Quasi-Periodic
tilings
A program to
calculate and display tilings of arbitrary rotational symmetry using a minimum
number of different rhombi. The 5-fold symmetric patterns are known as Penrose
patterns, so I like to call it generalized Penrose patterns.
The tilings are
produced from editable parameters in dual space. In this way an infinite number
of different patterns can be produced.
The construction
of these patterns using the dual method starts with a grid of lines, defined by
a star of vectors. Each family of lines is perpendicular to a vector, its
density proportional to the vector's length. To every open space between the
gridlines corresponds a vertex of a rhombus as follows:
If the area lies
between the i-th and i+1 th line of the j-th family then one finds a
corresponding point of the tiling by adding i steps along the j-th starvector.
This is to be repeated for every family of gridlines. The intersection of two
gridlines thus corresponds to a rhomb in the tiling as there are four spaces around
it defining four points in the pattern separated by single steps in one or both
of two grid vectors.
By varying number, length and orientation of
the star vectors an infinite number of tilings can be generated. Symmetry of
vectors and phases of the gridlines determine the symmetry of the result.
Penrose tilings
may be decomposed into patterns of rhombi as generated by the symmetric 5-fold
star. Many of the rhomb-patterns, however, cannot be transformed into kites and
darts, only if no more than two thin rombi meet alongside.
Further Reading:
http://library.thinkquest.org/16661/advanced.math/3.html
The program runs
under Windows9x, needs not to be installed and can be removed just by deleting.
It is free, but a contribution of e.g. $10 or 20 euro to an old age pensioner
would be welcome.
Frans C Mijlhoff
Master fan Loanstrjitte 24
NL 9123JP Metslawier
tel:+31 (0)519 241711
e-mail: fcm@chello.nl
homepage: http://surf.to/f.mijlhoff
e-art: http://community.webshots.com/user/frans_mijlhoff