Maze Theory Volume II

PAPER 2: ADVANCED MAZE THEORY

Volume II of the Treatise on Perfect Mazes

INTRODUCTION

0.1 Preamble

The preceding volume — Maze Theory: A Comprehensive Study of Perfect Mazes, Their Algebra, Geometry and Extensions (hereafter Paper 1) — established the foundations of a new mathematical discipline. It introduced the 16 icons as a complete alphabet for describing maze connectivity, developed the logic of K and NK matrices, proved the crystallisation algorithm and rotation method, and extended the theory to three-dimensional cubic mazes in two complementary models.

That work was, by design, an introduction. It focused on finite grids, primarily in two dimensions, and established the core theorems that make maze theory a coherent subject. It touched upon — but did not fully develop — the deeper algebraic structures, the infinite extensions, the non-cubic geometries, and the higher-dimensional generalisations that naturally follow from the foundational results.

The present volume — Maze Theory: Advanced Structures (Paper 2) — takes up these threads. It builds directly on Paper 1, assuming familiarity with its definitions and main theorems, and extends the theory in four principal directions:

To infinity: The constructive theory of infinite mazes, grounded in a heterodox philosophy of infinity that rejects set-theoretic actual infinity in favour of algorithmic, decidable structures.
To algebra: The full development of logic numbers — binary, ternary, and higher-order — as a numeric encoding of maze connectivity, with arithmetic operations, transformation laws, and algebraic structures (rings, modules, semigroups).
To non-cubic geometries: The extension of maze theory to lozenge tilings, rhombic grids, and Penrose tilings, with special attention to the three-colour structure of lozenge tilings and its implications for path theory.
To higher dimensions: The generalisation of maze theory to four dimensions and beyond, with detailed treatment of the three-dimensional case — the most important extension — including the proper 3D model (unique Hamiltonian path) and the icon-face model (multiple paths).

The volume concludes with a comprehensive list of open problems — 37 in total — charting the future of maze theory and its connections to topology, symbolic dynamics, quasicrystals, and cryptography.

0.2 The Scope of Paper 2

Paper 2 is not a replacement for Paper 1, nor is it a revision. It is a companion volume — a second pillar supporting the same edifice. Where Paper 1 is foundational, Paper 2 is structural. Where Paper 1 is introductory, Paper 2 is advanced. Where Paper 1 establishes the existence of perfect mazes, Paper 2 explores their properties, their algebra, and their generalisations.

The relationship between the two volumes is summarised in Table 1.

Table 1: Relationship Between Paper 1 and Paper 2

Domain	Paper 1 (Foundations)	Paper 2 (Advanced)
Grids	Finite (2D, some 3D)	Infinite, higher-dimensional, non-cubic
Icons	The 16 icons (visual)	Logic numbers (algebraic)
Connectivity	K and NK matrices	Composition, transformation, algebra
Paths	Existence and uniqueness	Classification, entropy, periodicity
Construction	Crystallisation, rotation	Generalised algorithms in all dimensions
Applications	Cryptographic preview	Deferred to Paper 4; pure theory here

0.3 The Heterodox Foundation

A word on the philosophical orientation of this work is necessary.

The theory of infinite mazes developed in Part I rests on a heterodox theory of infinity — a constructive approach that rejects the set-theoretic treatment of infinite sets as completed totalities. In this view, an infinite maze exists only if it can be generated by a finite rule, algorithm, or construction. Properties of infinite mazes must be decidable (or at least computably enumerable) to be meaningful.

This is not a mere philosophical preference. It has direct mathematical consequences:

Infinite mazes are classified by computational complexity, not by cardinality.
The "size" of an infinite maze is measured by growth rates and entropy, not by set-theoretic cardinality.
Decidability and computability are central concerns.
Aperiodic mazes (analogous to Penrose tilings) are of particular interest, as they exhibit infinite complexity without periodicity.

This heterodox foundation is developed in Part I and informs all subsequent parts of the work. It is, we believe, the correct framework for a constructive theory of mazes — one that is aligned with the traditions of computability theory, symbolic dynamics, and aperiodic order.

0.4 The Algebra of Logic Numbers

Part II develops the logic numbers — the algebraic core of maze theory. The 16 icons of Paper 1 are encoded as 4-bit binary numbers (0–15). Operations on these numbers — bitwise AND, OR, XOR, NOT; addition and multiplication modulo 16; rotation and reflection as bit permutations — give rise to a rich algebraic structure.

The logic numbers are not merely a convenient encoding. They are the language in which maze theory is most naturally expressed. The differentiation operator Δ (XOR of adjacent logic numbers) becomes a linear operation. The period of circular differentiation becomes a number-theoretic invariant. The composition of adjacent cells becomes an algebraic operation.

For three-dimensional mazes, the logic numbers become ternary (values 0, 1, 2), with 729 possible states. For higher dimensions, they generalise to arbitrary alphabets and arbitrary numbers of faces. The algebra of logic numbers is thus a universal algebra for maze connectivity — a single framework that encompasses all grid types and all dimensions.

0.5 Non-Cubic Geometries

Part III extends maze theory to lozenges and rhombic tilings — a natural generalisation of the square grid. The lozenge tiling has a three-colour structure (each lozenge belongs to one of three orientations), and adjacent lozenges always have different colours. This has profound implications for path theory: the Hamiltonian path of a lozenge maze must visit cells of all three colours in a constrained sequence.

The study of lozenge mazes also connects maze theory to Penrose tilings and quasicrystals. Penrose tilings are aperiodic lozenge tilings with two types of rhombi (thin and thick). A Penrose maze — a maze on a Penrose tiling — would be an aperiodic maze, with no translational symmetry but (potentially) a unique Hamiltonian path. This is one of the most exciting open problems in the field.

0.6 Higher Dimensions and 3D Mazes

Part IV generalises maze theory to arbitrary dimensions . The icon system in dimensions has binary states (or for alphabets of size ). The crystallisation algorithm and rotation method generalise naturally.

Part V provides a comprehensive treatment of three-dimensional mazes — the most important extension beyond 2D. The two models from Paper 1 are deepened:

The proper 3D maze (cell-centred, unique Hamiltonian path) is analysed through the lens of logic numbers.
The icon-face model (face-adjacency, multiple paths) is developed further, with the ternary logic numbers providing a complete algebraic description.

The 64 binary icons of proper 3D mazes are classified by symmetry into 10 orbits. The 729 ternary icons of the icon-face model are introduced, with the partial state (2) representing doors, portals, or one-way membranes. The non-uniqueness of Hamiltonian paths in 3D — a phenomenon that does not occur in 2D — is established and discussed.

0.7 The Algebra of Path Spaces

Part VI unifies the algebraic threads of the preceding parts. The path space — the set of all valid paths in a grid — is shown to be a monoid under concatenation, with a group of cycles and (over a field) a vector space structure. Operations on path spaces — concatenation, differentiation, integration, rotation, reflection — are defined and analysed.

The category of mazes is introduced, with perfect mazes as objects and boundary-preserving maps as morphisms. Invariants of path spaces — length, logic-number sum, period, entropy — are defined and discussed.

This algebraic framework provides the unifying structure for all of maze theory, connecting the combinatorial, geometric, and computational aspects of the subject.

0.8 Open Problems and Future Directions

Part VII collects 37 open problems across ten categories: enumeration, structural, algebraic, infinite mazes, non-cubic grids, cryptographic, connections to other fields, higher-order generalisations, computational, and philosophical.

These problems are not afterthoughts. They are the natural outgrowth of the theory developed in Papers 1 and 2 — questions that arise from the theorems we have proved and the structures we have discovered. They chart the future of maze theory and its connections to topology, symbolic dynamics, quasicrystals, and cryptography.

0.9 Relationship to the Other Papers

This volume is the second of four papers on maze theory and its applications:

Paper	Title	Focus
Paper 1	Maze Theory: Foundations	Basic theory, finite grids, 16 icons, crystallisation, 3D models
Paper 2	Maze Theory: Advanced Structures	Infinite mazes, logic numbers, lozenges, higher dimensions, algebra
Paper 3	Mazecoin	Cryptographic application (cryptocurrency) — complete, unaltered
Paper 4	Cryptographic Applications of Maze Theory	General cryptographic primitives from maze theory — to be written

Paper 2 assumes familiarity with Paper 1 and builds upon its results. It contains no cryptographic content, as that is deferred to Papers 3 and 4. It is, in the fullest sense, a work of pure mathematics — the development of a new branch of discrete geometry and algebra.

0.10 Summary

The present volume establishes the advanced theory of mazes. It extends the foundational results of Paper 1 to infinity, to algebra, to non-cubic geometries, and to higher dimensions. It provides a comprehensive treatment of three-dimensional mazes, the logic-number algebra, and the algebraic structure of path spaces. It concludes with 37 open problems that chart the future of the field.

Maze theory, as developed in these two volumes, is a unified framework for understanding connectivity, paths, and algebraic structure in discrete spaces. It is a new branch of mathematics — one that we believe will continue to grow and reveal new connections for many years to come.

PART I: INFINITE MAZES

1.1 Introduction to Infinite Mazes

Cross-reference to Paper 1: Paper 1, Parts 0–VI, established the theory of perfect mazes on finite grids. All mazes considered there were finite: an grid of cells, each bearing one of the 16 icons, with the path string traversing every cell exactly once.

We now generalise to infinite mazes — mazes on infinite grids, typically or , where the path may be infinite and the grid has no boundary.

This generalisation is not merely an academic exercise. It raises fundamental questions:

What does it mean for an infinite maze to be "perfect"?
Can an infinite maze have a unicursal path that visits every cell exactly once?
Are there infinitely many such mazes? If so, how do they grow?
Which infinite mazes are periodic? Which are aperiodic?
Can we decide, algorithmically, whether a given infinite maze is perfect?

These questions are rooted in the heterodox theory of infinity — a constructive approach that rejects set-theoretic actual infinity in favour of algorithmic, decidable, and computable structures (see §1.2).

1.2 The Heterodox Theory of Infinity: A Constructive Foundation

Before defining infinite mazes, we must establish our philosophical and mathematical orientation toward infinity itself.

1.2.1 The Orthodox View

In classical set theory (Cantor, ZFC), infinite sets are treated as completed totalities. The set of natural numbers is regarded as an actual, existing infinite object. Infinite mazes, under this view, would be arbitrary subsets of , existing independently of any construction.

1.2.2 The Heterodox View

The heterodox theory of infinity, as developed in the foundational papers accompanying this work, takes a different stance:

No actual infinity — Infinite objects are not treated as completed totalities.
Constructive existence — An infinite maze exists only if it can be generated by a finite rule, algorithm, or construction.
Decidability is central — Properties of infinite mazes must be decidable (or at least computably enumerable) to be meaningful.
Growth and asymptotics — The "size" of an infinite maze is measured not by cardinality but by growth rates, entropy, and asymptotic density.

This view aligns with traditions in:

Constructive mathematics (Bishop, Brouwer)
Computability theory (Turing, Post)
Symbolic dynamics (Hedlund, Morse)
Aperiodic order (Penrose, Robinson)

1.2.3 Implications for Maze Theory

Under the heterodox view:

An infinite maze is a computable function , where is the set of 16 icons, or a periodic/aperiodic tiling generated by a finite set of rules.
A "perfect" infinite maze is one whose path can be constructed algorithmically and visits every cell exactly once (a bi-infinite Hamiltonian path).
The classification of infinite mazes is by computational complexity, not by cardinality.

1.3 Defining Infinite Mazes

1.3.1 Grids

Let be an infinite grid. The primary cases are:

The square grid: or
The cubic grid: (or higher) — see Part V
Tessellations: Hexagonal, triangular, lozenge — see Part III

Each cell has a fixed set of neighbours (4 for square, 6 for hexagonal, etc.).

1.3.2 Icons and Connections

As in the finite case (Paper 1, Part I), each cell bears an icon — a 4-bit (square) or 6-bit (hexagonal) pattern indicating which walls are open.

For the square grid, the 16 icons are interpreted as:

Bit 0: North
Bit 1: East
Bit 2: South
Bit 3: West

A wall is open if the corresponding bit is 1; closed if 0.

1.3.3 Perfect Infinite Mazes

Definition 1.1 (Perfect Infinite Maze).
An infinite maze on a grid is perfect if:

Connectivity: Every cell is connected to every other cell via a unique path of open walls (the network is a tree).
Path Existence: There exists an infinite (or bi-infinite) path that visits every cell exactly once — a Hamiltonian path of the infinite tree.
Constructibility: The maze and its path are computable — there exists an algorithm that, for any finite region, can determine the maze's restriction and the path's behaviour within that region.

Note: Condition (3) is the heterodox addition. In classical set theory, one could assert the existence of such a path using the axiom of choice. Here, we require an explicit construction or algorithm.

1.3.4 Periodic Perfect Mazes

A special and important subclass is the periodic perfect maze.

Definition 1.2 (Periodic Perfect Maze).
A perfect maze is periodic if there exists a finite translation vector such that for all cells . That is, the icon pattern repeats indefinitely.

Periodic mazes are finitely representable (by a finite torus) and their properties are decidable by finite-state methods.

1.3.5 Aperiodic Perfect Mazes

Definition 1.3 (Aperiodic Perfect Maze).
A perfect maze is aperiodic if it is not periodic — no nonzero translation preserves the icon pattern.

Aperiodic mazes are of particular interest because:

They exhibit infinite complexity (no finite description by periodicity).
They may be generated by substitution rules or inflation rules (like Penrose tilings — see Part III).
They raise deep questions about decidability: can we determine whether an aperiodic maze is perfect?

1.4 Constructing Infinite Perfect Mazes

1.4.1 The Crystallisation Algorithm Extended to Infinity

In Paper 1, Part V, we developed the Crystallisation Algorithm — a deterministic method for transforming any even-sized binary field into a perfect maze.

Question: Can this algorithm be extended to the infinite case?

Construction 1.1 (Infinite Crystallisation).
Let be a computable binary field (e.g., a periodic or aperiodic sequence). The crystallisation algorithm, applied to each block and then extended to the limit, yields a perfect maze on , provided the algorithm terminates on every finite block and the extensions are consistent.

Open Problem 1.1: Characterise the binary fields for which this limit exists and yields a perfect infinite maze.

1.4.2 Periodic Constructions

The simplest infinite perfect mazes are periodic:

Example 1.1 (The Striped Maze).
Let the icon pattern alternate by rows:

Row even: all cells icon 5 (East+West open)
Row odd: all cells icon 10 (North+South open)

This yields an infinite maze where the path snakes indefinitely. It is periodic with period 2 in the vertical direction.

Example 1.2 (The Spiral Maze).
Let the icon pattern grow outward in a spiral:

The maze is defined by a recursive rule: draw a spiral corridor of width 1.
This is aperiodic (no finite period) but computable.
The path is the spiral itself.

1.4.3 Aperiodic Constructions

Example 1.3 (The Fibonacci Maze).
Let the rows be generated by the Fibonacci word:

Row has a pattern determined by the -th Fibonacci number.
The maze is aperiodic (the Fibonacci word is not periodic).
It may be perfect if the row patterns are chosen to connect uniquely.

Example 1.4 (Penrose Maze).
Using the Penrose rhombus tiling as a substrate:

Each rhombus becomes a cell with 4 or 6 neighbours.
The icon logic generalises to the rhombic case (see Part III).
The tiling is aperiodic, so the maze is aperiodic.
This connects maze theory to quasicrystals and aperiodic order.

1.5 Growth and Entropy

1.5.1 Growth Rates

For a perfect infinite maze, define:

The growth rate is:

(or in 1D).

Finite results (Paper 1, Part IV):

For : (K-matrices) or (E/J mazes)
For :
For : unknown (open)

Heterodox approach: Instead of seeking exact , we seek the growth rate , as this is the meaningful "size" of the infinite family.

Conjecture 1.1: For square mazes, exists and is finite, with .

Open Problem 1.2: Determine exactly.

1.5.2 Topological Entropy

In symbolic dynamics, the entropy of a subshift is:

For 2D mazes, this generalises to:

The entropy measures the exponential growth rate of the number of perfect mazes.

Open Problem 1.3: Compute for the space of perfect square mazes.

1.6 Decidability and Computability

1.6.1 The Perfection Problem

Definition 1.4 (The Perfection Problem).
Given a computable infinite maze (e.g., a periodic pattern or a substitution rule), decide whether is perfect.

Observations:

For periodic mazes, the problem is decidable: reduce to a finite torus and apply the finite perfection algorithms from Paper 1.
For aperiodic mazes, the problem is generally undecidable (by Rice's theorem / reduction from the halting problem).
For substitution mazes, the problem may be decidable for certain classes.

Theorem 1.1 (Undecidability of Infinite Maze Perfection).
The set of computable infinite mazes that are perfect is not decidable.

Proof sketch: Encode a Turing machine into a substitution rule for an infinite maze. The maze is perfect iff the Turing machine halts. Since the halting problem is undecidable, so is the perfection problem.

1.6.2 Semi-Decidability

Theorem 1.2 (Semi-Decidability).
The property "M is perfect" is semi-decidable: if M is perfect, there is an algorithm that will eventually confirm it (by checking larger and larger finite regions).

Proof: For any finite region, check if the induced finite maze is perfect and if the path is consistent. If M is perfect, every finite restriction is perfect, and the algorithm will never find a contradiction.

1.7 Classification of Infinite Mazes

We propose the following classification scheme for infinite mazes:

Class	Description	Examples
Periodic	Finite period; fully decidable	Striped maze, checkerboard maze
Substitution	Generated by inflation/substitution rules	Fibonacci maze, Penrose maze
Computable	Generated by a general algorithm (Turing machine)	Any decidable construction
Non-computable	No finite description; arbitrary set-theoretic	Existence only by choice axiom

Under the heterodox theory, only the first three classes are admissible.

1.8 Summary of Part I

We have established:

The heterodox theory of infinity as the philosophical foundation for infinite mazes.
A constructive definition of perfect infinite mazes.
Methods for constructing periodic and aperiodic perfect mazes.
Growth rates, entropy, and the existence of a finite entropy constant.
The undecidability of the general perfection problem.
A classification scheme for infinite mazes.

1.9 Open Problems (Part I)

Exact growth rate: Determine for square perfect mazes.
Entropy: Compute .
Decidable subclasses: Characterise the largest class of infinite mazes for which perfection is decidable.
Hamiltonian paths in infinite trees: Classify all infinite trees that admit a unique Hamiltonian path.
Aperiodic constructions: Construct an explicit aperiodic perfect maze using a finite substitution rule, analogous to Penrose tilings.

PART II: LOGIC NUMBERS — THE ALGEBRA OF MAZES

2.1 Introduction to Logic Numbers

Cross-reference to Paper 1: In Paper 1, Part I, we introduced the 16 icons as a graphical language for describing maze connectivity. Each icon corresponds to a 4-bit pattern indicating which of the four cardinal walls are open. This is a representation — a convenient visual encoding — but not yet an algebra.

Logic numbers elevate the icon system to a full algebraic structure. They provide:

A numeric encoding of maze connectivity.
Arithmetic operations (addition, multiplication, transformation) on connectivities.
A composition law for combining mazes or path segments.
A bridge between maze theory and other algebraic systems (rings, modules, semigroups).
A foundation for generalising to higher dimensions, non-cubic grids, and infinite mazes.

The term "logic number" reflects the dual nature of these objects:

They are numbers in the sense of having arithmetic operations.
They encode logical connectivity (which walls are open/closed).

2.2 Binary Logic Numbers

2.2.1 Definition

Definition 2.1 (Binary Logic Number).
A binary logic number is a 4-bit integer whose bits correspond to the four cardinal directions:

Bit	Position	Direction
0	1 (LSB)	North
1	2	East
2	4	South
3	8 (MSB)	West

The icon is open in direction if the corresponding bit is 1.

Thus, there is a bijection between the 16 icons and the binary logic numbers 0–15.

Example:

Icon E (East only): bits = 0100₂ = 4₁₀
Icon J (North+East): bits = 0011₂ = 3₁₀
Icon K (North+South): bits = 0101₂ = 5₁₀
Icon NK (East+West): bits = 1010₂ = 10₁₀
Icon F (all walls open): bits = 1111₂ = 15₁₀
Icon 0 (no walls open): bits = 0000₂ = 0₁₀

2.2.2 The Icon Table as Logic Numbers

Revisiting the 16 icons from Paper 1, Part I, we now encode them as logic numbers:

Icon	Name	Binary	Decimal	Open Walls
0	Empty	0000	0	None
1	N	0001	1	North
2	E	0010	2	East
3	J	0011	3	North+East
4	S	0100	4	South
5	K	0101	5	North+South
6	—	0110	6	East+South
7	—	0111	7	North+East+South
8	W	1000	8	West
9	—	1001	9	North+West
10	NK	1010	10	East+West
11	—	1011	11	North+East+West
12	—	1100	12	South+West
13	—	1101	13	North+South+West
14	—	1110	14	East+South+West
15	F	1111	15	All

Note: Some icons in the table above did not receive single-letter names in Paper 1; they are retained here by their logic numbers for completeness.

2.3 Arithmetic of Binary Logic Numbers

2.3.1 Bitwise Operations

The most natural operations on binary logic numbers are the bitwise logical operations inherited from binary arithmetic:

Bitwise AND (): — walls open in both.
Bitwise OR (): — walls open in either.
Bitwise XOR (): — walls open in exactly one.
Bitwise NOT (): — walls closed (complement), restricted to 4 bits.

These operations have direct maze-theoretic interpretations:

Operation	Interpretation
	Intersection of open walls
	Union of open walls
	Symmetric difference (toggle)
	Wall complement (swap open/closed)

2.3.2 Addition and Multiplication

Beyond bitwise operations, we can define arithmetic operations with carry:

Definition 2.2 (Logic Number Addition).
For binary logic numbers , their sum is:

i.e., standard integer addition modulo 16.

Definition 2.3 (Logic Number Multiplication).
For binary logic numbers , their product is:

i.e., standard integer multiplication modulo 16.

Remark: These operations are well-defined but may not have a direct maze-theoretic interpretation. They are useful for algebraic manipulation and for constructing logic-number rings.

2.3.3 Transformation Operations

The geometric transformations of mazes (rotation, reflection) correspond to permutations of the bits:

Definition 2.4 (Rotation by 90°).
For a binary logic number with bits , rotation by 90° clockwise maps:

Definition 2.5 (Reflection).
For a binary logic number , horizontal reflection maps:

Vertical reflection maps:

These operations are permutations of the 16 logic numbers.

2.3.4 Composition

Definition 2.6 (Composition of Logic Numbers).
Given two logic numbers and , representing the connectivity of two adjacent cells, their composition represents the connectivity of the combined region.

For the simplest case (two cells adjacent horizontally), composition corresponds to a wall cancellation: if both cells have an open wall between them, the wall remains open; otherwise, it is closed.

Formally:

where is the mask selecting the shared wall bit.

Open Problem 2.1: Fully characterise the composition operation for all adjacencies (horizontal, vertical, diagonal) and define the algebraic structure (semigroup, monoid, group?).

2.4 Ternary Logic Numbers

2.4.1 Motivation

The binary logic numbers (0–15) describe 2D square mazes where each wall is either open (1) or closed (0). For 3D mazes (see Part V), each face of a cubic cell has three states:

Open: the wall is absent.
Closed: the wall is present.
Partial: a door, portal, or semi-permeable membrane.

This suggests a ternary logic number with values in .

2.4.2 Definition

Definition 2.7 (Ternary Logic Number).
A ternary logic number is a 6-tuple , where the six positions correspond to the faces of a cube:

Position	Face
0	+X
1	-X
2	+Y
3	-Y
4	+Z
5	-Z

The value indicates:

0: Closed (wall present)
1: Open (wall absent)
2: Partial (door/portal)

Thus, there are possible ternary logic numbers, vastly more than the 16 binary ones.

2.4.3 Connection to the Cubic Die Model

In Paper 1, Part X, we developed the icon-face model of 3D mazes, where each face of a cubic cell may be open or closed. The ternary logic number generalises this by adding the partial state (2), which corresponds to:

A door that can be opened/closed by a key.
A portal that teleports to another face.
A one-way membrane allowing passage in only one direction.
A translucent wall allowing vision but not passage.

2.4.4 Arithmetic of Ternary Logic Numbers

The arithmetic of ternary logic numbers extends binary arithmetic in a natural way:

Addition: Componentwise addition modulo 3, with no carry.
Multiplication: Componentwise multiplication modulo 3.
Bitwise operations: Generalise AND, OR, XOR to ternary logic (min, max, symmetric difference).

Definition 2.8 (Ternary Addition).
For :

Definition 2.9 (Ternary Multiplication).
For :

2.4.5 Transformations in 3D

The rotation group of the cube (24 rotations) acts on ternary logic numbers by permuting the six faces. This yields 24 permutation operations, each corresponding to a rotation of the 3D maze.

2.5 Higher-Order Logic Numbers

2.5.1 Generalisation to Dimensions

Definition 2.10 (-Dimensional Logic Number).
A -dimensional logic number is a -tuple over a finite alphabet , where is a set of wall states.

For binary logic numbers:

, , size .

For ternary logic numbers:

, , size .

For general and alphabet size :

2.5.2 Non-Cubic Grids

For hexagonal grids (6 neighbours), the logic numbers are 6-tuples over :

For lozenge (rhombic) grids, the connectivity is more complex, and the number of faces/edges varies (see Part III).

2.6 The Algebra of Logic Numbers

2.6.1 Algebraic Structures

The set of binary logic numbers forms:

Structure	Operation	Status
Semigroup	Bitwise OR	Yes (idempotent)
Semigroup	Bitwise AND	Yes (idempotent)
Monoid	Bitwise OR with identity 0	Yes
Monoid	Bitwise AND with identity 15	Yes
Group	Bitwise XOR	Yes (each element self-inverse)
Ring	Addition mod 16, multiplication mod 16	Yes
Boolean algebra	AND, OR, NOT	Yes

The ternary logic numbers form:

Structure	Operation	Status
Semigroup	Componentwise min/max	Yes
Group	Componentwise addition mod 3	Yes
Ring	Componentwise addition/multiplication mod 3	Yes

2.6.2 Logic Numbers as a Module

Theorem 2.1 (Logic Numbers as a Module).
The set of binary logic numbers forms a module over under bitwise XOR. The set of ternary logic numbers forms a module over under componentwise addition.

2.6.3 Composition Laws

Open Problem 2.2: Define a composition law on logic numbers that corresponds to the concatenation of maze paths. Does this form a group or monoid? Is it associative?

2.7 Logic Numbers and Path Strings

2.7.1 Encoding Paths

A path string in Paper 1 is a sequence of icons (or logic numbers). Under the logic-number encoding:

where each is the logic number of the cell at position along the path.

2.7.2 Differentiation as an Operation on Logic Numbers

The differentiation operator defined in Paper 1, Part VI, applies XOR to adjacent logic numbers:

For circular differentiation:

This is a linear operation on the logic-number sequence, making it amenable to algebraic analysis.

2.7.3 Period Analysis

The period of circular differentiation for a path of length is the order of the linear transformation on . For binary logic numbers (4 bits), the period divides:

For , the period is (as established in Paper 1, Part VI).

For ternary logic numbers (6 trits), the period divides:

2.8 Summary of Part II

We have established:

The binary logic numbers (0–15) as a numeric encoding of the 16 icons.
Arithmetic operations: AND, OR, XOR, NOT, addition mod 16, multiplication mod 16.
Geometric transformations: rotation, reflection as bit permutations.
The composition operation for adjacent cells (wall cancellation).
Ternary logic numbers for 3D mazes (729 states).
Higher-order logic numbers for -dimensional and non-cubic grids.
Algebraic structures: semigroups, monoids, groups, rings, modules.
Connection to path strings and the differentiation operator.

Logic numbers provide the algebraic backbone for all subsequent developments in maze theory.

2.9 Open Problems (Part II)

Full characterisation of composition: Define composition for all adjacencies and prove associativity (or not).
Classification of finite algebras: Determine all algebraic structures arising from logic numbers.
Ternary logic numbers in 3D: Complete the enumeration of all 729 ternary logic numbers and classify them by symmetry.
Higher-order logic numbers: Develop the full theory for and arbitrary alphabets.
Applications to path algebra: Connect logic-number arithmetic to the algebra of path spaces (Part VI).

PART III: LOZENGES AND RHOMBIC TILINGS

3.1 Introduction to Lozenges

Cross-reference to Paper 1: The preceding parts of this work have focused exclusively on square grids — the natural Cartesian lattice with four cardinal neighbours. This choice is historically and computationally convenient, but it is not the only possible substrate for maze theory.

We now generalise to lozenge (rhombus) tilings, sometimes called rhombic tilings or diamond tilings. These are:

Two-dimensional tilings of the plane by congruent rhombi (lozenges).
Typically formed by three orientations of rhombi (acute angles and ), corresponding to the projection of a cubic lattice.
Closely related to hexagonal grids, triangular lattices, and Penrose tilings.

The study of mazes on lozenge tilings is motivated by several considerations:

Generality: Square grids are a special case of a broader class of tilings.
Richness: Lozenge tilings exhibit aperiodic order (Penrose) and complex connectivity.
Physical relevance: Rhombic structures appear in materials science, quasicrystals, and natural patterns.
Algebraic generalisation: The 16 icons of the square grid are a special case of a more general icon system for rhombic tilings.

3.2 Lozenge Tilings: Basic Concepts

3.2.1 Definition

Definition 3.1 (Lozenge Tiling).
A lozenge tiling of the plane is a tiling by congruent rhombi (lozenges), each with side length 1 and angles and .

The most important case for maze theory is the regular lozenge tiling with angles and , which is isomorphic to the hexagonal lattice.

3.2.2 The Hexagonal Lattice

The hexagonal lattice (or honeycomb lattice) is the dual of the triangular tiling. Each cell (hexagon) has 6 neighbours, arranged in a ring.

However, for lozenge tilings, it is more natural to treat the rhombi themselves as cells, rather than hexagons. Each rhombus has 4 neighbours (sharing edges), but the tiling as a whole has 3 rhombus orientations.

3.2.3 Orientations

In a regular lozenge tiling (angles ), there are three orientations of lozenges:

Orientation	Description
Type A	Rhombus with acute angle pointing up/down
Type B	Rhombus with acute angle pointing left/right (rotated 60°)
Type C	Rhombus with acute angle pointing at 120°/300° (rotated 120°)

Each lozenge shares edges with lozenges of the other two orientations, not with lozenges of the same orientation. This creates a 3-colouring of the lozenge tiling.

3.3 Maze Theory on Lozenge Tilings

3.3.1 Cells and Neighbours

In a lozenge tiling, each cell (rhombus) has exactly 4 neighbours, one across each of its four edges.

However, unlike the square grid, the neighbour directions are not orthogonal. The adjacency graph of a lozenge tiling is a triangular lattice (the dual of the hexagonal tiling).

Key observation: The adjacency graph of a lozenge tiling is bipartite (like the square grid) if and only if the tiling is balanced (equal numbers of each orientation). In general, it may not be bipartite, leading to odd cycles and different path properties.

3.3.2 Icon Logic for Lozenges

For a rhombic cell with 4 neighbours, we can define an icon system analogous to the square grid:

Each edge is open (1) or closed (0).
There are possible icons, exactly as in the square case.

However, the geometric arrangement of the four edges is different:

In the square grid, the four edges are arranged as two perpendicular pairs (N/S and E/W).
In a rhombus, the four edges form two pairs at angles and (for the acute and obtuse angles).

Thus, the 16 icons of the lozenge grid are isomorphic to the 16 icons of the square grid as abstract connectivity patterns, but their geometric interpretation differs.

3.3.3 Logic Numbers for Lozenges

Since there are still 4 edges, the binary logic numbers for lozenges are the same set . However:

The transformation operations (rotation, reflection) act differently because the symmetry group of a rhombus is smaller than that of a square.
The composition operation (wall cancellation) depends on the angles between edges.

Definition 3.2 (Lozenge Logic Numbers).
A lozenge logic number is a 4-bit integer where the bits correspond to the four edges of a rhombus, ordered cyclically around the cell.

3.4 The Three-Colour Structure

3.4.1 Orientation Colours

As noted above, each lozenge in a regular tiling belongs to one of three orientations. We can assign a colour to each cell indicating its orientation.

Theorem 3.1 (Colour Adjacency).
In a regular lozenge tiling, adjacent lozenges always have different colours. Moreover, the colour of a neighbour is determined by the edge crossed:

Edge 0 → colour
Edge 1 → colour
Edge 2 → colour
Edge 3 → colour

Thus, the adjacency graph is 3-colourable (each vertex has neighbours of the other two colours).

3.4.2 Mazes and 3-Colouring

The 3-colouring of lozenge tilings has deep implications for maze theory:

Path colour sequences: A path through a lozenge maze alternates colours in a constrained way (not simply alternating, but following the colour rules above).
Perfect mazes: In a perfect lozenge maze, the Hamiltonian path must visit cells of all three colours in a balanced way.
Higher-order logic numbers: The colour structure suggests a ternary logic number system where each cell's state depends on its colour and its neighbours.

Open Problem 3.1: Characterise the perfect mazes on lozenge tilings. Do they exist for all tilings? Are there multiple Hamiltonian paths?

3.5 Aperiodic Lozenge Tilings: Penrose Mazes

3.5.1 Penrose Tilings

The most famous aperiodic lozenge tiling is the Penrose tiling, consisting of two types of rhombi:

Thin rhombus: angles and
Thick rhombus: angles and

Penrose tilings are:

Aperiodic: no translational symmetry.
Non-periodic but repetitive: every finite pattern appears infinitely often.
Self-similar: can be generated by inflation/deflation rules.

3.5.2 Mazes on Penrose Tilings

Definition 3.3 (Penrose Maze).
A Penrose maze is a maze on a Penrose tiling, where each rhombus (thin or thick) is a cell with 4 neighbours, and each edge is open or closed according to an icon pattern.

Since there are two types of cells (thin and thick), the icon system must be augmented:

Thin rhombus: 16 icons
Thick rhombus: 16 icons
Total: 32 possible cell states (but not all combinations may be admissible due to tiling constraints).

3.5.3 Perfect Penrose Mazes

Question: Do perfect Penrose mazes exist?

Conjecture 3.1: Yes — there exist perfect mazes on Penrose tilings, generated by substitution rules.

Construction 3.1 (Substitution Maze).
Define a substitution rule that maps each rhombus in a Penrose tiling to a pattern of rhombi in the inflated tiling. Assign icons to the rhombi such that:

The inflated tiling is a maze.
The path is consistent across substitutions.
The limit (infinite) tiling is perfect.

This would yield an aperiodic perfect maze — a maze with no translational symmetry but with a unique Hamiltonian path.

Open Problem 3.2: Construct an explicit substitution rule for a perfect Penrose maze. Determine whether the resulting path is unique.

3.6 Logic Numbers for Lozenges and Rhombi

3.6.1 Generalised Logic Numbers

For a rhombus with 4 neighbours, the logic numbers are still 4-bit, but the symmetry group is smaller (order 2 for a general rhombus, order 4 for a square).

Definition 3.4 (Rhombic Logic Numbers).
The set of rhombic logic numbers is the same as , but with a different action of the dihedral group:

For a non-square rhombus, only identity and 180° rotation are symmetries.
The reflection operations (horizontal/vertical) are not symmetries unless the rhombus is a square or has additional symmetry.

This means that transformations on rhombic mazes are more restricted than on square mazes.

3.6.2 Composition on Rhombic Grids

The composition operation for adjacent rhombi depends on the angle between the shared edge and the cell axes. In general:

where is the mask selecting the shared edge.

Observation: The composition operation on rhombic grids is orientation-dependent: the same pair of logic numbers may compose differently depending on the relative orientation of the two rhombi.

3.7 Non-Cubic Grids as Logic-Number Algebras

3.7.1 Unification

We can now see a unified picture:

Grid Type	Neighbours	Logic Number Set	Size	Symmetry
Square (2D)	4		16	(order 8)
Rhombic (lozenge)	4		16	(order 2)
Hexagonal	6		64	(order 12)
Triangular	6		64	(order 12)
Cubic (3D)	6		729	(order 24)
Hypercubic (D)				Hyperoctahedral

The logic-number algebra generalises to any grid by taking the set of all subsets of edges incident to a cell.

3.8 Summary of Part III

We have established:

Lozenge tilings as a natural generalisation of square grids.
The three-colour structure of lozenge tilings and its implications for maze paths.
Penrose tilings as a source of aperiodic mazes.
The icon logic and logic numbers for rhombic grids.
The restricted symmetry of non-square rhombi.
A unified view of logic numbers across all grid types.

Lozenges provide a bridge between:

The algebraic theory of logic numbers (Part II)
The geometric theory of non-cubic grids (Part V)
The aperiodic order of quasicrystals and Penrose tilings.

3.9 Open Problems (Part III)

Perfect mazes on lozenge tilings: Characterise all perfect mazes on regular lozenge tilings. Do they exist for all tilings? What are the enumeration results for finite patches?
Penrose mazes: Construct an explicit perfect Penrose maze and determine whether the path is unique.
Colour constraints: Fully characterise the colour sequences of paths in lozenge mazes.
Composition on rhombic grids: Define composition in full generality and classify the resulting algebraic structures.
Higher-order lozenge tilings: Generalise to 3D rhombohedral tilings (e.g., the cubic lattice projected to 3D).

PART IV: HIGHER-DIMENSIONAL MAZES

4.1 Introduction to Higher Dimensions

Cross-reference to Paper 1: Paper 1, Parts IX–X, provided a preliminary treatment of three-dimensional cubic mazes. We now generalise to arbitrary dimensions .

Higher-dimensional mazes are not merely a mathematical curiosity. They arise naturally in:

Computational geometry: Pathfinding in high-dimensional configuration spaces.
Coding theory: Error-correcting codes as Hamiltonian paths in hypercubes.
Topology: Generalisations of the Jordan curve theorem and duality.
Physics: Lattice models in statistical mechanics (Ising model, percolation).
Cryptography: Key spaces with exponentially large dimensions (deferred to Paper 4).

The theory developed in Parts I–III provides the necessary algebraic and constructive foundations for this generalisation.

4.2 The -Dimensional Grid

4.2.1 Definition

Definition 4.1 (-Dimensional Grid).
The -dimensional grid is the set of all integer -tuples:

Two cells are adjacent if they differ in exactly one coordinate by . Each cell has exactly neighbours.

Finite grids: The finite -dimensional grid has cells.

4.2.2 Icons in Dimensions

In dimensions, each cell has faces (walls). An icon is a binary string of length :

where each bit indicates whether the corresponding wall is open (1) or closed (0).

Number of icons:

Dimension	Neighbours	Icons
2	4	16
3	6	64
4	8	256
5	10	1024

Observation: The number of icons grows exponentially with dimension, making exhaustive enumeration impractical beyond or 5.

4.3 Perfect Mazes in Higher Dimensions

4.3.1 Definition

Definition 4.2 (Perfect -Dimensional Maze).
A maze on a finite -dimensional grid is perfect if:

Connectivity: The network of open walls connects all cells in a single connected component.
Uniqueness: There is exactly one path between any two cells (the open-wall network is a tree).
Hamiltonian path: There exists a path that visits every cell exactly once, following open walls.

For infinite -dimensional grids, the definition extends as in Part I (constructive, periodic, or aperiodic).

4.3.2 The Spanning Tree Condition

As in 2D, the open-wall network of a perfect maze is a spanning tree of the grid graph. The maze is defined by choosing a spanning tree of the -dimensional grid graph and then assigning icons accordingly.

Key difference: The grid graph has exponentially many spanning trees, and their structure is much more complex than in 2D.

Theorem 4.1 (Existence).
For every finite -dimensional grid, there exists at least one perfect maze.

Proof: Take any spanning tree of the grid graph (e.g., a depth-first search tree) and construct the maze from it.

4.4 Hamiltonian Paths in Dimensions

4.4.1 Existence

Theorem 4.2 (Hamiltonian Paths in Hypercubes).
Every finite -dimensional grid with an even number of cells has a Hamiltonian path. If the number of cells is odd, a Hamiltonian path cannot visit every cell exactly once (parity argument).

Proof: The grid graph is bipartite. A Hamiltonian path alternates between the two bipartition classes. If the total number of cells is odd, the path would start and end in the same class, which is impossible.

Corollary: A perfect maze exists only if the grid has an even number of cells. For odd grids, no perfect maze exists.

4.4.2 Uniqueness of Hamiltonian Paths

In 2D perfect mazes, the Hamiltonian path is unique (a key theorem from Paper 1, Part IX). Does this generalise to higher dimensions?

Theorem 4.3 (Non-Uniqueness in ).
For , there exist perfect mazes with multiple Hamiltonian paths.

Proof sketch: Construct a 3D maze with a "branch" that can be traversed in two different orders without breaking the tree property. This is impossible in 2D due to planarity constraints.

Open Problem 4.1: Classify all perfect -dimensional mazes by the number of Hamiltonian paths. Is the maximum finite or infinite? Does it grow with grid size?

4.5 Logic Numbers in Higher Dimensions

4.5.1 Generalised Binary Logic Numbers

Definition 4.3 (-Dimensional Binary Logic Number).
A -dimensional binary logic number is a binary string of length :

with bits corresponding to the faces of a -dimensional hypercube.

Arithmetic: Componentwise AND, OR, XOR, NOT, as in 2D.
Transformations: The hyperoctahedral group (symmetry group of the -cube) acts on by permuting the bits.

4.5.2 Ternary and Higher-Order Logic Numbers

For -dimensional mazes with doors, portals, or partial walls, we generalise to alphabets of size :

where . The size is:

Example: For , , there are ternary logic numbers.

4.6 Constructing Higher-Dimensional Mazes

4.6.1 The Crystallisation Algorithm in Dimensions

The crystallisation algorithm from Paper 1, Part V, generalises naturally to dimensions:

Start with a -dimensional binary field (each cell has a 0/1 value).
Apply the crystallisation rules to produce a perfect maze.

Algorithm 4.1 (-Dimensional Crystallisation).
For each cell, the icon is determined by comparing its value with its neighbours:

where bit is 1 if the neighbour is in a different state, 0 otherwise.

Theorem 4.4 (Correctness).
For any -dimensional binary field with even side lengths, the crystallisation algorithm produces a perfect -dimensional maze.

Proof: The proof generalises the 2D argument: the algorithm creates a spanning tree of the grid graph by constructing the "boundary" between 0 and 1 regions.

4.6.2 Rotation Method in Dimensions

The rotation method (swapping primary and secondary networks) also generalises:

The -dimensional grid has directions.
A rotation in the -plane swaps the and axes, inducing a permutation of the faces.
Applying this rotation to a perfect maze yields another perfect maze.

Observation: The number of rotations in dimensions is the order of the hyperoctahedral group , which grows rapidly with .

4.7 Infinite Higher-Dimensional Mazes

4.7.1 Constructive Definition

As in Part I, infinite -dimensional mazes are defined constructively:

Definition 4.4 (Perfect -Dimensional Infinite Maze).
An infinite maze on is perfect if:

The open-wall network is a tree.
There exists a bi-infinite Hamiltonian path visiting every cell exactly once.
The maze is computable (there exists an algorithm generating it).

4.7.2 Periodic Constructions

The simplest infinite -dimensional mazes are periodic:

Example 4.1 (Striped -Dimensional Maze).
Let the icon pattern depend only on the first coordinate:

If is even: all walls in the direction are open; others closed.
If is odd: all walls in the direction are open; others closed.

This creates an infinite "layered" maze with a Hamiltonian path that snakes through the layers.

4.7.3 Aperiodic Constructions

Example 4.2 (Substitution Maze in Dimensions).
Generalise the substitution rules from 2D (e.g., Penrose) to dimensions using inflation rules on hypercubic grids.

Open Problem 4.2: Do aperiodic perfect mazes exist in ? If so, construct an explicit example and classify its properties.

4.8 Enumeration and Growth

4.8.1 Number of Perfect Mazes

For a -dimensional grid of size , the number of perfect mazes is the number of spanning trees of the grid graph:

where is the number of spanning trees.

Known results:

For , the number of spanning trees of the grid is known (exact formula using eigenvalues).
For , exact formulas are much more complex, but asymptotic results exist.

Theorem 4.5 (Asymptotic Growth).
For a -dimensional grid of side length :

where is a constant depending only on .

4.8.2 Entropy

The entropy of -dimensional perfect mazes is:

Open Problem 4.3: Compute exactly for .

4.9 Decidability in Higher Dimensions

4.9.1 The Perfection Problem

As in 2D, the perfection problem for infinite -dimensional mazes is undecidable in general (by the same Turing machine encoding argument).

Theorem 4.6 (Undecidability).
For , the set of computable infinite -dimensional mazes that are perfect is not decidable.

4.9.2 Decidable Subclasses

Periodic -dimensional mazes are decidable (reduce to a finite torus).

Conjecture 4.1: Substitution mazes in dimensions are decidable if the substitution rule has the "finite local complexity" property.

4.10 Topological Properties

4.10.1 Duality

The dual of a -dimensional grid graph is another -dimensional grid (the "dual lattice"). The primary and secondary networks of a perfect maze are complementary spanning trees.

Theorem 4.7 (Duality in Dimensions).
In any perfect -dimensional maze, the primary network (open walls) and secondary network (closed walls) are complementary spanning trees of the grid graph and its dual.

Proof: Generalises the 2D proof using planar duality. The -dimensional case requires -dimensional duality (Poincaré duality).

4.10.2 Hamiltonian Paths and Topology

In 2D, the Hamiltonian path of a perfect maze is a space-filling curve (a discrete Jordan curve). In dimensions, the path is a space-filling curve in -dimensional space.

Observation: The Hamiltonian path of a -dimensional perfect maze is a discrete analogue of a Hilbert curve or Peano curve.

4.11 Summary of Part IV

We have established:

The generalisation of maze theory to dimensions.
The icon system: possible icons.
Existence of perfect mazes for all finite grids with even cell count.
Non-uniqueness of Hamiltonian paths for .
Generalised logic numbers for higher dimensions.
The crystallisation algorithm and rotation method in dimensions.
Infinite higher-dimensional mazes (periodic and aperiodic).
Enumeration and entropy bounds.
Undecidability of the perfection problem.
Duality and topological properties.

Higher-dimensional maze theory reveals a rich structure that is fundamentally different from the 2D case — in particular, the failure of path uniqueness opens new avenues for both theory and applications.

4.12 Open Problems (Part IV)

Classification of Hamiltonian paths: Classify all perfect -dimensional mazes by the number of Hamiltonian paths. Is the number always finite? Does it grow polynomially or exponentially?
Exact enumeration: Compute the number of perfect mazes on small -dimensional grids ().
Entropy constant: Compute for .
Aperiodic constructions: Construct explicit aperiodic perfect mazes in .
Decidable subclasses: Characterise the largest class of infinite -dimensional mazes for which perfection is decidable.
Topological invariants: What topological invariants (e.g., homology, linking numbers) distinguish different Hamiltonian paths in the same maze?

PART V: THREE-DIMENSIONAL MAZES — A COMPREHENSIVE TREATMENT

5.1 Introduction to 3D Mazes

Cross-reference to Paper 1: Three-dimensional mazes are the most important extension of maze theory beyond 2D. In Paper 1, Parts IX–X, we introduced two complementary 3D models:

The proper 3D maze (cell-centred, unique path, Hamiltonian)
The icon-face model (face-adjacency, multiple paths)

We now deepen this treatment, integrating the logic-number algebra (Part II), the constructive framework (Part I), and the higher-dimensional generalisation (Part IV) to provide a complete theory of 3D mazes.

5.2 The 3D Cubic Grid

5.2.1 Definition

Definition 5.1 (3D Cubic Grid).
The 3D cubic grid consists of cells indexed by triples . Each cell has six neighbours, corresponding to the six faces of a cube:

Finite grids: The finite 3D grid has cells.

5.2.2 Icons in 3D

Each cell has 6 faces. An icon is a 6-bit string:

There are possible icons.

Notation: We denote icons by their 6-bit values, with bits ordered as:

Example:

Icon 0 (000000): all walls closed (isolated cell).
Icon 63 (111111): all walls open (full connectivity).
Icon with bits and only: a 1D corridor along the X-axis.

5.3 The Two 3D Models (Recap and Deepening)

5.3.1 Model 1: Proper 3D Maze (Cell-Centred)

Definition 5.2 (Proper 3D Maze).
A proper 3D maze is a maze on a 3D cubic grid where:

The primary network (open walls) connects all cells in a single tree.
The secondary network (closed walls) is also a tree (dual tree).
The unicursal path is the unique Hamiltonian path of the primary network.
The path visits cell centres, moving from cell to adjacent cell through open faces.

Key theorems (from Paper 1, Part IX):

The secondary network is a tree.
The unicursal path is unique (for 2D; see §5.5 for the 3D caveat).
The path is a Hamiltonian path of the cell graph.

New result (Paper 2):

Theorem 5.1 (Logic-Number Encoding of Proper 3D Mazes).
Every proper 3D maze corresponds to a ternary logic number for each cell, where:

The primary network determines which faces are open (1).
The secondary network determines which faces are closed (0).
The partial state (2) is not used (proper mazes have no partial walls).

Thus, proper 3D mazes are a subset of the 729 ternary logic numbers, specifically those with no 2s: icons.

Corollary: The set of proper 3D mazes is isomorphic to the set of binary 6-bit icons.

5.3.2 Model 2: Icon-Face Model (Face-Adjacency)

Definition 5.3 (Icon-Face Model).
In the icon-face model:

Faces, not cells, are the primary units of the path.
The path traverses open faces of the cubic cells.
Faces are adjacent if they share an edge (not a face).
The secondary network is the face-adjacency graph, which may contain cycles.
Multiple unicursal paths may exist.

Key theorems (from Paper 1, Part X):

The secondary network is a face-adjacency graph with cycles.
The unicursal path is not unique — multiple paths exist.
The path traverses open faces, not cell centres.

New result (Paper 2):

Theorem 5.2 (Logic-Number Encoding of Icon-Face Mazes).
Every icon-face maze corresponds to a ternary logic number for each cell, where:

1: Face is open (part of the primary network).
0: Face is closed (part of the secondary network).
2: Face is partial — a door, portal, or one-way membrane.

The partial state introduces cycles into the secondary network, allowing multiple paths.

5.4 Classification of 3D Mazes by Logic Numbers

5.4.1 The 64 Binary Icons (Proper 3D)

The 64 binary 6-bit icons are the proper 3D icons. They can be classified by the number of open faces:

Class	Description	Number
0-connected	All walls closed	1
1-connected	One face open	6
2-connected	Two faces open (adjacent, opposite, or at 90°)	15
3-connected	Three faces open	20
4-connected	Four faces open	15
5-connected	Five faces open	6
6-connected	All faces open	1
Total		64

Observation: This classification mirrors the 2D case (16 icons classified by number of open walls).

5.4.2 Symmetry Orbits of 3D Binary Icons

The full octahedral group (order 48, including reflections) acts on the 6 faces of the cube. The 64 binary 6-bit icons partition into 10 orbits, which we compute by considering the number of 1s (open faces) in the pattern:

Orbit	Size	Representative Pattern	Description (by number of 1s)
1	1	000000	0 ones (all closed)
2	6	100000	1 one
3	12	110000	2 ones (adjacent faces)
4	3	101000	2 ones (opposite faces)
5	8	111000	3 ones (meeting at a vertex)
6	12	110100	3 ones (one opposite pair + one adjacent)
7	3	111100	4 ones (band: two opposite closed)
8	12	111010	4 ones (adjacent closed faces)
9	6	111110	5 ones
10	1	111111	6 ones (all open)

Verification (sum of orbit sizes):

5.4.3 The 729 Ternary Icons (Icon-Face Model)

The 729 ternary 6-bit icons include:

64 binary icons (no 2s): proper 3D mazes.
665 icons with at least one partial wall: icon-face mazes with doors/portals.

The partial walls introduce new connectivity types:

Partial Walls	Effect
1 partial	One door/portal; creates a cycle in the secondary network
2 partial	Two doors/portals; multiple cycles possible
3+ partial	Rich connectivity; many possible paths

5.5 Proper 3D Mazes: Unique Hamiltonian Paths?

5.5.1 Existence

Theorem 5.4 (Existence of Proper 3D Mazes).
For every finite 3D grid with an even number of cells, there exists at least one proper 3D maze.

Proof: Construct any spanning tree of the 3D grid graph (e.g., a DFS tree). The tree defines the primary network; the dual tree is guaranteed by 3D duality.

5.5.2 Uniqueness in 3D

Cross-reference to Paper 1: In Paper 1, Part IX, we stated that the Hamiltonian path in a proper 3D maze is unique. This requires re-examination.

Theorem 5.5 (Non-Uniqueness in 3D Proper Mazes).
There exist proper 3D mazes with multiple Hamiltonian paths.

Proof sketch: Construct a 3D maze with a "branching" structure that allows the path to be traversed in different orders. This is possible in 3D because the grid is not planar, so the path is not constrained to be a simple space-filling curve in the same way as in 2D.

Open Problem 5.1: Classify all proper 3D mazes by the number of Hamiltonian paths. Is the number always finite? Does it grow with grid size?

5.6 Icon-Face Model: Multiple Paths

5.6.1 The Face-Adjacency Graph

Definition 5.4 (Face-Adjacency Graph).
The face-adjacency graph of a 3D maze has:

Vertices: the faces of the cubic cells.
Edges: between faces that share an edge (not a face).

The path in the icon-face model traverses faces, moving from one face to an adjacent face across a shared edge.

5.6.2 Cycles in the Secondary Network

Theorem 5.6 (Cycles in Icon-Face Model).
The secondary network of an icon-face maze contains cycles whenever there are partial walls (state 2) or when the primary network has a cycle.

Proof: The face-adjacency graph of a 3D cubic grid is not a tree; it contains cycles even in the simplest configurations. When walls are open (state 1), the face-adjacency graph includes cycles corresponding to loops around cells.

5.6.3 Multiple Hamiltonian Paths

Theorem 5.7 (Multiple Paths in Icon-Face Model).
In the icon-face model, there are multiple Hamiltonian paths of the face-adjacency graph.

Proof: The presence of cycles in the face-adjacency graph allows for different traversals of the same set of faces. The number of paths corresponds to the number of Hamiltonian paths in the face-adjacency graph (which may be exponentially large).

5.7 Logic Numbers and 3D Mazes

5.7.1 Binary Logic Numbers for 3D (Proper Model)

Definition 5.5 (3D Binary Logic Number).
A 3D binary logic number is a 6-bit integer , with bits corresponding to the six faces of a cube.

The set of 3D binary logic numbers is .

Operations: Bitwise AND, OR, XOR, NOT (componentwise).
Transformations: The octahedral group (order 48) acts on by permuting the 6 bits.

5.7.2 Ternary Logic Numbers for 3D (Icon-Face Model)

Definition 5.6 (3D Ternary Logic Number).
A 3D ternary logic number is a 6-tuple , where:

0: Closed wall.
1: Open wall.
2: Partial wall (door, portal, membrane).

The set of 3D ternary logic numbers is , with elements.

Operations: Componentwise addition modulo 3, componentwise multiplication modulo 3.
Transformations: The octahedral group acts on by permuting the 6 positions.

5.8 Constructing 3D Mazes

5.8.1 Crystallisation Algorithm in 3D

Algorithm 5.1 (3D Crystallisation).
Given a 3D binary field :

For each cell , consider its 6 neighbours.
For each neighbour , set bit if .
The resulting 6-bit number is the icon of cell .

Theorem 5.8 (Correctness in 3D).
For any 3D binary field with even dimensions, the crystallisation algorithm produces a perfect 3D maze.

Proof: The algorithm constructs the boundary between 0-regions and 1-regions. This boundary is a spanning tree of the dual graph, ensuring the maze is perfect.

5.8.2 Rotation Method in 3D

The 3D rotation method swaps the primary and secondary networks:

Take a perfect 3D maze .
Apply a rotation (permutation of the 3 axes).
Swap open and closed walls.
The result is another perfect 3D maze.

Theorem 5.9 (Rotation in 3D).
The rotation method maps perfect 3D mazes to perfect 3D mazes. It is an involution (applying it twice returns the original maze).

5.9 Infinite 3D Mazes

5.9.1 Periodic 3D Mazes

Example 5.1 (Layered 3D Maze).
Let the icon pattern depend only on the -coordinate:

If is even: all and walls open; walls closed.
If is odd: all walls open; and walls closed.

This creates an infinite "layered" 3D maze with a Hamiltonian path that snakes through the layers.

5.9.2 Aperiodic 3D Mazes

Open Problem 5.2: Do aperiodic perfect 3D mazes exist? If so, construct an explicit example using substitution rules.

5.10 Decidability in 3D

Theorem 5.10 (Undecidability in 3D).
The perfection problem for infinite 3D mazes is undecidable.

Proof: The Turing machine encoding from Part I generalises to 3D (and higher) by adding extra dimensions as "scratch space" for the computation.

5.11 Summary of Part V

We have provided a comprehensive treatment of 3D mazes:

The 64 binary icons for proper 3D mazes, classified by symmetry into 10 orbits.
The 729 ternary icons for the icon-face model.
The proper 3D model: unique Hamiltonian path (with caveats for branching).
The icon-face model: multiple Hamiltonian paths due to cycles.
Logic numbers for both models (binary and ternary).
Construction methods: crystallisation and rotation in 3D.
Infinite 3D mazes: periodic and aperiodic.
Decidability: undecidable in general.

3D mazes are the most important extension of maze theory, offering a rich structure with both unique and multiple path properties, and serving as the foundation for cryptographic applications (Paper 4).

5.12 Open Problems (Part V)

Classification of Hamiltonian paths: Classify all proper 3D mazes by the number of Hamiltonian paths. Is the number always finite? Does it grow with grid size?
Enumeration of proper 3D mazes: Count the number of proper 3D mazes on small grids (, , ).
Exact count of 3D perfect dice: Compute the number of perfect cubic dice (no F) beyond the 128 established in Paper 1.
Aperiodic 3D mazes: Construct explicit aperiodic perfect 3D mazes.
Face-adjacency path enumeration: Count the number of Hamiltonian paths in the face-adjacency graph for small 3D grids.
Logic-number composition in 3D: Fully characterise the composition operation for 3D logic numbers.

PART VI: ALGEBRAIC STRUCTURES — THE ALGEBRA OF PATH SPACES

6.1 Introduction

Cross-reference to Paper 1 and Paper 2 Parts I–V: Throughout this work, we have encountered algebraic structures at multiple levels:

The 16 icons (and their higher-dimensional analogues) form sets with operations: bitwise AND, OR, XOR, NOT.
Logic numbers (binary, ternary, higher-order) form rings, modules, and semigroups (Part II).
Path strings are sequences of logic numbers with operations: concatenation, differentiation, integration (Part II).
Mazes themselves form a category under composition (connecting two mazes along a boundary).

In this part, we unify these structures into a coherent algebra of path spaces — a formal framework that captures the algebraic essence of maze theory.

6.2 The Path Space

6.2.1 Definition

Definition 6.1 (Path Space).
For a given grid type and logic-number set , the path space is the set of all finite sequences of logic numbers that correspond to valid paths in the grid:

A path is valid if consecutive cells are adjacent and the open walls align (i.e., if cell has a wall open toward cell , then cell has a wall open toward cell ).

6.2.2 Paths as Words

A path string is a word over the alphabet :

The length of the path is .

6.2.3 The Empty Path

Definition 6.2 (Empty Path).
The empty path is the path of length 0. It is the identity element for concatenation.

6.3 Operations on Path Spaces

6.3.1 Concatenation

Definition 6.3 (Concatenation).
Given two paths and , their concatenation is:

provided that the last cell of is adjacent to the first cell of , and the walls align.

Concatenation is associative:

and has the empty path as identity:

Thus, the path space forms a monoid under concatenation.

6.3.2 Differentiation

Definition 6.4 (Differentiation Operator).
The differentiation operator maps a path to:

where is the bitwise XOR operation on logic numbers.

For circular differentiation, we define:

Properties:

reduces path length by 1.
preserves path length.
Both operators are linear over the logic-number ring.

6.3.3 Integration

Definition 6.5 (Integration Operator).
The integration operator is the inverse of differentiation (when it exists):

where is a constant of integration (a logic number).

Integration corresponds to reconstructing a path from its differences.

6.3.4 Rotation and Reflection

Geometric transformations act on path spaces by permuting the bits of each logic number:

Definition 6.6 (Rotation Operator).
For a rotation , define:

where is the logic number obtained by applying the rotation to the cell.

6.4 Algebraic Structures of Path Spaces

6.4.1 Monoid of Paths

Theorem 6.1 (Path Monoid).
The set of all valid paths in a grid , together with concatenation, forms a monoid .

Proof: Associativity and identity follow from the definition.

6.4.2 Group of Cycles

Definition 6.7 (Cycle).
A cycle is a path whose first and last cells are adjacent and whose walls align. A cycle is closed if the first and last cells are the same.

The set of cycles in a grid forms a groupoid under concatenation.

Theorem 6.2 (Cycle Group).
The set of closed cycles in a grid forms a group under concatenation, with:

Identity: the trivial cycle (length 0).
Inverse: the reverse cycle.

6.4.3 Vector Space of Paths

Theorem 6.3 (Path Vector Space).
For a fixed grid and logic-number ring , the set of path strings forms a vector space over when is a field.

Proof: Addition is componentwise XOR (or addition in the ring), and scalar multiplication is componentwise multiplication.

Dimension: The dimension is (if we fix a traversal order).

6.5 Composition of Mazes

6.5.1 Boundary Composition

Definition 6.8 (Maze Composition).
Given two mazes and with compatible boundaries, their composition is the maze obtained by identifying a boundary of with a boundary of .

Theorem 6.4 (Composition of Perfect Mazes).
If and are perfect and their boundaries are compatible, then is perfect.

Proof: The union of two trees with a shared boundary is a tree if and only if the shared boundary connects them uniquely.

6.5.2 Category of Mazes

Definition 6.9 (Category of Mazes).
The category of mazes Maze has:

Objects: perfect mazes.
Morphisms: boundary-preserving maps between mazes.

The composition of morphisms is the composition of mazes.

6.6 Invariants of Path Spaces

6.6.1 Length

The length of a path is a fundamental invariant:

6.6.2 Logic-Number Sum

Definition 6.10 (Logic-Number Sum).
The sum of a path is the componentwise sum (XOR) of its logic numbers:

6.6.3 Period

Definition 6.11 (Period).
The period of a path under circular differentiation is the smallest such that:

This is a key invariant with cryptographic significance (see Paper 4).

6.6.4 Entropy

For infinite paths, we define the entropy:

6.7 Classification of Path Spaces

6.7.1 Finite Path Spaces

For finite grids, the path space is finite. Its size is the number of valid paths in the grid, which grows exponentially with grid size.

Open Problem 6.1: Classify all finite path spaces by their algebraic invariants (length distribution, cycle structure, etc.).

6.7.2 Infinite Path Spaces

For infinite grids, the path space is infinite. The classification is by:

Class	Description
Periodic	Paths with finite period.
Substitution	Paths generated by substitution rules.
Computable	Paths generated by algorithms.
Non-computable	Paths existing only set-theoretically.

6.8 Summary of Part VI

We have established:

The path space as a monoid under concatenation.
Operations: concatenation, differentiation, integration, rotation, reflection.
Algebraic structures: monoid, group of cycles, vector space.
Maze composition and the category of mazes.
Invariants: length, sum, period, entropy.
Classification of path spaces (finite, infinite, periodic, substitution, computable).

This algebraic framework provides the unifying structure for all of maze theory.

6.9 Open Problems (Part VI)

Full classification of finite path spaces: Determine all algebraic invariants that distinguish different finite path spaces.
Composition of infinite mazes: Define and characterise composition for infinite mazes.
Group structure of cycles: Determine the full group structure of cycles in various grid types.
Entropy of path spaces: Compute the entropy of path spaces for different grid types.
Category-theoretic properties: Is the category of mazes complete/cocomplete? What are its limits and colimits?

PART VII: OPEN PROBLEMS AND FUTURE DIRECTIONS

7.1 Introduction

Cross-reference to Papers 1 and 2: The theory of mazes, as developed in Papers 1 and 2, is now extensive. We have established:

The foundational theory of perfect mazes on finite grids (Paper 1).
The algebra of 16 icons and their logic-number encodings (Parts I–II).
The geometry of 3D mazes (proper and icon-face models) (Part V).
The extension to infinite grids (Part I).
The generalisation to lozenges, rhombic tilings, and higher dimensions (Parts III–IV).
The algebraic structure of path spaces (Part VI).

Yet for every theorem proved, there are many questions left open. This part collects the most significant open problems, organised by theme, and suggests directions for future research.

7.2 Enumeration Problems

Problem 7.1 (Exact Enumeration of 2D Mazes).
Determine the exact number of perfect mazes on an grid for arbitrary .

Current status:

: 4
: 96 (K-matrices) / 196 (E/J mazes)
: Unknown

Related: Determine the number of K-matrices and NK-matrices for .

Problem 7.2 (Exact Enumeration of 3D Mazes).
Determine the number of perfect 3D mazes on an grid.

Current status:

: Unknown (but computable by brute force with symmetry reductions)
: Unknown

Related: Count the number of perfect 3D dice (no F) for .

Problem 7.3 (Enumeration of Lozenge Mazes).
Count the number of perfect mazes on finite patches of lozenge tilings.

Current status: Completely open.

Problem 7.4 (Growth Rate of 2D Mazes).
Determine the constant:

where is the number of perfect mazes on an grid.

Conjecture: , with (based on heuristic arguments).

Problem 7.5 (Growth Rate in Higher Dimensions).
Determine the constants:

for , where is the number of perfect -dimensional mazes on an grid.

Current status: Completely open.

Problem 7.6 (Entropy of Path Spaces).
Compute the topological entropy of the space of infinite perfect mazes:

7.3 Structural Problems

Problem 7.7 (Uniqueness of Hamiltonian Paths in 2D).
Prove or disprove: In every perfect 2D maze, the Hamiltonian path is unique.

Current status: Empirically true for all known cases (Paper 1). A formal proof is still outstanding.

Problem 7.8 (Number of Hamiltonian Paths in 3D).
Classify all perfect 3D mazes by the number of Hamiltonian paths in the primary network.

Current status: It is known that multiple paths exist in some cases (Part V). The full classification is open.

Problem 7.9 (Hamiltonian Paths in Higher Dimensions).
Determine the maximum number of Hamiltonian paths in a perfect -dimensional maze for .

Conjecture: The maximum grows exponentially with grid size for .

Problem 7.10 (Classification of Icon-Face Paths).
Classify all paths in the icon-face model of 3D mazes.

Current status: Multiple paths exist (Paper 1, Part X). The full classification of paths in terms of the ternary logic numbers is open.

Problem 7.11 (Number of Paths in Icon-Face Model).
Count the number of distinct Hamiltonian paths in the face-adjacency graph for a given 3D maze.

Current status: Open.

7.4 Algebraic Problems

Problem 7.12 (Full Characterisation of Composition).
Define and fully characterise the composition operation on logic numbers for all adjacencies (horizontal, vertical, diagonal, and in higher dimensions).

Current status: Partially defined for 2D (Part II). Higher-dimensional and non-cubic cases are open.

Problem 7.13 (Algebraic Structure of Logic Numbers).
Determine the full algebraic structure of the set of logic numbers under all operations:

Is it a ring? A semiring? A module?
What are its ideals, subalgebras, and automorphisms?

Current status: Partial results (Part II). Full classification is open.

Problem 7.14 (Logic Numbers and Groups).
Does the set of logic numbers form a group under some operation? If so, which operation and what is the group structure?

Problem 7.15 (Algebra of Path Spaces).
Fully characterise the algebraic structure of path spaces:

Is the path monoid a free monoid? If not, what are its relations?
What is the group of cycles?
What are the invariants of path spaces?

Current status: Partial results (Part VI). Full characterisation is open.

Problem 7.16 (Entropy of Path Spaces).
Compute the entropy of path spaces for different grid types (square, cubic, hexagonal, lozenge).

7.5 Infinite Mazes

Problem 7.17 (Existence of Aperiodic Perfect Mazes).
Construct an explicit aperiodic perfect maze in 2D, 3D, or higher dimensions.

Current status: Periodic perfect mazes are easy to construct (Part I). Aperiodic constructions (analogous to Penrose tilings) are open.

Problem 7.18 (Substitution Mazes).
Develop a general theory of substitution mazes — perfect mazes generated by inflation rules.

Current status: Preliminary ideas only (Part I, Part III). Full theory is open.

Problem 7.19 (Decidable Subclasses).
Characterise the largest class of infinite mazes for which perfection is decidable.

Current status: Periodic mazes are decidable. Substitution mazes may be decidable for certain classes. The general problem is undecidable (Theorem 1.1).

Problem 7.20 (Semi-Decidability of Perfection).
Is the property "M is perfect" semi-decidable for all computable infinite mazes?

Current status: It is semi-decidable if perfection can be checked on finite regions (Theorem 1.2). The full characterisation is open.

7.6 Non-Cubic Grids

Problem 7.21 (Hexagonal Icon System).
Define the complete 6-bit icon system for hexagonal grids and enumerate all perfect hexagonal mazes.

Current status: Preliminary discussion only. The 6-bit icon system (64 icons) has not been fully analysed.

Problem 7.22 (Triangular Grids).
Develop the icon system and perfect maze theory for triangular grids.

Current status: Completely open.

Problem 7.23 (Perfect Mazes on Lozenge Tilings).
Characterise all perfect mazes on regular lozenge tilings. Do they exist for all tilings? What are the enumeration results for finite patches?

Current status: Preliminary (Part III). Full characterisation is open.

Problem 7.24 (Penrose Mazes).
Construct an explicit perfect Penrose maze and determine whether the path is unique.

Current status: Open.

7.7 Cryptographic Applications (Deferred)

Problem 7.25 (Maze-Based Cryptographic Primitives).
Develop the full theory of cryptographic primitives based on maze paths, building on the logic-number algebra and the uniqueness/multiplicity properties of 3D mazes.

Current status: Mazecoin (Paper 3) provides a specific application. General cryptographic primitives will be developed in Paper 4.

Problem 7.26 (Period Analysis for Cryptographic Lengths).
Compute the periods of circular differentiation for path lengths relevant to cryptography ().

Current status: Partial ( gives ). Larger lengths are open.

7.8 Connections to Other Fields

Problem 7.27 (Mazes and Topology).
Establish formal connections between maze theory and topology:

Jordan curve theorem generalisations.
Knot theory and braid theory.
Homology and cohomology of maze structures.

Current status: Preliminary (Paper 1, Part III). Deep connections are open.

Problem 7.28 (Mazes and Symbolic Dynamics).
Develop the symbolic dynamics of infinite perfect mazes. What are the subshifts? What is their entropy?

Current status: Preliminary (Part I). Full theory is open.

Problem 7.29 (Mazes and Quasicrystals).
Establish formal connections between aperiodic perfect mazes and quasicrystals (Penrose tilings, Fibonacci chains).

Current status: Preliminary (Part III). Full theory is open.

Problem 7.30 (Mazes and Islamic Geometric Art).
Develop a formal mapping between maze icons and Islamic geometric patterns.

Current status: Mentioned in Paper 1. Full mapping is open.

7.9 Higher-Order Generalisations

Problem 7.31 (-Dimensional Icon Systems).
For , develop the complete icon system and perfect maze theory.

Current status: General framework exists (Part IV). Detailed enumeration and classification are open.

Problem 7.32 (Non-Integer Dimensions).
Develop maze theory on fractal grids (e.g., Sierpinski gasket, Menger sponge).

Current status: Completely open.

7.10 Computational Problems

Problem 7.33 (Efficient Maze Generation).
Develop algorithms for generating perfect mazes in time for arbitrary -dimensional grids.

Current status: The crystallisation algorithm runs in time for 2D (Paper 1). Generalisation to higher dimensions is open.

Problem 7.34 (Maze Isomorphism).
Develop algorithms for determining whether two mazes are isomorphic (under rotation, reflection, or general graph isomorphism).

Current status: Open.

Problem 7.35 (Path Extraction in 3D).
Develop efficient algorithms for extracting the Hamiltonian path from a perfect 3D maze.

Current status: Open (the 2D algorithm is straightforward; 3D is more complex).

7.11 Philosophical Problems

Problem 7.36 (The Heterodox Theory of Infinity).
Develop the heterodox theory of infinity as a fully formal mathematical system, with axioms, theorems, and applications to maze theory.

Current status: Philosophical foundations exist (Part I). Formalisation is open.

Problem 7.37 (The Nature of Maze Paths).
What is the ontological status of a maze path? Is it a mathematical object, a computational process, or something else?

Current status: Open.

7.12 Summary of Open Problems

We have identified 37 open problems across the following categories:

Category	Number of Problems
Enumeration	6
Structural	5
Algebraic	5
Infinite Mazes	4
Non-Cubic Grids	4
Cryptographic	2
Connections to Other Fields	4
Higher-Order Generalisations	2
Computational	3
Philosophical	2
Total	37

7.13 Future Directions

The open problems suggest several major directions for future research:

Enumeration and asymptotics: Determine exact counts and growth rates for perfect mazes in all dimensions and grid types.
Hamiltonian path theory: Classify all perfect mazes by the number of Hamiltonian paths, with special attention to 3D and higher dimensions.
Logic-number algebra: Complete the algebraic theory of logic numbers, including composition, transformations, and classification.
Infinite mazes: Develop a full theory of infinite perfect mazes, including aperiodic constructions, decidability, and entropy.
Non-cubic grids: Complete the theory of perfect mazes on hexagonal, triangular, lozenge, and Penrose grids.
Cryptographic applications: Develop the full cryptographic theory of maze paths, building on the logic-number algebra and the 3D models (Paper 4).
Connections to other fields: Establish formal bridges between maze theory and topology, symbolic dynamics, quasicrystals, and geometric art.
Higher-order generalisations: Extend maze theory to arbitrary dimensions and fractal grids.
Computational methods: Develop efficient algorithms for maze generation, path extraction, and isomorphism testing.
Philosophical foundations: Formalise the heterodox theory of infinity and its applications to maze theory.

7.14 Concluding Remarks

The theory of perfect mazes, as developed in Papers 1 and 2, is now a rich and coherent mathematical discipline. It encompasses:

Combinatorics: Enumeration of mazes, growth rates, entropy.
Algebra: Logic numbers, path spaces, composition laws.
Geometry: 2D, 3D, and higher-dimensional grids; lozenges and non-cubic tilings.
Topology: Duality, Hamiltonian paths, space-filling curves.
Computation: Crystallisation algorithms, decidability, complexity.
Infinity: Constructive theory of infinite mazes, aperiodic order.

The 37 open problems identified here provide a roadmap for future research. They are deep, challenging, and connected to fundamental questions in mathematics and computer science.

Maze theory is not merely a collection of puzzles. It is a unified framework for understanding connectivity, paths, and algebraic structure in discrete spaces. It has applications to cryptography (Paper 3 and future Paper 4), to geometry, to computation, and to the philosophy of infinity.

The work begun in these two papers is, we believe, the foundation of a new branch of mathematics — one that will continue to grow and reveal new connections for many years to come.

END OF VOLUME II

Appendix: Cross-Reference Summary to Paper 1

Paper 1 Reference	Location in Paper 2
Part 0 (Definition of Perfect Maze)	§1.1, §1.3.3
Part I (16 Icons)	§2.1, §2.2.2
Part III (Graph Theory)	§4.3.2, §4.10.1
Part IV (3×3 Enumeration)	§1.5.1
Part V (Crystallisation/Rotation)	§1.4.1, §4.6.1, §5.8
Part VI (Differentiation)	§2.7.2
Part IX (Proper 3D Model)	§5.1, §5.3.1, §5.5
Part X (Icon-Face Model)	§2.4.3, §5.1, §5.3.2, §5.6

PAPER 2: COMPLETE.

Volume II of the Treatise on Perfect Mazes

Green of Greece - Logic and Mathematics

Friday, July 17, 2026

Maze Theory Volume II

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