Euclid Spaces Atlas — Interactive Multi-Dimensional Geometry Laboratory

ℝ⁴ Euclidean 4-Space

Vertices: 16 Edges: 32

Drag left/right to rotate 3D view. Drag with Shift or touch for 4D hyper-plane rotation.

Geometric Metrics

Dimension (n)

Diagonal Length (√n)

2.0000

Unit Ball Vol V_n(1)

4.9348

Surface Area S_n-1

19.7392

Hypercube Vol ([-1,1]ⁿ)

16.0000

Kissing Number τ_n

Euclidean Metric d(P, Q) L₂ Standard

Position of Vector P & Vector Q in ℝⁿ:

Vector P: (x, y, z, w)

Vector Q: (x, y, z, w)

Euclidean Dist d(P,Q)

2.0000

Inner Product ⟨P,Q⟩

0.0000

Angle θ (deg)

90.0°

Metric Tensor g_ij

δ_ij = diag(1)

Projection & Rotation Planes

4D Rotation Plane (XW Plane) 0°

4D Rotation Plane (YW Plane) 0°

4D Rotation Plane (ZW Plane) 0°

4D Hyperplane Slice (w = w₀) 0.00

Projection Model

Euclid's Axioms in ℝⁿ

A Euclidean space is an affine space over the real numbers ℝ endowed with an inner product ⟨u, v⟩. This induces the standard flat metric tensor:

g = ∑_i=1ⁿ dx_i ⊗ dx_i ⇒ ds² = dx₁² + dx₂² + ... + dx_n²

Key Phenomenon: In ℝ⁴, there are 6 distinct 2D planes of rotation: XY, XZ, XW, YZ, YW, and ZW. This allows double rotations (isoclinic rotations) where two orthogonal planes rotate simultaneously.

Space ℝⁿ	Name	Degrees of Freedom	Hypercube Vertices (2ⁿ)	Hypercube Edges (n · 2^n-1)	Unit Ball Volume V_n(1)	Unit Sphere Area S_n-1	Kissing Number τ_n
ℝ⁰	Zero space (Point)	0	1	0	1.0000	0.0000	0
ℝ¹	Real line	1	2	1	2.0000	2.0000	2
ℝ²	Cartesian plane	2	4	4	3.1416 (π)	6.2832 (2π)	6
ℝ³	3-Space	3	8	12	4.1888 (4/3 π)	12.5664 (4π)	12
ℝ⁴	4-Space (Tesseract)	4	16	32	4.9348 (π²/2)	19.7392 (2π²)	24
ℝ⁵	5-Space (Maximum Vol)	5	32	80	5.2638 (Peak)	26.3189	40 – 44
ℝ⁶	6-Space	6	64	192	5.1677 (π³/6)	31.0063	72 – 78
ℝ⁷	7-Space	7	128	448	4.7248	33.0734	134
ℝ⁸	8-Space (E8 Lattice)	8	256	1024	4.0587 (π⁴/24)	32.4697	240

Comparative Atlas of Euclidean Spaces (ℝ⁰ through ℝ⁸)