The Science

Genus Skwedge and Phylum Apicalia: a formal taxonomy of apex-convergent solids

Five companion papers published February 2026. This page presents the definitions, theorems, and open problems in accessible form. Full proofs and derivations are in the papers linked below.

Classification

Phylum Apicalia and Genus Skwedge

Phylum Apicalia is the formal taxonomic home for all solids that taper from a base of positive area to an apex of lower dimension. The name derives from the Latin apex. Hand axes, chisels, cones, and pyramids are all Apicalia. The genus and species distinctions follow from the specific geometry of the taper.

Genus Skwedge is the genus within Phylum Apicalia whose members taper from a circular base to a linear apex. The base and apex are parallel, coaxial, and of the same diameter. Three species are well-characterized. A fourth remains an open problem.

Phylum Apicalia all apex-convergent solids; dim(apex) < dim(base)
Class
├─ Apexia monotonic tapering; cross-sections stay within base footprint
└─ Tumida bulging forms; domes, onion spires, gopuram towers
Order
├─ Apextoid circular bases; smooth or ruled lateral surfaces
└─ Prismatoid polygonal bases; flat faces; classical solid mensuration
Family
Order Prismatoid contains: Family Pyramis (pyramids; 0D apex) · Family Cuneus (wedges; 1D apex) · Family Prisma (prisms; 2D apex)
Family
├─ Hewel 1D linear apex; circle-to-line boundary archetype · Miluk hewel (trail, path)
└─ Puncta 0D point apex; cones, paraboloids, cone-type solids
Genus
├─ Skwedge canonical circle-to-line; diametral apex; four species
└─ Bulbus smooth NURBS/Bézier interpolations; future work
Species of Genus Skwedge
Species I Skwedge curvum ruled surface · conocuneus · k = π/2
Species II Skwedge convexum convex hull · stadium cross-sections · k = (π+2)/3
Species III Skwedge projectivum projective maximum · type species · k = π−4/3
Species IV Skwedge minimalis variational minimum · kIV open problem

The complete vertical path is: Phylum Apicalia → Class Apexia → Order Apextoid → Family Hewel → Genus Skwedge → Species I–IV. The taxonomy is introduced in [Anderson, 2026c] and surveyed historically in [Anderson, 2026d]. Family Hewel takes its name from the Miluk word hewel (trail, path), honoring the Indigenous geometric knowledge that motivated the research. The classification follows Linnaean convention: genus name capitalized and italicized, species epithet lowercase and italicized.

The Core Problem

The circle-to-line boundary problem

The boundary data for Genus Skwedge is simple to state. Given a circular disk D of radius R in the plane z = 0, and a line segment L of half-length R in the plane z = h, centered on the same axis, find a solid S whose lower boundary is D and whose upper boundary is L.

Admissibility conditions

An admissible solid S must: (1) have lower boundary equal to the closed disk D; (2) have upper boundary equal to the segment L; (3) be bounded, connected, and simply connected; (4) have piecewise smooth lateral surface. The boundary conditions do not determine S uniquely. The set of admissible solids forms an infinite-dimensional family.

This non-uniqueness is the central mathematical fact of the genus. The same boundary data admits at least three geometrically distinct, volumetrically different solutions. Each solution corresponds to a different geometric principle for filling the space between D and L: the ruling principle, the convexity principle, and the projection principle.

Prior to this taxonomy, the circle-to-line loft had been implemented in engineering and mathematics since Wallis (1684), but the family of solids it generates was not recognized as a genus admitting distinct species. CAD kernels necessarily implement a single construction strategy, because no such framework existed. The present work shows that these strategies correspond to distinct members of the same genus, and that the gap between them constitutes a measurable volumetric difference of 13.13%.

Species I, II, III

The three characterized species

Each species is defined by the geometric principle governing its cross-sections at each height. At height u = z/h (where u ranges from 0 at the base to 1 at the apex), the cross-section of each species is a different shape.

Species I
Skwedge curvum
the ruled surface · conocuneus
V = (π/2) R²h
kI = π/2 ≈ 1.5708
Cross-section at height u: ellipse with semi-axes R (along y) and R(1−u) (along x). The x-axis collapses to zero; the y-axis remains full diameter throughout. Gaussian curvature K < 0 everywhere on the lateral surface. A saddle surface, not developable.

Prior names: conocuneus (Wallis 1684; Guarini 1671). The oldest named member of the genus.
Species II
Skwedge convexum
the convex hull · stadium cross-sections
V = ((π+2)/3) R²h
kII = (π+2)/3 ≈ 1.7140
Cross-section at height u: Minkowski sum (1−u)D ⊕ uL. This is a stadium (rectangle capped by semicircles), with semicircle radius R(1−u) and center separation 2Ru. Gaussian curvature K = 0 almost everywhere (piecewise developable). The unique convex member of the genus.

Prior names: No established prior name. The CAD default loft approximates this species.
Species III
Skwedge projectivum
the projective maximum · ptsi·nɬ
V = (π−4/3) R²h
kIII = π−4/3 ≈ 1.8083
Cross-section at height u: clipped disk {(x,y): x²+y² ≤ R², |x| ≤ R(1−u)}. The full circle of radius R clipped by two vertical planes that approach the center as u increases. Gaussian curvature K = 0 on smooth patches. The projective maximum: largest admissible volume.

Prior names: ptsi·nɬ (Miluk); "cork plug" (Gardner 1958); "Visual Hull" (Laurentini 1994).

Volume derivation: the cross-section method

All three volumes follow from integrating the cross-sectional area A(u) over the height h. By the general formula V = h ∫01 A(u) du, each species reduces to a definite integral of its cross-section area function.

Theorem 3.7 (cross-section areas) [Anderson, 2026c]

At normalized height u = z/h, the lateral cross-sectional areas are:

AI(u) = πR²(1−u) AII(u) = πR²(1−u)² + 4R²u(1−u) AIII(u) = R²[2(1−u)√(2u−u²) + 2 arcsin(1−u)]

Integrating each from 0 to 1 and multiplying by h yields the volume coefficients kI, kII, kIII stated above.

Corollary: strict volume inequality

For any admissible solid of Genus Skwedge with base radius R and height h:

kI < kII < kIII (π/2) < (π+2)/3 < (π−4/3)

The three species are volumetrically distinct. The inequality is strict.

The Main Result

The Skwedge Disparity

The volumetric gap between the smallest and largest characterized species is 13.13% of the maximum admissible volume. This gap is not an approximation error or a consequence of smoothing. It is a mathematical consequence of the boundary problem being underdetermined: the same boundary data admits constructions of genuinely different volumes.

13.13%
volumetric disparity, curvum to projectivum
Δ = (kIII − kI) / kIII = ((π−4/3) − π/2) / (π−4/3)

In engineering terms: two CAD systems implementing different loft strategies for the same circle-to-line specification will produce solids that differ in volume by up to 13.13%. Prior to this taxonomy, there was no framework for identifying which strategy was being applied or what the gap implied.

Proposition 2.4 (disparity) [Anderson, 2026a]

Let SI and SIII be the curvum and projectivum respectively, with shared boundary data (R, h). Then:

V(SIII) − V(SI) = ((π−4/3) − π/2) R²h = (5π/6 − 4/3) R²h

The fractional disparity Δ = (VIII − VI)/VIII is exactly (5π/6 − 4/3) / (π − 4/3), approximately 0.1313.

The disparity paper is [Anderson, 2026a]. It establishes the gap, characterizes the admissible family, and demonstrates that curvum and projectivum are the extremes among a large class of interpolating constructions.

The π/2 Coincidence

The critical catenoid and Species I

The catenoid is the minimal surface of revolution: the surface of least area bounded by two parallel circles. When the two circles have the same radius R and the height h reaches a critical value h*, the catenoid reaches the Goldschmidt discontinuity, the threshold beyond which no minimal surface exists between the circles.

At this critical ratio h*/R, an unexpected identity holds.

Proposition 6.1 [Anderson, 2026e]

Let h* be the critical height at which the minimal catenoid of revolution bounded by two circles of radius R collapses (the Goldschmidt threshold). The volume enclosed by the critical catenoid is:

Vcat = (π/2) R²h*

This is exactly the volume coefficient kI of Skwedge curvum.

The critical height h* is determined by the fixed point of coth(x) = x (the Laplace limit constant, OEIS A033259). The identity Vcat = (π/2)R²h* connects two independently defined quantities: the volume of the oldest named species of Genus Skwedge, and the enclosed volume of the critical catenoid of revolution. The coincidence is exact, not approximate.

The full derivation, including the algebraic identity that yields (π/2), is in [Anderson, 2026e]. The paper also presents the geometric interpretation: why the critical catenoid, of all minimal surfaces, should enclose a volume equal to the conocuneus is not yet understood as anything other than an algebraic identity.

Open Problem

Species IV: Skwedge minimalis

The variational question motivates a fourth species: what is the admissible solid of Genus Skwedge with minimum lateral surface area? If it exists, it would be the area-minimizing surface bounded by the circle D and the segment L. This is Species IV, tentatively named Skwedge minimalis, and its volume coefficient kIV is an unsolved problem.

The open problem

Does an area-minimizing surface bounded by the circle DR and the line segment LR exist as a classical minimal surface? If so, what is its volume coefficient kIV?

Current status: Ken Brakke (Surface Evolver, Susquehanna University) has confirmed via sectional curvature argument that minimalis cannot exist as a classical minimal surface bounded by a true line segment: the curvature conditions required for a minimal surface cannot be satisfied at a boundary that degenerates to a line. All numerical experiments (Surface Evolver meshes across aspect ratios h/R from 0.5 to 3.0) result in the surface pinching off before reaching the boundary.

Epsilon-ellipse approach: Replacing the degenerate line segment apex with an ellipse of semi-minor axis ε and taking ε → 0 yields approximate coefficients. At h/R = 1.33, the epsilon-ellipse computation suggests kIV ≈ 1.27, but the result shows non-trivial dependence on aspect ratio and mesh parameters. This is not a settled value. Mesh independence has not been verified across all relevant parameter regimes.

What is known: If minimalis exists as a generalized surface (in the sense of geometric measure theory), it must satisfy kIV < kI = π/2 ≈ 1.5708, placing it below all three characterized species in volume.

The problem is described in [Anderson, 2026b]. The paper establishes the variational framework, the Brakke curvature argument, and the epsilon-ellipse convergence studies. The unsolved status of kIV is stated explicitly: this is an open problem, not a verification exercise.

Primary Sources

The five companion papers

All six papers are published on Zenodo under CC BY-SA 4.0. DOIs are stable and citable. Journal submissions are in progress.

All papers are licensed CC BY-SA 4.0. The taxonomy, definitions, and proofs are free to use, extend, and build upon with attribution. For correspondence and corrections, see miluk.org.