TECHNOLOGY
The Arithmetics of Forms
A non-anthropomorphic mathematical framework that decomposes physical structure into elementary building blocks, providing a language to describe and manipulate complex structures with absolute precision — powering spatial reasoning, multidimensional data indexing, and geolocation.
THE CHALLENGE
The Arithmetic Approach
Transmitting spatial knowledge to a computer requires choosing a theoretical model. Three classical approaches have been proposed:
Mechanistic
Turing machines — step-by-step procedural instructions. The path taken by most CAD systems.
Used by traditional CAD
Linguistic
Formal grammars — Chomsky, Post. Shape described through rewriting rules and production systems.
Grammar-based systems
Functional-Arithmetic
Recursive functions, lambda calculus — Skolem, Gödel, Kleene, Church. Spatial form as the output of pure recursive functions on natural integers.
The SGDL path
In the arithmetic approach, geometric and combinatorial models are not separate — they are unified through arithmetic, a pillar of metamathematics. All constructive and logical processes of spatial arrangement are formalized as algorithms that are exclusively and purely arithmetic.
“Arithmetic is transmuted into a metalanguage of spatial forms.”
Two Worlds That Need to Connect
Symbolic AI
- Reasons with logic and formal rules
- Based on programmed rules and knowledge
- Performant in structured, well-defined domains
- Explainable and traceable
- Limited when dealing with raw, unstructured data
Statistical AI
- Learns patterns directly from data
- Powered by LLMs and neural networks
- Performant for vision, translation, generation
- Scales with compute and data
- Approximative and black-box by nature
Neither approach alone can interact with the physical world. Their convergence gives rise to a new discipline: Physical AI.
The Emergence of Physical AI
Physical AI perceives the environment and acts in the real world. It powers robots, autonomous vehicles, surgical tools, and industrial automation — but also generative AI, synthetic cities, and low-altitude economy. It requires extreme reliability — there is no room for approximation when a robotic arm operates near a human body or a drone navigates urban airspace.
The Arithmetics of Forms provides exactly this: the deterministic, precise link between AI agents and deep-tech platforms that operate in physical space.
THE CORE MECHANISM
The Transmutation Principle
Geometry becomes arithmetic. Every spatial question becomes a calculation on integers.
Densities
Every point in space is assigned a natural integer encoding its topological position: density 0 means the point is outside, density 1 means it is on the boundary, density 2 means it is inside.
The topological question “is this point inside?” becomes the arithmetic question “what is this number?” When multiple forms overlap, densities accumulate — the intersection of two solids has density 4, their shared boundary density 2.
The Design Logic Operators
Geometric operations map exactly to arithmetic operations. Each spatial combination of forms reduces to a simple calculation on their density values — turning CAD into algebra.
DLadd
Union = Addition
Merge volumes — densities are added together.
DLmul
Intersection = Multiplication
Keep the common part — densities are multiplied.
DLdif
Difference = Subtraction
Remove material — densities are subtracted.
DLsab
Symmetric Diff. = Abs. Subtraction
Exclusive zone — absolute difference of densities.
DLgcd
Greatest Common Volume = GCD
Largest shared structure — GCD of densities.
DLlcm
Smallest Enclosing Vol. = LCM
Minimal container — LCM of densities.
The Boolean operations of classical CSG (Constructive Solid Geometry) appear as special binary cases of SGDL's ternary Kleene operators. SGDL does not replace CSG — it subsumes it.
The Paradigm Gap
Beyond B-Rep and CSG
Since the 1970s, computer geometry has relied on two systems. B-Rep (Boundary Representation) describes objects by listing their surfaces — an extensional catalog of faces, edges, and vertices. CSG (Constructive Solid Geometry) describes objects as Boolean operations on primitives — powerful, but limited to a fixed vocabulary of shapes.
Neither system can compute between geometry and arithmetic. Neither provides a computable bijection between integers and forms. The Arithmetics of Forms bridges this gap: every natural integer maps to a unique volumetric form, and every form maps back to its integer — with the full richness of arithmetic operations preserved as geometric operations.
B-Rep
Lists surfaces — extensional, no algebra
CSG
Boolean on primitives — limited vocabulary
sgdl
Integer ↔ Form bijection — full arithmetic on volumes
Key Properties
What makes the Arithmetics of Forms fundamentally different from statistical approaches.
Deterministic
Unlike probabilistic models, every computation produces exact, reproducible results with no randomness.
Precise
All computation uses exclusively natural integers — no floating-point numbers, no division, no rounding errors. Projective homogeneous coordinates ensure perfect precision through every calculation.
Frugal
CPU-native processing — no GPU clusters or data centers required. Orders of magnitude more efficient.
Economical
Dramatically lower infrastructure costs compared to GPU-dependent statistical approaches.
Unconditional
Projective geometry eliminates all singular configurations. The same algorithm works for every case — no IF/ELSE branching, no special-case handling. Parallel lines, degenerate surfaces, and edge cases are all treated uniformly.
Total
All operations are built on Primitive Recursive Functions (PRF), which are guaranteed to terminate. Every computation produces a result — no infinite loops, no undefined behavior, no crashes.
INTELLECTUAL PROPERTY
A Fortress of 21 Interconnected Patents
Since 2018, sgdl has focused its strategy on building a comprehensive intellectual property portfolio. Today, 21 patents form an interlocking system of protection covering the complete pipeline — from spatial encoding and volumetric computation to cryptographic applications.
Spatial Encoding & Indexation
Patents covering the MALVES coding system and multidimensional data indexing methods that transform physical objects into mathematical representations, enabling precise geolocation and spatial reasoning at any scale.
Cryptographic Applications
Patents on MCG (MetaCurve Gray) cryptographic stencils, providing unique encryption capabilities derived from space-filling curve signatures.
Volumetric Computation
Patents protecting the core Arithmetics of Forms pipeline — from projective quadric definitions through Design Logic operators to BigNum encoding.
All 21 patents are interconnected and mature, forming a unified system ready for industrial deployment across robotics, autonomous vehicles, medical imaging, defense, and materials science.
EXPLORE
The Platonic Solids
The Arithmetics of Forms operates on volumetric primitives. Explore the five Platonic solids — the foundational building blocks of geometric reasoning.