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!

Spiral of Theodorus from a tape

"Theaetetus. Theodorus here was drawing some figures for us in illustration of roots, showing that squares containing three square feet and five square feet are not commensurable in length with the unit of the foot, and so, selecting each one in its turn up to the square containing seventeen square feet and at that he stopped. Now it occurred to us, since the number of roots appeared to be infinite, to try to collect them under one name,..." 

(Plato, Theaetetus 147d)

This is a very famous passage from Theaetetus (Greek: Θεαίτητος) one of Plato's dialogues concerning the nature of knowledge. Unfortunately, we do not know what the geometric construction to which Plato refers.

Even if there is no historical basis, the geometric construction is traditionally associated with a spiral composed of contiguous triangles rectangles known as Spiral of Theodorus. 

Spiral of Theodorus (source wikipedia)

In the remarkable book Spiral: Origami | Art | Design  by Tomoko Fuse is shown as a spiral of Theodorus can fold from an ordinary strip of paper!

Tomoko Fuse model

It is amazing! 

Just with a strip of paper you can get an object so interesting from a mathematical point of view. However it is not a novelty that with a strip of paper you can easily get interesting geometric objects. A very famous example in this direction is the sangaku problem of the regular pentagon.

Captwist

The term hypar means a hyperbolic paraboloid shape, or more formally a partial hyperbolic paraboloid, cut from the full infinite surface. Erik Demaine pointed out that the term hypar was introduced by the architect Heinrich Engel in his 1967 book Structure Systems (page 215).

It is really easy and instructive to fold a hypar (see Figure 1) from a square piece of paper. If you fold the diagonals of a square, and several concentric squares in alternating direction (a square of mountain folds, then a square of valley folds, and so on), then the piece of paper naturally forms a pleated hyperbolic paraboloid shape. 

Figure 1. Hypar

Curiously, the four sides of the hypar represent a closed path, made of consecutive edges of a tetrahedron, which visit each vertex exactly once.  This closed path is called Hamiltonian circuit.  The crease pattern of the hypar is formed by the diagonals of the square sheet from which it has been bent plus a concentric family of squares. It is quite natural to ask what happens if we start folding from a not squared sheet of paper. 

Starting from a regular octagon Thomas Hull was able to fold a nice origami model, he called hyperbolic cube, which represents a surface with negative curvature, whose border is a Hamiltonian circuit on a cube. In this case the crease pattern is made of concetric octagons and the final shape has been got by wet-folding. 

A more amazing model, the hexagonal wrap, has been folded always by Hull starting from an hexagon. The model is basically a series of concentric hexagons with "zig-zag" creases coming from the center-most hexagon out to the midpoints of the paper's sides. It can be collapsed in many different ways. However, there is a twisted way which is mathematically very interesting to get an Eulerian circuit on an octahedron.

Working around this idea we have folded a model which represents a Hamiltonian circuit of a particular polyhedron called "capped tetrahedron". For this reason we have called it captwist. The crease pattern is a series of concentric squares with "zig-zag" creases coming from the center-most square out to the midpoints of the paper's sides. The model can be collapsed flat into a four-armed star shape. However, it can be much more funny to twist along a diagonal the model into an interesting shape, as we have done in Figure 2.

Figure 2. Captwist

The border of the model represents a Hamiltonian path on a "capped tetrahedron". This solid is obtained from a tetrahedron constructing on each face a pyramid as shown in Figure 3.

Figure 3. Capped tetrahedron