The Science

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

Five companion papers and a supplement explore these solids. This page presents the constructions, volume results, and open problems, with corrections and scope clarifications dated 16 September 2026. The papers are 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 uses a circular base and a diametral apex line in parallel planes, centered on the same axis. The intended proportions are fixed: base diameter = height = apex length, so h = 2R. The loft stays inside the cylinder-and-wedge envelope defined below and approaches the entire apex line. Curvum, convexum, and projectivum are three named constructions within an infinite family; minimalis remains provisional.

Phylum Apicalia all apex-convergent solids; dim(apex) < dim(base)
Class
├─ Apexia footprint-contained; sections need not narrow monotonically
└─ 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 h = 2R; diametral apex; cylinder-and-wedge envelope
└─ Bulbus separate bulging genus; detailed classification pending
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 provisional · existence and genus placement open

The complete vertical path is: Phylum Apicalia → Class Apexia → Order Apextoid → Family Hewel → Genus Skwedge → Species I–III, with IV provisional. 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

Let D be the disk x²+y² ≤ R² at z = 0, and L the segment x = 0, |y| ≤ R at z = h. For the intended Skwedge genus, h = 2R. Uniform scaling preserves these proportions; the unit normalisation is R = 1/2, h = 1.

Geometric scope

The base is D and the apex is the full line L. At each normalised height u = z/h, the horizontal section lies in E(u) = {(x,y): x²+y² ≤ R², |x| ≤ R(1−u)}. The sections approach D at the base and the entire L at the apex, continuously in Hausdorff distance. An isolated line attached to a point-ending taper does not meet this condition. These are geometric requirements; the detailed solid and surface regularity assumptions will be stated explicitly in the revised papers.

The boundary data and envelope still admit infinitely many lofts. The three named constructions below use the ruling principle, the convex-hull principle, and the projection principle. Containment does not require nested sections or monotonically decreasing widths or areas. A circle-to-line solid that stays in the cylinder but exceeds this wedge envelope belongs to the broader Hewel family, outside this Skwedge genus.

The published papers allow arbitrary positive R and h. Their three volume formulae remain valid at those general proportions; h = 2R selects the intended Skwedge subclass. The broader family's naming and the full classification of unnamed members remain open. A profile represented by a spline or Bézier curve is classified by its geometry, not by its encoding. CAD behaviour must likewise be specified for the particular construction and settings.

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. On the smooth lateral surface, Gaussian curvature K ≤ 0: it vanishes on four rulings and is negative elsewhere. The lateral surface is saddle-shaped and not developable; its horizontal sections are ellipses.

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 smallest convex solid containing D and L: every convex solid with these ends contains convexum. Projectivum is also convex, and there are infinitely many convex solids between them.

Prior names: No established prior name. This construction is the convex hull; no software-wide CAD default is asserted.
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: the full cylinder-and-wedge envelope E, hence the largest volume among solids contained in E. It is convex.

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.

Cross-section areas (proof of Theorem 3.7) [Anderson, 2026c]

At normalized height u = z/h, the horizontal 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 the three named constructions with the same base radius R and height h:

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

The three named constructions are volumetrically distinct. This ordering does not bound every other loft below by curvum.

The Main Result

The Skwedge Disparity

Projectivum has more volume than curvum by approximately 13.13% of projectivum's volume. This is an exact comparison of two specified constructions, rounded to a percentage. It illustrates how the same end data can admit 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, implementations of curvum and projectivum give this particular difference. It is not a universal upper bound on differences between arbitrary lofts or CAD outputs.

Curvum-to-projectivum 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 = (π/2 − 4/3) R²h

The fractional disparity Δ = (VIII − VI)/VIII is exactly (π/2 − 4/3) / (π − 4/3), approximately 0.1313.

Projectivum supplies the envelope-relative upper bound V ≤ (π−4/3)R²h. There is no positive lower bound for all lofts: elliptical sections with semi-axes R and R(1−u)p, p ≥ 1, give V = πR²h/(p+1), which tends to zero as p increases. Every finite p retains the full diametral apex line as its limit. Curvum is the p = 1 member, not a genus-wide volume minimum.

The π/2 Coincidence

The critical catenoid and Species I

A catenoid is a surface of revolution with zero mean curvature. For equal circular rings, its connected branches meet at the existence limit h*/R ≈ 1.32549. This is distinct from the earlier equal-area threshold h/R ≈ 1.055, where two separate disks match the stable catenoid's area. Beyond that earlier threshold, a connected catenoid can persist as a local area minimum until the existence limit.

At the existence limit, the following classical identity holds.

Critical-catenoid volume identity [Anderson, 2026e]

Let h* be the limiting separation for a catenoid between equal rings of radius R. The volume enclosed by that limiting catenoid and its end disks is:

Vcat = (π/2) R²h*

This is exactly the volume coefficient kI of Skwedge curvum.

History: Lindelöf printed the identity in 1863 (p. 364), attributing it to Plateau. Plateau restated it in 1873 (§§89–90) and described an experimental verification. Anderson's paper establishes the converse characterization and strict monotonicity of the normalised volume.

The positive root x* of coth(x*) = x* is approximately 1.19968 (OEIS A085984). The Laplace limit constant is the different number λ = 1/sinh(x*) ≈ 0.662743 (A033259), and h*/R = 2λ. The volume coefficient π/2 matches curvum's exactly. This is a coefficient comparison: the critical catenoid does not have the fixed Skwedge proportions h/R = 2.

The derivation is in [Anderson, 2026e]. For the distinction between the area crossing and the existence limit, see the reader's guide and Fold Explorer.

Open Problem

Provisional Species IV: Skwedge minimalis

The name minimalis is reserved provisionally for the variational investigation. Two problems must be distinguished: minimising lateral area among solids inside the prescribed envelope, and finding an unconstrained minimal surface or limiting surface with the prescribed ends. A constrained minimiser need not have zero mean curvature where it meets the envelope. An unconstrained solution need not stay inside it.

The open problem

Does the limiting variational construction yield a surface that bounds a genuine solid with the full diametral apex line? If so, is it uniquely determined, does it lie inside the Skwedge envelope at h = 2R, and what is its volume?

Current status: Existence as a nondegenerate limiting surface, uniqueness or canonical selection, and genus placement remain unresolved. Numerical pinch-off does not by itself prove non-existence. The earlier assertion of a confirmed non-existence theorem is withdrawn pending a citable argument.

Epsilon-ellipse approach: The paper studies limits as the upper ellipse's semi-minor axis ε tends to zero. The existence of a generalised limit does not establish that it bounds a positive-volume solid. Values at other h/R ratios cannot be assigned to the fixed-proportion Skwedge by stretching.

Conjectures: The proposed inequality kIV < π/2 is Conjecture 8.1 of [Anderson, 2026b], not a consequence of area minimisation. Departure from the wedge envelope is also conjectural (Conjecture 4.14). If a candidate leaves the envelope, it is outside this Skwedge genus.

The variational framework is described in [Anderson, 2026b]. Neither a catenary profile nor an area minimum within a chosen one-parameter loft family establishes the unrestricted minimalis.

Primary Sources

The five companion papers

The five companion papers and one supplement are available on Zenodo. Paper 5, The Volume of the Critical Catenoid, is also published in The Mathematical Intelligencer (journal DOI).

Revision status · 16 September 2026

This page incorporates corrections ahead of the next Zenodo versions. Revisions to papers 1–4 are queued. The author-corrected version of paper 5 is complete and serves as the gold copy; correction of the journal's typesetting errors is pending. The revised papers 1–5 will be released on Zenodo after the corrected journal article appears online or in print and has been checked against that gold copy. Existing DOI links remain in place until that release.

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.