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