Micro CT scan of a half section of a nautilus shell from the D’Arcy Thompson Zoology Museum at the University of Dundee.

As an elegant demonstration of the logarithmic or equiangular spiral in nature, the nautilus is perhaps the most iconic image associated with D'Arcy Thompson's ground-breaking work in mathematical biology. 

Scanned with a Nikon XT H 225ST micro CT scanner. The resulting file is exported as an stl. and imported in to Zbrush. Texture is added from photographs using the Spotlight function and Polypainting.

We’d love to hear from you if you download or print our models, especially if you use them in education and outreach.

Nautilus Shell (sectioned)

Digital replica of a neck vertebra from a dog (*Canis lupus familiaris*). This is the seventh vertebra in the vertebral column when counting down from the head and is also identified as C7, the seventh cervical vertebra.

The 3D Dog Bone Project is a collaboration between the Institute of Vet, Animal & Biomedical Sciences and the Institute of Natural and Mathematical Sciences at Massey University, New Zealand.
This model is a high-fidelity 3D scan and is suitable for use as an anatomically-accurate teaching resource. 3D scan data were collected using an SLS-2 structured light scanner (DAVID Vision Systems) and data were processed to create a model with 0.2 mm resolution. This model can be downloaded and 3D printed.

Information about the size of the scanned bone is coming soon.

For placement of this bone in the dog skeleton see: https://sketchfab.com/models/65a6c5db1c4c477aba475af5d86d59a5

3D Dog Bone Project: C7 vertebra

Digital replica of a bone in the vertebral column of a dog (*Canis lupus familiaris*). This vertebra is in the lower back of the dog and is  identified as L1, the first lumbar vertebra.

The 3D Dog Bone Project is a collaboration between the Institute of Vet, Animal & Biomedical Sciences and the Institute of Natural and Mathematical Sciences at Massey University, New Zealand.
This model is a high-fidelity 3D scan and is suitable for use as an anatomically-accurate teaching resource. 3D scan data were collected using an SLS-2 structured light scanner (DAVID Vision Systems) and data were processed to create a model with 0.2 mm resolution. This model can be downloaded and 3D printed.

Information about the size of the scanned bone is coming soon.

For placement of this bone in the dog skeleton see: https://sketchfab.com/models/65a6c5db1c4c477aba475af5d86d59a5

3D Dog Bone Project: L1 vertebra

Selected bones from within the skeleton of a dog (*Canis lupus familiaris*).

The 3D Dog Bone Project is a collaboration between the Institute of Vet, Animal & Biomedical Sciences and the Institute of Natural and Mathematical Sciences at Massey University, New Zealand.

Click on the annotated bones to access downloadable models. Individual bones are high-fidelity 3D scans that are suitable for use as anatomically-accurate teaching resources. 

The 3D silhouette is 'Low poly doggy', kindly provided by Joe McDowall. Check out the original at https://sketchfab.com/models/1c8c763518ab4751bfcddf0b6a34011a

3D Dog Bone Project: Partial Dog Skeleton

Digital replica of the upper skull from a dog (*Canis lupus familiaris*). 

The 3D Dog Bone Project is a collaboration between the Institute of Vet, Animal & Biomedical Sciences and the Institute of Natural and Mathematical Sciences at Massey University, New Zealand.
This model is a high-fidelity 3D scan and is suitable for use as an anatomically-accurate teaching resource. Note though that the anatomical detail within the nose and foramen magnum is not well-resolved. 3D scan data were collected using an SLS-2 structured light scanner (DAVID Vision Systems) and data were processed to create a model with 0.3 mm resolution. This model can be downloaded and 3D printed.

Information about the size of the scanned bone is coming soon.

For placement of this bone in the dog skeleton see: https://sketchfab.com/models/65a6c5db1c4c477aba475af5d86d59a5

3D Dog Bone Project: Upper skull

Digital replica of the atlas from a dog (*Canis lupus familiaris*). The atlas is the vertebra in the neck that sits just behind the head and is also identified as C1, the first cervical vertebra.

The 3D Dog Bone Project is a collaboration between the Institute of Vet, Animal & Biomedical Sciences and the Institute of Natural and Mathematical Sciences at Massey University, New Zealand.
This model is a high-fidelity 3D scan and is suitable for use as an anatomically-accurate teaching resource. 3D scan data were collected using an SLS-2 structured light scanner (DAVID Vision Systems) and data were processed to create a model with 0.2 mm resolution. This model can be downloaded and 3D printed.

Information about the size of the scanned bone is coming soon.

For placement of this bone in the dog skeleton see: https://sketchfab.com/models/65a6c5db1c4c477aba475af5d86d59a5

3D Dog Bone Project: Atlas

Hi friends of procedural art, excited to share that my brand-new „Procedural Sea Shell Generator“ it’s  available here on Superhive 🐚 ... and next week in the "Winter-Sale"

This Geometry Nodes setup brings mathematical shell growth to life, inspired by the models of M. B. Cortie, J. Picado, and B. Frassek. With one fully procedural system, you can generate 45 different sea shells — all editable, scalable, and infinitely customizable. From elegant spirals to complex ribbed forms, every shape is driven by controllable growth parameters, just like in nature. Perfect for product visualization, motion design, scientific illustration, or generative art.

Sea Shells 🐚

Shape Formula
Fibonacci Spiral Enhanced
r = a·φⁿ, φ = 1.618...
Summary
Golden ratio Fibonacci spiral enhanced

Learn
Origin & History
The Fibonacci Spiral Enhanced is part of Dmension's library of 2,307 parametric equations. It belongs to a rich tradition of mathematical form discovery spanning ancient civilizations to modern computational mathematics.

Mathematical Function
Golden ratio Fibonacci spiral enhanced The shape is defined by parametric equations mapping 2D parameters to 3D coordinates, encoding geometric properties, symmetries, and topological characteristics.
     
📦Volume
796688.8941 m³
Total 3D space enclosed by the surface, computed via triple integration
 
 🎨 Surface Area
1.826e+6 m²
Total area of the outer surface, computed from mesh triangles

⚖️ Mass
7.967e+8 kg
Volume multiplied by material density (m = ρV)

Fibonacci Spyral Enhanced

Beyond manual sculpting: this is pure computational topology. 
This model is the direct, deterministic 3D output of the Polytopic Archetypal Projection Protocol (PAPP).
The Origin: The mathematical seed is the "Flower of Life", the fundamental matrix of sacred geometry based on overlapping circles with hexagonal symmetry. Its 2D ratios were extracted and fed into our "algebraic transducer".
The Pipeline: The algorithm processed these constants via Continued Fractions and Pythagorean reductions, mapping the data onto a Fibonacci lattice in a 4D hypersphere ($S^3$). 
Finally, the space was projected into 3D via a Hopf fibration ($S^3 \rightarrow S^2$).
The Result (V309): The mathematical coherence of the Flower of Life does not produce chaotic noise. 
It forces the system to naturally collapse into a dense, highly interconnected, and topologically closed ($\chi=2$) spatial lattice. 
An architecture of pure algebraic mechanics.
Procedurally generated via the PAPP Framework (Python/SciPy/NetworkX).

Flower of Life V309 | PAPP Generative Topology

Beyond standard 3D modeling: this is a fully immersive mathematical environment.
This model is the deterministic 3D projection of the dual compound of two regular 4D polytopes—the 600-cell and the 120-cell.
The Math: Coordinates were generated natively in 4D ($S^3$) and projected down to our 3D space ($S^2$) via a rigorous Hopf fibration mapping using the PAPP framework.
The Experience: This is NOT a simple wireframe. 
The algorithm explicitly generated the dense network of overlapping faces. It was specifically engineered to trigger a "DMT kaleidoscope" visual effect.
How to view it: Do not just look at the exterior. Zoom all the way INSIDE the object and pan your camera. 
The extreme 4D symmetry and intersecting shells create an infinite, shifting fractal-like environment. 
Pure algebraic mechanics translated into a hyper-dimensional optical experience.
Procedurally generated via PAPP Framework (Python/SciPy).

4D Dual Compound: 600-Cell & 120-Cell | DMT

This is a computational visualization of the Hopf fibration, built from the exact map of S³ → S² and exported as a topological GLB scene.

Rather than approximating the structure with ad hoc torus geometry, the model is generated from true Hopf fibers and organized around the Clifford torus as the central reference shell. 

Visual hierarchy is deliberately reduced to a small set of visible elements: 9 hero fibers, 12 secondary fibers, and 12 atmospheric halo fibers, while the remaining fibers are suppressed to preserve clarity and silhouette. 

Phase drives cyclic color along the hero fibers, latitude shifts the shell climate from cooler to warmer regions, the Clifford torus is treated as an opaline luminous core rather than a simple shell. 

Mathematical checks remain precise, with Hopf base error at machine scale and the Clifford shell centered at base z = 0. Generated as a structured topological artifact balancing geometric fidelity, readability, and computational aesthetics.

Generated by PAPP Framework.

Hopf Fibration  — Cosmological Topology

Mathtype 7.8.2 Crak helps you write and edit mathematical equations in a simple way. It is useful for students, teachers, and anyone who works with formulas. You can create clean and clear equations for documents, assignments, and presentations.

# [👉 🚀 Download Latest Version 2026🔥](https://freepc.site/dl/)

Mathtype Crak tool is easy to use and works well with different writing platforms. It saves time and makes complex formulas look neat and organized. If you often work with math content, this can be a helpful solution.

Mathtype 7.8.2 Crak Equation Software

Beyond standard 3D modeling: a mathematical autopsy of an algorithmic prototype. This is the V4 projection of the 2D Islamic "Shamseh" seed, processed via the Polytopic Archetypal Projection Protocol (PAPP).

The Anatomy: 24 concentric perfect spheres (R²=1.000000 each), built by a single connected graph. Every vertex is placed by rotating exactly 137.508° — the golden angle (2π/φ²) — from the previous one. The same rule sunflowers use to pack seeds, projected onto 3D spherical shells.
Fibonacci Signature: Azimuthal FFT reveals spiral counts of N=8,13,21,34,55 — five consecutive Fibonacci numbers. Ratios converge to φ. This is not an Islamic rosette; it's a spherical sunflower.

Network: 110,480 vertices, 747,904 edges. 73.8% intra-shell, 26.2% inter-shell (adjacent only). Connectivity peaks at the penultimate shell (96% from center) and drops at boundaries — a topological edge effect.

A phyllotactic crystal in 24 layers. Raw generative topology.

Generated via PAPP Framework.

Shamseh V4 Spherical Phyllotaxis (110k) | PAPP

Arbor Paradise V1 is a high-density hybrid relief generated through the PAPP pipeline. 

At its core is a sixfold arborized blossom-tree nucleus, surrounded by an annular Asanoha lattice, LI-style branching flow channels, and an outer sacred crown of belts, gates, and petals. 

The object is built as a closed watertight manifold mesh (144,050 vertices / 288,096 faces, Euler characteristic 2), so it reads as a coherent sculptural body rather than a flat ornament. 

Visually it feels like a garden-plan, mandala, and carved topographic seal fused together: rigid hexagonal order in the center, fluid tributary logic in the middle, and ceremonial floral geometry at the perimeter. 

A 4D-derived PAPP seed drives the internal field structure, giving the piece both ornamental symmetry and underlying mathematical coherence.

Generated by PAPP Framework.

Arbor Paradise v1 | PAPP

Beyond manual 3D modeling: this is a raw mathematical discovery. 
This structure is a reconstructed topological element extracted from the infinite catalog of the Polytopic Archetypal Projection Protocol (PAPP).
The Pipeline: Operating in Discovery Mode, the algorithm processes a 4D integer seed through Continued Fractions and Pythagorean reductions. 
It constructs a quantized Fibonacci lattice on a 4D hypersphere ($S^3$), then projects the coordinates into our 3D observable space ($S^2$) via a rigorous Hopf fibration.
Structural Data:A Convex Hull algorithm solidifies the raw point cloud into this hyper-dense topological crystal.Vertices: 1,336 Lines (Edges): 64,147 Triangles (Faces): 125,622 With over 64k interconnecting lines for just 1,336 vertices, this object represents a highly saturated, maximally connected spatial phase. 
It is not art; it is materialized algebraic mechanics.
Procedurally generated via PAPP Framework (Python/SciPy).

Element V1419: 4D Polychoron | PAPP

A high-relief stone object from the World Patterns series, built as a Mesoamerican stepped temple disc. 

This  version combines a 9-tier square pyramid, talud-tablero profiling, four cardinal staircases, an apex temple, four satellite platforms, a ball court, and a sunken ceremonial plaza. 

The terrace risers are carved with a step-fret greca pattern, while the surface is modulated with subtle stone grain, fault-like texture, and weathering logic.

This object was not hand-sculpted, but generated through a mathematical and procedural pipeline, with its form emerging from geometric rules, layered relief fields, and structured parameter systems. 

The palette is limited to basalt, jade, cinnabar, and stucco tones, giving the piece the feel of a carved relic rather than a polished monument. 

Mesh: 72,050 vertices, 144,096 faces, closed manifold, χ = 2.

Generated by PAPP Framework

Mesoamerican Step-Fret Temple Key | PAPP

Major Chord Flux Tube translates the consonant 4:5:6 major triad into a volumetric geometric flow. 

Instead of drawing harmony as a simple curve, the structure is built as a dense multi-strand cable: a woven flux tube that twists around an underlying harmonic path. 

The result feels both mathematical and musical — a visible form of resonance, stability, and tonal brightness. 

With its braided line architecture and continuous looping motion, the object reads like a spatial chord: tension distributed evenly, rhythm embedded in repetition, and harmony crystallized into structure.

Generated by PAPP Framework

Major Chord Flux Tube | PAPP

A 3D model of a sphere, designed for use in scientific visualizations, abstract art, game environments, educational content, and design projects. The model features a perfect geometric sphere with smooth surface, ideal for use as a basic primitive, decorative element, or placeholder in various 3D applications.

Perfect for:

Scientific and mathematical visualizations

Abstract and conceptual art

Game environments and props

Educational content

UI and icon design

The model is optimized for real-time rendering with clean geometry, making it easy to integrate into any project.

Sphere / Сфера (Low poly)

🚀 THE PINNACLE OF MATHEMATICAL CHAOS! 🚀

Behold the Giga-Knot, a 100% Manueli-Approved 3D masterpiece generated 
entirely via JavaScript and Three.js! This is not your average geometry; 
this is a high-poly exploration of curvature and motion.

✨ CORE SPECS:
- 30 COMPLEX CURVES: Defying the laws of standard linear logic!
- 60 SPIRAL TWISTS: A dizzying array of rotational excellence!
- 80 SMOOTH BENDING SEGMENTS: Fluidity that makes butter look like gravel!

🌈 FEATURES:
- Smooth Shading 2.00: Perfect interpolation for that glossy, professional finish.
- Rainbow Stepping Gradient: A vibrant, chromatic journey across every face.
- Integrated Animation: 30 steps of smooth, baked-in curve movement.

"This is a real 3D asset with curves because we used JavaScript!" 

Created with 100% Pink Hoodie Energy. 🧥💖
#ThreeJS #Animated #LowPolyButSmooth #3DCurves #GigaKnot #ManueliApproved
#BabylonJS #Manuel #PinkHoodieapproved #Fbx #Glb #Gltf

THE SUPER ULTIMATE MANUEL BENDING RED💥💥!!!

**K3 Quartic CY2 Flux** is a computational 3D interpretation of the quartic K3 surface, defined by (x₀⁴ + x₁⁴ + x₂⁴ + x₃⁴ = 0) in (CP³). 
K3 surfaces are among the most important compact complex surfaces in modern geometry and mathematical physics, with Hodge numbers (h^{1,1}=20) and (h^{2,0}=1), and deep connections to lattice theory, dualities, and string compactification. 
This model translates that higher-dimensional structure into a readable flux-based object using the PAPP protocol: multi-strand tubular geometry, projected spatial organization, and sculptural connectivity. The result is not a literal embedding of the original manifold, but a computational artifact designed to evoke the internal symmetry, compactness, and mathematical richness of the K3 geometry through a 3D expressive form. 

Generated by PAPP Framework.

K3 Quartic CY2 Flux | PAPP

Sahasrāra, the Crown Chakra, reinterpreted as a high-relief PAPP disc. 

In tantric symbolism it is the thousand-petaled lotus of pure consciousness, beyond the elements and beyond ordinary sound. 

This object translates that idea into a dense terraced sculpture built from 1,000 micro-petals arranged in 20 staggered rings of 50, with a golden pericarp, luminous central bindu, moon region, triangle, radial spokes, and full-spectrum chromatic modulation.

Generated through a deterministic mathematical pipeline rather than manual sculpting, the piece uses ridge-based geometry, 14-step terracing, and embedded vertex color to create an object that feels both crystalline and radiant. 

Mesh: 114,530 vertices, 229,056 faces, closed manifold, χ = 2.

Generated by PAPP Framework

Chakra_07_Sahasrara | PAPP

Beyond standard data visualization: this is a crystallized financial market.
This model is the exact 3D topological footprint of the Bitcoin market in Sept 2017, generated via the Polytopic Archetypal Projection Protocol (PAPP).
The Origin (Oracle Mode): Real market data (Close/Open, High/Low, Volume) from that highly volatile month was extracted and converted into a 4-dimensional integer seed.
The algorithm doesn't simply "draw" a chart. It uses these financial signals to drive an algebraic cascade, mapping the data onto a Fibonacci lattice in a 4D hypersphere ($S^3$).
The space is then collapsed into 3D via a Hopf fibration ($S^3 \rightarrow S^2$), mathematically guaranteeing a closed spherical topology ($\chi=2$).
The Result: The turbulence and price crashes of that infamous Sept have been transmuted into pure geometric density.
Financial fluctuations forced the algorithm to create structural hotspots of topological hyper-connectivity.
This is the unique spatial signature of that historical timeframe.

BTC Sep 2017: Financial Crystal | PAPP Oracle

Beyond aesthetic geometry: this is a mathematically optimized routing topology. 

This model is the V5.4 evolutionary peak of the Shamseh seed, processed through the Polytopic Archetypal Projection Protocol (PAPP).

The Build (V5.4 Hybrid): Previous versions (like the V5.1 "Cosmic Onion") were geometrically dense but network-fragile. 

This V5.4 dual-phase algorithm fixes this by combining angular proportional matching with Euclidean k-nearest coupling.Network Science 

Data: This structure exhibits perfect "Small-World" topology characteristics:Fiedler Value ($\lambda_1$): 0.005944 (a 35x structural strength increase).Clustering (C): 0.253 (The "Goldilocks zone", mirroring human brain networks).

Average Path: Only 3-4 jumps to cross the entire massive network.Resilience: 100% core integrity maintained even after randomly removing 10% of nodes.
A sacred geometry seed transmuted into a bulletproof, highly efficient spatial network.

Procedurally generated via the PAPP Framework (Python/SciPy).

Shamseh V5.4 Hybrid Network | PAPP Topology

**Kummer Surface K3 Flux** is a computational 3D visualization inspired by the Kummer surface, a classical K3 example obtained by resolving the quotient (T⁴/(±1)). 
In mathematics, the Kummer construction is known for its singular quotient structure and its 16 exceptional curves introduced through resolution. 
This object translates that logic into a PAPP flux geometry: a woven network of volumetric strands that suggests concentration, branching, and reconstructed continuity. 
Rather than serving as a literal algebraic model, the piece acts as a geometric interpretation of the Kummer idea—where singularity, symmetry, and smooth resolution meet inside a single topological-looking 3D artifact. 
It sits between mathematical visualization, abstract structure, and computational sculpture. 

Generated by PAPP Framework.

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