Formal Reference

Let γ : [0, L] → ℝ² be a simple closed plane curve parameterized by arc length, with signed curvature κ(s). The curve γ is a toastangle if and only if its curvature function satisfies the following structural conditions.

The curvature κ(s) vanishes on exactly three disjoint open intervals of [0, L], corresponding to the base and two sides of the figure, and is nonzero on exactly three disjoint open intervals, corresponding to the top curve and two shoulder curves. The six boundary points between these intervals are points of tangent discontinuity — the six vertices of the figure. The cyclic ordering of segments along γ is (base, left side, left shoulder, top curve, right shoulder, right side). The top-curve interval has constant sign of κ throughout — the curve does not inflect — though that sign may be positive (a convex top, as in Standard) or negative (a concave top, as in Saddle).

The six primary subtypes are generated by a 3 × 2 combinatorial structure on shoulder configuration (both concave, both convex, or mixed) crossed with top-curve sign (convex or concave). The three modifiers — Leaning, Splayed, Flared — are continuous deformations that do not affect these signs and compose freely with any subtype.

The architectural round trefoil — a three-lobed opening profile used in Gothic, Islamic, and Romanesque architecture — is a Standard toastangle with high τ. Its lateral lobes are the shoulder curves; its upper lobe is the top curve; its door jambs and threshold are the three straight segments. This identification has not previously appeared in any geometric or architectural reference. The round trefoil has appeared on standard arch-type charts for over eight centuries, alongside shapes that are fully formalized in plane geometry (semicircular, parabolic, segmental), without being recognized as an instance of a formally definable plane figure.

The architectural shouldered arch (Caernarvon arch) is the flat-top precursor of the toastangle: it shares the shoulder transitions but terminates in a flat lintel (a fourth straight segment) rather than a curve, producing four zero-curvature intervals and two nonzero intervals — a distinct curvature signature. When the lintel is replaced by a curve, the shouldered arch becomes a toastangle.

The toastangle is not a new entry in the architectural glossary. It is a formalization in plane geometry of a shape that the architectural glossary has only informally described.

The toastangle number τ measures the shoulder dominance of the figure. It vanishes (τ = 0) for the limiting case of an arch, which has no shoulders, and lies strictly between zero and one for any toastangle. As τ approaches zero, the shoulders shrink and the figure approaches an arch — the Sharp limiting family. As τ approaches one, the top curve shrinks to a vanishing arc and the figure becomes shoulder-dominated.

The toastangle number is invariant under uniform scaling but not under non-uniform scaling. It does not, by itself, distinguish among the six primary subtypes — a low-shoulder Standard and a high-shoulder Inverse may share the same τ — and full subtype classification requires the additional curvature-sign invariants described above. τ is best understood as the simplest and most direct invariant separating the toastangle from its nearest neighbor in the curvilinear shape taxonomy.