Synthia - Complete Documentation

Overview

Synthia is a Blender addon that lets you turn math into 3D. You can visualize formulas, algorithms, and shapes instantly, making math easier to understand and more fun to explore.

Installation

Download synthia.zip
Open Blender (4.0.0 or later)
Go to Edit → Preferences → Add-ons
Click Install and select synthia.zip
Enable Synthia by checking the checkbox
Access from N-Panel → Synthia in the 3D Viewport

Quick Start

Open the N-Panel (press N in 3D Viewport)
Navigate to Synthia tab
Choose a category: Equations, Algorithms, or Geometry
Click any preset to generate the visualization
Adjust parameters in Visualization Controls panel

Animate by keyframing parameters on the timeline.

Main Interface

Category Selector

Switch between three main categories:

Equations - Mathematical functions and surfaces
Algorithms - Sorting, pathfinding, fractals, data structures
Geometry - Theorems, vectors, transformations, topology

Search Bar

Filter presets by name - type to find specific visualizations quickly

Preset Buttons

Click any button to instantly generate that mathematical concept as 3D geometry

Categories & Presets

📊 EQUATIONS (30+ presets)

Basic Functions

Sine Wave - sin(x) wave function
Cosine Wave - cos(x) wave function
Tangent - tan(x) function with asymptotes
Parabola - x² quadratic function
Exponential - e^x growth curve
Logarithm - log(x) function

Advanced Surfaces

Saddle Surface - Hyperbolic paraboloid
Ripple - Concentric wave pattern
Gaussian Bell Curve - Normal distribution
Paraboloid - 3D parabolic surface
Hyperboloid - Double-cone surface
Helicoid - Spiral staircase surface
Catenoid - Minimal surface (soap film)
Enneper Surface - Self-intersecting minimal surface
Möbius Strip - Non-orientable surface with one side
Klein Bottle - 4D bottle in 3D space
Seashell - Logarithmic spiral

Parametric Curves

Helix - 3D spiral
Spiral - 2D/3D spiral curve
Torus Knot - Knotted curve on torus
Lissajous Curve - Harmonic motion patterns

Attractors

Lorenz Attractor - Chaotic butterfly pattern
Rössler Attractor - Chaotic spiral system

🔢 ALGORITHMS (35+ presets)

Sorting Algorithms

Bubble Sort - Animated bar sorting
Quick Sort - Divide-and-conquer sorting
Merge Sort - Merge-based sorting
Insertion Sort - Insert elements in order
Selection Sort - Select minimum repeatedly
Heap Sort - Binary heap sorting

Controls: Play/Pause/Reset buttons, adjustable speed

Pathfinding

A Pathfinding* - Heuristic-based search (cone toward target)
Dijkstra's Algorithm - Uniform cost search (circular wave)
Breadth-First Search - Level-by-level exploration (circular wave)
Depth-First Search - Deep exploration (snake pattern)

Visual: Cubes grow/rise as algorithm explores grid

Fractals

Mandelbrot Set - Iconic fractal boundary
Julia Set - Related fractal pattern
Koch Snowflake - Recursive snowflake curve
Sierpinski Triangle - Triangle with holes
Dragon Curve - Fractal dragon pattern

Data Structures

Binary Tree - Hierarchical tree with spheres and connections
Heap Structure - Pyramid layout with heap properties
Graph Network - Nodes in circle with random edges

Cellular Automata

Conway's Game of Life - Cells live/die based on neighbors
Langton's Ant - Single ant creating highway pattern
Elementary CA - 1D Rule 30 scrolling down

Number Theory

Prime Spiral (Ulam) - Primes arranged in spiral showing diagonal patterns
Sieve of Eratosthenes - Prime numbers visualized as grid
Fibonacci Spiral - Golden ratio spiral curve

Procedural Geometry

Perlin Noise - Terrain-like surface
Voronoi Diagram - Nearest-neighbor cells
Delaunay Triangulation - Connected triangular mesh

📐 GEOMETRY (40+ presets)

Theorems

Pythagorean Theorem - Visual proof: a² + b² = c² with squares
Circle Theorems - Inscribed angles and chords
Triangle Inequality - Valid vs invalid triangle sides
Thales' Theorem - 90° angle in semicircle

Vectors

Vector Addition - Parallelogram law
Vector Subtraction - Difference of vectors
Dot Product - Projection visualization
Cross Product - Perpendicular result with plane
Vector Projection - Component along direction

Transformations

Matrix Rotation - Before/after rotation
Matrix Scaling - Stretch/compress transformation
Matrix Shear - Skew transformation
Matrix Reflection - Mirror across axis
Eigenvectors - Vectors that don't change direction

3D Shapes

Platonic Solids - All 5 perfect polyhedra (tetrahedron, cube, octahedron, dodecahedron, icosahedron)
Euler Characteristic - V - E + F = 2 proof with labeled cube
Cube Net - Unfolded cross pattern

Rotation

Euler Angles - Roll, pitch, yaw rotations demonstrated

Constants

Golden Ratio - Phi spiral with Fibonacci rectangles
Pi Visualization - Circumference/diameter ratio

Calculus

Tangent Line - Derivative as slope on parabola
Area Under Curve - Integral as Riemann sum rectangles
Gradient Vector Field - Arrows showing steepest ascent

Topology

Trefoil Knot - Simplest nontrivial knot

Visualization Controls

When a visualization is selected, the Visualization Controls panel appears:

General

Resolution - Mesh detail level (4-256)
- Low (4-16): Fast, blocky
- Medium (32-64): Balanced
- High (128-256): Detailed, slower

Parameters (Equations only)

Amplitude - Wave height/scale (0.01-10.0)
Frequency - Wave repetition rate (0.01-10.0)
Phase - Wave offset (-6.28 to 6.28 radians)
Scale - XYZ multipliers (0.01-10.0)

Note: Special surfaces (Möbius, Klein, Attractors) have fixed geometry and don't support parameter changes - they display an info message instead.

Algorithm Controls

Play/Pause - Start/stop animation
Reset - Return to initial state and regenerate
Step - Current algorithm step number (read-only)
Speed - Animation playback speed (0.1-5.0)
Progress Bar - Visual indicator of completion

Actions

Regenerate - Rebuild mesh with current parameters
Delete - Remove visualization from scene.

Custom Formulas - The Power Feature

This is Synthia's most powerful feature! Create unlimited mathematical surfaces by writing your own formulas.

Understanding What Custom Formulas Create

Custom formulas in Synthia create mathematical surfaces defined by the equation:

z = f(x, y)

Where:

x and y are horizontal coordinates
z is the height (vertical output)
The formula defines the height at each (x,y) point

What You CAN Create.

✅ Surfaces - Any function of x and y
✅ Terrains - Hills, valleys, mountains
✅ Waves - Ripples, interference patterns
✅ Mathematical Art - Spirals, fractals, patterns
✅ Geometric Shapes - Pyramids, cones, curved surfaces
✅ Organic Forms - Smooth, flowing shapes

What You CANNOT Create!

❌ Closed 3D Solids - Can't make a cube, sphere (requires volume definition)
❌ Implicit Surfaces - Can't use f(x,y,z) = 0 equations
❌ Multiple Disconnected Parts - Single continuous surface only
❌ Self-Intersecting Volumes - Surface-based, not volume-based

Why? Synthia evaluates formulas as z = f(x,y) - a height field over a grid. For each (x,y) position, there's only one z value. This creates surfaces like landscapes, not enclosed volumes.

How to Use Custom Formulas

Open Custom Formula panel
Enter formula in the text field
Click Create Custom to generate new visualization
Or select existing visualization and click Apply Changes

Note: A dedicated guide containing how to get started with custom formulas is included in the files. (PDF file)

...........................................................................................................

Feel free to reach out on my inbox for more information and support.

- Egret.