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Intrinsic Crystallographic Organization and Projected Crystal Habit in Coherence Geometry

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Internal ID: CGI-RSR-000044
Author(s): Barry L. Petersen
Document Type: Research Paper
Publication Date: September 30, 2026
Original Creation Date: September 29, 2026
Revised Document Date: September 29, 2026
Status: Public
Domains: Physics, Chemistry, Crystallography, General
Sub-Domain: Crystallographic Projection / Structural Realization
Research Topics: Coherence Geometry, Crystallography, Crystal Habit, Phase-Gradient Geometry, Gram Geometry, Structural Projection, Crystal Growth, Environmental Modulation, Rotational Covariance, Polarity, Observable Morphology

Abstract

This paper develops a coherence-geometric representation of crystalline form in which intrinsic crystallographic organization, interfacial habit, growth realization, and observable morphology are treated as distinct but coupled geometric layers.

Crystallographic organization is encoded through the Gram geometry of phase-gradient fields. This intrinsic organization determines relational crystal directions independently of laboratory orientation, while interfacial habit, environmental weighting, growth realization, and observable morphology enter through subsequent projection and realization layers.

The resulting architecture separates five stages:

intrinsic crystallographic organization → habit selection → environment-dependent growth realization → projected crystalline amplitude → observable morphology

The construction is tested numerically through two-dimensional fourfold and sixfold crystal organizations, three-dimensional cubic-family habits, a non-centrosymmetric tetrahedral realization, and environmental modulation of a fixed cubic crystal.

The calculations examine rotational covariance, characteristic support ratios, projection refinement, shape-preserving growth, polarity, and the distinction between intrinsic crystal directions and environment-dependent rates of expression.

The work is structural rather than a complete material-specific theory of crystal growth. Dynamic transport, interface-environment feedback, morphological instability, and dendritic realization remain subsequent extensions.

Available Document

DOI: 10.5281/zenodo.23056140

Citation:
Petersen, B. L. (2026). Intrinsic Crystallographic Organization and Projected Crystal Habit in Coherence Geometry(Version 1.0). Coherence Geometry Institute. Zenodo. https://doi.org/10.5281/zenodo.23056140

Source Code and Supporting Materials

The Zenodo record includes a reproducibility archive containing seven Python scripts used for the crystallographic projection, growth, environmental-modulation, and figure-generation calculations.

Primary file:
Intrinsic_Crystallographic_Organization_and_Projected_Crystal_Habit.pdf

Version 1.0 archival research paper.

Supporting computational archive:
Crystallographic_Projection_Scripts.zip

The archive contains the seven Python scripts used for the reported two-dimensional and three-dimensional crystallographic realizations, environmental-modulation studies, numerical validation, and figure generation.

The scripts preserve the computational realizations used for:

  • two-dimensional fourfold crystal organization;
  • two-dimensional sixfold crystal organization;
  • three-dimensional cubic-family realizations;
  • rhombic-dodecahedral habit;
  • octahedral habit;
  • non-centrosymmetric tetrahedral realization; and
  • environmental asymmetry of a fixed cubic crystal.

The scripts are provided in the form used for the published research and may require a compatible Python scientific-computing environment to run. They are archival research artifacts rather than a maintained general-purpose crystallographic software package.

A README file accompanying the Zenodo deposit records document provenance, scope, citation information, file integrity, and archival metadata.

Summary and Notes

Scope:

The paper develops a structural distinction between intrinsic crystallographic organization and the observable shape ultimately expressed by a crystal.

Rather than identifying crystal class directly with visible morphology, the construction separates several levels of organization.

At the intrinsic level, crystallographic relationships are encoded through phase-gradient fields. If

$$
q_a=\nabla\theta_a,
$$

their pairwise geometric relationships may be represented through the Gram structure

$$
G_{ab}=q_a\cdot q_b.
$$

This relational geometry describes crystallographic organization without requiring the crystal to be tied to one laboratory orientation.

The full realization is organized conceptually as:

intrinsic crystallographic organization → habit selection → environment-dependent realization → projected crystalline amplitude → observable interface

This distinction allows several effects that would otherwise be conflated to remain separate.

Crystal class describes intrinsic directional organization.

Habit selection determines which geometrically compatible interfaces are preferentially expressed.

Environmental weighting modifies rates or strengths of realization without necessarily changing the underlying crystal class.

Polarity permits directional asymmetry in non-centrosymmetric realizations.

Observable morphology is the final projected interface and therefore need not be identified directly with the complete intrinsic crystallographic structure.

The numerical program tests this architecture through several cases:

  • fourfold two-dimensional organization;
  • sixfold two-dimensional organization;
  • cubic three-dimensional organization;
  • rhombic-dodecahedral and octahedral habits;
  • non-centrosymmetric tetrahedral organization;
  • rotational covariance;
  • characteristic geometric support ratios;
  • projection refinement;
  • shape-preserving growth; and
  • environmental modulation of a fixed cubic crystal.

A particularly important distinction concerns environmental influence. The environment may alter the rate or degree with which intrinsic directions are expressed without redefining the underlying crystallographic organization itself.

Thus, a single intrinsic crystal organization can support different observable realizations depending on projection, habit selection, environmental weighting, and growth conditions.

Within the broader Coherence Geometry framework, the paper provides a concrete structural realization of the Source-to-projection architecture. Intrinsic organization belongs upstream of the final observable morphology, while the observable crystal appears through a sequence of projection and realization operations.

The model is intentionally structural. It is not presented as a complete material-specific crystal-growth theory. Dynamic transport, interface-environment feedback, morphological instability, dendritic growth, and detailed constitutive material physics remain outside the scope of the present construction.

Related Work

Foundational framework:

Petersen, B. L. (2026). Coherence Geometry Foundations, Part I: Orientation, Closure, and Algebraic Foundations(Version 0.1). Zenodo.
https://doi.org/10.5281/zenodo.20156531

Petersen, B. L. (2026). Coherence Geometry Foundations, Part II: Physical Projections. Zenodo.
https://doi.org/10.5281/zenodo.20156996

Source and projection architecture:

Petersen, B. L., Johnson, R. K., & Bruno, A. E. (2026). A Conceptual Overview of the Coherence-Geometric Framework: Organizational Structure, Terminology, and Projection (Version 1.0). Coherence Geometry Institute. Zenodo.
https://doi.org/10.5281/zenodo.22720726

Petersen, B. L. (2026). Projection-Covariant Dynamics in Coherence Geometry: Presymplectic Reduction, Mapping-Space Evolution, and Observable Descent (Version 1.0). Coherence Geometry Institute. Zenodo.
https://doi.org/10.5281/zenodo.23040342

Structural realization and crystal projection:

Petersen, B. L. (2026). Coherence Geometry Foundations, Volume II: Source Curvature, Book II: Projection Archetypes and Structural Realizations (Version 0.1). Coherence Geometry Institute. Zenodo.
https://doi.org/10.5281/zenodo.23057738

Petersen, B. L. (2026). Pattern Formation via Curvature-Driven Amplitude Relaxation in Coherence Geometry. Zenodo.
https://doi.org/10.5281/zenodo.20121185

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