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The Internal Space and Boundary Growth of the Spectre and Tsiokos’s Chair44 Tilings

This paper provides an exact computational comparison of the hierarchical aperiodic monotiles Spectre and Chair44, analyzing their distinct inflation dynamics, boundary fractals, and internal spaces while proving that all lattice-registered Chair44 tilings belong to the substitution hull with specific symmetry groups and demonstrating that the Spectre's hierarchy cannot function as a positional clock.

Original authors: Brent Hartshorn, Tristan Romanov

Published 2026-10-28

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Brent Hartshorn, Tristan Romanov

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet corners of geometry, there exists a class of shapes known as aperiodic monotiles. These are single, rigid tiles that can cover an infinite floor without ever leaving a gap, yet they can never form a pattern that repeats itself in a regular, predictable way. For decades, mathematicians believed such shapes were impossible, or at least required a complex set of different pieces to work. The discovery of the "hat" and the "spectre" shattered that belief, proving that a single shape could force a non-repeating order. This phenomenon is not just a curiosity of shapes; it connects to deep questions about how order emerges from simple rules, how crystals form, and even how the universe might be structured at its most fundamental level. When a shape inflates, or grows larger by a fixed rule, it reveals hidden layers of arithmetic and symmetry that determine whether the resulting pattern is chaotic or perfectly ordered.

Two researchers, Brent Hartshorn and Tristan Romanov, have taken a deep dive into two specific examples of these shapes: the Spectre, a two-dimensional tile, and the Chair44, a three-dimensional solid. Their work is a precise comparison of how these two shapes grow and how their edges behave as they expand into infinity. While both shapes share the ability to tile a space without repeating, the researchers found that they operate on entirely different mathematical principles. The Spectre grows in a way that involves a complex, irrational expansion factor, creating a boundary that becomes increasingly jagged and fractal-like with every step. In contrast, the Chair44 grows by a simple, whole-number factor, resulting in a smooth, ordinary surface that behaves much more like a standard geometric object.

The researchers began by examining how the Spectre expands. They discovered that with every level of growth, the area of the shape increases by a specific, irrational number, but the length of its outer edge grows by a different, even more complex number. This mismatch means the edge does not grow in a straight line; instead, it becomes a fractal curve, a shape that is infinitely detailed and rough. The boundary of the Spectre's growing structure has a dimension that is not a whole number, sitting somewhere between a line and a flat surface. This is a rare and specific property, indicating that the edge is far more complex than a simple outline. Furthermore, the researchers mapped out the internal structure of the Spectre, finding that its vertices fit into a four-dimensional space that acts as a hidden "window" for the pattern. This internal space is bounded and finite, acting as a container that keeps the pattern organized despite its infinite growth.

In stark contrast, the Chair44 behaves with a simplicity that the Spectre lacks. This three-dimensional shape, which resembles a chair made of seven cubes, expands by a factor of exactly two at every step. Because it grows by a whole number, it does not have a complex, fractal boundary; its surface remains a standard, smooth two-dimensional sheet. The researchers found that the Chair44 does not have a hidden Euclidean internal space like the Spectre. Instead, its internal structure is defined by a different kind of mathematics related to powers of two, which keeps the pattern ordered without the need for a complex geometric window. The Chair44 also lacks the "twist" found in the Spectre; while the Spectre rotates and mirrors itself at every level of growth, the Chair44 simply scales up without changing its orientation or flipping its mirror image.

A major part of the study focused on the rules that govern how these tiles fit together. For the Chair44, the researchers proved that any valid arrangement of these tiles must follow a strict, hierarchical structure. They showed that every possible tiling of space using this shape belongs to a single, unified family of patterns. They also determined the exact symmetries these patterns can have, finding that they can only be rotated or flipped in very specific ways, such as a half-turn or a quarter-turn, but never in a way that would create a repeating grid. This classification provides a complete map of the possible symmetries for this shape. However, the researchers were careful to note that while they proved the structural rules for the Chair44, the claim that it is a true "einstein" (a single tile that forces a non-repeating pattern) relies on a separate proof by another mathematician that has not yet been fully peer-reviewed.

The paper also tested whether the growth of these shapes could serve as a clock or a record of time. Some theories suggest that the level of growth in a tiling could tell an observer how "old" a specific part of the pattern is, much like reading the rings of a tree. The researchers found that for the Spectre, this is not the case. While the pattern does have a hierarchy, there is no single arrow of time that can be read from the arrangement of the tiles. The position of a tile does not reveal its level in the growth sequence, and the pattern does not show a clear distinction between past and future. The alternation of mirror images in the Spectre's growth is a rule of the construction, not a dynamic process that evolves over time.

Ultimately, this work highlights the profound differences that can exist between two shapes that seem to do the same thing. The Spectre and the Chair44 both create non-repeating patterns, but they do so through completely different mechanisms. One relies on complex, irrational numbers and fractal boundaries, while the other uses simple, whole-number scaling and smooth surfaces. By comparing them side by side, the researchers have provided a clearer picture of the diverse ways in which order can emerge from simple rules. Their findings confirm that the universe of aperiodic tilings is rich and varied, with each shape offering its own unique mathematical signature. The study does not claim to have solved every mystery of these shapes, but it has established a rigorous foundation for understanding how they grow, how they fit together, and how their boundaries behave, separating fact from speculation with exact computation.

Technical Summary: The Internal Space and Boundary Growth of the Spectre and Tsiokos's Chair44 Tilings

Problem Statement
This research article provides an exact computational comparison of two aperiodic monotiles: the planar Spectre (Smith et al.) and the three-dimensional Chair44 (Tsiokos). While both tiles generate hierarchical tilings, they differ fundamentally in their arithmetic properties, inflation mechanisms, and internal spaces. The paper aims to quantify these differences through exact computation, specifically analyzing:

The inflation factors and the number fields they generate.
The growth rates of tile counts versus boundary perimeters.
The existence and nature of "internal spaces" (Euclidean vs. $p$ -adic).
The symmetry groups and hull properties of the resulting tilings.
Whether the substitution hierarchy can serve as a physical "clock" (providing an arrow of time or look-back mechanism).

Methodology
The authors employ a rigorous mix of exact algebraic computation, finite patch enumeration, and finite certificate verification.

Exact Arithmetic: Computations are performed using exact rational arithmetic and algebraic number fields ( $\mathbb{Z}[\sqrt{15}]$ , $\mathbb{Q}(\sqrt{5})$ , $\mathbb{Q}(\sqrt{3}, i)$ ) to avoid floating-point errors.
Matrix Iteration: The substitution matrices for both tiles are analyzed for eigenvalues, Jordan blocks, and spectral properties to determine growth rates and frequency convergence.
Boundary Operators: A finite set of contact types (93 for Spectre, derived from slot pairs) is used to construct a boundary transfer operator. This allows for the exact derivation of perimeter recurrence relations.
Finite Certificates: Key theorems regarding the hierarchy and symmetry of the Chair44 are verified using independent code implementations that check finite certificates (e.g., admissible blocks, contact rules) rather than relying solely on asymptotic arguments.
4D Lift: The Spectre vertices are lifted to a four-dimensional integer lattice $\mathbb{Z}^4$ to analyze the internal space and window geometry.

Key Contributions and Results

1. Spectre (Planar) Analysis

Inflation and Arithmetic: The Spectre inflates by the unit $\lambda^2 = 4 + \sqrt{15}$ in the ring $\mathbb{Z}[\sqrt{15}]$ . The linear inflation factor is $\lambda = \frac{1}{2}(\sqrt{6} + \sqrt{10})$ .
Boundary Growth: A major finding is the decoupling of area and perimeter growth. While the area grows by $\lambda^2$ $λ^{2}$ , the perimeter grows by the unit $\nu = 2 + \sqrt{5}$ $ν = 2 + 5$ (the cube of the golden ratio, $\phi^3$ $ϕ^{3}$ ).
- The perimeter $P(k)$ at order $k$ is exactly $2F_{3k+5} + 4$ , where $F_n$ is the Fibonacci sequence.
- This results in a fractal boundary with box dimension $D_b = \frac{\log \nu}{\log \lambda} \approx 1.3993$ , strictly greater than the topological dimension of 1.
Internal Space: The Spectre possesses a Euclidean internal space. The vertices lie in a rank-4 module $\mathbb{Z}[\zeta_{12}]$ . The inflation acts with two expanding and two contracting directions. The "window" (the projection of the lattice onto the internal space) is a fractal attractor with boundary dimension $\approx 1.111$ .
Twist: The Spectre supertile undergoes a twist at every level. The limiting twist angle $\theta_S \approx 11.565^\circ$ is irrational (not a root of unity), and the supertile alternates between mirror images.

2. Chair44 (3D) Analysis

Inflation and Arithmetic: The Chair44 inflates by the integer $2$ on $\mathbb{Z}^3$ .
Boundary Growth: The boundary grows by exactly $\lambda^2 = 4$ (since $\lambda=2$ ). The boundary dimension is $D_b = 2$ , meaning the supertile boundary is an ordinary surface with no fractal excess.
Internal Space: There is no Euclidean internal space. The internal space is 2-adic ( $\mathbb{Z}_2^3$ ). The diffraction of the center set is pure point, derived from a modular coincidence and 2-adic structure rather than a cut-and-project scheme.
Symmetry and Hull:
- Theorem 4: Every tiling by the Chair44 solid (obeying the 44 contact rules) lies within the substitution hull. This relies on Kamenetsky's unique-hierarchy criterion and an independently verified finite certificate of admissible blocks.
- Theorem 5: The symmetry group of any such tiling is conjugate to one of four groups: $1$, $C_2$ (half-turn about a face diagonal), $C_4$ , or $D_4$ . No 3-fold axes or Klein four-groups occur.
Chirality: The Chair44 is chiral; its legal contacts admit no mirror images, forcing the tiling to be strictly oriented.

3. Comparative Analysis

Fields: The Spectre involves a divide between the area field ( $\mathbb{Q}(\sqrt{15})$ ) and the perimeter field ( $\mathbb{Q}(\sqrt{5})$ ). The Chair44 operates entirely within $\mathbb{Q}$ .
Orientation Sums: The sum of orientation vectors for the Chair44 grows exactly as $2^n$ . For the Spectre, the sum grows as $c \lambda^n$ , where the constant $c$ is identified but not proven to be algebraic in the text.
Hierarchy as a Clock: The paper tests if the substitution level can serve as a physical clock (an arrow of time).
- Positional Clustering: Shows no excess over a lattice control; no arrow.
- Look-back: The radius required to identify an ancestor grows by a factor of $\approx 3$ per level, but this is a generic property of self-similar hierarchies, not a unique clock.
- Parity: The mirror alternation is visible in the outlines of supertiles but not in the tiles themselves. The paper concludes the hierarchy does not supply a physical clock.

Significance and Claims
The paper claims to provide the first exact, finite-computation-based comparison of these two specific aperiodic monotiles.

Main Result: The classification of the symmetry groups of Chair44 tilings and the proof that they lie in the substitution hull (conditional on Tsiokos's registration theorem and Kamenetsky's criterion).
Boundary Fractality: The demonstration that the Spectre's boundary is a fractal with dimension $\approx 1.4$ , distinct from the linear inflation rate, a property not shared by the Chair44.
Internal Space Distinction: The clear delineation between the Spectre's Euclidean internal space (cut-and-project) and the Chair44's 2-adic internal space (modular coincidence).

Limitations and Caveats
The authors are explicit about the scope of their claims:

The Einstein property (aperiodicity) of the Chair44 is taken from Tsiokos's claimed proof, which has been audited but not refereed.
The pure-point diffraction of the Spectre is a published theorem, not re-proved here.
The hierarchy tests (clock, look-back) are numerical computations on finite patches and are presented as negative or generic results, not as theorems about the infinite tiling.
The closed form for the Spectre orientation constant is identified but not rigorously proved.
The four-dimensional analogue of the Chair44 (Kamenetsky's Problem 9.1) remains open; the authors' search found no closed assignments.

In summary, the paper establishes that while both tiles are hierarchical aperiodic monotiles, they belong to distinct arithmetic and geometric classes: the Spectre is a Pisot-unit system with fractal boundaries and Euclidean internal space, while the Chair44 is an integer-inflation system with a 2-adic internal space and non-fractal boundaries.

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