History · what stands behind the present work Lineage
No work begins in 2026. Every result inherits a question and the work of those who asked it first.
introductory note
Substrate Geometry inherits two distinct lineages: a thirty-year arc of mono-monostatic body research originating with Arnold's 1990s Hamburg seminar, and a century-old line of developable-surface geometry originating with Paul Schatz's 1929 discovery of the oloid.
The mono-monostatic lineage
1990s · Hamburg
Vladimir I. Arnold
Posed the existence question at a Hamburg seminar: does there exist a convex, homogeneous, three-dimensional body with exactly two equilibria, one stable and one unstable? The question framed the problem as one about the equilibrium structure of the center-of-mass height function on the body's surface, opening the path to subsequent analytical and computational work.
2006 · Budapest
Gábor Domokos & Péter Várkonyi
Constructed the first such body at the Budapest University of Technology and Economics and named it the Gömböc. The construction directly answered Arnold's existence question and remains the foundational result of the field. The two-parameter family characterized the regime in which mono-monostatic equilibrium emerges from departures from spherical symmetry.
2023 · Austin
M. L. Sloan
Published the first analytical parameterization of a smooth-Gömböc class of surfaces, casting the body as a perturbation of the sphere governed by a phase function with two surface critical points. Sloan's parameterization is the analytical foundation that made systematic computational extension possible; previous work had relied on numerical optimization over arbitrary control points.
2026 · Present
Catalog work
Computationally extends Sloan's phase function with a Fourier term, optimized via differential evolution to identify mono-monostatic instances. Thirteen verified members of the catalog at the resolution required to distinguish surface critical points from center-of-mass height minima. The mathematical contribution: surface critical points are necessary but not sufficient for mono-monostatic equilibrium. The catalog is the family of solutions in which the necessary condition is also sufficient.
The catalog is positioned as a sharpening of Domokos and Várkonyi 2006, not a refutation. The Gömböc remains the foundational object. What the catalog adds is the empirical observation that the center-of-mass height landscape on the unit sphere can carry equilibrium structure that the surface's critical-point structure alone does not predict, and the Fourier extension that closes the gap. Recent adjacent work shaping the current landscape includes Bogosel 2024, numerically confirming the Meissner conjecture restricted to the Meissner polyhedra subclass; the 2024 construction of a previously unknown constant-width body U3 with tetrahedral symmetries; and Domokos–Kovács 2023 on the discrete twenty-one-point-mass Gömböc.
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The oloid lineage
1929 · Switzerland
Paul Schatz
Discovered the oloid while investigating inversion of the cube, observing that the body constructed from two perpendicular circles of equal radius with the shared chord equal to the common radius exhibits a developable surface and a uniform-rolling motion. Schatz's work was conducted largely outside the academic mainstream and remained marginal in formal geometry literature for decades.
1997 · TU Wien
Hans Dirnböck & Hellmuth Stachel
Formally proved the developable property of the oloid surface and characterized its rolling kinematics analytically. The 1997 result established the mathematical foundation that subsequent computational work could extend. Stachel's continued engagement in the geometry community provided the credentialing chain that endorsed the present program's first paper.
2026 · Present
Paper I & the CDS invariant
Introduces the Contact Distribution Score (CDS) as the first computable operational invariant for which the oloid demonstrably occupies a local optimum within the developable roller family. The 58× discrimination between oloid and cylinder at matched mesh resolution, the 1,430-genome parametric sweep, the two-tier invariant transfer structure. Endorsed for arXiv by Georg Nawratil at Hellmuth Stachel's referral.
The endorsement chain for the program's first paper is direct, traceable, and inside the geometry community's standard pattern of how an outside contribution enters the literature. The path is not new; the contribution is.
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For the formal claim structure of the present work, see papers and methodology. For the visual map of the program's territory, see atlas.