GeoMath

GeoMath is a harmonic mathematical framework that defines the fundamental spatial relationships, transformations, and resonance patterns governing geometric structures in the DragonFire ecosystem, enabling operations that preserve mathematical beauty and coherence.

Introduction

GeoMath represents the mathematical soul of the DragonFire system—a comprehensive framework for harmonic mathematics of space that defines how geometric forms relate, transform, and resonate with one another. While GeoCode provides the symbolic encoding of geometric structures, GeoMath supplies the mathematical operations and principles that govern these structures and their transformations.

At its core, GeoMath implements a system of mathematical operations based on fundamental constants (φ, π, √2, √3) that maintain harmonic relationships during transformations. This approach enables mathematical operations that preserve the inherent beauty and coherence of geometric structures across transformations and dimensional shifts.

GeoMath views geometry not as a static construct but as a dynamic, resonant system where forms can transform between states while maintaining mathematical relationships. This perspective aligns with both ancient mathematical traditions and cutting-edge approaches to geometric computation.

Core Principle: GeoMath operates on the principle that mathematical beauty and coherence can be preserved through transformations by adhering to harmonic relationships based on fundamental constants. These relationships create resonance patterns that govern the behavior of geometric structures across different dimensions and transformational states.

Key Concepts

Harmonic Relationships

Mathematical connections between geometric structures based on ratios derived from fundamental constants, creating coherent patterns across transformations.

Resonance Patterns

Geometric configurations that exhibit heightened stability or coherence at specific ratios, mimicking the physical concept of resonance in mathematical space.

Transformation Preservation

Mathematical operations that maintain key geometric properties during transformations between different forms and dimensions.

Constant-Based Scaling

Scaling operations based on mathematical constants (φ, π, √2, √3) that preserve harmonic relationships between geometric structures.

Vector Form Mathematics

Mathematical framework that treats geometric structures as dynamic vector forms capable of continuous transformation rather than discrete entities.

Harmonic Wave Functions

Mathematical functions that describe the wave-like behaviors of geometric transformations across dimensional spaces.

Harmonic Constants

GeoMath is fundamentally built upon a set of mathematical constants that create harmonic relationships in geometric space:

Phi (φ)

1.618033988749895...

The golden ratio, which governs the most harmonious proportions in nature and art. In GeoMath, φ is used for:

Icosahedral/dodecahedral transformations
Golden spiral vector paths
Natural growth and scaling patterns
Optimal boundary conditions between geometric states

// Apply φ-based transformation
void applyPhiTransform(vector_t* vectors, size_t count, uint8_t iterations) {
    // Apply phi-based scaling to vectors
    for (size_t i = 0; i < count; i++) {
        for (uint8_t j = 0; j < iterations; j++) {
            // Apply golden ratio scaling
            vectors[i].magnitude *= PHI;
            
            // Apply golden angle rotation (137.5 degrees)
            vectors[i].theta += 2.0 * PI * (1.0 - 1.0 / PHI);
        }
    }
}

Pi (π)

3.14159265358979...

The ratio of a circle's circumference to its diameter. In GeoMath, π is used for:

Circular and spherical harmonics
Wave-based transformations
Rotational symmetry operations
Cyclical pattern generation

// Apply π-based harmonic wave
void applyPiHarmonicWave(vector_t* vectors, size_t count, 
                        double frequency, double amplitude) {
    // Apply pi-based wave transformation
    for (size_t i = 0; i < count; i++) {
        // Generate wave pattern based on position
        double phase = vectors[i].magnitude * frequency;
        
        // Apply wave transformation
        vectors[i].magnitude += amplitude * sin(phase * PI);
        vectors[i].direction.x += amplitude * cos(phase * PI) / 2.0;
        vectors[i].direction.y += amplitude * sin(phase * PI) / 2.0;
    }
}

Root 2 (√2)

1.4142135623731...

The diagonal of a unit square. In GeoMath, √2 is used for:

Cubic/octahedral transformations
Diagonal relationship calculations
Boundary transitions between geometric domains
Square-root-based harmonic progressions

// Apply √2-based transformation
void applySqrt2Transform(vector_t* vectors, size_t count) {
    // Apply sqrt2-based scaling and rotation
    for (size_t i = 0; i < count; i++) {
        // Apply sqrt2 scaling for cubic transformation
        vectors[i].magnitude *= SQRT2;
        
        // Apply 45-degree rotation (π/4)
        vectors[i].theta += PI / 4.0;
        
        // Apply proper diagonal relationship
        double temp = vectors[i].direction.x;
        vectors[i].direction.x = (vectors[i].direction.x + vectors[i].direction.y) / SQRT2;
        vectors[i].direction.y = (temp - vectors[i].direction.y) / SQRT2;
    }
}

Root 3 (√3)

1.73205080756888...

The height of an equilateral triangle with side length 2. In GeoMath, √3 is used for:

Tetrahedral/triangular transformations
Hexagonal grid systems
Triangular harmonic relationships
Three-fold symmetry operations

// Apply √3-based triangular transformation
void applySqrt3Transform(vector_t* vectors, size_t count) {
    // Apply sqrt3-based scaling and triangular transformation
    for (size_t i = 0; i < count; i++) {
        // Apply sqrt3 scaling for triangular transformation
        vectors[i].magnitude *= SQRT3 / 2.0;
        
        // Apply 60-degree rotation (π/3)
        vectors[i].theta += PI / 3.0;
        
        // Apply triangular transformation
        vectors[i].direction.z = SQRT3 / 2.0 * 
            sqrt(vectors[i].direction.x * vectors[i].direction.x + 
                 vectors[i].direction.y * vectors[i].direction.y);
    }
}

Root 6 (√6)

2.449489742783178...

A composite constant with special properties in geometric transformations. In GeoMath, √6 is used for:

Rhombic dodecahedron constructions
Complex 3D rotational systems
Optimal packaging structures
Cross-dimensional transformations

// Apply √6-based rhombic transformation
void applySqrt6Transform(vector_t* vectors, size_t count) {
    const double SQRT6 = 2.449489742783178;
    const double SQRT6_HALF = SQRT6 / 2.0;
    
    // Apply sqrt6-based scaling and rhombic transformation
    for (size_t i = 0; i < count; i++) {
        // Apply sqrt6/2 scaling for rhombic dodecahedron radius
        vectors[i].magnitude *= SQRT6_HALF;
        
        // Apply rhombic transformation
        double norm = sqrt(vectors[i].direction.x * vectors[i].direction.x +
                          vectors[i].direction.y * vectors[i].direction.y +
                          vectors[i].direction.z * vectors[i].direction.z);
        
        if (norm > 0) {
            vectors[i].direction.x = vectors[i].direction.x / norm * SQRT6_HALF;
            vectors[i].direction.y = vectors[i].direction.y / norm * SQRT6_HALF;
            vectors[i].direction.z = vectors[i].direction.z / norm * SQRT6_HALF;
        }
    }
}

Combined Harmonic Operations

GeoMath enables sophisticated combined operations that utilize multiple constants:

// Apply combined harmonic transformation
void applyCombinedHarmonicTransform(vector_t* vectors, size_t count, 
                                  harmonic_type_t type, double factor) {
    // Get harmonic parameters
    double phi_factor = 0.0, pi_factor = 0.0;
    double sqrt2_factor = 0.0, sqrt3_factor = 0.0;
    
    switch (type) {
        case HARMONIC_ICOSAHEDRAL:
            phi_factor = 0.7;
            pi_factor = 0.2;
            sqrt2_factor = 0.0;
            sqrt3_factor = 0.1;
            break;
            
        case HARMONIC_OCTAHEDRAL:
            phi_factor = 0.0;
            pi_factor = 0.3;
            sqrt2_factor = 0.6;
            sqrt3_factor = 0.1;
            break;
            
        case HARMONIC_TETRAHEDRAL:
            phi_factor = 0.1;
            pi_factor = 0.2;
            sqrt2_factor = 0.1;
            sqrt3_factor = 0.6;
            break;
            
        case HARMONIC_BALANCED:
            phi_factor = 0.25;
            pi_factor = 0.25;
            sqrt2_factor = 0.25;
            sqrt3_factor = 0.25;
            break;
    }
    
    // Apply combined harmonic transformation
    for (size_t i = 0; i < count; i++) {
        // Apply weighted combination of harmonic transformations
        if (phi_factor > 0) {
            applyPhiComponent(&vectors[i], phi_factor * factor);
        }
        
        if (pi_factor > 0) {
            applyPiComponent(&vectors[i], pi_factor * factor);
        }
        
        if (sqrt2_factor > 0) {
            applySqrt2Component(&vectors[i], sqrt2_factor * factor);
        }
        
        if (sqrt3_factor > 0) {
            applySqrt3Component(&vectors[i], sqrt3_factor * factor);
        }
        
        // Normalize result if needed
        normalizeVector(&vectors[i]);
    }
}

GeoMath Harmonic Constants Relationships

Geometric Operations

GeoMath defines a comprehensive set of geometric operations that preserve harmonic relationships:

Harmonic Transformations

Transformations that preserve harmonic relationships between geometric structures:

// Define harmonic transformation types
typedef enum {
    TRANSFORM_SCALING,        // Harmonic scaling
    TRANSFORM_ROTATION,       // Harmonic rotation
    TRANSFORM_WAVE,           // Wave-based transformation
    TRANSFORM_JITTERBUG,      // Fuller's jitterbug transformation
    TRANSFORM_RESONANCE,      // Resonance-based transformation
    TRANSFORM_DIMENSIONAL     // Dimensional shift transformation
} harmonic_transform_t;

// Apply harmonic transformation
void applyHarmonicTransform(geometry_t* geometry, 
                          harmonic_transform_t transform_type,
                          transform_params_t* params) {
    // Prepare transformation
    prepareHarmonicTransform(geometry, transform_type);
    
    // Apply transformation based on type
    switch (transform_type) {
        case TRANSFORM_SCALING:
            applyHarmonicScaling(geometry, params->scaling_factor);
            break;
            
        case TRANSFORM_ROTATION:
            applyHarmonicRotation(geometry, params->rotation_angle, 
                                 params->rotation_plane);
            break;
            
        case TRANSFORM_WAVE:
            applyHarmonicWave(geometry, params->wave_amplitude, 
                             params->wave_frequency, params->wave_phase);
            break;
            
        case TRANSFORM_JITTERBUG:
            applyJitterbugTransform(geometry, params->jitterbug_phase, 
                                   params->transformation_factor);
            break;
            
        case TRANSFORM_RESONANCE:
            applyResonanceTransform(geometry, params->resonance_type, 
                                   params->resonance_frequency);
            break;
            
        case TRANSFORM_DIMENSIONAL:
            applyDimensionalShift(geometry, params->target_dimension);
            break;
    }
    
    // Post-process transformation
    finalizeHarmonicTransform(geometry, transform_type);
}

Vector Field Operations

Operations that apply mathematical transformations to vector fields:

// Apply harmonic vector field operation
void applyVectorFieldOperation(vector_field_t* field, field_operation_t operation) {
    // Get field dimensions
    size_t width = field->width;
    size_t height = field->height;
    size_t depth = field->depth;
    
    // For each vector in the field
    for (size_t z = 0; z < depth; z++) {
        for (size_t y = 0; y < height; y++) {
            for (size_t x = 0; x < width; x++) {
                // Get vector at position
                vector_t* v = getVectorAt(field, x, y, z);
                
                // Apply operation based on type
                switch (operation.type) {
                    case FIELD_CURL:
                        applyCurlOperation(field, v, x, y, z);
                        break;
                        
                    case FIELD_DIVERGENCE:
                        applyDivergenceOperation(field, v, x, y, z);
                        break;
                        
                    case FIELD_GRADIENT:
                        applyGradientOperation(field, v, x, y, z);
                        break;
                        
                    case FIELD_HARMONIC_FLOW:
                        applyHarmonicFlowOperation(field, v, x, y, z, 
                                                 operation.params);
                        break;
                        
                    case FIELD_RESONANCE_PATTERN:
                        applyResonancePatternOperation(field, v, x, y, z, 
                                                     operation.resonance_type);
                        break;
                }
            }
        }
    }
    
    // Normalize field if needed
    if (operation.normalize) {
        normalizeVectorField(field);
    }
}

Geometry Morphing

Operations that morph between different geometric structures while preserving harmonic relationships:

// Morph between geometric structures
void morphGeometricStructures(geometry_t* source, geometry_t* target, 
                            geometry_t* result, double morph_factor) {
    // Validate structures
    if (source->vertex_count != target->vertex_count) {
        fprintf(stderr, "Error: Structures must have same vertex count\n");
        return;
    }
    
    // Prepare result structure
    initGeometryStructure(result, source->vertex_count);
    
    // Determine optimal morphing path
    morph_path_t path = determineMorphPath(source, target);
    
    // Apply harmonic morphing
    for (size_t i = 0; i < source->vertex_count; i++) {
        // Get source and target vertices
        vector_t* src_vertex = &source->vertices[i];
        vector_t* tgt_vertex = &target->vertices[i];
        
        // Apply harmonic interpolation based on path
        switch (path.type) {
            case MORPH_LINEAR:
                // Simple linear interpolation
                linearInterpolate(src_vertex, tgt_vertex, 
                                 &result->vertices[i], morph_factor);
                break;
                
            case MORPH_SPHERICAL:
                // Spherical interpolation (SLERP)
                sphericalInterpolate(src_vertex, tgt_vertex, 
                                    &result->vertices[i], morph_factor);
                break;
                
            case MORPH_HARMONIC:
                // Harmonic interpolation based on golden ratio
                harmonicInterpolate(src_vertex, tgt_vertex, 
                                   &result->vertices[i], morph_factor, PHI);
                break;
                
            case MORPH_JITTERBUG:
                // Fuller jitterbug interpolation
                jitterbugInterpolate(src_vertex, tgt_vertex, 
                                    &result->vertices[i], morph_factor);
                break;
        }
    }
    
    // Update result structure properties
    result->structure_type = (morph_factor < 0.5) ? 
        source->structure_type : target->structure_type;
    
    // Update topological properties
    updateTopology(result);
}

Resonance Patterns

GeoMath identifies and leverages fundamental resonance patterns in geometric structures:

Geometric Resonance

Specific configurations where geometric structures exhibit heightened stability or coherence:

// Define geometric resonance types
typedef enum {
    RESONANCE_PHI,            // Golden ratio resonance
    RESONANCE_PI,             // Pi-based circular resonance
    RESONANCE_SQRT2,          // Sqrt2-based diagonal resonance
    RESONANCE_SQRT3,          // Sqrt3-based triangular resonance
    RESONANCE_COMPOSITE       // Composite resonance pattern
} resonance_type_t;

// Calculate resonance strength at given point
double calculateResonanceStrength(geometry_t* geometry, 
                                vector_t* point, 
                                resonance_type_t resonance_type) {
    double strength = 0.0;
    
    // Calculate base resonance value
    switch (resonance_type) {
        case RESONANCE_PHI:
            strength = calculatePhiResonance(geometry, point);
            break;
            
        case RESONANCE_PI:
            strength = calculatePiResonance(geometry, point);
            break;
            
        case RESONANCE_SQRT2:
            strength = calculateSqrt2Resonance(geometry, point);
            break;
            
        case RESONANCE_SQRT3:
            strength = calculateSqrt3Resonance(geometry, point);
            break;
            
        case RESONANCE_COMPOSITE:
            // Weighted combination of resonance types
            strength = 0.4 * calculatePhiResonance(geometry, point) +
                      0.3 * calculatePiResonance(geometry, point) +
                      0.2 * calculateSqrt2Resonance(geometry, point) +
                      0.1 * calculateSqrt3Resonance(geometry, point);
            break;
    }
    
    return strength;
}

Harmonic Nodes

Points of maximum resonance within geometric systems:

// Find harmonic nodes in geometry
node_set_t* findHarmonicNodes(geometry_t* geometry, 
                            resonance_type_t resonance_type, 
                            double threshold) {
    // Create node set
    node_set_t* nodes = createNodeSet();
    
    // Get geometry bounding box
    bounding_box_t bounds = calculateBoundingBox(geometry);
    
    // Sample points within bounding box
    const size_t SAMPLES_PER_DIMENSION = 10;
    vector_t sample_point;
    
    for (size_t z = 0; z < SAMPLES_PER_DIMENSION; z++) {
        sample_point.z = bounds.min.z + (bounds.max.z - bounds.min.z) * 
                        ((double)z / (SAMPLES_PER_DIMENSION - 1));
        
        for (size_t y = 0; y < SAMPLES_PER_DIMENSION; y++) {
            sample_point.y = bounds.min.y + (bounds.max.y - bounds.min.y) * 
                           ((double)y / (SAMPLES_PER_DIMENSION - 1));
            
            for (size_t x = 0; x < SAMPLES_PER_DIMENSION; x++) {
                sample_point.x = bounds.min.x + (bounds.max.x - bounds.min.x) * 
                               ((double)x / (SAMPLES_PER_DIMENSION - 1));
                
                // Calculate resonance strength at this point
                double strength = calculateResonanceStrength(
                    geometry, &sample_point, resonance_type);
                
                // If above threshold, add to node set
                if (strength >= threshold) {
                    harmonic_node_t* node = createHarmonicNode();
                    node->position = sample_point;
                    node->strength = strength;
                    node->resonance_type = resonance_type;
                    
                    addNodeToSet(nodes, node);
                }
            }
        }
    }
    
    // Refine node positions for maximum resonance
    refineNodePositions(nodes, geometry, resonance_type);
    
    return nodes;
}

Wave Interference Patterns

Complex patterns formed by the interference of harmonic waves in geometric space:

// Generate harmonic wave interference pattern
wave_pattern_t* generateWaveInterferencePattern(
    wave_params_t* waves, size_t wave_count, 
    vector3d_t volume_min, vector3d_t volume_max,
    size_t resolution) {
    
    // Create wave pattern
    wave_pattern_t* pattern = createWavePattern(resolution);
    
    // Calculate cell size
    double cell_size_x = (volume_max.x - volume_min.x) / resolution;
    double cell_size_y = (volume_max.y - volume_min.y) / resolution;
    double cell_size_z = (volume_max.z - volume_min.z) / resolution;
    
    // For each point in the volume
    for (size_t z = 0; z < resolution; z++) {
        double z_pos = volume_min.z + z * cell_size_z;
        
        for (size_t y = 0; y < resolution; y++) {
            double y_pos = volume_min.y + y * cell_size_y;
            
            for (size_t x = 0; x < resolution; x++) {
                double x_pos = volume_min.x + x * cell_size_x;
                
                // Create position vector
                vector3d_t pos = {x_pos, y_pos, z_pos};
                
                // Calculate combined wave amplitude at this point
                double amplitude = 0.0;
                
                for (size_t i = 0; i < wave_count; i++) {
                    // Calculate distance from wave source
                    double distance = calculateDistance(&pos, &waves[i].source);
                    
                    // Calculate wave phase
                    double phase = distance * waves[i].frequency;
                    
                    // Add wave contribution
                    amplitude += waves[i].amplitude * 
                               sin(phase * 2.0 * PI + waves[i].phase_offset);
                }
                
                // Store amplitude in pattern
                setPatternValue(pattern, x, y, z, amplitude);
            }
        }
    }
    
    return pattern;
}

Vector Forms

GeoMath implements a sophisticated system of vector forms that represent geometric structures:

Vector Form Representation

Mathematical representation of geometric structures as vector forms:

// Define vector form structure
typedef struct {
    uint32_t dimension;          // Form dimension
    uint32_t vector_count;       // Number of vectors
    vector_t* vectors;           // Array of vectors
    uint32_t* connectivity;      // Connectivity information
    form_type_t type;            // Type of vector form
    form_properties_t properties; // Form properties
} vector_form_t;

// Create vector form from geometry
vector_form_t* createVectorForm(geometry_t* geometry) {
    // Allocate vector form
    vector_form_t* form = (vector_form_t*)malloc(sizeof(vector_form_t));
    
    // Set basic properties
    form->dimension = geometry->dimension;
    form->vector_count = geometry->vertex_count;
    
    // Allocate and copy vectors
    form->vectors = (vector_t*)malloc(form->vector_count * sizeof(vector_t));
    memcpy(form->vectors, geometry->vertices, 
          form->vector_count * sizeof(vector_t));
    
    // Set up connectivity based on geometry type
    setupConnectivity(form, geometry);
    
    // Determine form type
    form->type = determineFormType(geometry);
    
    // Calculate form properties
    calculateFormProperties(form);
    
    return form;
}

Vector Flow

Dynamic flows of vectors that transform geometric structures:

// Define vector flow structure
typedef struct {
    vector_field_t* field;         // Vector field
    flow_type_t type;              // Type of flow
    double flow_rate;              // Flow rate
    double time;                   // Current flow time
    flow_params_t params;          // Flow parameters
} vector_flow_t;

// Update vector flow
void updateVectorFlow(vector_flow_t* flow, double delta_time) {
    // Update flow time
    flow->time += delta_time * flow->flow_rate;
    
    // Update field based on flow type
    switch (flow->type) {
        case FLOW_CURL:
            updateCurlFlow(flow->field, flow->time, flow->params);
            break;
            
        case FLOW_DIVERGENCE:
            updateDivergenceFlow(flow->field, flow->time, flow->params);
            break;
            
        case FLOW_HARMONIC:
            updateHarmonicFlow(flow->field, flow->time, flow->params);
            break;
            
        case FLOW_SPIRAL:
            updateSpiralFlow(flow->field, flow->time, flow->params);
            break;
            
        case FLOW_JITTERBUG:
            updateJitterbugFlow(flow->field, flow->time, flow->params);
            break;
    }
    
    // Normalize field if needed
    if (flow->params.normalize) {
        normalizeVectorField(flow->field);
    }
}

Form Transformations

Transformations between different vector forms:

// Transform between vector forms
vector_form_t* transformVectorForm(vector_form_t* source, 
                                 form_type_t target_type,
                                 transform_params_t* params) {
    // Create target vector form
    vector_form_t* target = createEmptyVectorForm(
        source->dimension, source->vector_count);
    
    // Determine transformation type
    transform_type_t transform_type = determineTransformationType(
        source->type, target_type);
    
    // Apply transformation
    switch (transform_type) {
        case TRANSFORM_OCTAHEDRAL_TO_ICOSAHEDRAL:
            transformOctahedralToIcosahedral(source, target, params);
            break;
            
        case TRANSFORM_ICOSAHEDRAL_TO_OCTAHEDRAL:
            transformIcosahedralToOctahedral(source, target, params);
            break;
            
        case TRANSFORM_OCTAHEDRAL_TO_TETRAHEDRAL:
            transformOctahedralToTetrahedral(source, target, params);
            break;
            
        case TRANSFORM_TETRAHEDRAL_TO_OCTAHEDRAL:
            transformTetrahedralToOctahedral(source, target, params);
            break;
            
        case TRANSFORM_DIMENSIONAL_SHIFT:
            transformDimensionalShift(source, target, params);
            break;
            
        default:
            // Generic transformation
            transformGeneric(source, target, params);
            break;
    }
    
    // Set target form type
    target->type = target_type;
    
    // Calculate target form properties
    calculateFormProperties(target);
    
    return target;
}

Implementation Guide

GeoMath API

The GeoMath API provides interfaces for working with harmonic mathematics of space:

// Initialize GeoMath system
GeoMath* geomath_init() {
    GeoMath* gm = (GeoMath*)malloc(sizeof(GeoMath));
    
    // Initialize constants
    gm->constants.phi = 1.618033988749895;
    gm->constants.pi = 3.14159265358979;
    gm->constants.sqrt2 = 1.4142135623731;
    gm->constants.sqrt3 = 1.73205080756888;
    gm->constants.sqrt6 = 2.449489742783178;
    
    // Initialize transformation system
    gm->transformations = initTransformationSystem();
    
    // Initialize vector form system
    gm->vector_forms = initVectorFormSystem();
    
    // Initialize resonance system
    gm->resonance = initResonanceSystem();
    
    // Set default configuration
    gm->default_dimension = 3;
    gm->default_resonance_type = RESONANCE_PHI;
    
    return gm;
}

// Create geometric structure
geometry_t* geomath_createGeometry(GeoMath* gm, geometry_type_t type) {
    geometry_t* geometry = NULL;
    
    // Create geometry based on type
    switch (type) {
        case GEOMETRY_OCTAHEDRON:
            geometry = createOctahedron(gm);
            break;
            
        case GEOMETRY_ICOSAHEDRON:
            geometry = createIcosahedron(gm);
            break;
            
        case GEOMETRY_TETRAHEDRON:
            geometry = createTetrahedron(gm);
            break;
            
        case GEOMETRY_CUBOCTAHEDRON:
            geometry = createCuboctahedron(gm);
            break;
            
        case GEOMETRY_DODECAHEDRON:
            geometry = createDodecahedron(gm);
            break;
            
        case GEOMETRY_RHOMBIC_DODECAHEDRON:
            geometry = createRhombicDodecahedron(gm);
            break;
    }
    
    return geometry;
}

// Apply harmonic transformation
void geomath_applyTransformation(GeoMath* gm, geometry_t* geometry, 
                               transform_type_t transform_type,
                               transform_params_t* params) {
    // Validate geometry
    if (!geometry) {
        fprintf(stderr, "Error: Invalid geometry\n");
        return;
    }
    
    // Apply transformation based on type
    applyHarmonicTransform(geometry, transform_type, params);
    
    // Update geometry properties
    updateGeometryProperties(geometry);
}

// Calculate resonance pattern
resonance_pattern_t* geomath_calculateResonance(GeoMath* gm, geometry_t* geometry,
                                             resonance_type_t resonance_type) {
    // Create resonance pattern
    resonance_pattern_t* pattern = createResonancePattern(geometry);
    
    // Calculate resonance values
    calculateResonancePattern(pattern, geometry, resonance_type);
    
    return pattern;
}

// Create vector form
vector_form_t* geomath_createVectorForm(GeoMath* gm, geometry_t* geometry) {
    // Create vector form from geometry
    return createVectorForm(geometry);
}

Using Different Resonance Patterns

GeoMath supports multiple resonance patterns for different applications:

Resonance Type	Primary Constant	Best For	Usage Example
Phi Resonance	Golden Ratio (φ)	Natural patterns, growth sequences	Spiral structures, organic forms
Pi Resonance	π	Circular patterns, wave phenomena	Oscillating systems, sound waves
Sqrt2 Resonance	√2	Cubic structures, grid systems	Urban planning, architectural layouts
Sqrt3 Resonance	√3	Triangular structures, 3-fold symmetry	Molecular structures, crystals
Composite Resonance	Multiple constants	Complex systems, hybrid structures	Biological systems, adaptive networks

Performance Optimization

For optimal GeoMath performance:

Resonance Caching: Cache resonance patterns for frequently used geometries
Transformation Precomputation: Precompute common transformations for reuse
Resolution Control: Adjust resonance calculation resolution based on need
Batch Processing: Process multiple geometric operations in batches
Dimension Optimization: Use the minimum necessary dimensions for operations

// Optimize GeoMath performance
void optimizeGeoMathPerformance(GeoMath* gm) {
    // Enable resonance caching
    gm->cache_resonance = true;
    gm->resonance_cache_size = 64;
    gm->resonance_cache = createResonanceCache(gm->resonance_cache_size);
    
    // Precompute common transformations
    gm->precompute_transformations = true;
    precomputeCommonTransformations(gm);
    
    // Set resonance resolution based on performance need
    gm->resonance_resolution = RESONANCE_RESOLUTION_MEDIUM;
    
    // Enable batch processing
    gm->batch_size = 32;
    gm->batch_buffer = allocateBatchBuffer(gm->batch_size);
    
    // Configure dimension optimization
    gm->optimize_dimensions = true;
    gm->min_working_dimension = 3;
    
    // Initialize optimization subsystem
    initOptimizationSubsystem(gm);
}

Integration with DragonFire Ecosystem

GeoMath integrates with other DragonFire components to provide harmonic mathematical capabilities:

GeoCode Integration

GeoMath builds on GeoCode's encoded geometry to apply harmonic operations:

// Integrate GeoMath with GeoCode
void integrateWithGeoCode(GeoMath* gm, GeoCode* gc) {
    // Connect GeoMath's harmonic operations to GeoCode's bit encodings
    gm->bit_encoder = gc;
    
    // Set up geometric encoding functions
    gm->encode_geometry = gc->convert_to_bits;
    gm->decode_geometry = gc->create_from_bits;
    
    // Map GeoMath's harmonic spaces to GeoCode's geometric structures
    mapHarmonicSpacesToGeometries(gm, gc);
    
    // Set up transformation mapping
    setupTransformationMapping(gc, gm);
}

4D Mathematics Integration

GeoMath works with 4D Mathematics for higher-dimensional operations:

// Integrate GeoMath with 4D Mathematics
void integrateWith4DMath(GeoMath* gm, FourDMath* math) {
    // Connect GeoMath's harmonic operations to 4D Math's transformations
    gm->math_library = math;
    
    // Set up 4D rotation functions
    gm->rotate_4d = math->rotate_vector;
    
    // Map GeoMath's harmonic spaces to 4D Math's vector spaces
    mapHarmonicSpacesToVectorSpaces(gm, math);
    
    // Set up transformation conversions
    setupTransformationConversions(gm, math);
    
    // Initialize higher dimensional operations
    initHigherDimensionalOperations(gm, math);
}

DragonHeart Integration

GeoMath provides the mathematical foundation for DragonHeart's harmonic processing:

// Integrate GeoMath with DragonHeart
void integrateWithDragonHeart(GeoMath* gm, DragonHeart* heart) {
    // Connect GeoMath's resonance patterns to DragonHeart's processing
    heart->geomath = gm;
    
    // Set up harmonic constants
    heart->phi = gm->constants.phi;
    heart->pi = gm->constants.pi;
    heart->sqrt2 = gm->constants.sqrt2;
    heart->sqrt3 = gm->constants.sqrt3;
    
    // Map resonance patterns to processing modes
    mapResonancesToProcessingModes(gm, heart);
    
    // Set up harmonic transformation functions
    setupHarmonicTransformations(gm, heart);
}

Unified Mathematical Framework

GeoMath provides a unified harmonic mathematical framework across the DragonFire ecosystem:

Key Integration Insight: GeoMath serves as the mathematical soul of the DragonFire ecosystem, providing harmonic mathematical operations that preserve beauty and coherence across transformations. While GeoCode encodes geometric structures and 4D Mathematics handles higher-dimensional operations, GeoMath ensures that all mathematical operations maintain harmonic relationships and resonance patterns, creating a coherent mathematical framework across the entire system.

Examples

Basic GeoMath Usage

#include "geomath.h"

int main() {
    // Initialize GeoMath
    GeoMath* gm = geomath_init();
    
    // Create octahedron geometry
    geometry_t* octahedron = geomath_createGeometry(gm, GEOMETRY_OCTAHEDRON);
    
    printf("Created octahedron with %d vertices\n", octahedron->vertex_count);
    
    // Configure transformation parameters
    transform_params_t params;
    params.transform_type = TRANSFORM_SCALING;
    params.scaling_factor = gm->constants.phi;  // Golden ratio scaling
    
    // Apply harmonic transformation
    geomath_applyTransformation(gm, octahedron, TRANSFORM_SCALING, ¶ms);
    
    printf("Applied phi scaling transformation\n");
    
    // Calculate resonance pattern
    resonance_pattern_t* pattern = geomath_calculateResonance(
        gm, octahedron, RESONANCE_PHI);
    
    printf("Calculated phi resonance pattern with %d nodes\n", 
           pattern->node_count);
    
    // Create vector form
    vector_form_t* form = geomath_createVectorForm(gm, octahedron);
    
    printf("Created vector form with dimension %d\n", form->dimension);
    
    // Free resources
    geomath_freeResonancePattern(gm, pattern);
    geomath_freeVectorForm(gm, form);
    geomath_freeGeometry(gm, octahedron);
    geomath_free(gm);
    
    return 0;
}

Harmonic Transformation Example

// Demonstrate harmonic transformations
void demonstrateHarmonicTransformations() {
    // Initialize GeoMath
    GeoMath* gm = geomath_init();
    
    // Create octahedron geometry
    geometry_t* geometry = geomath_createGeometry(gm, GEOMETRY_OCTAHEDRON);
    
    printf("Created octahedron with %d vertices\n", geometry->vertex_count);
    
    // Configure jitterbug transformation parameters
    transform_params_t params;
    params.transform_type = TRANSFORM_JITTERBUG;
    params.jitterbug_phase = JITTERBUG_ICOSAHEDRON;
    params.transformation_factor = 0.5;  // 50% transformation
    
    // Apply jitterbug transformation
    geomath_applyTransformation(gm, geometry, TRANSFORM_JITTERBUG, ¶ms);
    
    printf("Applied jitterbug transformation to icosahedron (50%%)\n");
    printf("Resulting structure has %d vertices\n", geometry->vertex_count);
    
    // Calculate geometry properties
    geometry_properties_t props;
    geomath_calculateGeometryProperties(gm, geometry, &props);
    
    printf("Geometry properties:\n");
    printf("- Volume: %.2f\n", props.volume);
    printf("- Surface area: %.2f\n", props.surface_area);
    printf("- Symmetry groups: %d\n", props.symmetry_groups);
    printf("- Harmonic coefficient: %.4f\n", props.harmonic_coefficient);
    
    // Calculate phi resonance at center
    vector_t center = {0, 0, 0};
    double resonance = geomath_calculateResonanceAtPoint(
        gm, geometry, ¢er, RESONANCE_PHI);
    
    printf("Phi resonance at center: %.4f\n", resonance);
    
    // Free resources
    geomath_freeGeometry(gm, geometry);
    geomath_free(gm);
}

Vector Form Example

// Demonstrate vector forms
void demonstrateVectorForms() {
    // Initialize GeoMath
    GeoMath* gm = geomath_init();
    
    // Create icosahedron geometry
    geometry_t* icosahedron = geomath_createGeometry(gm, GEOMETRY_ICOSAHEDRON);
    
    printf("Created icosahedron geometry\n");
    
    // Create vector form
    vector_form_t* form = geomath_createVectorForm(gm, icosahedron);
    
    printf("Created vector form with %d vectors\n", form->vector_count);
    
    // Configure vector flow
    flow_params_t flow_params;
    flow_params.type = FLOW_SPIRAL;
    flow_params.rate = 0.5;
    flow_params.center.x = 0;
    flow_params.center.y = 0;
    flow_params.center.z = 0;
    flow_params.normalize = true;
    
    // Create vector flow
    vector_flow_t* flow = geomath_createVectorFlow(gm, form, &flow_params);
    
    printf("Created spiral vector flow\n");
    
    // Simulate flow for 10 steps
    for (int i = 0; i < 10; i++) {
        // Update flow
        geomath_updateVectorFlow(gm, flow, 0.1);  // 0.1 second delta time
        
        // Update form from flow
        geomath_updateFormFromFlow(gm, form, flow);
        
        printf("Updated flow step %d\n", i + 1);
        
        // Calculate form energy
        double energy = geomath_calculateFormEnergy(gm, form);
        printf("Form energy: %.4f\n", energy);
    }
    
    // Transform form to octahedral type
    transform_params_t params;
    params.transform_type = TRANSFORM_FORM;
    params.target_form_type = FORM_OCTAHEDRAL;
    params.transformation_factor = 1.0;
    
    vector_form_t* transformed = geomath_transformVectorForm(
        gm, form, ¶ms);
    
    printf("Transformed to octahedral form\n");
    
    // Free resources
    geomath_freeVectorFlow(gm, flow);
    geomath_freeVectorForm(gm, form);
    geomath_freeVectorForm(gm, transformed);
    geomath_freeGeometry(gm, icosahedron);
    geomath_free(gm);
}

View more examples in our SDK Examples section or try the Interactive GeoMath Demo.

Next Steps

Explore the complete GeoMath API Reference
Download the GeoMath SDK
Try the Interactive GeoMath Demo
Learn about GeoCode for encoded geometry as symbolic logic
Explore 4D Mathematics for higher-dimensional operations

Getting Started

Mathematical Architecture

GeoMath Guide

GeoMath

Introduction

Key Concepts

Harmonic Relationships

Resonance Patterns

Transformation Preservation

Constant-Based Scaling

Vector Form Mathematics

Harmonic Wave Functions

Harmonic Constants

Phi (φ)

Pi (π)

Root 2 (√2)

Root 3 (√3)

Root 6 (√6)

Combined Harmonic Operations

Geometric Operations

Harmonic Transformations

Vector Field Operations

Geometry Morphing

Resonance Patterns

Geometric Resonance

Harmonic Nodes

Wave Interference Patterns

Vector Forms

Vector Form Representation

Vector Flow

Form Transformations

Implementation Guide

GeoMath API

Using Different Resonance Patterns

Performance Optimization

Integration with DragonFire Ecosystem

GeoCode Integration

4D Mathematics Integration

DragonHeart Integration

Unified Mathematical Framework

Examples

Basic GeoMath Usage

Harmonic Transformation Example

Vector Form Example

Next Steps