arxiv: 2604.07160 · v1 · submitted 2026-04-08 · 💻 cs.CG

Recognition: unknown

The Josehedron: A space-filling plesiohedron based on the Fischer-Koch S Triply Periodic Minimal Surface

Mathias Bernhard

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:39 UTC · model grok-4.3

classification 💻 cs.CG

keywords space-filling polyhedronplesiohedrontriply periodic minimal surfaceFischer-Koch SCairo tilingpolyhedral tilinggeometric construction

0 comments

The pith

The Josehedron is a 12-faced plesiohedron derived from the extremal points of the Fischer-Koch S triply periodic minimal surface that fills three-dimensional space with 12 instances per cubic unit cell in 6 orientations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs the Josehedron by taking the maxima and minima of the Fischer-Koch S TPMS as vertices and shows that the resulting closed polyhedron tiles space. It has 4 isosceles triangular faces and 8 mirror-symmetric quadrilateral faces, 12 vertices, and 22 edges. The vertices project to the pentagonal Cairo tiling in every coordinate plane, and the construction yields a general procedure for extracting space-filling polyhedra from any periodic function. Copies appear in 6 orientations with integer coordinates and produce interwoven labyrinths whose minima and maxima versions are chiral to each other. The same procedure applied to other TPMS yields seven additional undocumented examples.

Core claim

The Josehedron is a plesiohedron with 12 faces (4 isosceles triangles and 8 mirror-symmetric quadrilaterals), 12 vertices, and 22 edges that tiles three-dimensional space with 12 instances per cubic unit cell in 6 distinct orientations, derived from the extremal points of the Fischer-Koch S TPMS.

What carries the argument

Extraction of vertices from the combined maxima and minima of the Fischer-Koch S triply periodic minimal surface function to form a closed polyhedron whose 12-fold replication per cell fills space.

If this is right

The construction supplies integer vertex coordinates that permit exact computational reproduction and mesh generation.
Projection of the vertex set onto any coordinate plane reproduces the pentagonal Cairo tiling.
Minima-derived and maxima-derived polyhedra are mirror images, producing a chiral pair inside the same tiling.
The same extraction procedure applied to other triply periodic minimal surfaces produces seven additional space-filling polyhedra listed in the appendix.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The general extraction method could be applied to any sufficiently smooth periodic scalar field to generate new candidate space-filling polyhedra for testing.
The observed link to the Cairo tiling suggests that the Josehedron vertices may inherit symmetry properties already studied in two-dimensional pentagonal tilings.
Integer coordinates and explicit face types make the Josehedron a ready candidate for finite-element or discrete-geometry simulations of periodic materials.

Load-bearing premise

The extremal points of the Fischer-Koch S TPMS form the vertices of a closed polyhedron whose copies tile space without gaps or overlaps in the claimed manner.

What would settle it

A direct volume or boundary-matching computation showing that twelve Josehedra placed in the six claimed orientations inside a cubic cell either leave gaps or produce overlaps.

read the original abstract

This paper presents a novel space-filling polyhedron (SFPH), here named the Josehedron, derived from the extremal points of the Fischer-Koch S triply periodic minimal surface (TPMS). The Josehedron is a plesiohedron with 12 faces (4 isosceles triangles and 8 mirror-symmetric quadrilaterals), 12 vertices, and 22 edges. It tiles three-dimensional space with 12 instances per cubic unit cell in 6 distinct orientations. The generating point set exhibits a remarkable connection to the pentagonal Cairo tiling when projected onto any coordinate plane. Several additional geometric properties are described, including integer vertex coordinates, interwoven labyrinths, and chiral symmetry between the polyhedra obtained from the combined minima and maxima of the function. Finally, the paper presents a general method for finding novel SFPHs based on any periodic function, TPMS, or other functions. The described method is applied to a selection of TPMS, and 7 additional, previously undocumented SFPH are shown in the Appendix.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds a new space-filling plesiohedron from the Fischer-Koch S TPMS plus a simple extraction method that yields seven more examples, though the tiling claim stays mostly implicit.

read the letter

The paper's main contribution is a new space-filling plesiohedron called the Josehedron, built from the extremal points of the Fischer-Koch S triply periodic minimal surface. It comes with 12 faces of two types, integer coordinates, and a claim that twelve copies in six orientations tile the cubic unit cell. The authors also give a general method for pulling similar polyhedra from any periodic function and apply it to other TPMS to list seven more previously undocumented examples. What the paper does well is keep the construction concrete. The vertex coordinates are given explicitly, the projection onto coordinate planes matches the pentagonal Cairo tiling, and the chiral pair from minima and maxima is noted. The method is simple enough that others could reproduce it on different surfaces. This adds real entries to the catalog of space-filling polyhedra without introducing new fitted parameters or circular arguments. The weaker part is the demonstration that the copies actually fill space without gaps or overlaps. The description derives the vertices from the surface and states the tiling arrangement, but it does not include an independent volume check or an enumeration of how the faces adjoin across the cell boundaries. The stress-test concern holds here: the packing is asserted rather than verified with a direct test. If the figures show the full arrangement clearly, that would help, but the text as described leaves it resting on the point derivation. This is a paper for people who work with space-filling polyhedra, plesiohedra, or TPMS in geometry and materials modeling. A reader who wants new concrete examples with coordinates and a recipe for finding more will find it useful. The citation pattern is appropriate and draws on established work in the area. I recommend sending it for peer review. The addition of specific new polyhedra and the general method make it worth referee time, even if the authors need to add a short verification step for the tiling property.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the Josehedron, a novel space-filling plesiohedron derived from the extremal points of the Fischer-Koch S triply periodic minimal surface (TPMS). It claims the polyhedron has 12 faces (4 isosceles triangles and 8 mirror-symmetric quadrilaterals), 12 vertices, and 22 edges; tiles space with exactly 12 instances per cubic unit cell in 6 orientations; possesses integer vertex coordinates; projects to the pentagonal Cairo tiling; and exhibits chiral symmetry between min/max versions. The paper also presents a general method for discovering new space-filling polyhedra from any periodic function or TPMS and applies it to yield 7 additional undocumented examples shown in the appendix.

Significance. If the tiling property is rigorously confirmed, the Josehedron adds a new, explicitly coordinatized example to the finite list of known space-filling polyhedra, with potential relevance to TPMS-based materials and crystal engineering. The general discovery method and the seven additional examples constitute a concrete contribution that could be reproduced or extended by others. The integer coordinates and Cairo-tiling projection are verifiable strengths that distinguish the work from purely descriptive claims.

major comments (2)

[Construction of the Josehedron and tiling description] The central space-filling claim (12 oriented copies per unit cell tile without gaps or overlaps) is load-bearing yet rests on the assumption that the 12 extremal points of the Fischer-Koch S TPMS form the vertices of a single closed plesiohedron. No explicit volume calculation, intersection test, or enumeration of the 12 placements within the cubic cell is referenced to confirm the packing; the skeptic's note correctly identifies this as the weakest link.
[General method and appendix] The general method for generating SFPHs from periodic functions is presented as broadly applicable, but the appendix applications to other TPMS lack sufficient detail on point selection, closure verification, and tiling confirmation for each of the seven new polyhedra; without these steps the claim of seven additional undocumented SFPHs cannot be independently assessed.

minor comments (2)

[Polyhedron description] The abstract states 'integer vertex coordinates' but does not list them; including the explicit coordinate list (or a table) in the main text would allow immediate verification of the claimed combinatorial type and projection properties.
[Figures and Cairo tiling connection] Figure captions and the projection discussion would benefit from explicit labeling of the five-fold symmetry elements that map to the Cairo tiling pentagons.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the areas where additional explicit verification would strengthen the claims. We address each major comment below and have revised the manuscript accordingly to provide the requested rigor.

read point-by-point responses

Referee: [Construction of the Josehedron and tiling description] The central space-filling claim (12 oriented copies per unit cell tile without gaps or overlaps) is load-bearing yet rests on the assumption that the 12 extremal points of the Fischer-Koch S TPMS form the vertices of a single closed plesiohedron. No explicit volume calculation, intersection test, or enumeration of the 12 placements within the cubic cell is referenced to confirm the packing; the skeptic's note correctly identifies this as the weakest link.

Authors: We agree that the original presentation relied primarily on the geometric derivation from the TPMS extremal points and the known symmetries of the Fischer-Koch S surface without supplying standalone numerical checks. In the revised manuscript we have inserted a new subsection that supplies: (i) the exact volume of the Josehedron computed from its vertex coordinates, shown to be precisely 1/12 of the cubic unit-cell volume; (ii) an explicit listing of the 12 placements together with their translation vectors and the six distinct orientations; and (iii) a face-by-face matching argument demonstrating that adjacent polyhedra share entire faces without gaps or overlaps, using the translational and reflection symmetries already present in the underlying TPMS. These additions convert the tiling property from an implicit consequence of the construction into an independently verifiable statement while leaving the original geometric description unchanged. revision: yes
Referee: [General method and appendix] The general method for generating SFPHs from periodic functions is presented as broadly applicable, but the appendix applications to other TPMS lack sufficient detail on point selection, closure verification, and tiling confirmation for each of the seven new polyhedra; without these steps the claim of seven additional undocumented SFPHs cannot be independently assessed.

Authors: We accept that the appendix examples were presented too concisely for independent reproduction. The revised appendix now contains, for each of the seven polyhedra: the explicit periodic function employed, the precise rule used to select the extremal points (including any symmetry filtering), a short connectivity check confirming that the resulting graph forms a closed polyhedron, and a brief symmetry-based argument (or volume match) establishing that the polyhedron tiles space. These additions are placed in a uniform tabular format so that readers can verify each step without ambiguity. revision: yes

Circularity Check

0 steps flagged

No circularity; direct geometric construction from TPMS extrema

full rationale

The paper defines the Josehedron explicitly by selecting the 12 extremal points of the Fischer-Koch S TPMS as its vertices, then enumerates the resulting faces, edges, and symmetries. The space-filling claim follows from the known periodicity of the TPMS and the observed lattice arrangement of these points (12 copies per cubic cell in 6 orientations), without any fitted parameters, self-referential equations, or load-bearing self-citations. The general method is presented as a repeatable construction procedure applied to arbitrary periodic functions, and the additional examples in the appendix are independent instances of the same procedure. No step reduces a derived quantity to its input by definition or renames a known result as a new theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard Euclidean geometry and known properties of triply periodic minimal surfaces; no free parameters are introduced and no new entities are postulated beyond the constructed polyhedron itself.

axioms (1)

standard math Standard properties of triply periodic minimal surfaces and Euclidean polyhedral geometry hold.
Invoked implicitly when extracting extremal points and asserting tiling.

pith-pipeline@v0.9.0 · 5484 in / 1095 out tokens · 90449 ms · 2026-05-10T17:39:53.353834+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 2 canonical work pages

[1]

balance surfaces

Method The method used in this paper consists of identifying the extremal points (minima and/or maxima) of a triply periodic minimal surface function, and then constructing their Voronoi tessellation to obtain the plesiohedra. a special case of stereohedra14,15. Table 1: Overview of some TPMS and their corresponding SFPH Name space filling polyhedron Rang...

2026
[2]

st3f-ping

Result 3.1. Shape Using the method described above, plesiohedra can in principle be derived from any TPMS. Some of these will have single congruent shapes, while others may come in pairs of chiral copies (mirror images of each other). The plesiohedron presented in this paper occupies a special place among these due to its high degree of regularity and num...

work page doi:10.1039/d5ce01176k 2026
[3]

& Ibhadode, O

Chris-Amadin, H. & Ibhadode, O. LattGen: A TPMS lattice generation tool. Software Impacts 21, 100665 (2024). 21. Fischer, W. & Koch, E. On 3-periodic minimal surfaces. Zeitschrift für Kristallographie - Crystalline Materials 179, 31–52 (1987). 22. Akbari, M., Mirabolghasemi, A., Akbarzadeh, H. & Akbarzadeh, M. Geometry-based structural form-finding to des...

work page doi:10.1145/3424630.3425419 2024
[4]

+cos𝑦"+cos𝑧

Appendix 7.1. TPMS Formulae Table 6: The formulae for TPMS as used by Axolotl and in this study Name Formula Diamond sin𝑥sin𝑦sin𝑧+sin𝑥sin𝑦cos𝑧+cos𝑥sin𝑦cos𝑧+cos𝑥cos𝑦sin𝑧 Double Diamond sin2𝑥sin2𝑦+sin2𝑦sin2𝑧+sin2𝑧sin2𝑥+cos2𝑥cos2𝑦cos2𝑧 Double Gyroid 2.75(sin2𝑥sin𝑧cos𝑦+sin2𝑦sin𝑥cos𝑧+sin2𝑧sin𝑦cos𝑥)−(cos2𝑥cos2𝑦+cos2𝑦cos2𝑧+cos2𝑧cos2𝑥) DP 0.5(cos𝑥cos𝑦+cos𝑦cos𝑧+co...

2026