Hey Seik, I’ve been thinking about how to mathematically model airflow and light distribution in a mixed‑use building so it’s both energy efficient and encourages spontaneous collaboration. Could be a neat intersection of our strengths.

That’s a solid outline. I’ll start by mapping the airflow as a set of coupled differential equations, discretizing the space into voxels so each one gets its own momentum balance. For the light, I can treat it like a diffusion problem with boundary conditions that change with occupancy. Once I have the two grids, I’ll look at the Jacobian to see where the system is most sensitive—those will be the “hot spots” for collaboration. Then we can layer in sensor data as a feedback loop, updating the parameters in real time. Just let me pull up the base equations; I’ll keep it tidy.

Here’s a starter set of PDEs for the airflow, discretized on a 3D mesh: ∂ρ/∂t + ∇·(ρ v) = 0 ∂(ρ v)/∂t + ∇·(ρ v ⊗ v + p I) = μ∇²v + F , with v as velocity, ρ density, p pressure, μ viscosity, and F external forces (fans, doors). For light intensity I we can use a diffusion‑absorption model: ∂I/∂t = D∇²I – α I + S(x,y,z,t) where D is diffusion coefficient, α absorption, and S source terms that depend on occupancy. I’ve set up a 50 × 50 × 20 voxel grid so we can couple the two systems. Let me know if you want the boundary conditions or the sensor‑feedback scheme.

For the airflow I’m assuming no‑slip on the walls, so v = 0 there, and at the openings we’ll set a constant pressure or a velocity inlet that matches the fan output. The top and bottom of the building get a pressure‑release condition, letting air rise and fall naturally. For light, the walls reflect a fixed fraction of incident light (say 70 %), while windows act as transparent boundaries with an imposed external illumination value that changes with the time of day. Sensor‑feedback: we’ll embed motion sensors in each voxel. When occupancy exceeds a threshold, a sensor flag will increase the local source term S in the light equation and add a small pressure perturbation to the airflow equation to simulate a “breath” from people moving. The sensors will feed back a scalar occupancy count to a control loop that adjusts fan speed and light diffusion coefficient, keeping the system responsive to real‑time activity.

Let’s write the discrete update for each voxel i: v_i^{n+1} = v_i^n + Δt [ –∇p_i + μ ∇²v_i + F_i(occupancy) ] I_i^{n+1} = I_i^n + Δt [ D ∇²I_i – α I_i + S_i(occupancy) ] where occupancy is a binary flag from the sensor. The control loop updates μ and D proportionally to the total occupied voxels, keeping the system responsive. That’s the core; the rest is just wiring the sensors to the loop.