

Moreover, there is two possible ways to replace a face of the base polyhedron with a pyramid. The first one is to have the pyramid pointing outwards making the pointy thing that one would expect when finding about augmented (or starred) polyhedron. The second one is to have the pyramids pointing inwards, towards the inside of the polyhedron, making a hole-y (but not sacred) shape. However, this second option would not work with any edge-to-base ratio as the tips of the pyramids could possibly intersect each other (take the tetrahedron and an edge-to-base ratio of 1 for instance). Obviously, there is two pointing choices for each face so many different solids are possible, 2^f to be exact with f the number of faces of the base polyhedron.

About the folding

The origami folding that I decided to use has an edge-to-base ratio of around 1.3 which is "good" for augmenting triangles, squares and pentagons but I had no other choice anyway.

The exact number can be deduced by taking a look at the crease lines of the folding, see Figure 2. The original square paper is first folded in half forming segments EG and FH that intersects at point I. Then, point B (bottom right corner) is brought onto I and forms the crease GH making a 45 degrees angle with the bottom square side. At last, HG is folded to coincide with EG resulting in a crease bisecting the angle and intersecting the horizontal line FH at J. Finally, the same is done with the top left corner resulting in J′. In the final module, the two isosceles faces are EJJ′ and GJJ′. Their base length is b and their side length is a. The edge-to-base ratio M = is related to the sine of α:

(3)

so

(4)

My origami constructions have all the pyramids pointing outwards. I have not tried making some inward-pointing pyramids with this module but since the edge-to-base ratio is large, having an full inward-pointing augmented cube is not possible. Though, I think it could be done with a dodecahedron.

References