What I'm about to write is a fascinating story that connects the hook to the scientific world and in particular to mathematics. I never thought that my scientific studies of high school could bind to my passion for crochet. And here, instead, I met, on the network, the experience of Daina Taimina, a mathematics scholar Lithuanian, with its models crocheted, has succeeded to explain to his students the difficult concept of the hyperbolic plane, making it a reality with a model in three dimensions.
Daina refers in his wonderful book (Crocheting Adventures with Hyperbolic Planes) how she came to use the crochet hook to make his model of the hyperbolic plane.
To understand his reasoning and scope of his work is useful to a small introduction about Hyperbolic Geometry.
We define, Meanwhile, the concept of zero curvature, positive or negative of a surface.
(There is help in the Prof.. Lazzarini, who in his very useful blog gives us a very intuitive explanation, below that carry the entire).
The idea is to “crush” the surface on the floor. When we try to “flatten” a curved surface are given three options:
We can flatten the surface, operate without tearing or overlapping. We will say in this case that the surface has zero curvature (ie in all points of the surface curvature is nothing). For example any region of the cylindrical surface can be made perfectly flat and therefore has zero curvature. Reflect on the fact that a cylindrical surface can be achieved by rolling a sheet of paper. May regret but things are just so: there are areas that we are accustomed to consider curves but, technically, are considered to have no curvature.
Riusciano not to flatten the surface, why should we make of tears. It’ what happens, eg, with a region of the spherical surface; we can think, trusting intuition, that in this case there is “surface phenomenon” of what it takes to be flattened. In this case we say that the surface has positive curvature.
In the next picture you see a balloon that has been cut in half, along a maximum circumference. Try to flatten: you will not succeed.
The only way is to make radius cuts as you see in the picture below (the greater the number of cuts, the greater the adhesion to the plane).
As you see between one cut and the other of the surface are created spaces that correspond to surface missing.
Riusciano not to flatten the surface, why should we work overlaps. It’ what happens, eg, with a region of surface saddle-shaped; we can think, entrusting again intuition, that in this case there is “more surface” how much will be in the plane. In this case we say that the surface has negative curvature.
In the picture below you see a saddle surface.
You can easily get it by proceeding in this way. Drawn on a sheet of paper a circle and a circular sector with the same radius of the circle and an amplitude, say, of 60 degrees (see figure below). Cut out the circle and sector. Cut the circle along one of its radius so that it has a slit. Inserted into the slot and secure the area at the edge of the crack with clear tape. Of course, this insert operation is not possible if you stay in the plan (We are expecting to add more 60 degrees in a full circle); but we can do it if we leave flex the surface in the third dimension. So you will get a saddle surface (is also appropriate to apply the tape on the other face of the surface). The realization of this model is very instructive: you realize that a saddle invades “more surface” how much will be in the plane.
Now try to flatten your seat on the plane, for example relying on a book: you will realize that form folds, overlap, As you can see in the picture below.
Rieman in 1854 gives this definition:"Hyperbolic geometry can be considered the intrinsic geometry of a surface with curvature constantly negative that extends indefinitely in all directions". This assumption is based on all the research mathematicians who over the years have dedicated themselves to find the complete hyperbolic surface. In fact, while you can find examples in nature of surfaces constantly negative
Such surfaces, unfortunately, not have the characteristic of stretch to infinity. The mathematicians for a long time, therefore, asserted, that it was not possible to obtain three-dimensional Euclidean space, a total area of a hyperbolic plane (a surface with constant negative curvature extended indefinitely).
In 1954 Kuiper (a German mathematician) suggested that such a surface could exist, but did not explain how you could build. Fu William Thurston nel 1970 to have the idea of using strips of paper (definite “anuli”) to describe a hyperbolic plane in three dimensional space.
Taimina, in his book explains how to make the paper model with "anuli" , as described in the above picture.
From paper model Daina was inspired to create his model or crocheted; studying the anuli paper, in fact, she realized that, working so as to increase the number of meshes in a constant way from line to line, and following a definite proportionality, was obtained in the form of a hyperbolic plane model or crocheted.
The concept is simple and revolutionary at the same time, because through the use of a technique housewife and within everyone's reach, how to crochet, you can create a three-dimensional model which was considered by scholars impossible.
Whoever can use the hook can be home a hyperbolic object. I found explanations in English here, in a pdf downloadable, but equally useful is the Italian translation of points used described in this post my friend Cosmosicula (I thank him for the inspiration of this wonderful topic that is the hyperbolic crochet).
These beautiful models closely resemble natural forms Mushroom, coral sea slugs etc..
Ovviamentemi I ventured execution crochet some of these forms and I got these results:
In the book of Daina there are a lot of images and you'll find many ideas about the nature of a hyperbolic plane that can be verified through the crochet patterns; I'm going to write more posts on this topic I find so fascinating.
In addition, in a few days I'll be able to show even more of my creations with the hyperbolic crochet that, I'm sure, inspire your future work, as it has been for the Moebius strip.