Ganja Wedge Game

ganja-wedge-game

Compositional game theory meets Geometric Algebra via ganja.js

Version: 1.0.0
Trit: +1 (PLUS - generative/extension)
Color: #8F8 (from wedge game UI)

Overview

The Wedge Game by Steven De Keninck teaches Projective Geometric Algebra (PGA) through direct manipulation. This skill extracts its implicit player mechanics and maps them to compositional game theory.

Core Algebra: PGA2D = Cl(2,0,1)

var PGA2D = Algebra(2,0,1);  // 2 positive, 0 negative, 1 zero dimension

// Basis: 1, e0, e1, e2, e01, e02, e12, e012
// Metric: e0²=0, e1²=1, e2²=1

// Points are bivectors (grade 2)
var point = (x,y) => 1e12 - x*1e01 + y*1e02;

// Lines are vectors (grade 1)  
var line = (a,b,c) => a*1e1 + b*1e2 + c*1e0;

Game Operations as Player Skills

Skill 1: Join (∧ Wedge for Points → Lines)

Drag points APART → Join them into a line

P₁ & P₂ = P₁ ∨ P₂ (regressive product = dual of wedge of duals)

Skill 2: Meet (∧ Wedge for Lines → Points)

Drag lines TOGETHER → Meet them at a point

l₁ ^ l₂ = l₁ ∧ l₂ (exterior product)

Skill 3: Add (+ When Same Grade)

Drag same-type elements in SAME direction → Add them

P₁ + P₂ (point addition)
l₁ + l₂ (line addition)

Skill 4: Dot (· Contraction)

HOLD a point, drag a line TO it → Project line through point

l << P = l · P (left contraction)

Grade ↔ Trit Mapping

Implicit Player Skills (What Players Learn)

Level 1: Incidence

• Points lie on lines (bivector ∧ vector = 0 test)
• Lines pass through points (same test, dual view)

Level 2: Construction

• Orthocenter: ((A&B)<<C) ^ ((B&C)<<A)
• Circumcenter: ((A&B)<<(A+B)) ^ ((B&C)<<(B+C))
• Euler line: orthocenter & circumcenter

Level 3: Transformations (Not in basic game)

• Rotors: cos(θ/2) + sin(θ/2)*bivector
• Translators: 1 + 0.5*ideal_line

Open Games Mapping

From open-games and unwiring-arena:

-- Wedge Game as Open Game
WedgeGame : Lens_{(Ω,℧)}(State, Costate)

where:
  Ω = Strategy profiles (drag gestures)
  ℧ = Reward vectors (score: 1000 - time/1000 - moves*10)
  State = Current level elements
  Costate = Target (red) elements

Play/Coplay Structure

// Play: Forward pass (player action)
play: (gesture, state) => {
  if (gesture.type === 'apart') return state.Vee(target);  // Join
  if (gesture.type === 'together') return state.Wedge(target);  // Meet
  if (gesture.type === 'hold+drag') return state.Dot(target);  // Project
  return state.Add(target);  // Add
}

// Coplay: Backward pass (score feedback)
coplay: (gesture, state, reward) => {
  // Check if result matches required (red) elements
  const match = required.some(r => 
    state.Sub(r).Length < 0.01 || state.Add(r).Length < 0.01
  );
  return match ? reward + 100 : reward - 10;
}

Arena Protocol Integration

{
  "arena_id": "wedge-game-pga2d",
  "seed": 1069,
  "players": [
    {
      "id": "student",
      "trit": 0,
      "skills": ["join", "meet", "add", "dot"]
    }
  ],
  "channels": {
    "play": "gesture → element construction",
    "coplay": "score → skill reinforcement"
  }
}

Dependencies

GF(3) Triads

open-games (-1) ⊗ ganja-wedge-game (+1) ⊗ bisimulation-game (0) = 0 ✓
unwiring-arena (0) ⊗ ganja-wedge-game (+1) ⊗ temporal-coalgebra (-1) = 0 ✓

Implementation Sketch

// ganja.js Wedge Game with Open Game semantics
Algebra(2,0,1, function() {
  const origin = 1e12, EX = -1e02, EY = 1e01;
  
  // Player skill interface
  class PlayerSkill {
    constructor(name, trit, operation) {
      this.name = name;
      this.trit = trit;  // GF(3): -1, 0, +1
      this.operation = operation;
    }
    
    apply(a, b) {
      return this.operation(a, b);
    }
  }
  
  // Implicit skills from gesture detection
  const skills = {
    join: new PlayerSkill('join', +1, (a,b) => a.Vee(b)),
    meet: new PlayerSkill('meet', +1, (a,b) => a.Wedge(b)),
    add:  new PlayerSkill('add',  0,  (a,b) => a.Add(b)),
    dot:  new PlayerSkill('dot', -1,  (a,b) => a.Dot(b))
  };
  
  // Gesture → Skill mapping (from wedge_game.html)
  function detectSkill(gesture, selection) {
    const [a, b] = selection;
    const apart = gesture.direction === 'divergent';
    const together = gesture.direction === 'convergent';
    const held = gesture.held !== null;
    
    if (apart && a.grade === 2 && b.grade === 2) return skills.join;
    if (together && a.grade === 1 && b.grade === 1) return skills.meet;
    if (held) return skills.dot;
    return skills.add;
  }
  
  // Score function (from wedge game)
  function score(moves, timeMs) {
    return 1000 - timeMs/1000 - moves*10;
  }
  
  return { skills, detectSkill, score };
});

Levels as Skill Progression

Commands

# Open wedge game in browser
open https://enkimute.github.io/ganja.js/examples/example_game_wedge.html

# Run local ganja.js
npm install ganja.js

# Julia equivalent via Grassmann.jl
julia -e 'using Grassmann; @basis ℝ^(2,0,1)'

References

• ganja.js GitHub - Steven De Keninck
• Wedge Game
• bivector.net - GA community
• Ganja.jl - Julia interface
• Grassmann.jl - Native Julia GA

Autopoietic Marginalia

The interaction IS the skill improving itself.

Every use of this skill is an opportunity for worlding:

• MEMORY (-1): Record what was learned
• REMEMBERING (0): Connect patterns to other skills
• WORLDING (+1): Evolve the skill based on use

Add Interaction Exemplars here as the skill is used.

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