AIFS Language Reference

AIFS is the program format used by IFStile and the ifslib rendering library. It describes iterated function systems ranging from simple 2D contractions to high-dimensional algebraic IFS projected onto the plane. The language is documented formally in Mekhontsev (2019), Appendix E .

File Structure

@@import other.aifs    # optional: include another .aifs file

@ID:ParentID           # start of a block; ID and ParentID are optional
# comment
$dim=2                 # dimension of the rational space (required per block)
...

A file contains any number of blocks separated by @ID:ParentID lines. Each block is a GIFS that may define multiple attractor sets. Each block inherits all definitions from its parent and can override them. ID may be empty (@); ParentID may be empty (no parent).

Rendering selection

• ifslib (used in this site) renders the first visible block and within it the first visible attractor set, unless $root overrides this.
• IFStile lets you browse all blocks and all sets within a block via its IFS List UI.

Variable Types

Operators

Precedence note: 2*[1,0] = 2(x+[1,0]) = 2x+[2,0], but [1,0]*2 = 2x+[1,0]. The general rule is [t]*M gives f(x) = M·x + t (standard affine form).

Vectors and Matrices

[a1, a2, ..., an] defines a vector of n elements, automatically converted to a translation or matrix by context:

• For $dim=2: a 2-vector is a translation, a 4-vector is a 2×2 matrix (row-major).
• For $dim=n: an n-vector is a translation, n²-vector is an n×n matrix.
• Element access: if v=[a1,..,an] then v[0]=a1, v[n-1]=an.

Affine Maps

The standard 2D affine map form is:

f=[tx,ty]*[a,b,c,d]   # f(x) = [[a,b],[c,d]]·x + [tx,ty]
[tx,ty]               # pure translation by [tx,ty]
[0.5,0,0,0.5]         # pure scaling by 0.5 (no translation)

Special Matrix Constructors

Example: s=$companion([1,-1,1,-1]) builds the companion matrix for x⁴−x³+x²−x+1 (the 10th cyclotomic polynomial), whose eigenvalues are primitive 10th roots of unity — two complex conjugate pairs. Index 0 selects the first complex pair (a 2D real Jordan cell), so $subspace=[s,0] alone projects to a 2D plane where the action of s is a rotation by π/5 = 36°.

Special Variables

When $dim=2 no $subspace is needed — the rational and rendering spaces coincide. For higher-dimensional rational forms use $subspace to select the 2D eigenplane where the attractor lives.

Mathematical Functions

Available for scalar expressions: sin, cos, tan, asin, acos, atan, exp, log, floor, ceil, arg, if — conditional (three forms):

• if(cond) → 1 if cond > 0, else 0
• if(cond, val1, val2) → val1 if cond > 0, else val2
• if(cond1, val1, cond2, val2, default) → multi-branch: first positive condition wins

Array Operations

Auto-generation syntax — a single array can contain any number of generator segments interleaved with static elements:

[static..., $, termCond, genExpr, static..., $, termCond, genExpr, static...]

• Static elements before, between, and after generator segments are appended as-is.
• $ — marks start of a generator segment; $ is shorthand for $(0) — a reference to the current array being built
• $() = $(0)() = current length of the array being built (equals current index during generation)
• $[-1] = $(0)[-1] = last element of the current array (i.e. the previously appended element)
• $(1), $(2), … — references to enclosing (parent, grandparent, …) arrays; useful when building nested arrays
• termCond — generate while termCond > 0; N-$() generates until total count reaches N
• genExpr — expression for each generated element
• tail... — zero or more static elements appended after generation stops

Examples:

• [0, 1, $, 7-$(), $[-1]+$[-2]] → 7 Fibonacci values [0,1,1,2,3,5,8]
  Stops by count: 7-$() becomes 0 once 7 elements have been appended.
• [1, $, 1000-$[-1], $[-1]*2] → powers of 2 while last element < 1000: [1,2,4,8,...,512]
  Stops by value: 1000-$[-1] goes non-positive once the last element reaches 1000.

Block Functions

Any block (@name) can be used as a reusable function:

@funcName
param1 = defaultVal    # parameter with default value
param2 = defaultVal
ret = expr             # return value

@
result = funcName(arg1, arg2)         # call by position
result = funcName()                   # call with all defaults
result = $new(funcName, param2, val)  # named override
field  = result.param1                # field access on block instance

• The last variable defined in the block is the return value. Naming it ret is a readable convention — the name has no special meaning.
• Block inheritance: @child:parent — child overrides parent variables; any block can be a parent.
• @:ParentID — anonymous child block, inherits from parent.

JavaScript Interop

AIFS files can embed ES module JavaScript before the first @@ separator:

export function funcName(arg) { return arg + 1; }
export const constArr = [11, 12];
@@

@
$init = funcName        # call JS function once at init
result = funcName(3)    # call JS function from AIFS
val    = constArr[1]    # access JS constant

• JS functions callable from AIFS must be exported.
• console.log(...) is available inside JS functions.
• $init = jsFunc calls jsFunc once during block initialization; the function can set this.varName = value to inject AIFS variables.

Attractor Equations

Ordinary IFS

S=(f1|f2|f3)*S    # S is the unique compact attractor of {f1,f2,f3}

The attractor equation is mandatory. Without it ifslib_init returns 0 and nothing renders.

GIFS — Generalized IFS (multiple attractors)

When a self-similar system defines two or more interleaved attractors, each gets its own equation. Attractor variables appear on both sides:

A1=g^-1*(h1*A1|h2*A1|h3*A2)
A2=g^-1*(h4*A1|h5*A2)

Here h*A denotes the image of attractor set A under map h. g must be an expansion (spectral radius > 1 after projection) to guarantee that the attractor exists; if g is a contraction, existence depends on the GIFS dependency graph. See the Robinson Triangles entry for a worked example.

Templates (Finder / Search)

Templates describe families of maps for use with the IFStile Finder module. They are not required for rendering with ifslib:

T can be a $real(a,b) or $integer(a,b) uniform distribution, or a number (log-variance of a normal distribution on ℝ, i.e. σ = e^(T/2)), or omitted (IFStile uses its Finder “log variance” search parameter as default).

Complete Examples

Sierpiński Triangle — simple 2D IFS

@
$dim=2
f1=[0.5,0,0,0.5]
f2=[0.5,0]*[0.5,0,0,0.5]
f3=[0.25,0.5]*[0.5,0,0,0.5]
S=(f1|f2|f3)*S

See: Sierpiński Triangle

Rauzy Fractal — 3D rational form with companion matrix

@
$dim=3
$subspace=[g,0]
g=$companion([-1,1,1])   # companion for x³+x²+x-1 (tribonacci)
A=(g^-1|g^-2*[0,1,0]|g^-3*[0,1,1])*A

See: Rauzy Fractal

Robinson Triangles — GIFS with 4D rational form

@
$dim=4
$subspace=[s,0]
s=$companion([1,-1,1,-1])   # 10th cyclotomic polynomial
r=$exchange()
g=s-s^4
h1=s^4*[-1,0,-1,0]
h2=s^2*r*[-1,0,0,0]
h3=s^9
h4=s^6*r*[0,-1,0,-1]
h5=s^3*[1,-1,0,-1]
A1=g^-1*(h1*A1|h2*A1|h3*A2)
A2=g^-1*(h4*A1|h5*A2)

See: Robinson Triangles

Reference

• Mekhontsev, D. (2019). An algebraic framework for finding and analyzing self-affine tiles and fractals . Doctoral thesis, Ernst-Moritz-Arndt-Universität Greifswald. Appendix E contains the full language specification.
• IFStile — the software package that implements the AIFS language.