3D: Cylinder ⊃ Sphere
4D: Hypercube ⊃ Hypersphere
🎯 Geometric Parameters
📊 Volume Analysis
3D Analysis
Cylinder: V = π r² (2r) = 14.14
Sphere: V = (4/3) π r³ = 14.14
Inverse Volume: 0.00
Ratio: 1.50 (Archimedes' 3:2!)
Ratio: 1.50 (Archimedes' 3:2!)
4D Analysis
Hypercube: V = (2r)⁴ = 81.00
Hypersphere: V = (π²/2) r⁴ = 49.35
4D Inverse Volume: 31.65
4D Ratio: 1.64
4D Ratio: 1.64
🔄 Visualization Mode
🧠 Key Insight
Archimedes' Discovery: The ratio of cylinder to sphere volume is exactly 3:2 in all cases!
4D Extension: What happens to this ratio in 4D? The hypercube to hypersphere ratio approaches a completely different limit...
The Pattern: As dimensions increase, hyperspheres become increasingly "lonely" in their hypercubes!