Compare triangle angles and circumferences in positive, negative, and flat geometries—explore the spatial curvature of the universe
In cosmology and general relativity, space itself can have intrinsic curvature. This toy lets you explore the three fundamental types of constant-curvature geometries that appear in the Friedmann-Lemaître-Robertson-Walker (FLRW) models of cosmology.
A spherical geometry like the surface of a sphere. Key properties:
In cosmology, this corresponds to a closed universe with Ω_total > 1. If our universe has positive curvature, you could (in principle) travel in a straight line and return to your starting point.
Euclidean geometry—the familiar geometry of everyday experience. Properties:
Observations suggest our universe is very close to flat, with Ω_total ≈ 1.00 ± 0.01. This is the "flatness problem"—why is the universe so precisely flat? Cosmic inflation provides one explanation.
Hyperbolic geometry, like a saddle surface. Characteristics:
This corresponds to an open universe with Ω_total < 1. In such a geometry, there's "more room" than in flat space—exponentially more as you move away from any point.
The FLRW metric describing our universe's geometry includes a curvature parameter k that can be +1 (closed), 0 (flat), or -1 (open). The curvature is determined by the total density parameter Ω_total:
Current observations from the cosmic microwave background (CMB) place tight constraints on spatial curvature. The Planck satellite finds Ω_total = 1.000 ± 0.002, implying our universe is spatially flat to within measurement precision. However, this flatness is a puzzle: why should Ω_total be so close to exactly 1? Inflation theory predicts that any initial curvature would be exponentially diluted during the inflationary epoch, leaving the universe extremely flat today.
Triangle angles: In positive curvature, make the triangle larger—watch the interior angles increase. Each angle can exceed 90°! In the extreme, a triangle can have three right angles (270° total).
Geodesic deviation: Switch to geodesics mode with positive curvature. Start with small separation and watch how initially parallel paths converge. This is the geometric meaning of tidal forces in general relativity.
Hyperbolic space: In negative curvature, notice how there's exponentially more "room" as you move outward. Geodesics diverge rapidly, and triangles have increasingly acute angles.
Compare to flat: Toggle between geometries with the same triangle size to see the differences. The flat case serves as the baseline—it's what our intuition expects.
The intrinsic curvature is encoded in the Riemann curvature tensor. For spaces of constant curvature, the Ricci scalar R is constant everywhere. The Gaussian curvature K relates to the curvature parameter:
The Gauss-Bonnet theorem connects geometry and topology: for a triangle on a surface of constant curvature K with area A, the sum of interior angles α + β + γ = π + KA. This is why larger triangles deviate more from 180°.
In general relativity, spacetime curvature is determined by the Einstein field equations: G_μν + Λg_μν = 8πG T_μν. For a homogeneous, isotropic universe (the FLRW model), these equations reduce to the Friedmann equations, which relate the spatial curvature k to the matter and energy content. Geometry and physics are inseparable.