These are the four fundamental flat finite forms of modular origami.

Coasters are simply flat polygonal forms, Stars are flat forms with symmetrical points, Rotors flat forms with assymetrical points and Rings are any of the above with hollow centres. Complex Coasters are sometimes called Mandalas. Rings are sometimes called Wreaths. Removing one ore more modules from these forms will often turn them into dish-like forms.

The design on the left is my 3-Fold Star, a minimalist modular design.

Quilts and Lattices

In origami a quilt is a form in which modules are used to create a flat surface. A Lattice is simply a quilt with holes in it. Quilts and Lattices are usually infinitely extendable forms but can, of course, be edged so that they are finite forms instead.

The lattice on the right is an unpublished design related to Windfarm.

Modular origami can be used to make attractive open and lidded containers of many shapes.

The triangular lidded box pictured here is a design by Tomoko Fuse.

Modular origami provides an excellent way to model many Platonic, Archimedean, Rhombic and other polyhedra. In order to achieve a flat-faced design the geometry of the modules must match (or at least closely approximate) the geometry of the polyhedron being modelled.

The design shown to the right is my full-faced dodecahedron from Mathematical Origami.

It is sometimes possible to model a flat-faced polyhedron in such a way that some of the faces are entirely missing as in the case of this truncated icosahedron design.

This is an as yet unpublished design of my own.

Pimpled forms

Pimpling of the faces of a polyhedral model occurs when (because the geometry of the modules does not match the geometry of the underlying polyhedron) there is too much paper to form a flat face and the excess paper is configured to point outwards away from the centre of symmetry of the design, thus forming pimples (or pyramids) instead.

The form shown here is an 8-point Stubby Star made from Sonobe modules. The underlying polyhedron is an octahedron.

Conversely dimpling occurs when the excess paper is configured to point inwards towards the centre of symmetry of the underlying polyhedron.

This picture shows my Tricorne design which is a dimpled silverhexahedron.

Open-frame forms

An open-frame form is simply a polyhedron with holes in the centre of the faces. Forms of this kind are sometimes referred to as pierced designs. Like unpierced forms open-frame polyhedra may be flat-faced, pimpled or dimpled.

This design is an open-frame dodecahedron from Mathematical Origami.

Nolids ( that is solids of no volume) can be seen as polyhedra whose faces have been dimpled to the point where the apex of the dimple lies at the centre of symmetry of the underlying polyhedron. They are also sometimes referred to as skeletal or planar designs, though the latter description may be used of any design in which finite planes are represented by layers of paper irrespective of whether the design is a nolid or not.

This design is my nolid icosidodecahedron.

Hybrid forms

It is frequently possible to combine many of these modelling tecxhniques withi a single design. This design, for instance, is my Proteus sculpture which has flat triangular faces combined with open-frame dimpled pentagonal ones.

The surface of many polyhedra can be distorted (by folding) to create forms that look entirely different but are nevertheless in modular origami terms closely related designs.

The design shown to the left is my 6-part Enigma Cube which is an analogue of the cube and is made by altering the way in which the six simple modules that are used to make a Blintz Cube are configured without altering their relationship to each other within the finished modular assembly.

Sculptural forms

Any of these types of polyhedral forms can be augmented with additional flaps, extra modules etc to create forms that, while based on the form of an underlying polyhedron, are in final appearance more akin to decorative ornaments or sculpture.

Forms of this kind which resemble a ball of flowers are known as kusudama.

Modular designs can also be created by combining several, or many, basic polyhedral forms to produce a more complex form. Sometimes this kind of modular structure is not valid mathematically, but works in terms of modular origami because the inaccuracies are small enough to be absorbed in the assembly process.

My 20-Cubes design pictured here is a good example of such an unmathematical form.

Modular Action Designs

Modular designs are not necessarily rigid. Many can be manipulated to change the relationship of the parts to each other. Modules may rotate or slide into different positions, the whole design my rotate through its centre of symmetry or it may collapse flat in one or several directions. The possibilities for such designs are probably unlimited.

The design shown here is my Rotating Ring of Rhombic Polyhedra from Mathematical Origami.

Stacks (or towers) are macromodular structures in which the macromodules sit directly on top of each other. They may be held together by joining-pieces or just by gravity. Walls are formed by connecting or arranging several stacks in a linear fashion, forests by connecting them into larger non-linear groups. Pyramids are formed where the macromodules are not stacked directly on top of each other. Pyramids may also be linear or non-linear in form.

The picture shows a forest of my Columbus Cubes.

Macromodular Lattices

In contrast to stacks and pyramids macro-modular lattices are structures that are firmly linked together in all directions. Many lattices have open spaces within the structure but that is not a necessary defining characteristic.

The picture to the right shows my Borromean Lattice design.

Nested forms can be created by building several complete modular polyhedral structures inside each other. In this case a solid tetrahedron nestles inside an open-frame cube which in turn sits within an open-frame dodecahedron.

Tomoko Fuse has published several designs for structures of this kind.

Interwoven Macromodular Forms

Forms of this type can be created by linking several (or many) open-frame modular polyhedral designs to and through each other.

The design shown here is Tom Hull's Five Intersecting Tetrahedra. The open-frame tetrahedra which Tom has used as macromodules are a design by Francis Ow.