Curler units

The curler unit by Hermann Van Goubergen is one of the most interesting modules to build amazing origami models.  This unit is a simple but ingenious variation of the waterbomb base. 

Actually it is really easy to fold and you can find the diagram here!

Figure 1. Cuboctahedron with curler units

Using the curler units several models of polyhedra can be folded . With only twelve modules you can get the elegant cuboctahedron of Figure 1.

The cuboctahedron is an Archimedean solid and also a uniform polyhedron. Many people mistakenly believe that Archimedean and uniform are equivalent concepts.

The Archimedean polyhedra can be defined using the following local criterion

Local criterion: All faces are regular polygons, and the cyclic arrangement of the faces around each vertex is the same.

The uniform polyhedra are defined using a global criterion 

Global criterion: All faces are regular polygons, and all vertices form one orbit under isometric symmetries of the polyhedron.

There are fourteen convex polyhedra that satisfy the local criterion and should be called "Archimedean", but only thirteen that satisfy the global criterion and are appropriately called "uniform" (or "semiregular").

The interested reader can delve into this aspect by reading the interesting paper of Branko Grünbaum entitled An enduring error (avaliable online).

Origami boxes and tetrahedron volume

A few days ago I attended a short course on origami boxes. Among the various models I have been hit by Hildegards Schächtele.

Figure 1. Hildegards Schächtele

The diagrams of this easy modular model (two units) are available at the website of the author Carmen Sprung.

The shape of the box is that of an irregular tetrahedron formed by four faces that are congruent isosceles triangles. The opening of the box is an edge.

I wondered if this model had some special mathematical property or some educational interest.

A good exercise for young students consists of calculating the volume of the box given 

1. the length L of the side of the square of the sheet of paper; 
2. the possibility to manipulate the model; 
3. the crease pattern of the model.

Figure 2. Geometry of the crease pattern

The solution of the problem is left to the reader!

Before concluding, it is interesting to note that the problem of calculating the volume of an irregular tetrahedron has very interesting mathematical implications. See here for example.

Proof without words

The interesting book  Spiral: Origami | Art | Design  by Tomoko Fuse contains an entire chapter devoted to spirals made of a succession of similar polygons.

I folded one of these models to obtain the result of Figure 1.

Figure 1. Triangular spiral

It is interesting to note that this model is an example of what is known in mathematics as "proof without words". Let me explain better!

Figure 2.

In Figure 2, I reproduced with GeoGebra the succession of triangles of the origami model. If we assume that the area of ​​the initial equilateral triangle is unitary. Then, the areas of the different triangles appearing in the picture follow a geometric progression of common ratio 1/4. 

It is well known that

So Figure 2 "makes visible" and intuitive these facts and it can be used to introduce the idea of series at an elementary level.

From an origamistic point of view it is interesting to underline that the model is folded starting from a preprinted crease pattern which contains only mountain and valley folds.

Figure 3. Crease pattern of the origami model

I have created the crease pattern with GeoGebra, but it could be a good exercise in programming to do the same writing a PostScript code!