In Loop Quantum Gravity, space itself is not continuous — geometry is built from a spin network: a graph whose nodes represent quanta of volume and whose links represent quanta of area between adjacent chunks of space. Each link carries a spin label j ∈ {½, 1, 3/2, 2, …}, an SU(2) representation, exactly like the spin of an electron.
The area operator's eigenvalues (Rovelli–Smolin / Ashtekar–Lewandowski) give the area punctured by a link of spin j:
A_j = 8π γ l_P² √(j(j+1))
γ ≈ 0.2375 Barbero–Immirzi parameter
l_P = √(ħG/c³) Planck length (set to 1 here)
The seed network here is a quantum tetrahedron: one central node (a chunk of volume) with 4 links to boundary nodes — the tetrahedron's 4 faces, each an independent area quantum. Pick a link and drag its spin to see its area jump between discrete allowed values — area in LQG is not continuous, it comes in a quantized spectrum.
Node volume: a real, exactly provable LQG fact is that a 3-valent node carries zero volume — three link vectors that close (sum to zero) are automatically coplanar, so the volume operator's triple product vanishes identically. For a 4-or-more-valent node the exact eigenvalues require diagonalizing the volume operator on the node's recoupling ("intertwiner") space, which is not a simple closed-form function of the spins. The "Node volume" readout here is therefore an illustrative scaling estimate — zero on trivalent nodes, and growing with the average spin on higher-valence nodes as V ≈ l_P³ √((2/9) j̄(j̄+1)(j̄+2)) — not the literal operator spectrum, but it correctly shows volume switching on only once a node has enough links to enclose a region.
Evolve performs a 1→4 Pachner move on a random node: it splits one chunk of volume into four smaller ones, wiring the new sub-nodes into a small tetrahedral cluster and handing each of the split node's old links to one of the four children. This is the combinatorial move that generates the sum-over-triangulations in spin foam models — the path-integral formulation of LQG's dynamics — refining the discrete geometry the way subdividing a mesh refines a surface.
- Link selector / spin slider — pick any link, set its spin, watch the quantized area and the network total update live.
- Node selector — pick any node to read its valence and estimated volume.
- Evolve — one Pachner refinement step (capped so the graph stays readable).
- Reset — back to the single quantum tetrahedron.