Manifoldsweeper

Minesweeper on two-dimensional manifolds

Play the latest browser version here

Abstract

Manifoldsweeper is a variant of Minesweeper that adapts the original game's gridded playing board to different surfaces while preserving its existing rules. Such surfaces are constructed by gluing opposite edges of the game's grid in different configurations. These conditions allow for four different surfaces (up to certain deformations): the cylinder, torus, Möbius strip, and Klein bottle.

Gameplay Guide

Starting

1. Select a manifold—determines how the playing board will connect with/glue to itself (see § Manifolds)
2. Select a difficulty—determines the number of horizontal and vertical grid cells and the number of mines
   • Easy, Medium, or Difficult presets have fixed values that depend on the chosen manifold
   • Custom allows for custom values
3. Select a mapping—determines how the playing board will be presented (see § Mappings)
4. Click "Generate"
5. Click anywhere on the surface to start a new game

Playing

The rules of Manifoldsweeper are identical to that of ordinary Minesweeper (guides for which can be found elsewhere).

Manifoldsweeper was originally designed to be played with a mouse and keyboard.

Binding	Action
Left mouse button	Reveal tile
Right mouse button	Flag tile
Middle mouse button + move cursor	Rotate camera
Scroll wheel	Zoom camera
esc key	Open main menu/pause
r key	Reset game

Mechanics

Manifolds

Each manifold can be constructed by gluing together the edges of an ordinary rectangular Minesweeper playing board in different configurations. These edges can be represented by pairs of colored arrows that are glued so that they lie on top of each other pointing in the same direction.

Cylinder	Torus	Möbius strip	Klein bottle

This "gluing" operation can be described more formally in terms of quotient spaces.

Further Information: Quotient Manifolds

The surfaces of Manifoldsweeper are the quotient manifolds of ordinary Minesweeper's two-dimensional Euclidean plane. In other words, each manifold is the quotient space of the unit square $I^2 / \sim$, where $\sim$ is the corresponding equivalence relation(s) from the table below.

Manifold	Equivalence relations
Cylinder	$(0,\ t) \sim (1,\ t)$
Torus	$(0,\ t) \sim (1,\ t)$ and $(s,\ 0) \sim (s,\ 1)$
Möbius strip	$(0,\ t) \sim (1,\ 1-t)$
Klein bottle	$(0,\ t) \sim (1,\ 1-t)$ and $(s,\ 0) \sim (s,\ 1)$

These manifolds can be equivalently described as the quotients of the plane by certain wallpaper groups. (Further explanation pending.)

Mappings

There are different ways that two-dimensional manifolds can be represented in two- and three-dimensional space. Therefore, Manifoldsweeper allows the player to choose a particular mapping into these spaces when setting up a game and to re map any time after. There are two main categories of mappings to choose from: (1) those that map to a universal covering space depicted in two-dimensional space and (2) those that embed or immerse the manifold in three-dimensional space.

1. Each manifold has the "Flat" mapping chosen by default which represents the playing board as a flat rectangle, like original Minesweeper, but with the rectangle repeating infinitely at its glued edges if any are present. Any action completed on one tile will also be completed on all of its copies.

The "Flat" mapping of the Klein bottle

2. All four manifolds have different embeddings or immersions in three-dimensional space that represent the playing board as a flat rectangle deformed such that its glued edges are attached to each other in the correct configuration.

The "Bottle" immersion of the Klein bottle

The manifold can be re mapped at any time by entering the main menu. Here, you can select a new map from the dropdown and click the "Map" button to re map.

Torus transitioning from the "Flat" map to the "Torus" map

Acknowledgements

Manifoldsweeper was developed as my senior project at Red Bank Regional High School's Academy of Information Technology.