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.
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.
Each variable can be assigned only once, see Static single assignment form, but the order of the definitions is not important.

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.

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.

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 (row-major) [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([a1, a2, a3,…, an)])

Exchange matrix $exchange()

Any number x can be implicitly converted to a matrix x*I where I is an n-dimensional Identity matrix.

$charpoly(A) computes the Characteristic polynomial (as an n-dimensional vector) for linear part of the affine map A.

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= $integer - represents the number of empty ($e) 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.

Editor

There are 4 editing modes:

Controls - block-specific parameters (if available, see Examples/IR3).
Special - additional information that will be stored in the block (camera, background, etc).
Source - source code of the current block.
JS - Javascript code located at the beginning of the current file.

Buttons:

Apply - write changes back to original block.
+Block - add the modified copy to the main list.
Rand - randomize available parameters.

Mouse drag modifiers for numeric controls:

Shift - increase change magnitude.
Alt - decrease change magnitude.
Control - switch to keyboard editing.

Animation

Batch tab

“Render” button - automatically renders specific blocks to a user defined folder.

Interpolation tab

“Create frames” button - creates intermediated blocks and adds them to the list - without rendering. It uses checked blocks as keyframes (see Examples/Animation). Created blocks can be rendered later using “Batch” tab.

Location

The location window can be used to control screen/camera position and section settings.

There are additinal useful controls:

Toolbar buttons: “Fit to screen” and “Zoom out”.
Tools menu: “Zoom box”, “Rotate” and “Pan” items to map the corresponding action to the first mouse button. Other actions will be automatically mapped to other buttons.
Mouse wheel: to move camera forward or backward.

3D

There are 4 paramters that fully control position:

Camera location
Target point
Vertical direction
Field of view

Distance from the camera location to the target point is a very important for animation interpolation and 3D navigation. This distance can be considered as a “zoom factor”. To support this, there are 2 modes: Zoom (default) and Fly. The distance to the target is locked in the ‘Fly’ mode. In simple animation scenarios, it is better to separate zoom and “Fly” at a fixed zoom.

When the mouse is hovered over a 3D point on the screen, the relative depth (Zt) of the point is displayed in the status bar. This value is equal to the distance from the camera plane to the point divided by the distance from the camera to the target.

In the ‘Fly’ mode, a ‘Sensitivity’ parameter can be used to adjust speed of move controls. Mouse click to any 3D point on the screen will automatically set this parameter (equal to ‘Zt’).

“Pick” button can be used to pick target point from the screen point by mouse click. Also, in the “pick” mode you can see 3D point coordinates under the mouse cursor.

At any time, you can quickly save and load camera and section parameters using the “Save” and “Load” buttons at the top of the window.

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

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
F11 switch full screen
Ctrl R reload 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
F1 show context help
Ctrl Q reload current help

Numeric controls modifiers

Shift - fast change
Alt - very slow change
Alt+Shift - slow change
Ctrl - keyboard input