Periodic and Aperiodic Lattices in R²

Once upon a time I did some research into aperiodic crystals in the context of photonic bandgaps. I ended up needing some figures, and thought it fun to write them directly in PostScript.

Here are the are five regular lattice types in R², known as Bravis lattices. They are (click on the link to go to the source)

• Oblique
• Rectangular
• Rhomb
• Square
• Triangular

Additionally, I provide the Fibonacci and the random lattice.

• Fibonacci
• Random

Fibonacci

Download Fibonacci PostScript

%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Created from bitmaplib by Paul Bourke
%%BoundingBox: 0 0 100 100
%%LanguageLevel: 2
%%Pages: 1
%%DocumentData: Clean7Bit

% size of lattice
/sx 100 def
/sy 100 def

% radius of object
/b 0.25 def

% object (a "circle") at each lattice point
/circle{
    gsave
    currentpoint translate
    newpath 0 0 b 0 360 arc fill stroke
    grestore
} def

% golden ration
/theta 1 5 sqrt add 2 div def

% returns the kth fibonacci number
/fib {
    dup theta exch exp 5 sqrt div exch neg theta neg exch exp 5 sqrt div sub cvi
} def

% proper mathematical modulo 1
/mod1 {
    dup truncate exch sub
} def

/k 18 def
/i 0 def
/n k fib def % how man iterations
sx sy translate
n {
    %100 i ne {
    i n div mod1 sx mul 
    i n div k 1 sub fib mul mod1 sy mul
    moveto circle
    %} if
    /i i 1 add def
} repeat

showpage

%%EOF

Oblique

Download Oblique PostScript

%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Created from bitmaplib by Paul Bourke
%%BoundingBox: 0 0 100 100
%%LanguageLevel: 2
%%Pages: 1
%%DocumentData: Clean7Bit

% makes a triangular lattice

% lattice constant
/a 2 def

% radius of lattice entities
/r 0.25 def

% size of the grid x and y
/sx 100 def
/sy 100 def

% a circle shape exists at each lattice element
/circle{
    gsave
    currentpoint translate
    newpath 0 0 r 0 360 arc fill stroke
    grestore
} def

% go to an origin well below this dingle
sx neg 0 moveto
% iterate over all columns by the lattice constant
0 a sy 2 mul {
    a 0 rmoveto
    gsave
    % make the lattice diagonally
    0 a sx 3 sqrt mul { a 3 div a 5 sqrt 2 div mul rmoveto circle } for
    grestore
} for


%%EOF

Random

Download Random PostScript

%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Created from bitmaplib by Paul Bourke
%%BoundingBox: 0 0 100 100
%%LanguageLevel: 2
%%Pages: 1
%%DocumentData: Clean7Bit
%100 100 scale

% makes a square or rectangular lattice

% lattice constant
/ax 10 def
/ay 10 def

% radius of lattice entities
/r 1.00 def

% size of the grid x and y
/sx 100 def
/sy 100 def

/circle{
    gsave
    currentpoint translate
    newpath 0 0 r 0 360 arc fill stroke
    grestore
} def
newpath
0 0 moveto
0 sy lineto
sx sy lineto
sx 0 lineto
closepath clip
newpath
0 0 moveto
0 sy lineto
sx sy lineto
sx 0 lineto
closepath stroke


% go to origin
0 0 moveto
% columns
0 ay sy { 
0 ax sx { 
    gsave
        rand 100 mod rand 100 mod moveto circle
    grestore
} for 
} for 

showpage

%%EOF

Rectangular

Download Rectangular PostScript

%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Created from bitmaplib by Paul Bourke
%%BoundingBox: 0 0 100 100
%%LanguageLevel: 2
%%Pages: 1
%%DocumentData: Clean7Bit
%100 100 scale

% makes a square or rectangular lattice

% lattice constant
/ax 2 def
/ay 4 def

% radius of lattice entities
/r 0.25 def

% size of the grid x and y
/sx 100 def
/sy 100 def

/circle{
    gsave
    currentpoint translate
    newpath 0 0 r 0 360 arc fill stroke
    grestore
} def

% go to origin
0 0 moveto
% columns
0 ay sy { 
    0 ay rmoveto
    gsave
        % rows
        0 ax sx { ax 0 rmoveto circle } for
    grestore
} for 

showpage

%%EOF

Rhomb

Download Rhomb PostScript

%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Created from bitmaplib by Paul Bourke
%%BoundingBox: 0 0 100 100
%%LanguageLevel: 2
%%Pages: 1
%%DocumentData: Clean7Bit

% makes a triangular lattice

% lattice constant
/a 2 def

% radius of lattice entities
/r 0.25 def

% size of the grid x and y
/sx 100 def
/sy 100 def

% a circle shape exists at each lattice element
/circle{
    gsave
    currentpoint translate
    newpath 0 0 r 0 360 arc fill stroke
    grestore
} def

% go to an origin well below this dingle
sx neg 0 moveto
% iterate over all columns by the lattice constant
0 a sy 2 mul {
    a 0 rmoveto
    gsave
    % make the lattice diagonally
    0 a sx 2 mul { a 2 div a rmoveto circle } for
    grestore
} for


%%EOF

Square

Download Square PostScript

%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Created from bitmaplib by Paul Bourke
%%BoundingBox: 0 0 100 100
%%LanguageLevel: 2
%%Pages: 1
%%DocumentData: Clean7Bit
%100 100 scale

% makes a square or rectangular lattice

% lattice constant
/ax 2 def
/ay 2 def

% radius of lattice entities
/r 0.25 def

% size of the grid x and y
/sx 100 def
/sy 100 def

/circle{
    gsave
    currentpoint translate
    newpath 0 0 r 0 360 arc fill stroke
    grestore
} def

% go to origin
0 0 moveto
% columns
0 ay sy { 
    0 ay rmoveto
    gsave
        % rows
        0 ax sx { ax 0 rmoveto circle } for
    grestore
} for 

showpage

%%EOF

Triangular

Download Triangular PostScript

%!PS-Adobe-3.0 EPSF-3.0
%%Creator: Created from bitmaplib by Paul Bourke
%%BoundingBox: 0 0 100 100
%%LanguageLevel: 2
%%Pages: 1
%%DocumentData: Clean7Bit

% makes a triangular lattice

% lattice constant
/a 2 def

% radius of lattice entities
/r 0.25 def

% size of the grid x and y
/sx 100 def
/sy 100 def

% a circle shape exists at each lattice element
/circle{
    gsave
    currentpoint translate
    newpath 0 0 r 0 360 arc fill stroke
    grestore
} def

% go to an origin well below this dingle
sx neg 0 moveto
% iterate over all columns by the lattice constant
0 a sy 2 mul {
    a 0 rmoveto
    gsave
    % make the lattice diagonally
    0 a sx 3 sqrt mul { a 2 div a 3 sqrt 2 div mul rmoveto circle } for
    grestore
} for


%%EOF