The question. As a city gains people, does it spread out or build up? The Alonso-Muth-Mills monocentric model answers with one idea: everyone commutes to the centre, so land close to it is worth more, and the price must fall exactly fast enough to cancel the extra commuting cost. This lab grows a city in 3D and shows the footprint, the skyline and the commute.
Bid-rent. Households pay land rent L(r) = Rf + tau (Rb - r) / a at distance r (km), where Rf is farmland rent, tau is the yearly commuting cost per km of distance, a is the floor area per household (km2) and Rb is the city edge, where rent falls to the farmland value. Builders add floors as rent rises: f(r) = min(fmax, 1 + kp (Rb - r)) with kp = kappa tau / (a Rf) (1 per km). This reduced form replaces the full capital-land substitution of the textbook model.
Population fixes the radius. Households living inside radius Rb: N = (2 pi / a) integral of f(r) r dr. Without a height limit this gives a cubic, (pi / a)(Rb^2 + kp Rb^3 / 3) = N; if every home is one floor it is the familiar Rb = sqrt(N a / pi). The page solves the general (height-limited) case by bisection on an exact piecewise integral. Population follows logistic growth dN/dt = g N (1 - N / Nmax).
Commuting and emissions. The mean one-way commute is the floor-weighted mean of r. Commute CO2 per household is 2 x 250 days x commute x emission factor (a round trip, illustrative factors).
Checks on the page. (1) the flat-city radius against sqrt(N a / pi); (2) the bisection radius against Newton's solution of the cubic; (3) and (4) Simpson integration of the floor profile, with and without a height limit, must return N; (5) RK4 logistic growth against its closed form. They re-run each time the page loads.
Honest simplifications. One centre, a circular city, no roads or congestion, identical households, no land for parks or industry, no mixed or foreign rents, a linear height response. Every parameter is an order-of-magnitude illustration, not a calibrated city. In the picture each block stands for a 350 m cell averaged over many buildings, and heights are exaggerated about five times. The view clips the drawn city at 22 km radius.