The Sonobe module is one of the many units used to build

modular origami Modular origami or unit origami is a paperfolding technique which uses two or more sheets of paper to create a larger and more complex structure than would be possible using single-piece origami techniques. Each individual sheet of paper is fol ...

. The popularity of Sonobe modular origami models derives from the simplicity of folding the modules, the sturdy and easy assembly, and the flexibility of the system.

The history of the Sonobe module

The origin of the Sonobe module is unknown. Two possible creators are Toshie Takahama and Mitsunobu Sonobe, who published several books together and both members of Sosaku Origami Group 67. The earliest appearance of a Sonobe module was in a cube attributed to Mitsunobu Sonobe in a Sosaku Origami Group book published in 1968, however it does not reveal whether he invented the module or used an earlier design: the phrase "finished model by Mitsunobu Sonobe" is ambiguous. Its next appearance was "Toshie's Jewel", which appeared in 1974. However neither folder took advantage of the full potential of the module. This potential was discovered in the 1970s by other folders – particularly Steve Krimball, who created the 30-unit ball – as part of a sudden period of development in modular origami. Despite the module's importance and continued popularity, its designer remains uncertain.

The unit

Each individual unit is folded from a square sheet of paper, of which only one face is visible in the finished module; many ornamented variants of the plain Sonobe unit that expose both sides of the paper have been designed. The Sonobe unit has the shape of a

parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

with 45 and 135 degrees angles, divided by creases into two diagonal tabs at the ends and two corresponding pockets within the inscribed center square. The system can build a wide range of three-dimensional geometric forms by docking these tabs into the pockets of adjacent units. Three interconnected Sonobe units will form an open-bottomed triangular pyramid with an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

for the open bottom, and

isosceles right triangle A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...

s as the other three faces. It will have a

right-angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. T ...

apex (equivalent to the corner of a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

) and three tab/pocket flaps protruding from the base. This particularly suits

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...

that have equilateral triangular faces: Sonobe modules can replace each notional edge of the original

deltahedron In geometry, a deltahedron (plural ''deltahedra'') is a polyhedron whose face (geometry), faces are all equilateral triangles. The name is taken from the Greek language, Greek upper case delta (letter), delta (Δ), which has the shape of an equ ...

by the central diagonal fold of one unit and each equilateral triangle with a right-angle pyramid consisting of one half each of three units, without dangling flaps. The pyramids can be made to point inwards; assembly is more difficult but some cases of encroaching can be obviously prevented. The simplest shape made of these pyramids, often called "Toshie's Jewel" (shown on the right), is named after origami enthusiast Toshie Takahama. It is a three-unit hexahedron built around the notional scaffold of a flat

(two "faces", three edges); the protruding tab/pocket flaps are simply reconnected on the underside, resulting in two triangular pyramids joined at the base, a

triangular bipyramid In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, i ...

. The most popular intermediate model is the

triakis icosahedron In geometry, the triakis icosahedron (or kisicosahedronConway, Symmetries of things, p.284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron. Cartesian coordinates Let \phi be the golden ratio. The 12 po ...

, shown below. It requires 30 units to build.

Models made with a Sonobe unit

The table below shows the correlation between three basic characteristics – faces, edges, and vertices – of polygons (composed of Toshie's Jewel sub-units) of varying size and the number of Sonobe units used: The model made of three units results in a

. Building a pyramid on each face of a regular

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

, using six units, results in a

(the central fold of each module lays flat, creating square faces instead of isosceles right triangular faces, and changing the formula for the number of faces, edges, and vertices), or

triakis tetrahedron In geometry, a triakis tetrahedron (or kistetrahedron) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron. The triakis tetrahedron can be see ...

. Building a pyramid on each face of a regular

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

, using twelve Sonobe units, results in a

triakis octahedron In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedronConway, Symmetries of things, p. 284) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube. It can be seen as an octahedron with triangular ...

. Building a pyramid on each face of a regular

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

requires 30 units, and results in a

. Uniform polyhedra can be adapted to Sonobe modules by replacing non-triangular faces with

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...

s having equilateral faces; for example by adding pentagonal pyramids pointing inwards to the faces of a dodecahedron a 90-module ball can be obtained. Arbitrary shapes, beyond symmetrical polyhedra, can also be constructed; a deltahedron with 2N faces and 3N edges requires 3N Sonobe modules. A popular class of arbitrary shapes consists of assemblies of equal size cubes in a regular cubic grid, which can be easily derived from the six unit cube by joining multiple ones at faces or edges. There are two popular variants of the main assembly style of three modules in triangular pyramids, both using the same flaps and pockets and compatible with it: * Joining four modules together (instead of three), forming a flattened square pyramid that can become part of a quilt or a larger polyhedral face, e.g. in 12 and 24 modules large cubes. Such a square lacks structural integrity because without the diagonal folds the flaps are not restrained to stay in the far corner of the pockets. * Joining only two modules, forming a triangular fin that can be used as an ornament for suitable models and to make a 1 module triangle (one fin, made with the two halves of the same module) or a 2 module square (two fins).

Notes and references

Bibliography

* Takahama, Toshie, and

Kunihiko Kasahara (born 1941) is a Japanese origami master. He has made more than a hundred origami models, from simple lion masks to complex modular origami, such as a small stellated dodecahedron. He does not specialize in what is known as "super complex origami ...

. ''Origami for the Connoisseur.'' Japan Publications, Tokyo, 1987. * Takahama, Toshie, "Creative Life with Creative Origami" Volume I (1974) (original source for Toshie's jewel) * Sosaku Origami Group 67, Magazine 2 (Mitsunobu's original cube)

External links

Sonobe unit folding instructions

Sonobe unit folding video

Additional sonobe unit folding instructions (WikiHow)

30-unit stellated icosahedron assembly instructions
{{Authority control Paper art Origami Polyhedra