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3 years ago
Mathematical knitting After Habr's separation into two resources it turned out, I think not only at me, so, that articles remained on Habré, and to Giktayms transferred to a profile only comments and a rating. At the same time as the user without publications I have no right to vote for the author's rating the pleasant articles that it would be desirable to correct. Ideas for articles if were, then belonged to Habr's subject. But the comment in article "Two weeks later emergence of Geektimes: alas..." suggested a strange idea — to write about mathematical knitting. That is about the different mathematical models connected by a hook or spokes. Well and it is a little about models too.



Strange attractor


Attractor (English attract — to attract, attract) — a compact subset of phase space of dynamic system which all trajectories from some neighborhood aim at it at time aiming at infinity. One of nominal examples of an attractor is the Lorenz attractor.

The Lorenz attractor was found in numerical experiments of Lorentz investigating behavior of trajectories of nonlinear system:

Mathematical knitting

at values of parameters: σ=10, r=28, b=8/3. The system arises in the following physical questions and models: convections in the closed loop, rotation of a water wheel, model of the mono-mode laser, the dissipative harmonic oscillator. Lorentz's model is a real physical example of dynamic systems with chaotic behavior.

Mathematical knitting

The mathematician Hinke Osinga has quite mathematical hobby - knitting. Including loops in off-duty time, she has a rest. Somehow her research supervisor, professor Krauskopf imprudently threw: "You would connect something useful!". And doctor Osinga connected.

So there was a chaos model. Now the only knitted chaos in the world turns under a ceiling in an office of mathematicians at university of Bristol.

Hinke Osinga, doctor of mathematics, university Bristol: "I knitted each free minute. Generally in the evenings. Approximately for 2 hours a day for nearly two months. In total, 85 hours. More than 25 thousand loops, the chaos also turned out. And, very nice". Now, when it can be touched, it is easier for mathematics to study it. They are engaged in it 2 years (interview of 2004), modeling "Lorentz's equations" which just and describe the chaotic movements on the computer. Mathematicians promised a champagne bottle to the one who will offer the first other connected model. The first letters with photos came only in two weeks.

Mathematical knitting

Mathematical knitting

Hyperbolic plane


Lobachevski geometry (hyperbolic geometry) — one of non-Euclidean geometriya, the geometrical theory based on the same main sendings as normal Euclidian geometry, except for an axiom about parallel which is replaced with an axiom about parallel Lobachevsky. In Lobachevski geometry the following axiom is accepted: pass through the point which is not lying on this straight line at least two straight lines which are lying from this straight line in one plane and not crossing it.

Pseudosphere (Beltrami's surface) — the surface of constant negative curvature formed by rotation of a tractrix about its asymptote. The name underlines similarity and distinction to the sphere which is an example of a surface with curvature, also permanent, but positive. The name "pseudosphere" of a surface was given by Beltrami.

He paid attention that the pseudosphere implements local model of Lobachevski geometry.

Mathematical knitting

Daina Tayminya resolved a centenary problem of non-Euclidean geometry on visualization of the hyperbolic planes. The hyperbolic planes are related to non-Euclidean geometry which is traditionally difficult for visualizing. Daina Taymina managed to make it with use of knitted fabrics. It connected the first model of the hyperbolic plane by a hook in 1997 to use in a studio course of non-Euclidean geometry. Since then it connected more than one hundred geometrical models.

Mathematical knitting

Mathematical knitting

To her technician use in ecology. Margaret Vertkheym heads the project on a reconstruction of inhabitants of a coral reef, using equipment crumble (knittings by a hook), invented by the mathematician — glorifying an udivitelnost of a coral reef, and being submerged in hyperbolic geometry which is the cornerstone of creation of a coral.
Video on TED: Margaret Vertkheym about beautiful mathematics of a coral (and knittings by a hook) where the simple explanation of Euclidean and hyperbolic space is offered.

Klein's bottle


Klein's bottle — the nonorientable (one-sided) surface for the first time described in 1882 by the German mathematician F. Klein. It is closely connected with the Moebius band and the projective plane. The name, apparently, comes from the wrong transfer of the German word Fläche (surface) which in German is close on writing to the word Flasche (bottle); then this name returned in such type to German.

Klein's surface in the form of "a figure 8", shown in drawing below, can be presented in the form of system of equations with parameters which looks much more simply, than for a classical bottle of Klein:

Mathematical knitting
Mathematical knitting
Mathematical knitting

Mathematical knitting

If rassech Klein's bottle on two halves along a plane of symmetry, then two mirror Moebius bands, one — with a turn half-turned to the right, another — with a turn half-turned to the left turn out. Actually, perhaps rassech Klein's bottle so that one Moebius band will turn out. Otherwise, Klein's bottle can be presented in the form of two Moebius bands connected with each other by a normal double-sided tape. In drawing below the internal surface of this tape is colored in the white color, and external — blue.

Mathematical knitting

Knitted bottle of Klein:

Mathematical knitting

As it is possible to see, the Acme firm also does glass bottles.

Fractals


Fractal (Latin fractus — shredded, broken, broken) — the mathematical set possessing property of self-similarity, that is uniformity in different scales of measurement (any part of a fractal is similar to all set entirely). In mathematics understand the sets of points in an Euclidean space having fractional metric dimension as fractals (in Minkowski or Hausdorff's sense), or metric dimension, excellent from topological therefore they should be distinguished from the other geometrical figures limited to a finite number of links.

Since the end of the 19th century, in mathematics examples of self-similar objects with properties, pathological from the point of view of the classical analysis, appear. A Serpinski triangle — a fractal, one of two-dimensional analogs of a great number of the Cantor offered by the Polish mathematician Václav Sierpinski in 1915. It is also known as "grid" or Sierpinski's "napkin". The middle of the parties of an equilateral triangle of T0 connect segments. 4 new triangles turn out. From an initial triangle the interior of a median triangle is removed. The set of T1 consisting of 3 remained triangles of "the first rank" turns out. Arriving in the same way with each of triangles of the first rank, we will receive the set of T2 consisting of 9 equilateral triangles of the second rank. Continuing this process infinitely, we will receive infinite sequence of T0⊃T1 ⊃⋯⊃ Tn ⊃ … which intersection of members is a Serpinski triangle.

Mathematical knitting

Doctor David Uilstrom though the man, but too sometimes knits. In knitting it was trained at one of textile seminars, and since then it does interesting things of threads in free time.

Mathematical knitting

And, finally, some more knitted fractals.

Pythagoras's tree

Mathematical knitting

These plaids were created by Woollythoughts firm. They also do unusual panels on which image is visible only under a certain corner.

Set of Zhyulia

Mathematical knitting

Sources:


About a knitted attractor
Hinke Osinga and Bernd Krauskopf's website
Hyperbolic knitting
Daina Tayminya
Hyperbolic knitting by a hook
About Daina Taymini's exhibition
Klein's bottle
Acme klein bottle company
Mathematical Knitting Network
Pythagoras's tree
Article about mathematical knitting, but images are unavailable

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