AT512060A2 - METHOD FOR PRODUCING A PATTERN STRUCTURE - Google Patents

Abstract

The present invention includes three different recursive methods to generate aperiodic and asymmetric tilings based on a single tile (= prototeil) structured with sub-parts specified to be non-invariant in rotation, design objects, and architecture -Structures, images, moving images, music, textiles and materials to create. For each iteration of the recursive process, a pivot point is chosen around which one figure (output figure = prototype) is rotated in two, three or five steps. Due to the precise overlapping of the figures, these processes simultaneously create a second parallel hidden aperiodic and asymmetric tiling - a so-called plane background tiling, which lies exactly behind it and is different from the tiling visible in the foreground.

Description

1 10 2011/01515

BACKGROUND OF THE INVENTION / PRIOR ART

The invention - called the inductive rotation (IR) method - has its origin in that field of geometry / known as tiling, which deals with the coverage of given areas with prototypes ("tiles") of prescribed shape. This field has been used in particular in the production of design objects and architectural structures (paving, wall coverings), but also in the creation of new materials {quasicrystals).

In the general field of tiling the symmetry plays an essential role. Lattices with bizarre, threefold, five-count, and six-fold axes are in particular related to tilings, but result in noticeable periodic repetitions. It has already been proposed to generate aperiodic tilings by using a single protoparticle (i.e., covering the area using a single face), but the practical application is very cumbersome and, moreover, three-dimensionally periodic. The existence of a single prototype, which generates only aperiodic tilings in higher dimensions or simply generated in two dimensions without an investment rule, has so far been an unsolved problem. SUMMARY:

The present invention includes three different recursive methods to generate aperiodic and asymmetric tilings based on a single tile (= prototeil) structured with sub-parts specified to be non-invariant in rotation, design objects, and architecture -Structures, images, moving images, music, textiles and materials to create. For each iteration of the recursive process, a pivot point is chosen around which one figure {output figure = prototeil) is rotated in two, three, or five steps. Due to a precise overlapping of the figures, these methods simultaneously create a second parallel hidden aperiodic and asymmetric tiling - a so-called plane background tiling, which lies exactly behind it and

SUBSEQUENT

Claims (8)

2 is different from the visible in the foreground tiling. CLAIMS: There are three so-called inductive rotation (IR) methods: 1. The three-step IR method, which executes a rotation in three steps starting from a single square basic figure (= start prototype): 90 °, 180 ° and 270 °. 2. The two-step IR method, which executes a rotation in two steps, starting from a single star-shaped basic figure (= start prototype): 120 ° and 240 °. 3. The five-step IR method, which executes a rotation in five steps starting from a single hexagonal basic figure (= starting prototype): 60 °, 120 °, 180 °, 240 ° and 300 °, whereby the start ( Original) prototeil is structured with sub-parts that are not invariant upon rotation to recursively generate aperiodic, asymmetric flat patterns or tilings. For each iteration of the recursive process, a pivot point is chosen around which the figure (initial figure = prototeil) is rotated in two, three or five steps according to the respective IR method used. Due to a precise overlapping of the figures, the IR method simultaneously creates a second, parallel, concealed, aperiodic and asymmetric tiling - the so-called background tiling, which lies exactly behind it and is different from the tiling visible in the foreground. 4. Design objects and architectural structures produced according to claim 1. 5. images prepared according to claim 1. 6. Moving images produced according to claim 1. 7. Music produced according to claim 1. REPLACED 3 8. Textiles and other materials made according to claim 1. DESCRIPTION: The invention is formally described in mathematical definitions to clearly illustrate its geometric and practical applications by developing examples, also to simplify the explanation. These definitions and examples are only special cases to apply the IR method, e.g. In this case, the rotation is oriented in a clockwise direction and the axis of rotation is defined as the central, most eastern point of the figure. By its nature, the IR method can also be performed using the counterclockwise rotation or defining the axis of rotation as the central westerly point of the figures. As a result, different tilings are produced which have similar aperiodic and asymmetric structures. FORMAL DEFINITIONS: Let a, b be figures, X = (x, y) rotation axis, w an integer; for a, X, w define OEa, X, w]: = rotate a by X clockwise by w degrees. For figures a, b define a U b: = Figure b is positioned behind the figure a. It is clear that for all figures a, b, c: (a U b) U c = a U (b U c) = a U b U c. 1. The Three-Step IR Method: The initial basic figure (= prototeil) of the recursion R is the quadratic figure Q-s, Fig. 1; without limiting the generality Q should have the side length = 2 with the center o = (0,0) and be formed of four quadratic sub-parts with side length = 1, which should not be invariant on rotation. For n = 0, set R (0): = Q. For each iteration of the recursion R, the rotation point is READY