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| 1. | One Sonobe module | |
|---|---|---|
| 3. | Opened (left) and completed (right) Toshie's Jewel(triangular bipyramid) | |
| 6. | Cube (triakis tetrahedron) | |
| 12. | Triakis octahedron | |
| 30. | Triakis icosahedron |
The Sonobe module is one of the many units used to build modular origami. 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 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. [2]
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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 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 for the open bottom, and isosceles right triangles as the other three faces. It will have a right-angle apex (equivalent to the corner of a cube) and three tab/pocket flaps protruding from the base. This particularly suits polyhedra that have equilateral triangular faces: Sonobe modules can replace each notional edge of the original deltahedron 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 above), is named after origami enthusiast Toshie Takahama. It is a three-unit hexahedron built around the notional scaffold of a flat equilateral triangle (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.
The most popular intermediate model is the triakis icosahedron, shown below. It requires 30 units to build.
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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:
| Number of Sonobe Units | Faces | Edges | Vertices |
|---|---|---|---|
| s | 2s | 3s | s + 2 |
| 3 | 6 | 9 | 5 |
| 6 | 12 | 18 | 8 |
| 12 | 24 | 36 | 14 |
| 30 | 60 | 90 | 32 |
| 90 | 180 | 270 | 92 |
| 120 | 240 | 360 | 122 |
| 270 | 540 | 810 | 272 |
The model made of three units results in a triangular bipyramid. Building a pyramid on each face of a regular tetrahedron, using six units, results in a cube (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. Building a pyramid on each face of a regular octahedron, using twelve Sonobe units, results in a triakis octahedron. Building a pyramid on each face of a regular icosahedron requires 30 units, and results in a triakis icosahedron.
Uniform polyhedra can be adapted to Sonobe modules by replacing non-triangular faces with pyramids 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:
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.
In geometry, an octahedron 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 each vertex.
Modular origami or unit origami is a two-stage paperfolding technique in which several, or sometimes many, sheets of paper are first folded into individual modules, or units, and then assembled into an integrated flat shape or three-dimensional structure, usually by inserting flaps into pockets created by the folding process. These insertions create tension or friction that holds the model together.
In geometry, a deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many deltahedra, all having an even number of faces by the handshaking lemma. Of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces. The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra.
In geometry, a triakis octahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.
In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid, with 60 isosceles triangle faces. Its dual is the truncated dodecahedron. It has also been called the kisicosahedron. It was first depicted, in a non-convex form with equilateral triangle faces, by Leonardo da Vinci in Luca Pacioli's Divina proportione, where it was named the icosahedron elevatum. The capsid of the Hepatitis A virus has the shape of a triakis icosahedron.
The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.
The Tamatebako (玉手箱) is an origami model named after the tamatebako of Japanese folk tale. It is a modular cube design that can be opened from any side. If more than one face of the model is opened, the cube falls apart and cannot easily be reconstructed. The model, and the directions for creating it, had been lost for centuries and only recently rediscovered.
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces. The precise number of near-misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons.
In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.
In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.
In geometry, the truncated triakis tetrahedron, or more precisely an order-6 truncated triakis tetrahedron, is a convex polyhedron with 16 faces: 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps.
In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a simplicial polyhedron in three dimensions contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph.
In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty', and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" or "icosahedrons".
The skeleton of a cuboctahedron, considering its edges as rigid beams connected at flexible joints at its vertices but omitting its faces, does not have structural rigidity and consequently its vertices can be repositioned by folding at edges and face diagonals. The cuboctahedron's kinematics is noteworthy in that its vertices can be repositioned to the vertex positions of the regular icosahedron, the Jessen's icosahedron, and the regular octahedron, in accordance with the pyritohedral symmetry of the icosahedron.