Shape Analysis: From Finite Modes to Infinite Dimensions

Infinite Dimensions, Infinite Expressions: Geometry, Measurement and Language in High-Dimensional Worlds

Abstract

This book explores the hypothesis that intelligent organisms inhabiting spaces with many spatial dimensions would develop mathematical concepts, measurement systems, and languages fundamentally different from ours. We synthesise recent advances in geometric morphometrics showing that shape is inherently infinite dimensionalbiorxiv.org, typological surveys revealing the cultural variability of numeral baseswals.info, cognitive‐linguistic studies demonstrating that space is semantically and phonologically loaded in sign languagespmc.ncbi.nlm.nih.gov, research on classifier languages where morphemes encode shape, animacy and measurementpmc.ncbi.nlm.nih.gov, historical analyses of metrological units tied to human anatomy and socio economic relationsscirp.org, mathematical developments in surreal numbers that unify real, infinite and infinitesimal magnitudesen.wikipedia.org, and statistical principles such as Zipf’s law of abbreviationpmc.ncbi.nlm.nih.gov. Through a combination of theoretical discussion and computational simulations, the book examines how high dimensional geometry could drive the emergence of numeral systems, measure words, classifier paradigms, and spatial grammars, and how social and economic factors would shape these systems. The volume is written for researchers in linguistics, mathematics, cognitive science and anthropology, but is accessible to curious readers interested in the interplay between space, form and language.

Scientific Article: http://dx.doi.org/10.13140/RG.2.2.31403.96802

Naming · Measuring · Shaping Toward a Unified Science of Quantified Form (Etymology × Metrology × Morphometrics)

Abstract

This book establishes a new integrative field—the science of quantified form—that studies how human communities turn lived magnitudes and shapes into stable, transmissible systems of words, standards, and forms. We braid three mature disciplines: etymology (how measure-words and number names arise and change), metrology (how units and standards are defined, transmitted, and governed), and morphometrics (how shapes and artifacts are compared with statistical rigor). The core thesis is that these systems co-evolve under cognitive, technological, and social constraints: psychophysical scaling shapes the lexicon of magnitude; production technologies and base systems fix modular geometries; and named categories feed back to stabilize both tolerances and forms. Methodologically, the field unites (i) geometric morphometrics (landmarks, Procrustes analysis, thin-plate splines), (ii) corpus linguistics and historical semantics, (iii) dimensional analysis and standardization theory, and (iv) information-theoretic and phylogenetic models of cultural diffusion. We formulate testable hypotheses—e.g., that cultures with body-based measurement etymons show unit modules aligned to anthropometric distributions; that linguistic regularization in measure-terms correlates with tighter manufacturing tolerances; and that term frequency and word length obey efficiency laws that also govern prevalent artifact geometries. While maintaining strict academic standards (formal definitions, proofs where appropriate, reproducible workflows), the exposition is designed for broad accessibility: every chapter includes a plain-language overview, minimal math boxes, annotated examples, and open datasets. Applications span archaeology (reconstructing lost standards), heritage science (authenticating artifacts via joint linguistic–geometric signatures), design and UX (aligning unit vocabularies with perceptual salience), and public policy (transparent standardization). The book closes with a research agenda, benchmarks, and protocols for building cumulative evidence in this new domain.

Scientific Article: http://dx.doi.org/10.13140/RG.2.2.32642.59843

From Finite Modes to Infinite Dimensions: A Practical Pipeline for Morphometric Statistics," "Fourier Limits: Fast, Reproducible Infinite-D Shape Analysis," "Infinite-D Shape Statistics Made Practical

We present a simple, reproducible pipeline for statistical analysis of shapes modeled as elements of an infinite-dimensional Hilbert space. Closed planar shapes are represented as periodic curves with Fourier truncation at modes; increasing emulates the passage to infinite dimension while keeping computation tractable. On this ambient space we define an metric on sampled curves with circular-shift alignment (a pragmatic surrogate for reparameterization), enabling iterative Fréchet-like means and a linearized principal geodesic analysis (PGA) via PCA around the mean. We couple these with dimension-agnostic observables—perimeter, enclosed area, and bending energy —and provide convergence diagnostics versus , demonstrating stabilization of observables and principal modes. The method is illustrated on synthetic families of closed curves (ellipses + structured Fourier roughness) and a small real-outline vignette; code and data are released for full reproducibility. We also show how a “unified number system” container can carry auxiliary descriptors (exploded/compressed/supernatural labels) while confining geometry and probability to the real Hilbert component. This lightweight pipeline offers a practical bridge between infinite-dimensional shape theory and everyday morphometric workflows, and serves as a baseline for future extensions to elastic/Sobolev metrics and surface geometry.

Scientific Article: http://dx.doi.org/10.13140/RG.2.2.20426.68800

Beyond the Horizon: Infinitely Many-Dimensional Space and the Foundations of Reality

Abstract

This book explores the profound implications of conceptualizing infinitely many-dimensional spaces with real spatial extension, presenting a bold reinterpretation of foundational physics. By challenging traditional notions of spacetime, singularities, and dimensionality, this work connects higher-dimensional thought experiments to key unsolved problems in physics, such as the information paradox, quantum gravity, and pre-Big Bang cosmology. Using a unique combination of visual analogies, cutting-edge simulations, and mathematical models, the book proposes new pathways for reconciling quantum mechanics with general relativity, solving singularities, and expanding human understanding of the universe. Designed for both experts and curious readers, it offers scientific depth alongside accessible explanations and tools for further exploration.

Scientific Article: http://dx.doi.org/10.13140/RG.2.2.18083.16162

Integrating Surreal, Exploded, Compressed, and Supernatural Numbers

Abstract

This book explores the development of a novel number system that synthesizes the properties of surreal numbers, exploded numbers, compressed numbers, and supernatural numbers. The integration of these diverse number systems aims to provide a robust framework for solving complex algebraic and geometric problems. The text delves into theoretical foundations, arithmetic operations, algebraic structures, and applications, complemented by extensive formulas and programming examples. By leveraging the unique characteristics of each constituent number system, this comprehensive approach facilitates advanced mathematical modeling and problem-solving.

Scientific Article: http://dx.doi.org/10.13140/RG.2.2.23855.34727