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Universal Grammar

The Angle sibling — rules governing constructions. Why meanings emerge from geometry through the Erlangen Program.

“A geometric object IS its symmetry group.”

— Felix Klein, Erlangen Program (1872)

Two Classification Systems

Syntax and Grammar classify the same objects differently — and both are needed.

SystemBasisCategoriesUsed By
SortΣ_UL algebraic signaturee, r, m, aSyntax
SymmetryErlangen Program (Klein)Noun, Verb, Adjective, DeterminerGrammar

Sort tells you what algebraic operations apply. Symmetry tells you how the symbol behaves linguistically.

Symmetry → Parts of Speech

The Grammar classifies symbols by their symmetry properties, just as natural languages classify words by their grammatical role.

Noun

High rotational symmetry

A circle (○) or point (•) looks the same from every angle. High symmetry = entity/object.

○ (circle)• (point)□ (square)

Verb

Low rotational symmetry

An arrow (→) has a preferred direction. Low symmetry = action/relation with directionality.

→ (directed line)⌒ (curve)

Adjective

Reflection symmetry

An angle (∠) is symmetric about its bisector. Reflection symmetry = modifier/quality.

∠ (angle)Modifiers

Determiner

Scale invariance

Quantification is realized by scaling an entity relative to its frame. Scale = scope.

∀ (fills frame)∃ (point in frame)

The Erlangen Hierarchy

Meaning in UL operates at three geometric levels. Each level preserves different properties and contributes different kinds of meaning.

1

Level 1: Topology

Group: Homeomorphisms
Preserved: Connectedness, containment, adjacency

The most abstract structural properties — is something inside, outside, or connected?

2

Level 2: Affine

Group: Affine transformations
Preserved: Parallelism, ratios, betweenness

Order and proportion — which comes first, which is between, what ratio relates them?

3

Level 3: Euclidean

Group: Isometries (rigid motions)
Preserved: Distances, angles, shapes

The full geometric content — specific angles, lengths, and shapes carry precise meaning.

Word Formation Procedure

  1. 1

    Decompose: Break your thought into sorts (Entity, Relation, Modifier, Assertion)

  2. 2

    Select primitives: Choose the geometric primitives that realize each sort

  3. 3

    Apply operations: Use the 11 Σ_UL operations to compose primitives

  4. 4

    Check well-formedness: Verify all 6 Syntax rules are satisfied

  5. 5

    Read symmetry: Verify the symmetry classification produces the intended linguistic role

  6. 6

    Verify at each Erlangen level: Does topology, affine, and Euclidean meaning all align?

Interactive Grammar Tools

Circle

NounRotation symmetry
Rotational order∞
Reflection axes∞

Infinite rotational symmetry — looks the same from every angle. Maximum symmetry → entity/object (Noun).

Relationship to Siblings

Grammar explains why meanings emerge — the semantic theory. Lexicon records the results — the canonical definitions. Symbology shows what the atoms look like. Syntax governs which compositions are valid, and the Thesaurus maps paths between related meanings.