Generated with sparks and insights from 9 sources

Based on my research into computational geometry, 3D art terminology, and polyhedral naming conventions, here are the existing and emerging terms for your specific type of geometric art:

Existing Terminology

Core Polyhedral Terms

• Symmetrohedron - A high-symmetry polyhedron containing regular polygons with gaps filled by irregular polygons Virtual Polyhedra1
• Polyhedral Kaleidoscope - Art using mirrors to create repeating geometric patterns within polyhedra Instructables2
• Mirrored Polyhedra - Polyhedra with reflective interior faces creating infinite reflections Geometry Code3

Technical Process Terms

• Shrinkwrap Projection - Mapping from regular to curved geometric forms Parametric Monkey4
• Displacement Mapping - Using pixel lightness values to create 3D surface extrusion NVIDIA Developer5
• Polyhedral Pictures - Images projected onto polyhedra faces Simon Tatham6
• Cube-to-Sphere Projection - Mathematical transformation from cubic to spherical geometry JCGT7

Artistic Movement Terms

• Computational Geometry Art - Art created through algorithmic geometric processes George W. Hart8
• Parametric Art - Art defined by algorithmic parameters and rules Parametric Monkey4
• Generative 3D Art - Art created through form-finding algorithms Parametric Monkey4

Emerging/Proposed Terminology

For Your Specific Art Form

Given your description of mirror-symmetrical pictures on curvilinear polyhedra with pixel-based extrusion, I suggest these terms:

1. Specular Polyhedron - A polyhedron with mirror-symmetrical imagery on each face
2. Hypermedia Polyhedron - A polyhedron containing interactive/layered visual information
3. Luminance-Extruded Polyhedron - A polyhedron with surfaces extruded based on pixel brightness values
4. Kaleidoscopic Spheroid - A sphere-like form with kaleidoscopic patterns on curved faces
5. Morphic Polyhedron - A polyhedron that has been topologically transformed (shrinkwrapped) from its original form

Technical Compound Terms

• Curvilinear Symmetrohedron - A symmetrohedron projected onto curved surfaces
• Displacement-Mapped Kaleidoscope - A kaleidoscopic pattern with 3D surface displacement
• Exponential-Perspective Polyhedron - A polyhedron with exponentially-scaled perspective effects
• Mirrored Extrusion Art - Art combining mirror symmetry with pixel-based extrusion

Process-Based Terms

• Shrinkwrap Mapping - The process of projecting regular polyhedra onto curved surfaces
• Luminance Extrusion - Creating 3D relief based on image brightness values
• Polyhedral Texture Synthesis - Generating complex textures across polyhedra faces
• Spherical Facet Mapping - Applying imagery to spherically-projected polyhedra faces

Naming Pattern Suggestions

For Individual Artworks

Following established patterns from geometric sculpture George W. Hart8, combine descriptive and poetic elements:

Technical Pattern: [Process] + [Base Form] + [Modifier]

• "Luminance-Extruded Cubic Sphere"
• "Mirror-Mapped Truncated Icosahedron"
• "Shrinkwrapped Kaleidoscopic Dodecahedron"

Artistic Pattern: [Evocative Name] + [Technical Descriptor]

• "Digital Cosmos (Exponential-Perspective Spheroid)"
• "Mirror City (Displacement-Mapped Polyhedron)"
• "Pixel Mandala (Luminance-Extruded Symmetrohedron)"

For the Art Movement

Based on computational art terminology Bridges MathArt Archive9:

• Specular Computational Geometry
• Hypermedia Polyhedric Art
• Luminance-Displacement Sculpture
• Kaleidoscopic Morphic Geometry

Academic/Technical Classification

For formal documentation, your art form could be classified as:

Primary Category: Computational Geometric Art
Subcategory: Polyhedral Mirror-Mapping with Luminance Displacement
Technique: Curvilinear Projection with Exponential-Perspective Mirroring
Medium: 3D Hypermedia Infographic Sculpture

This terminology bridges traditional polyhedra naming conventions with contemporary computational art language, providing both technical precision and artistic expressiveness for your innovative geometric art form.

Appendix: Supplementary Video Resources

<div class="-md-ext-youtube-widget"> { "title": "How to MIRROR in Valence3D // 3D Modeling // Tutorial", "link": "https://www.youtube.com/watch?v=ZAEA57qy5Ao", "channel": { "name": ""}, "published_date": "May 9, 2024", "length": "6:27" }</div>

<div class="-md-ext-youtube-widget"> { "title": "VALENCE | Polygonal 3d modeling app | How to mirror", "link": "https://www.youtube.com/watch?v=9bBp_MvN4BU", "channel": { "name": ""}, "published_date": "May 18, 2024", "length": "7:26" }</div>