File Format

General

AIFS is a text format for representation of the Iterated Function Systems (IFS). Any AIFS file consists of an arbitrary number of blocks (even millions). Usually each block corresponds to a family or a particular IFS. Every block starts with @ID:ParentID, where ID and ParentID are the unique identifiers of the block and its parent block. ID can be empty if we do not refer to the block from elsewhere. ParentID can be empty if the block has no parent. A block can use all definitions from its parent, and can override some of them. It is possible to call one block from another one as a function by ID. In that case the first variables in the called block are replaced by arguments, and the last variable is used as return value. @@import file_to_include.ext can be used to include another file with a relative path. Comments start with a #, and the program will ignore the rest of the line: #This is a comment.

Mathematical operations

To perform calculation with numbers and affine maps it is possible to use ordinary operations like +,-,*,/,^ and brackets. For numbers sin, cos, tan, asin, acos, atan, exp, log, floor, ceil, arg are defined. Also ‘if’ function defined: if (cond, val1, val2) is equivalent to (cond > 0)?val1 : val2 in C-like languages.

Identifiers

An identifier is a case-sensitive string that denotes a variable, an operator or a block. An identifier can consist of symbols [a-z], [A-Z], [0-9] and “_”, but cannot start with a digit. Identifiers beginning with $ and & symbols have a special meaning. Variable with a name beginning with ‘$’ symbol is considered as built-in variable and has special meaning. Variable with a name beginning with ‘&’ symbol is considered as substitution and recalculated in every place where used.

Vectors

Like in many other languages, v=[a1, a2, .., an] defines the vector of elements (a1, a2, .., an). We can access to the elements of a vector using square brackets, so a1 can be accessed as v[0], and an = v[n-1]. A vector of appropriate length can be automatically converted to a matrix or to a translation.

Affine maps

When $dim=n the following definitions of maps in R^n are available:

Translation [t1,t2, .., tn]

|x1’| |x1+t1|

|x2’| |x2+t2|

|…| = | … |

|xn’| |xn+tn|

Transformation matrix [a11,a12, .., a1n, .., an1, an2, .., ann]

|x1’| |a11 a12 … a1n| |x1|

|x2’| |a21 a22 … a2n| |x2|

|…| = |……………| |..|

|xn’| |an1 an2 … ann| |xn|

Companion matrix $companion([a0, a1, a2,…, a(n-1)]) for the polynomial a0 + a1*x + … + a(n-1)x^(n-1) + x^n.

Diagonal matrix $diagonal([a0, a1, a2,…, a(n-1)])

Exchange matrix $exchange()

Identity matrix $i

Operators

Composition of the maps (or operators): f1*f2*…*fn x => f1(f2(..(fn(x)))

Examples in R²: 2(z+[1,0]) == 2*[1,0]

2z+[1,0] == [1,0]*2

x’=a11*x+a12*y+b1 y’=a21*x+a22*y+b2 can be expressed as [b1,b2]*[a11,a12,a21,a22]

Union of the sets (or operators): S1|S2|…|Sm S1 U S2 U … U Sm. Examples:

  • (f1|f2)*A means f1(A) U f2(A)
  • H=f1|f2|…|fn is a Hutchinson operator
  • A=H*A is a definition of the attractor A

Power function f^n Composition f with itself n times f(f(f(…)))

Inverse map f^-1

Empty set $e

Templates

Templates describe infinite sets of affine maps together with a random distribution that can be used in the search procedure and in the editor window. Templates can be used in the same places where affine maps appear: within compositions, unions, etc.

$number(T) Real or integer number. T= $integer or $real : [optional] type and a random distribution of the number.

$vector(L, T) Vector of numbers, L = length of the vector, L = 0 or L=empty means L = $dim. T= $integer or $real : [optional] type and a random distribution for the vector entries.

$binary(L, T) Vector of empty sets ($e) and identity maps ($i). L = length of the vector, L = 0 or L=empty means L = $dim. T= $integer - represents the number of non-empty ($i) components.

$semigroup([g1, g2,…,gm], T) Element of the semigroup (possibly infinite) generated by the affine maps g1, g2, …, gm. T=$integer - [optional, for infinite semigroups only] represents normal distribution for the length of compositions of the generators.

$real(a, b), $integer(a, b) Type and uniform distribution for real or integer numbers from a to b.

$real(v), $integer(v) Type and (half-)normal distribution for real or integer numbers. |v| - variance. if v < 0 then distribution is normal. if v > 0 then distribution is half-normal.

Special variables

$n=… User-defined name of the IFS

$a=… Attributes, can be c(hecked) and/or h(idden)

$dim=n Dimension of all affine maps defined in the block. If $subspace is not used, then equal to dimension of the Euclidean space R^n where IFS will be generated.

$subspace=M i1 i2 … Invariant subspace of the matrix M where IFS will be generated i1, i2, … - indices of the Jordan blocks from Real Jordan Form of M. Resulting subspace is a product of the subspaces that corresponds to the blocks. If omitted, then whole space ($dim) will be used.

How to use Creator

Creator can be used to create/find a family of IFS. You can open creator from the main menu “File”->”New” or from the toolbar.

Usually you should:

  • Fill the “Graph” field
  • Optionally change other parameters
  • Press “Search” button
  • Press “Stop” button
  • Select a group (and polynomial) from the list
  • Press the “Create” button

After it the program will switch to the IFS search mode.

When you find sufficient number of IFS (that can have dimension less than you are expected), you can switch to another search mode (in the “Finder” window) by:

  • setting “#var maps” field to >=1
  • decreasing “#empty maps”

Sometimes you need many hours to find something interesting, sometimes only several seconds!

Examples

Chair or Square family One set: “A” that consists from 4 smaller pieces of itself with the same coefficients k We denote graph as: a.a.a.a or 4a Group=D4 Dihedral, 360/4=90 degree rotation + reflection, Poly=2 ***

Rauzy tile family One set: A that consists from 3 smaller copies of itself with coefficients k,k^2,k^3 We denote graph as: a.a2.a3 Group=C2 Cyclic, 360/2=180 degree rotation, Poly=x^3-x^2+x+1 ***

Robinson triangle family Two sets: A,B A consists from 3 smaller copies of A,A,B with coefficient k B consists from 2 smaller copies of A,B with coefficient k We denote graph as: a.a.b-a.b or 2a.b-a.b Group=D10 Dihedral , 360/10=36 degree rotation + reflection, Poly=-x^3+x^2+1

Shortcuts

IFS List

  • Home scroll to the first IFS
  • End scroll to the last IFS
  • PgUp scroll one screen up
  • PgDown scroll one screen down
  • Up move one IFS up
  • Down move one IFS down
  • Left build the previous set of the current IFS
  • Right build the next set of the current IFS
  • Shift Up move to the next unique IFS up (according to the latest sort column)
  • Shift Down move to the next unique IFS down (according to the latest sort column)

Other

  • Ctrl Tab switch two latest panes
  • Ctrl L switch full screen
  • Ctrl R reload the current file from disk
  • Ctrl S save file
  • Ctrl O open file
  • Ctrl = add current view to the list
  • Esc close/cancel current modal window
  • Shift F4 show/hide the current pane