⊕

Glyph Composition

How primitives combine in space to create meaning

You know the five geometric primitives and the eleven operations. Now learn how they combine in two-dimensional space — the spatial grammar that turns individual marks into composite meanings.

The Five Spatial Relationships

Two marks on a surface can relate in exactly five ways. No more, no fewer.

Why only five?

Why only five? By the Jordan Curve Theorem, each closed mark partitions the plane into interior and exterior. Two partitions interact as: containment, intersection, adjacency, or separation. Connection adds a bridging mark to separation. Five cases, exhaustive.

Relationship	Operator	Topology	Semantic
Containment	⊕	A ⊂ B (proper subset)	Member of / instance of / specification of
Intersection	⊗	A ∩ B ≠ ∅, A ⊄ B, B ⊄ A (partial overlap)	Shared properties, logical AND, meaning blending
Adjacency	⊙	∂A ∩ ∂B ≠ ∅, int(A) ∩ int(B) = ∅ (boundaries touch, interiors disjoint)	Alternatives (OR) or sequential
Separation	‖	A ∩ B = ∅, ∂A ∩ ∂B = ∅ (completely disjoint)	Independent / unrelated; gap size encodes degree of independence
Connection	—(r)	A ∩ B = ∅, but a mark has one endpoint in A and one in B	Explicit relation; the connecting mark IS the relation

From Operations to Space

The 11 Σ_UL operations each manifest as a specific spatial arrangement.

11 Operations (Σ_UL)

Spatial Manifestation

Select an operation to see its spatial relationship

When you write conjoin(a, b), you’re drawing a ⊗ b — overlapping frames. The shared area IS the conjunction. This is what ‘a construction’s meaning IS its geometry’ means in practice.

Composition Parameters

Beyond which spatial relationship you choose, five parameters control the nuance of meaning.

Relative Scale

Dominance ↔ Subordination

A >> B— A is primary; B is detail/modifier

A ≈ B— Co-equal participants

A << B— A is subordinate to B

The relative sizes of composed marks carry meaning. Scale ratio is continuous — there are no discrete breakpoints. Larger marks dominate; smaller marks are subordinate detail.

Relative Orientation

Agreement ↔ Opposition

0° (aligned)— Agreement, identity, parallelism

60°— Harmony

90° (perpendicular)— Independence, orthogonality

120°— Complementarity

180° (opposed)— Contradiction, contrast

The angular relationship between composed marks. Orientation preserves or breaks symmetry — two identical shapes at 0° maintain symmetry; at other angles, new emergent symmetries can arise.

Z-Ordering

Salience ↔ Background

A on top— A is foreground — more salient, more recent

Equal— Both equally present — true blending

B on top— B is foreground — more salient, more recent

When marks overlap, which is visually on top carries meaning. In handwritten UL, z-ordering follows drawing order — what you draw last is on top. In digital UL, z-ordering must be explicit.

Alignment

Structural relationship

Center-aligned— Symmetric relationship — about the center

Offset vertically— Hierarchical — upper = prior/cause, lower = posterior/effect

Offset horizontally— Sequential — left = before, right = after

Offset diagonally— Both hierarchical and temporal

Where composed marks sit relative to each other's centers. Alignment encodes the structural nature of the relationship between the components.

Grouping

Evaluation order / Scope

Enclosed— Everything inside composes first, then the unit composes outward

Unenclosed— Default left-to-right, outside-in evaluation

Enclosure boundaries mark groups — everything inside composes first (forms a unit), then the unit composes with what's outside. Enclosures ARE parentheses — they determine evaluation order.

Every composite glyph occupies a point in a 5-dimensional semantic space: spatial relationship × scale × orientation × z-order × alignment × grouping.

Three Ways to Build

There are exactly three strategies for increasing a glyph’s definition.

Deepening

Vertical — add a nesting layer

Add an enclosing layer around the existing glyph. Each layer adds categorization or specification. This is vertical composition — increasing depth.

• → △{•} → □{△{•}} → ○{□{△{•}}} — from "exists" to "universally organized structural fundamental existence"

Broadening

Horizontal — add adjacent or connected marks

Add marks at the same depth level via adjacency or connection. This relates the symbol to other concepts without deepening the nesting hierarchy.

• → • → • → • → • ← • — from "exists" to "two things act on a mediator"

Blending

Overlap — intersect with another symbol

Overlap the glyph with another symbol. The intersection zone creates emergent meaning that neither component has alone.

△ → △ ⊗ ○ → △ ⊗ ○ ⊗ □ — from "fundamental" to "fundamental AND universal AND structural"

Composition Depth Ladder

0Atomic≈ Phoneme / morpheme•, ─, ∠, ◠, ○

1Simple compound≈ Root word△{•}, •→•, ∠60°, ○{◠}

2Complex compound≈ Compound word / phrase○{△{•}}, [•→•], •→○{•}

3Embedded structure≈ Clause[FACT[•→•]→•], ○{[•→•] ∠60° [•→•]}

4Recursive compound≈ Sentence○{○{○{•}}}, [FACT[FACT[•→•]→•]→•]

Monotonic Definition Principle

Each layer of composition can only increase specificity, never decrease it. Deeper means more specific, never vaguer.

The Formal Algebra

Show formal definition

GCA = (M, ⊕, ⊗, ⊙, ‖, —, ε)

M = M = set of all marks (glyphs) on the glyph space GS

⊕ = A ⊕ B = "B inside A"

⊗ = A ⊗ B = "A overlapping B"

⊙ = A ⊙ B = "A touching B"

‖ = A ‖ B = "A apart from B"

—(r) = A —r— B = "A related to B by r"

ε = ε = empty mark (Void). ε ⊕ A = A (void inside anything = just the thing).

Property	⊕	⊗	⊙	‖	—(r)
Commutative	✗	✓	✗	✓	✗
Associative	✓	✓	✓	✓	✓
Identity (ε)	✓	—	—	—	—
Idempotent	✗	✓	✓	✓	✗
Depth-increasing	always	sometimes	sometimes	no	sometimes

Containment (⊕) is non-commutative: A ⊕ B ≠ B ⊕ A. ‘Triangle inside Circle’ means something fundamentally different from ‘Circle inside Triangle.’ The container determines the category.

Interaction Laws

1. Containment absorbs connection

If A ⊕ B and B —r— C, then A ⊕ (B —r— C)

A contains [B related to C]. The connection is internalized within the container.

2. Intersection distributes over adjacency

A ⊗ (B ⊙ C) = (A ⊗ B) ⊙ (A ⊗ C)

"A overlapping [B next to C]" = "[A overlapping B] next to [A overlapping C]."

3. Containment is transitive

If A ⊕ B and B ⊕ C, then A ⊕ C

C inside B inside A implies C is inside A. Nesting chains compose.

4. Separation is broken by connection

If A ‖ B and we add A —r— B, then ¬(A ‖ B)

Connection overrides separation. Adding a bridge removes independence.

5. Adjacency promotes to intersection

If A ⊙ B and overlap → ε⁺, then A ⊗ B

Adjacency is the limiting case of intersection as overlap approaches zero. Thickening the shared boundary promotes adjacency to intersection.

Compound Taxonomy

Composite glyphs fall into five natural families — one for each spatial relationship.

Nested glyph

Category hierarchy built through containment layers

• ∈ △ ∈ ○ — a fundamental instance of a universal concept

Blended glyph

Meaning fusion through overlap — the shared region is an emergent concept

△ ⊗ ○ — that which is both fundamental and universal

Sequential glyph

Compound word or disjunction through touching boundaries

△ ⊙ ○ — fundamental or universal; or fundamental then universal

Linked glyph

Relational structure through explicit bridging marks

A —r→ B — A relates to B through relation r

Constellation glyph

Field of independent concepts — related by shared context, not topology

△ ○ □ — three autonomous concepts in a shared field

What's Next?

Ready to practice? Build compositions with the Expression Builder, or see spatial relationships at work in the worked examples.

Practice composing →See it in action →

Symbology

The five primitives

Syntax

Well-formedness rules