Convex pentagonal monotiles in the 15 Type families

The properties of convex pentagonal monotiles in the 15 Type families and their tilings are summarized. The Venn diagrams of the 15 Type families are also shown.

💡 Research Summary

The paper provides a comprehensive survey of the fifteen known families of convex pentagonal monotiles—tiles that can cover the Euclidean plane by congruent copies without gaps or overlaps. Each family, labeled Type 1 through Type 15, is defined by algebraic relationships among edge lengths and interior angles; some families retain degrees of freedom beyond uniform scaling, while Types 14 and 15 are completely rigid. The author first clarifies terminology: a monotile is a prototile that admits a monohedral tiling, and reflected copies (the “posterior side”) are considered identical to the original.

Representative tilings for each type are displayed (Figure 1) together with a minimal “translation unit” (the smallest set of tiles that generates a periodic pattern by translation). This demonstrates that every type can generate a periodic tiling, but the use of reflected tiles is required for many families. Types 2 and 7–15 rely on reflections in their canonical tilings; if reflections are prohibited, those families cannot tile the plane at all.

A central distinction is made between edge‑to‑edge tilings (where adjacent tiles share a full edge or a vertex) and non‑edge‑to‑edge tilings. Theorem 1, originally derived by the author and later independently by Bagina, states that a convex pentagon capable of an edge‑to‑edge monohedral tiling must belong to at least one of the families Type 1, 2, 4, 5, 6, 7, 8, or 9. Consequently, Types 3, 13 and the rigid Types 14, 15 never admit edge‑to‑edge tilings, regardless of how their free parameters are chosen.

To visualize the overlap among families, a Venn diagram (Figure 3) is constructed. Several intersections contain tiles with remaining degrees of freedom (e.g., T₁∩T₂, T₁∩T₄, T₁∩T₅, T₂∩T₄, T₂∩T₅), allowing a continuum of shapes. Other intersections are uniquely determined (e.g., T₁∩T₅∩T₆, T₁∩T₇). Notably, the tile lying in T₁∩T₇ can realize both the Type 1 and Type 7 representative tilings; in this case the tiling can be constructed without reflected copies, showing that some tiles belonging to a “reflection‑dependent” family can nevertheless produce reflection‑free isohedral patterns.

The paper proceeds to a symmetry analysis. For a tiling ℑ, Sym(ℑ) denotes its Euclidean isometry group, and the number k of transitivity classes of tiles defines a k‑isohedral (or tile‑k‑transitive) tiling. The representative tilings of Types 1–5 are 1‑isohedral, Types 6–9 and 11–13 are 2‑isohedral, and Types 10, 14, 15 are 3‑isohedral. Hence any convex pentagon belonging to at least one of Types 1–5 can generate an isohedral tiling; tiles outside these families are generally anisohedral, although several intersections (e.g., T₁∩T₇, T₁∩T₈, T₁∩T₉) produce non‑anisohedral examples.

The discussion concludes with two recent conjectures. Mann, McLoud‑Mann, and von Derau (2018) conjecture that every unmarked convex pentagon that tiles the plane admits at least one periodic tiling, implying the existence of an i‑block transitive tiling for all such pentagons. Rao (2017) claims that every convex pentagonal monotile belongs to one of the fifteen known families, which would automatically guarantee periodicity. Both statements remain unproven, but all known examples satisfy them, supporting the view that the fifteen families likely exhaust the universe of convex pentagonal monotiles.

In summary, the paper catalogues the fifteen convex pentagonal monotile families, clarifies which families can produce edge‑to‑edge, isohedral, periodic, or non‑periodic tilings, illustrates the overlaps via Venn diagrams, and situates the results within current conjectures about completeness and periodicity of convex pentagonal tilings. Future work should aim at rigorously confirming Rao’s exhaustiveness claim and Mann et al.’s periodicity conjecture, as well as searching for any yet‑undiscovered families.