Primary Design Co.

What Are High-Level Space FIelds?

Begin with a Complete Graph

All dimensions can be projected into lower dimensions, manifesting as encodings within basic geometric points, lines, and planes derived from a sphere. The inner symmetries of polygons can be seen as the 2D scaffoldings of higher dimensional spaces.

Go to Higher Levels

By using symmetrical radial duplication, the complete graph is transposed into a super-graph. This duplication process can occur symmetrically at any point on the radius or apothem of the space field, creating infinitely various dimensional levels.

Render in Raster Space

Higher-dimensional geometry is inherently vector-based. When exported as an image and algorithmically adjusted in bitmap, raster space we create artistic designs that imperfectly render colorful multidimensional images.

Color-Rendered Art Prints: [Collection-B]

A-20D

O20CC_RGBY+pp_(0-lb30-hl-drkn)(light(gb,30)-cb(gb,30)-hue(gb,30)-lum(gb,30))-(C([0,0[,[18,255],[41,0],[64,255],[93,0],[121,255],[152,0],[179,255],[208,0],[231,255],[255,0]))x2-BC(150,100)-I

A-24D

O24CC_RGBYBlWLPOPiC+pp_GF-lb24(Effs(BE,IS,IG,OG,DS))-CB(-100,0,0)

A3D-8D

O8CC-3D_RGB+pp_0-m4-sl-hl(rbz,50)4-C([0,255],[21,184],[37,163],[62,224],[73,157],[98,17],[124,255],[151,17],[158,248],[191,19],[200,247],[235,80],[246,241],[255,0])-CB(-32,-37,-78)-BC(-38,-50)-L(0,1.08,255,0,255)-V(+100,-8)-HS(-120,+100,0)

All geometry begins with a sphere.

Sphere-Based Design Theory (SBDT): The Geometry of Universal Order

At Primary Design Co., we recognize all form as an emergent expression of a deeper geometric logic, encoded within the sphere. Every structure—whether architectural, urban, computational, or artistic—traces back to this fundamental archetype. Sphere-Based Design Theory (SBDT) is the foundation of our work, driving breakthroughs in recursive geometry, High-Level Space Fields (HLSFs), and Radial-Wave Tessellation (RWT).

The Sphere: The Universal Generator of Space

The sphere is not just a shape—it is the organizing principle of spatial order. Its seamless curvature, boundless symmetry, and recursive structure encode the rules of form generation. By projecting, slicing, and tessellating the sphere, we reveal the geometric frameworks that govern both physical and digital reality:

Triangles (3-sided): The irreducible unit of spatial stability, forming the core of all polyhedral structures.
Quadrilateral Systems (4-sided): The Cartesian framework of computation, modular repetition, and pixel-based reality.
Higher n-gons (5, 6, 7+ sided): Recursive geometries that move beyond Euclidean constraints, forming the foundations of fractal, multi-dimensional, and non-linear space.

A defining realization of SBDT is that a square tessellation of the sphere forms the same logical structure as digital grids, pixel arrays, and computational systems—bridging the divide between geometry, information processing, and simulation-based modeling.

High-Level Space Fields (HLSFs): The Recursive Architecture of Form

If SBDT establishes the fundamental rules of spatial logic, then High-Level Space Fields (HLSFs) are its functional application. HLSFs are not static forms; they are recursive space networks that evolve through layered adjacency, revealing hidden orders of symmetry and emergence.

Key Principles of HLSFs:

Dimensional Adjacency – Each n-gon has an angular division (360°/n) that determines its proportional relationships within a space field. These adjacencies guide recursive layering.
Recursive Expansion – HLSFs evolve iteratively, where each stage builds on prior formations, generating self-replicating networks of interlocking geometries.
Emergent Entities – Within these recursive fields, distinct yet interconnected bilaterally symmetrical forms emerge—expressing pattern recognition, symbolic cognition, and structured adjacency.
Geodesic Projection – HLSFs naturally integrate with spherical overlays, transitioning seamlessly between flat, curved, and multi-dimensional frameworks.

HLSFs as a Computational Framework

By applying HLSF principles programmatically, we unlock new capabilities:

Real-time recursive geometry simulations across multiple scales.
Generative design tools that ensure symmetry while maintaining adaptability.
AI-assisted adjacency modeling, bridging the gap between intuitive form-finding and rigorous computational logic.

HLSFs function both as artistic expressions and applied systems, from symbolic design to AI-driven spatial intelligence to full-scale urban morphologies.

Radial-Wave Tessellation (RWT): The Future of Urban Design

Traditional cities rely on Cartesian grids—a rigid, rectilinear logic unsuited to the complexities of natural landscapes, fluid mobility, and emergent urban forms. Radial-Wave Tessellation (RWT) offers a next-generation framework for adaptive, human-scaled, and multidimensional urbanism.

Why RWT?

Geometric Efficiency – RWT dynamically adapts block, street, and parcel layouts to geographic constraints, reducing inefficiencies of rectilinear zoning.
Multimodal Integration – RWT supports pedestrian-first mobility, VTOL infrastructure, and transit-oriented development, eliminating the need for auto-dependent planning.
Spherical Adaptation – The wave-based nature of RWT aligns with natural topographies, hydrological flows, and organic spatial expansion.
Recursive Zoning & Multi-Scale Order – RWT introduces a hierarchical land-use framework where small-scale adjacency builds up into higher-level city structures—mirroring recursive HLSF logic in urban space.

RWT can be implemented both manually and algorithmically—allowing human designers to shape city plans through explicit RWT rules or enabling AI-driven simulations to generate adaptable, self-organizing urban fabrics.

Sphere-Based Design in Action: From AI to Architecture

At Primary Design Co., SBDT, HLSFs, and RWT converge across multiple disciplines, from computational geometry to future city planning.

1. AI & Generative Design Systems

Custom parametric HLSF generators simulate recursive spatial networks in real time.
AI-driven adjacency mapping bridges human intuition with mathematical precision.

2. Architectural & Urban Development

Sphere-Based Architecture creates multi-scale modular environments that feel organically structured yet mathematically rigorous.
Recursive urban zoning aligns built environments with topographical, ecological, and human-scale proportions.

3. VTOL & Next-Generation Infrastructure

HLSF and RWT-based planning optimizes VTOL-ready cities with elevated skyways, interlinked towers, and transit hubs.
Multi-level circulation networks enhance urban walkability and high-efficiency transport flows.

4. Symbolic Cognition, Art, & Product Design

HLSFs reveal hidden symmetry structures in corporate logos, national flags, and religious iconography.
Fourier Frequency analysis bridges geometric cognition with human perception, unlocking AI-driven symbolic design.

5. The Future of Urbanism: HLSFs & RWT at Scale

Global Capital Megacity & Great Rift Valley Urban Core – SBDT applied at the urban scale, defining next-generation city structures.
Orbital Megastructures & Extraterrestrial Settlements – HLSFs provide a scalable framework for AI-driven off-world urban planning.

From Sphere to City, From Symbol to Structure

At Primary Design Co., we are not merely designing spaces—we are revealing the universal geometries that structure reality.

Sphere-Based Design Theory (SBDT) defines the logic.
High-Level Space Fields (HLSFs) expose the hidden order of form.
Radial-Wave Tessellation (RWT) transforms urbanism into a dynamic, adaptive, and human-scaled system.

Through computational precision, recursive logic, and geometric innovation, we are developing the blueprint for the next era of design.

The Pool Ball ADU: A Spherical Leap in Sustainable Micro-Living

Introduction: The Future of Compact, Self-Sustaining Living

The Pool Ball is a next-generation Accessory Dwelling Unit (ADU) that fuses cutting-edge geometry with sustainable technology. Designed as a pool house, guest suite, or off-grid micro-residence, the Pool Ball embodies efficiency, adaptability, and a deep commitment to environmental stewardship. This revolutionary structure utilizes a rectilinear-geodesic dome combined with a ringed torus foundation and an integrated central service spine, forming a living system that maximizes resource conservation, occupant comfort, and architectural ingenuity.

Structural and Architectural Innovations

1. Geodesic Dome: A High-Performance Enclosure

At the core of the Pool Ball’s design is a lightweight geodesic dome constructed from an aluminum strut framework reinforced with carbon-fiber nodes. The dome’s envelope consists of triple-layered polycarbonate panels, embedded with:

Current Technology;
- Micro-Prism Etching: Diffuses sunlight, reducing glare while enhancing thermal efficiency.
- Phase-Change Materials (PCMs): Regulate indoor temperatures by absorbing and releasing heat as needed.
- Solar Panel Optimization: Flat facets support high-efficiency monocrystalline photovoltaic cells, ensuring maximal energy generation.
- Rainwater Collection & Filtration: Panel contours direct rainwater into concealed tubing, feeding into an integrated filtration and hydrology system.

Future Technology:
- Micro-Turbine Nodes: Cylindrical junctions could accommodate mini vertical-axis wind turbines (VAWTs) to harness airflow for supplementary energy.
- Transparent Solar Cells: Future developments in transparent solar technology could enable energy generation from all dome surfaces without affecting aesthetics.

2. Ringed Torus Foundation: Multi-Functional Base Layer

Beneath the dome, the Pool Ball’s ringed torus foundation acts as a stabilizing, multi-functional platform incorporating:

Current Technology:
- Closed-Loop Hydrology: A toroidal reservoir captures and stores rainwater for irrigation, greywater recycling, and evaporative cooling.
- Thermal Mass Regulation: A ferrocement and aerated concrete composite buffers extreme temperature fluctuations, reducing heating and cooling loads.
- Soil and Hydroponic Integration: An embedded subsurface planter ring enables on-site food cultivation and biophilic integration.

Future Technology:
- Graphene-Based Water Filtration: Advances in graphene filtration could enhance the efficiency and longevity of water purification systems.
- Self-Healing Concrete: Innovations in bio-concrete could provide enhanced structural longevity by autonomously repairing minor cracks and damage.

3. Central Service Spine: The Functional Core

The vertical service spine acts as the technological nucleus of the Pool Ball, seamlessly integrating structure, circulation, and essential systems:

Current Technology:
- Vertical Circulation: A retractable spiral staircase allows access to the loft level. The central shaft can be configured as a pneumatic elevator for ADA-compliant access.
- Mechanical & Energy Systems: The shaft houses critical systems, including an air-to-water heat pump, ERV ventilation, greywater filtration, and power distribution.
- Hydroponic Air Purification: Vertical hydroponic towers provide fresh herbs and microgreens while simultaneously purifying indoor air.

Future Technology:
- AI-Optimized Environmental Control: Predictive AI could autonomously regulate indoor climate and resource usage based on real-time conditions.
- Quantum Dot Lighting: This emerging technology could offer highly efficient, tunable indoor lighting with minimal energy consumption.

Sustainability & Adaptive Living Systems

1. Off-Grid Energy Autonomy

The Pool Ball is designed to be an energy-positive dwelling, operating independently through a network of renewable energy systems:

Current Technology:
- Solar Photovoltaics: High-efficiency monocrystalline solar panels provide the primary power source.
- Smart Microgrid Management: AI-driven systems optimize energy flow between solar, battery storage, and hydrogen reserves.

Future Technology:
- Micro-Wind Generation: Cylindrical nodes could support miniature vertical-axis wind turbines, harvesting airflow from any direction.
- Hydrogen Fuel Cell Backup: Excess energy could power an electrolysis system, storing hydrogen for nighttime or low-sun conditions.
- Solid-State Batteries: Future developments in solid-state battery technology could significantly enhance energy storage capacity and efficiency.

2. Climate Adaptability & Resilience

The Pool Ball is engineered to adapt to diverse climates, ensuring comfort and resilience:

Current Technology:
- Fire Resistance: The aluminum framework is treated with intumescent coatings, providing a two-hour fire rating.
- Geothermal Integration: Subterranean geothermal loops stabilize temperatures, reducing heating energy demands in cold climates.

Future Technology:
- Shape-Memory Alloys: These materials could allow the structure to dynamically adjust its form in response to temperature changes.
- Passive Atmospheric Water Harvesting: Nanostructured coatings could enhance passive moisture collection from the air for enhanced water self-sufficiency.

A Vision for the Future of Sustainable Living

The Pool Ball ADU is more than just a compact dwelling—it is an integrated living system designed to harmonize with its environment, respond to climatic conditions, and maximize sustainability through advanced energy systems, closed-loop hydrology, and biophilic design principles.

By combining current cutting-edge technology with future advancements in material science, energy systems, and adaptive design, the Pool Ball redefines what’s possible in micro-living, sustainable housing, and autonomous off-grid habitats. Whether functioning as a high-tech urban retreat, an off-grid wilderness refuge, or an eco-conscious guest suite, the Pool Ball represents a new paradigm in adaptive, self-sustaining living.

From the fundamental elegance of the sphere emerges a living solution that is smart, sustainable, and scalable—a true embodiment of design for the future.

Specifications

Height (max): 23 feet
Diameter: 30 feet
Base n-gon: 16-sided polygon
Total Polygonal Panels: 38

Integrated Solar Panels: 16 (high-efficiency, flexible PV)
Mini-VAWT (Vertical Axis Wind Turbines): 16 (micro-grid integration, low-noise operation)

Loft Smart Windows: 16 (dynamic tinting, heat gain reduction)
Body Windows: 6 (strategic placement for passive solar gain & cross-ventilation)

Cylindrical Structural Nodes: 64 (lightweight, reinforced for seismic stability)
Hollow Aluminum Struts: 142 (precision-engineered for strength-to-weight optimization)

Passive Cooling Vents: 8 (positioned for natural convection airflow)

Rainwater Collection System: Integrated into exterior curvature

Integrated Thermal Mass Flooring: Phase-change material for heat regulation

Adaptive Smart Home System: AI-controlled lighting, HVAC, and security

Modular Interior Configurations: Convertible work/living spaces

About Primary Design Co.

Founded in 2021 by Alex London in Los Angeles, Primary Design Co. (PDCo) pioneers Sphere-Based Design Theory (SBDT)—a groundbreaking framework that redefines geometry, architecture, and urban planning through recursive adjacency modeling and higher-dimensional spatial logic.

During the COVID-19 pandemic, Alex developed SBDT in response to traditional design limitations, revealing how spatial organization follows universal geometric principles encoded within the sphere. This exploration led to the discovery of High-Level Space Fields (HLSFs)—a geometric framework theorized as the 2D/3D projection of higher-dimensional structures.

Our Work & Vision

PDCo’s research drives innovation across architecture, urban planning, and AI-assisted computational design. Our key areas of study and development include:

Radial-Wave Tessellation (RWT): A next-generation urban grid integrating recursive zoning, pedestrian-first mobility, and VTOL-ready infrastructure.
Parametric & AI-Driven Architecture: Adaptive, sphere-based structures shaped by recursive geometries and dynamic environmental inputs.
Symbolic Cognition & Geometric Perception: Investigating HLSFs, Fourier Frequency signatures, and cognitive resonance in design, branding, and symbolic systems.
Computational Tools & Generative Systems: Developing interactive AI-driven models to simulate, visualize, and refine high-level spatial frameworks in real time.

At Primary Design Co., we merge mathematics, nature, and technology to unlock the next era of recursive design, multidimensional urbanism, and AI-optimized spatial systems.

Exploring the Geometry of Existence.

Mapping the Symmetry of Higher Dimensions

High-level Space Fields (HLSFs)