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Style Usage

Given a domain schema (specifying the domain of the diagram) and a substance program (specifying what to draw), the style schema describes the recipe of drawing the objects and relations on a canvas.

A style schema is composed of blocks, of which there are two types:

  • namespaces, which can be used to help program organization;
  • style blocks, which carry out the meat of the diagram generation.


The syntax for a namespace is as follows:

namespace_name {
    ... (the namespace body)

Refer to this section for a detailed explanation of what may appear in the body of a namespace.

Values declared within a namespace can be read outside of the namespace using the "dot" operator:


Hence they are also called global variables. Overwriting these values is not allowed.

Canvas Preamble Block

Each style schema must contain a canvas preamble block, a special type of namespace which describes the width and height of the canvas. For example, preamble block

canvas {
    width = 800
    height = 700

tells Penrose that the drawing canvas should have a width of 800 pixels and a height of 700 pixels.

Style Blocks

Style blocks are the most important component in a style schema, since they actually describe how to draw elements of a diagram. The syntax for style blocks is as follows:

forall list_object_declarations
where list_relations {


  • list_object_declarations is a semicolon-separated list of object declarations, similar to the object declarations in the substance schema. Each object declaration has syntax type_name object_name. The names declared in list_object_declarations are referred to as style variables.
  • list_relations is a semicolon-separated list of constraints (about objects in list_object_declaration) that must be satisfied in order for this style block to be triggered.
  • list_body_expressions is the body of this style block, containing statements that represent the computational and graphical aspects of the diagrams that are triggered when this style block is triggered. Refer to this section for a detailed explanation of what may appear in the body of a style block.

If list_relations is empty, then the clause where ... needs to be omitted.

In the set-theory example, a style block may look like

forall Set x {


forall Set x; Set y
where IsSubset (x, y) {

Matching style block against substance program in general

Penrose functions by matching a style block against a substance program. In a nutshell, given a style block

forall Set x; Set y
where IsSubset (x, y) {

the Penrose compiler searches through the substance program to find sets of objects consistent with Set x; Set y such that IsSubset(x, y) is satisfied. This is done through generating mappings from style variables to substance variables, which are the objects in the substance program.

For instance, consider a simple set-theory substance program that works with the previous style block:

Set A, B, C
IsSubset (A, B)
IsSubset (B, C)

By matching the style block against the substance program, we essentially consider six possible mappings (note that repeated elements are not allowed), some of which are valid and some are invalid:

MappingIsSubset(x, y) becomesSatisfied by substance
x -> A; y -> BIsSubset(A, B)Yes
x -> A; y -> CIsSubset(A, C)No
x -> B; y -> AIsSubset(B, A)No
x -> B; y -> CIsSubset(B, C)Yes
x -> C; y -> AIsSubset(C, A)No
x -> C; y -> BIsSubset(C, B)No

Here, Penrose filters out mappings which do not satisfy the constraints listed in the style block, and keeps a list of good mappings (in this example, two mappings are kept). For each good mapping, the body of the style block (list_body_expressions) is executed, where each instance of the style variables (x and y) is substituted with the corresponding substance variables (once with A and B, once with B and C).

Object Declarations

In the list of object declarations in a style block, we can declare two types of objects, which are matched differently by the Penrose compiler.

Substance objects

We can declare a substance object, whose object name is surrounded by backticks. For instance,

forall Set `A` {

can only be mapped to the substance object with the exact same name (A) provided that the types match (subtyping allowed). In other words, given substance program

Set A, B, C

matching the style block against the substance block yields only one valid mapping: `A` -> A.

Style objects

If an object name is not surrounded by backticks, then this object is a style object with a style variable. As seen before, the Penrose compiler will try to map style variables to any substance objects, provided that their types match (subtyping allowed).

Allowed Relations

A style block supports three types of relations, two of which can also be seen in the substance program.

Predicate Applications

Just like in the substance program, each predicate application has syntax

predicate_name (argument_list)

where elements of argument_list can refer to objects declared in list_object_declarations, or be other predicate applications. The types must still match, allowing subtyping.

Optionally, one can give an alias to a predicate application:

predicate_name (argument_list) as alias_name

If such an alias is set, then alias_name will be accessible in the style block body, and it will always refer to the version of the predicate application within the substance program.


If a predicate is declared as symmetric, then it gets special treatment. Suppose we have the following domain schema:

type Atom
type Hydrogen <: Atom
type Oxygen <: Atom
symmetric predicate Bond (Atom, Atom)

and the following style block:

forall Hydrogen h; Oxygen o
where Bond (h, o) {

The style block will successfully match the following substance schema:

Hydrogen H
Oxygen O
Bond (O, H)

where Bond (h, o) in the style block matches against Bond (O, H) in the substance schema. Because Bond is declared symmetric, when Penrose looks for Bond (h, o), it also looks for Bond (o, h) and finds a match. In other words, the matching algorithm handles the equivalence between Bond (h, o) and Bond (o, h) correctly.

Function and Constructor Applications

Each function or constructor application has syntax

object_name := function_name (argument_list)

We do not allow aliasing for function and constructor applications. Arguments in argument_list must have types that match the domain argument types, similar to the substance schema.

Object Property Relations

Aside from predicate applications and function (constructor) applications, Penrose also supports a predicate-like relation that checks whether an object has a certain property, say label. For instance, we may write

forall Set s
where s has label {
    ... some code that uses s.label

If a certain Set A in the substance program does not have a label (perhaps due to NoLabel declarations), then s will not be mapped to A, thus preventing an access of nonexistent properties.

We can further distinguish between math labels and text labels (see substance labeling): where p has math label matches math labels, whereas where p has text label matches text labels.

Matching Deduplication

The matching algorithm is designed to avoid duplicated mappings. If two mappings give us the same set of matched objects (in the substance program) and the equivalent set of matched substance relations (predicate applications and function or constructor applications), then the algorithm only triggers on one of them.

For instance, say Penrose tries to match the style block

forall Set x; Set y {

against substance program

Set A, B

Then, only one of mappings x -> A; y -> B and x -> B; y -> A triggers the Style block.

Reserved Variables

Within a style block body, some variable names are reserved for metadata purposes:

  • match_count is an integer that refers to the number of times that this style blocks will be triggered (or matched) in total; and
  • match_id is the 1-indexed ordinal of this current matching.

These values can directly be read or overwritten within the style block body if needed.

Block Body

The body of a block contains declarations of variables, shapes, and the relationship between objects.


We can assign an expression to a field:

type_annotation field = expression


  • type_annotation is an optional field denoting the type of the variable,
  • field is a path to the variable being assigned, and
  • expression is the expression to be assigned to field.

field can either be

  • A single identifier, which denotes a local assignment, not accessible outside of this matching; or
  • An object name (defined in list_object_declarations) or predicate application alias, followed by a dot operator and an identifier, which denotes an assignment bound to a substance instance of object or predicate application after we substitute in the mapping. These assignments are accessible if the same substance object or predicate application is matched again.

For example, consider the following style block:

forall MyType t1; MyType t2
where MyPredicate (t1, t2) as r1 {
	x = ... // this is a local assignment not accessible outside of this substitution or this block
	t1.a = ... // this is bound to the substance instance of `MyType t1`
	r1.c = ... // this is bound to the substance instance of `MyPredicate (t1, t2)`

Refer to this section for a detailed explanation of the available expressions and their associated types.

Override and Deletion

The style language allows users to modify fields that are previously declared. The override keyword changes the value of the field. As an example,

forall Set X {
    shape X.shape = Circle {
        x: X.x
        r: 100

forall Set `A` {
    override `A`.shape.r = 200

the radius of the circle for every Set is 100, except if the Set has name A, then the radius is 200.

Deletion of fields works similarly, with the delete keyword. This feature can be helpful for, e.g., removing visual elements for a subtype. For instance,

-- by default, draw a circle for all instances of type T
forall T x {
    x.widget = Circle { ... }

-- but don't draw this circle for instances of a subtype S <: T
forall S x {
    delete x.widget

Note that one must be careful not to reference deleted attributes in a later generic block. For instance, the following block will produce an error if invoked for an instance of S:

forall T x {
    shape x.newWidget = Circle {
        center : -- not defined for instances of S

Constraints and Objectives

A good diagram must satisfy some basic constraints, while trying to optimize upon some objectives (specifying diagram beauty). We declare these constraints and objectives within the style blocks. A constraint declaration has syntax

ensure constraint_name (argument_list)

and an objective declaration has syntax

encourage objective_name (argument_list)

where argument_list may refer to constant values, global / local variables, and other variables bound to substnace instances of objects and predicate applications. A full list of available constraints and objectives can be found here.

We also provide syntax sugar expressions for some commonly-used objectives and constraints. In particular,

  • a > b is the syntax sugar for the constraint / objective greaterThan(a, b),
  • a == b is the syntax sugar for the constraint / objective equal(a, b), and
  • a < b is the syntax sugar for the constraint / objective lessThan(a, b).


We can specify the layering between two shapes (particularly useful when two shapes overlap) using layering statements: either

layer shape_1 above shape_2


layer shape_1 below shape_2

where shape_1 and shape_2 can be variables assigned to shapes.

We have special handling of layering statements for Group shapes, found here.

Expressions and their Types

The list of supported Style types is:

  • scalar
  • int
  • bool
  • string
  • path
  • color
  • file
  • style
  • shape
  • vec2, vec3, vec4
  • mat2x2, mat3x3, mat4x4
  • function
  • objective
  • constraint

These are what may appear in the optional type_annotation field of field assignments.


Shape declarations have syntax

shape_name {
    property_name_1 : value_1
    property_name_2 : value_2

Once declared, the value of each property can be accessed using


For example,

forall Set x {
    -- declares a circle with radius 50
    x.shape = Circle {
        r : 50
    -- set its center to be (50, 100)[0] = 50[1] = 100

Each property of a shape has a default value. A full list of available shapes and their properties (and their default values) can be found in the Shape Library.

Unknown Scalar

The ? expression evaluates to a scalar whose value is automatically determined by the Penrose engine.


Strings have type string and string literals are delimited by double quotes. Strings can be concatenated using the + operator. For instance, to put parentheses around the label associated of x, write

string fancyLabel = "(" + x.label + ")"

Vectors and matrices

Style supports dense n-dimensional vector and matrix types, and standard operations on these types. These types behave largely like small, dense matrix types found in other languages (such as GLSL), with some specific differences noted below. Note that these types are meant largely for manipulating small 2D, 3D, and 4D vectors/matrices in the context of standard graphics operations (transformations, perspective projection, etc.), and may not perform well for larger matrix manipulations. Like all other objects in Style, the value of any vector or matrix entry can be declared as unknown (?) and determined automatically via optimization by the layout engine.

Vector and matrix types

Style is designed to support n-dimensional dense vectors of type vecN, and square n-dimensional matrices of type matNxN, where in both cases N is an integer greater than or equal to 2. E.g., types commonly used for diagramming are vec2 and mat3x3. Some library functions may be available only for vectors or matrices of a specific size (e.g., the function cross(u,v), which computes a 3D cross product, assumes that both u and v have type vec3).

Initializing vectors and matrices

A vector is constructed by specifying its components. For instance,

vec2 u = (1.23, 4.56)

constructs a 2-dimension vector with x-component 1.23 and y-component 4.56. As noted above, unknown values can be used as components, e.g.,

vec2 p = (?, 0.0)

specifies a point p that sits on the x-axis with an unknown x-coordinate which is determined by the optimizer, according to any constraints and objectives involving p. More advanced initializers (e.g., initializing a 3-vector from a 2-vector and a scalar) are currently not supported, but are planned for future language versions. In most cases, the same functionality can currently be emulated by directly referencing components of a vector, e.g.,

vec3 a = ( b[0], b[1], 1.0 )

A matrix is constructed by specifying a list of vectors. Each vector corresponds to a row (not a column) of the matrix. For instance,

mat2x2 A = ((1,2),(3,4))

initializes a 2x2 matrix where the top row has entries 1, 2 and the bottom row has entries 3, 4. Rows can also reference existing vectors, e.g.,

vec2 a1 = (1, 2)
vec2 a2 = (3, 4)
mat2x2 A = (a1, a2)

builds the same matrix as above. As with vectors, matrix entries can be unknown. E.g.,

scalar d = ?
mat3x3 D = ((d, 0, 0), (0, d, 0), (0, 0, d))

describes a 3x3 diagonal matrix, where all three diagonal entries take the same, undetermined value d.

Vector and matrix element access

Individual elements of a vecN can be accessed using square brackets, and an index i between 0 and N-1 (inclusive). For instance,

vec3 u = (1, 2, 3)
scalar y = u[1]

will extract the y-coordinate of u (i.e. y=2). Matrix entries are similarly accessed:

mat2x2 M = ((?, ?), (?, ?))
scalar trM = M[0][0] + M[1][1]

constructs an expression for the trace of M. In this case, since the elements of M are declared as unknown scalars, the value of the trace will depend on the entry values of the optimized matrix.


Colors have type color, and include an alpha (i.e., opacity) channel. Colors can be specified via two different color models:

  • rgba(r, g, b, a) defines a color via the RGB color model, with red, green, blue, and alpha values in the range [0, 1].
    • If the required color is fixed and known, one can also use the hexadecimal representation #rrggbbaa, which gets converted to the corresponding color in rgba(r, g, b, a). If the alpha value is not provided in the hexadecimal representation, it defaults to 1.0.
  • hsva(h, s, v, a) defines a color via the HSV color model, with hue in the range [0, 360], saturation and value in the range [0, 100], and alpha in the range [0, 1].

To specify that a color should be omitted altogether, you can also use none(). E.g.,

fillColor: none()
strokeColor: none()

Note that none() is different from using a 100% transparent color: it really prevents the fill or stroke from being drawn altogether.

Computation Functions

Computation functions take in multiple values, and return a new value. The aforementioned functions rgba, hsva, and none are examples of computation functions. Other computation functions also exist, including many mathematical functions.

See the list of computation functions.

Operations between Numerical Expressions

Aside from concatenation of strings using the operator +, the style language supports the operations listed below on scalars, vectors, and matrices. Here we assume that c and d have type scalar, u and v have type vecN, and A and B have type matNxN (all for the same N).

Note that elementside multiplication .* and division ./ get applied independently to each entry of a vector or matrix. For instance, if u = (6, 8, 9) and v = (3, 2, 3), then the elementwise dicision operation u ./ v yields the vector (2, 4, 3) (i.e. six divided by three, eight divided by two, and nine divided by three).


  • c + d - sum of c and d
  • c - d - difference of c and d
  • c * d - multiplies c and d
  • c / d - divides c by d


  • c * v — scales v by c (from the left)
  • v * c — scales v by c (from the right)
  • v / c — divides v by c


  • c * A — scales A by c (from the left)
  • A * c — scales A by c (from the right)
  • A / c — divides A by c


  • u + v — sum of u and v
  • u - v — difference of u and v
  • u .* v — elementwise product of u and v
  • u ./ v — elementwise quotient of u and v


  • A*u — matrix-vector product Au
  • u*A — matrix vector product uᵀA


  • A * B — matrix-matrix product AB
  • A + B — sum of A and B
  • A - B — difference of A and B
  • A .* B — elementwise product of A and B
  • A ./ B — elementwise quotient of A and B
  • A' — matrix transpose Aᵀ