Mathematics • Games • Systems Thinking

Using game puzzles as a laboratory for geometry, topology, and algorithmic reasoning.

This project explores how portal-style games can be understood through coordinate transformations, momentum conservation, graph search, and mathematical modeling.

Try the Simulator Read the Research Notes

Core Idea

From game mechanics to mathematical models

Coordinate Transformations

When a player enters one portal and exits another, their position and direction are transformed by rotation, translation, and coordinate-frame changes.

Momentum Conservation

A moving object keeps its speed through a portal, but its direction changes according to the exit portal’s orientation.

Graph Search

A puzzle can be modeled as a graph of states and actions. Algorithms such as BFS or A* can search for a valid solution path.

Topology

Portals connect distant locations as if space were folded, suggesting ideas from topology and non-Euclidean geometry.

Interactive Demo

Portal Coordinate Transformation Sandbox

Change the entry position, entry velocity, and exit portal angle. The simulation shows how a vector is transformed when it exits the portal.

Model:
p' = Rθ · p + t
v' = Rθ · v

Mathematical Structure

What the model demonstrates

1. Rotation Matrix

A portal can rotate a vector from the entry frame into the exit frame. In two dimensions, this can be modeled using a rotation matrix: 

[ x' ]   [ cosθ  -sinθ ] [ x ]
[ y' ] = [ sinθ   cosθ ] [ y ]

2. Translation

After rotation, the object is placed near the exit portal. This is modeled by adding a translation vector t. 

p' = Rθp + t

3. Momentum Direction

Speed is preserved in the simplified model, but direction changes according to the portal rotation: 

v' = Rθv

4. Puzzle State Graph

A puzzle can be modeled as states connected by actions. A shortest solution can be searched using BFS: 

start → place portal → move cube → press button → exit

Research Notes

Possible written explanation for university applications

This project began with a simple question: why do portal-based puzzles feel so mathematically strange? The answer is that the game mechanic combines several deep ideas: transformations from linear algebra, conservation principles from physics, state-space search from computer science, and spatial identification from topology.

Instead of treating a game as only entertainment, this project uses it as a modeling environment. The goal is to make invisible mathematical structures visible through interactive simulations.

Future versions could add 3D transformations, real puzzle maps, a graph-based puzzle solver, and a more formal explanation of how portals resemble topological gluing operations.