Determine a coordinate system to uniquely represent points in a hyperbolic square tiling with order 5 | Step-by-Step Solution

Understanding Hyperbolic Coordinate Systems for Order 5 Square Tilings

What We're Solving:

We need to create a coordinate system using just two real numbers that can uniquely identify every point in a hyperbolic plane tiled with squares, where exactly 5 squares meet at each vertex.

The Approach:

This is a beautiful problem that connects geometry, group theory, and coordinate systems! In hyperbolic geometry, we can create regular tilings that are impossible in flat (Euclidean) geometry. An order 5 square tiling means 5 squares meet at each vertex, and this only works in hyperbolic space. We need to find a clever way to "address" each point using just two numbers.

Step-by-Step Solution:

Step 1: Understand the Hyperbolic Model

• Choose a model for hyperbolic geometry. The Poincaré disk model is excellent here
• In this model, the entire hyperbolic plane fits inside a unit circle
• The square tiling appears curved from our Euclidean perspective, but the "squares" are actually regular in hyperbolic terms

Step 2: Establish a Fundamental Domain

• Pick one square as your "home base" or fundamental domain
• This square will be your coordinate origin
• Every other point in the tiling can be reached by following a specific path from this origin

Step 3: Create the Coordinate System

• First coordinate (m): Count how many steps you take in one direction (say, horizontally) from your origin square
• Second coordinate (n): Count how many steps you take in the perpendicular direction (vertically) from your origin square
• These can be positive or negative integers, giving you coordinates like (m,n)

Step 4: Handle the Hyperbolic Geometry

• Unlike regular grid coordinates, these coordinates must account for the hyperbolic curvature
• The actual position in the Poincaré disk can be calculated using hyperbolic translation formulas
• Each unit step corresponds to a hyperbolic transformation (isometry)

Step 5: Verify Uniqueness

• Every point gets exactly one (m,n) coordinate pair
• Every (m,n) pair corresponds to exactly one point in the tiling
• The group structure of the tiling transformations guarantees this works perfectly

The Answer:

The coordinate system uses two integers (m,n) representing the "hyperbolic grid coordinates" - the number of square-edge steps in two perpendicular directions from a chosen origin square. These coordinates are then mapped to actual positions in the hyperbolic plane using the group action of hyperbolic isometries that generate the tiling.

Memory Tip:

Think of it like giving directions in a city: "Go 3 blocks east, then 2 blocks north" - except your "city" is curved hyperbolic space where 5 square "blocks" meet at each intersection instead of the usual 4! The coordinates are just your "directions from home" in this curved world.