About sgdl
Pioneering the Mathematics of Physical AI
sgdl is a Swiss company founded by mathematicians and researchers, pioneering non-anthropomorphic exolanguages for spatial reasoning, multidimensional data indexing, and geolocation — based on the Arithmetics of Forms.
Our Mission
Transforming Research into Reality
sgdl is at a pivotal moment, structuring and financing to transform 40 years of fundamental research into a concrete, scalable industrial offering. Our mathematical framework — the Arithmetics of Forms — enables machines to perceive, reason about, and interact with the physical world with mathematical precision.
From surgical robotics and space exploration to synthetic cities, robotic teleoperation, and digital twins, our technology provides the spatial intelligence layer that Physical AI demands: deterministic, CPU-native, and mathematically validated.
Key Facts
Founded in Switzerland
Headquartered in the heart of European innovation
40+ Years of R&D
Continuous research since the early 1980s
21 Patents
Comprehensive IP portfolio covering the full pipeline
Active Partnerships
CERN, CSA, AOUSD, Spineart
Intellectual Heritage
Sieur Girard Desargues Lyonnois
The name sgdl pays homage to Girard Desargues (1591–1661), the Lyon-born architect and mathematician who founded projective geometry. In 1639, Desargues proposed a revolutionary idea: by adding points at infinity to ordinary geometry, two lines always intersect in exactly one point — and parallelism becomes simply the case where that point lies at infinity.
This insight launched a five-century intellectual lineage — from the perspective paintings of Renaissance Florence through the formalization of projective space, space-filling curves, and computability theory — culminating in Jean-François Rotgé's Arithmetics of Forms: a formal isomorphism between natural integers and three-dimensional volumetric forms.
The name carries layered meaning. For specialists in volumetric modeling accustomed to CSG, SGDL stands for Solid Geometry Design Logic — a system that subsumes CSG as a special case. For language theorists, it reads as Solid Geometry Design Language. As a homage to the founder of projective geometry, S.G.D.L. are the initials of Sieur Girard Desargues Lyonnois. And as a final tribute to Kurt Gödel, whose arithmetization of logic made the entire enterprise conceivable, SGDL is also a Système GöDeLien.
Rotgé's thesis reunites arithmetic, geometry, grammar, and logic — four of the disciplines of the medieval quadrivium. The fecundity of this theoretical blending, as he writes, “was not to be demonstrated — it was to be re-demonstrated in a computational framework different from that of the ancients, who had known how to marry the genres around the problem of graphical representation.”
1435
Leon Battista Alberti
Founds linear perspective in De Pictura — the seed of projective geometry, where parallel lines converge at a vanishing point
1639
Girard Desargues
Publishes the Brouillon projet, founding projective geometry by adding points at infinity to Euclidean space
1822
Jean-Victor Poncelet
Reconstructs projective geometry from memory in a Russian prison, publishes the Traité des propriétés projectives
1872
Felix Klein
Presents the Erlangen Program, unifying all geometries under the umbrella of transformation groups
1891
David Hilbert
Constructs his recursive space-filling curve, demonstrating that a 1D path can faithfully cover 2D space while preserving locality
1920s
Thoralf Skolem
Champions Primitive Recursive Functions — a constructive, computable approach to mathematics that avoids the paradoxes of the infinite, seeking only operations that always terminate
1931
Kurt Gödel
Shows that complex logical statements can be encoded into unique numbers (Gödel numbering), allowing a system to reason about its own structure — the philosophical ancestor of arithmetizing geometry
1936
Alonzo Church
Defines computation through pure function abstraction and application (Lambda Calculus), proving that complex operations can be built from simple, elegant rules — the foundation of functional programming
1947
Frank Gray
Invents the reflected binary code at Bell Labs — consecutive integers differ by exactly one bit, enabling error-free traversal of binary spaces
1997
Jean-François Rotgé
Defends his doctoral thesis The Arithmetics of Forms at the Université de Montréal, establishing a computable bijection between natural integers and 3D volumetric forms — unifying the work of Desargues, Hilbert, Gödel, Church, and Gray into a single formal system
Leadership
Our Team
A. Kummerman
Founder & CEO
Mathematician (NYU, MIT), leading sgdl's strategic vision and organizational development.
J.F. Rotgé
Founder & Chief Scientific Officer
PhD, Université de Montréal. Inventor of all sgdl patents, 40+ years of research on the Arithmetics of Forms. Mentored by Janos Baracs (structural topology), Gert Sabidussi (graph theory), and Henry Crapo (matroid theory).
L. Daniel
AI Engineer, MSc
Brings essential expertise to the engineering team, focusing on the development and optimization of spatial reasoning capabilities. Co-developer of the original sgdlscript since 1983.
A. Tawil Kummerman
CFO
Finance and administration, ensuring sound financial governance and operational efficiency.
M. Aubert
COO
Coordination and institutional communication, bridging research and organizational strategy.
J. Farret
CTO (SGDL Innovation)
Software architecture, leading the technical implementation and platform development.
J.M. Le Goff
Director (SGDL Foundation)
Technology transfer and industry-academia collaboration (35 years at CERN).
P.O. Dehaye
AI Director (SGDL Labs)
PhD Mathematics (Stanford, Oxford), IP portfolio valorization and AI strategy.
M. Bernard
Head of Industrial Partnerships
Focused on industrial junction and acceptability — demonstrating how sgdl technology integrates within existing CAD, BIM, and OpenUSD ecosystems.
Structure
The sgdl Ecosystem
Three complementary entities working together to protect, advance, and deploy sgdl's breakthrough technologies.
SGDL Innovation
IP Management: Patents, source codes, algorithms
Protecting fundamental research and transforming it into exploitable assets. SGDL Innovation manages a portfolio of 21 interconnected patents spanning spatial encoding, cryptographic applications, and volumetric computation — a comprehensive IP fortress built through 40 years of continuous R&D.
SGDL Foundation
Open Science: Exolanguage Ariadne, partnerships, education
Managing fundamental research, academic partnerships, and scientific dissemination. The Foundation ensures that sgdl's discoveries contribute to the broader scientific community.
SGDL Labs
Industrial Projects: Thinking Crystal, commercial prospection
Deploying sgdl technologies through R&D collaboration contracts and licenses. Labs bridges research and market, delivering project tracking and commercial development.
Governance
Board of Directors
A. Kummerman
Chairman
J.F. Rotgé
Member
M. Kummerman
Member
D. Akzam
Member
A. Tawil Kummerman
Member
S. Mugnier-Jacob
Member
Milestones
Four Decades of Discovery
Introduction to computing, courses in structural topology and constructive geometry
Development of sgdlscript with Laurent Daniel, MSc — a collaboration that continues to this day
Michel Kummerman begins long-term support
Foundational discoveries at GRCAD, Université de Montréal — the transmutation principle, Design Logic operators, and density-based topology
First international presentation at CSG-96 conference, Winchester UK — Principles of Solid Geometry Design Logic
Doctoral thesis defense on the Arithmetics of Forms
Industrial sgdl development
ISS Canadarm simulator for CSA
Four consecutive issues of Tangente magazine covering arithmetic, algorithmics, algebra, and projective geometry
Canadian/US patents on MALVES coding system
HAL publication series on Gray curves, Canadian patent on MCG cryptographic stencils
Book on Boustrophedonic Geometry published
Exolanguage Ariadne presentation
Member of the Alliance for OpenUSD (announced alongside Qualcomm, Booz Allen Hamilton following Core Spec 1.0 ratification), CERN collaborations active
Want to learn more about sgdl?
Explore our technology, read the science, or get in touch with our team.