Fundamental Concepts
Explore the core principles and theoretical foundations that govern the MootFlip universe.
Toroidal Topology
The MootFlip game board is structured as a hexagonal lattice wrapped onto a torus (donut shape), creating a continuous space with no edges. This unique topology creates fascinating mathematical properties that influence gameplay.
Toroidal Space Properties
- Continuous Surface: A torus has no edges or boundaries, creating a seamless playing field
- Wrapping Coordinates: Moving beyond any edge of the board wraps to the opposite side
- Constant Curvature: Every point on the torus has identical geometric properties
- Self-Interaction: Patterns can wrap around the board and potentially interact with themselves
- Concentric Rings: The board is organized into rings that represent different energy levels and time scales
Mathematical Foundation
The toroidal structure of MootFlip is based on several mathematical principles:
Modular Arithmetic
Coordinates on the board follow modular arithmetic, where position (x,y) is equivalent to (x+n, y+m) for any integers n and m that are multiples of the board dimensions. This creates the wrapping effect where moving off one edge brings you to the opposite edge.
(x, y) ≡ (x mod w, y mod h)
Where w and h are the width and height of the board in hexes.
Hexagonal Tiling
The board uses a regular hexagonal tiling, which is the most efficient way to divide a plane into regions of equal area with minimal total perimeter. Each hex has exactly six neighbors, creating consistent connectivity across the board.
For any hex at (x,y), its six neighbors are at:
(x+1,y), (x+1,y-1), (x,y-1), (x-1,y), (x-1,y+1), (x,y+1)
Axial Coordinate System
MootFlip uses an axial coordinate system for hexagonal grids, where each hex is identified by two coordinates (q,r). This system simplifies calculations for distance, line-of-sight, and pattern interactions.
Distance between hexes: d((q1,r1), (q2,r2)) = (|q1-q2| + |r1-r2| + |q1+r1-q2-r2|) / 2
Strategic Implications
Toroidal Visualization
The image above shows a 2D representation of the MootFlip board. While displayed as a flat hexagonal grid, the board actually wraps around both horizontally and vertically. This means that patterns at the left edge are adjacent to those at the right edge, and patterns at the top edge are adjacent to those at the bottom edge.
The colored hexes represent different pattern types deployed on the board. Notice how some patterns appear to be disconnected but are actually part of the same network when considering the toroidal wrapping.
Concentric Ring Structure
The board is organized into five concentric rings, each with unique properties:
Center Ring
Highest energy, slowest time scale (0.5×)
Inner Ring
High energy, slow time scale (0.75×)
Middle Ring
Standard energy, normal time scale (1×)
Outer Ring
Low energy, fast time scale (1.25×)
Edge Ring
Lowest energy, fastest time scale (1.5×)
Distance and Connectivity
On a toroidal board, distance calculations must account for wrapping:
Shortest Path Distance
The distance between two hexes is the minimum number of steps required to move from one to the other, considering all possible paths including those that wrap around the board.
Maximum Distance
On a standard 91-hex board, the maximum distance between any two hexes is 6 steps, significantly less than it would be on a non-toroidal board of the same size.
Connectivity Implications
The toroidal structure means that every hex has exactly the same connectivity properties, with six neighbors and the same potential for pattern interactions. This creates a perfectly balanced playing field where position is determined purely by strategic placement rather than inherent board advantages.