Underlying Theory and Mechanics

© 2026, Arindam Jana. All rights reserved. Please do not cite or reproduce without permission.

1. Background

In an ongoing body of research, I have been studying the evolution of built geometries of cities over time. The motive for this ongoing work is that extant literature, both in urban and spatial economics, recognises the welfare costs of non-compact urban forms, but why and to what extent geometrically inefficient layouts persist remains less understood. The analytical framework developed finds that: first, incremental growth concentrates on established corridors, but this observed persistence is an artefact of durability, with a majority of new building densifying standing built stock; and second, outward growth at the fringe is primarily direction-agnostic, geared more towards coalescing fragmented built-up, which in turn presents conflicting interpretations of welfare costs.

2. What This Simulator Does

The analytical framework described above, and the retrospective analysis on which it is built, deciphers observed patterns of growth. What it does not do is analyse and visualise the mechanics that result in those observations.

This simulator is built for that.

It defines the equations that govern how a city grows, takes a hypothetical city, and applies those rules to track how a city can grow. It then uses the instruments developed in the analytical framework to evaluate the simulated growth.

In doing so, the simulator stress-tests the emergent findings from the ongoing retrospective analysis. As described above, one of the findings is that buildings last, so a city’s growth lands where its buildings already are, and that this alone can make growth look directed. If that is true, a rule with no term for direction should still put more growth in the existing arms than the ground in them would warrant. It does: section 7 describes this further.

The cities here are hypothetical. By allowing the user to define initial conditions that are taken as given in the retrospective analysis, it strengthens the stress test. All parameters are seeded from real-world scenarios, but users have the ability to alter them as they deem fit. The simulator runs on these initial values, not any real city.

What Grows

The city is represented as built-up surface on a grid. Each cell is 100 m square, the resolution of the satellite record, so that a simulated city is read at the same resolution as a real one. Buildings may cover at most the whole of a cell. The first 100 × 100 cells (demarcated by a black line) represent a 10 km square container.

Drawing on GHS-BUILT-S, a cell is defined as built once buildings cover 5 percent of it. For scale: in 2020 the median built cell inside India’s delineated urban boundaries was 27 percent covered, and one at the ninetieth percentile 50 percent.

Four quantities on every cell describe the city completely at any time t: its built surface, its floor space, its terrain, and its zoning. The simulation is a first-order Markov process: given the state at t, what is built at t + 1 does not depend on how the city came to that state. Version 1 leaves out prices formed in expectation of the future, land held for speculation, and the sorting of households; prices determined within the model come in Version 2.

Built surface and population grow at rates set outside the model, each compounding on its own:

Here S is the city’s total built surface, which the rule then places, and N its population. In 2D the population only decides the city’s size class, and so which planning norms apply; it never enters the rule that places growth. At the default g of 2.9 percent a year, built surface doubles every 24 years, and built surface per person grows at [(1 + g)/(1 + n)] − 1 a year, 1.5 percent at the defaults.

The readout reports the rate of land consumption against the rate of population growth, the ratio behind SDG 11.3.1. Measured on the footprint, the cells counted as built, rather than on built surface, the ratio is higher when growth takes new ground and lower when it goes onto cells already built.

The defaults are medians over 13,743 five-year increments in Indian settlements between 1975 and 2020: built surface grew at 2.9 percent a year, annualised, and the population inside the same boundaries at 1.4 percent.

The default town starts with 2.7 km² of built surface, the 90th percentile of Indian settlements in 1975. At 20 m² of built surface a person, the figure for India’s delineated settlements taken together in 2020, it houses about 135,000 people. A 100-year run at the default rates gives each person 87 m² of building, more than four times as much; a run that long tests the allocation rule at its limit and is not a projection of any town.

Temporal Discretisation and Observational Artefacts

The rule places growth every year (Δt = 1), and the instruments read the city every five years, as the GHSL satellite record reads real settlements.

Each cell’s class (infill, extension, or leapfrog) is judged against the footprint at the previous reading. A cell built in year 3 beside a cell first built in year 1 therefore counts as a leapfrog against the footprint of year 0, although it extended the edge in the year it was built. The simulator records the classes year by year as well as over five years, and the gap between the two shows how much of what a five-yearly record calls leapfrogging is fast extension at the edge.

3. First and Second Nature Geography: Initial Conditions and Siting

Where a settlement sits is taken as given by its history. Following Krugman (1993), its advantages of location are of two kinds. First nature is the physical ground: slopes, coasts, and rivers. Second nature is what people built to reach the place, here trunk roads and railways. Both are fixed when a city is seeded, and the model traces how the city’s form spreads outward from them.

Seed a city draws the ground and then places the first nucleus where that ground favours a town: at a river confluence, on the coast, where transport lines cross, or at a railway station, away from steep slopes. The coast, a river, and a range of hills each appear with probability one half, independently of one another. The trunk road and the railway are on by default, because a formal urban settlement usually sits on the lines that reach it. Each feature can be switched on or off, or redrawn, by hand.

The nucleus is also moved from the container’s centre by a random offset of up to ten cells (± 1 km), so that the centre from which directions are read does not sit on the city’s core by construction. The same gap separates the centroid of a delineated boundary from the functional core of the settlement inside it.

Transport Cost Surfaces and Network Accessibility

Each cell has a travel cost to the nucleus, c_i, in effective kilometres: a kilometre off the network counts as 1.0 and a kilometre along a trunk line as 0.33, by default.

The journeys are routed under three conditions:

• Refraction at the Network. A traveller does not head at right angles for the nearest point on the road. They join it at the angle that makes the whole journey cheapest, where cos θ equals the cost of travel on the road relative to travel off it (1/3), the same rule by which light is refracted.
• Rail Access at Stations. The railway lowers the cost of a journey only if it boards at a station.
• Rivers as Barriers. A river can be crossed only at a bridge, so a cell just across the water can still be a long way off.

Every distance is a straight-line (Euclidean) distance, not a count of grid steps as in the Manhattan or Chebyshev metrics. Counting steps would make some directions cheaper than others because of the grid alone, and would give the city arms the rule never produced.

Baseline Geometries at Initialisation (t = 0)

A city sited by geography grows for twenty years before year 0, under the same rule and with no plan in force, so that by year 0 its form already reflects its ground and its transport lines.

During those years, large projects can site themselves on well-connected stretches of the trunk road and at stations, and start detached nuclei of their own. The town is sized so that it holds exactly 2.7 km² of built surface at year 0, whatever growth rate the user has chosen, and the master plan is drawn on the town as it then stands.

Three standard test shapes can replace it at year 0, on flat ground with no features, so that the effect of the shape can be seen on its own:

• Concentric Disc. The most compact form there is. It starts at a disconnection index of D = 1.0, so any change in the index afterwards comes from growth.
• Linear Strip. A strip four times as long as it is wide, which has arms by construction and no road holding them there. The durability test in section 7 runs on it.
• Bicentric System. Two centres 4 km apart, which should grow into each other; it shows how the corridor between them closes over time.

Run it: the concentric disc Run it: the linear strip Run it: the bicentric system

Adaptive Spatial Boundaries

The instruments read the city inside a fixed container of 100 × 100 cells (100 km²) and take every direction from the container’s centre, which never moves. A delineated administrative boundary is read in the same way.

Growth is not stopped at the container’s edge, and the canvas grows to hold it. When a patch of at least ten connected built cells comes within five cells of the canvas’s edge, the canvas adds 50 cells on that side, up to 300 × 300 cells (900 km²) in all; a lone project far out does not move it. Because the canvas grows only on the side that needs room, the container and its centre stay where they were.

Every instrument is computed twice: inside the container, and on the whole canvas.

4. The Spatial Allocation Mechanism

Four conditions define the allocation rule:

• Directional Isotropy. Nothing in the probability of choosing a cell depends on its bearing from the centre. Any direction the city takes therefore comes from buildings lasting, from barriers such as the coast or a river, or from where the trunk lines run.
• Endogenous Growth Typologies. The rule never decides whether a project densifies, fills a gap, extends the edge, or leapfrogs; each class is read afterwards from where the project went.
• Microeconomic Coherence. Every planning instrument enters as a limit on quantity or a charge on price, the two forms regulation takes in standard urban economics.
• Unified Counterfactual Framework. The comparisons the city is read against, such as growth scattered in random directions or packed by a central planner, are the same rule at particular parameter values.

Microeconomic Specification: Discrete Choice and Infrastructure Lumpiness

Each year’s growth arrives as projects of five sizes, S ∈ {500, 2000, 8000, 32000, 128000} m². The smallest, a single increment of 500 m², is enough to turn an empty cell into a built one.

Where a project goes is a discrete choice, modelled as random utility. The project weighs five things about each cell:

• Marginal Capacity (Headroom). The room left in the cell under its coverage limit. It counts the sites a project could take there, and so multiplies the cell’s chances rather than adding to its utility.
• Footprint Proximity. How near the cell is to the built footprint, since a site beyond it must be connected at a cost.
• Core Accessibility. The cost of the journey to the nucleus, on and off the transport network.
• Topographic Suitability. The slope, which counts against the cell and rules it out above a 30 percent grade.
• Land Conversion Premiums. The charge on converting land that is not yet urban.

The rule’s main economic mechanism is that infrastructure is indivisible. Connecting a development to the trunk networks (water, drainage, road access) costs a fixed amount, K, for each 100 m beyond the built edge, whatever the size of the project. Spread over a single 500 m² plot, the connection cost K/S makes an isolated site at the fringe a poor choice. Spread over a township of 128,000 m², the same cost is 256 times smaller per square metre, so a large project can pay for its own connection and site itself where access is good, at a highway junction or a railway station on open ground. Leapfrog development arises this way.

Once a project has chosen its site, its remaining increments are built around the first, where they share its infrastructure and benefit from agglomeration economies.

Mathematical Formulation

The probability P_i that a project of size S chooses cell i is:

Headroom h_i multiplies the cell’s chances, since a cell with twice the room offers twice the sites. The utility u_i is the cell’s advantage, a_i, less the connection cost spread over the project, (K/S)e_i.

Headroom and advantage are:

Equation (1) is the conditional logit of discrete choice (McFadden, 1974). Divide each cell’s headroom into sites of 500 m², and give every site the utility of its cell plus an independent random draw from a Gumbel distribution with scale 1/θ. The probability that the best site of all lies in cell i is then equation (1), and headroom enters it linearly because it counts the sites a cell has.

The parameter θ sets how much utility decides the choice against chance. As θ → 0, every site with room left is equally likely, and growth spreads in proportion to headroom, as it does in the random-directions comparison. As θ → ∞, every project goes to the cell with the highest utility.

The Role of Project Lumpiness (λ)

Earlier cellular models of urban growth added increments of one size everywhere, and so gave growth no economic reason to jump beyond the edge. In the simulator, project size takes five values, 500 × 4^k m² for k ∈ {0, …, 4}, each four times the last, and λ sets how the year’s building is shared among them. At λ = 0 every project is a single 500 m² increment, and nothing is built far from the edge. At the default of 0.1, about 90 percent of the built surface arrives as single increments and about 1 percent in projects of 8,000 m² or more, which is enough to start a satellite near a junction or a station now and then.

Raising λ from 0.1 to 0.3 cuts densification’s share of new growth from 64 percent to 36, raises leapfrogging’s from 7 percent to 17, and turns the disconnection index from falling over the run to rising.

Continuous-Space Mean-Field Representation

Averaged over the random draws, and with the grid treated as continuous, the rule becomes an equation for how the density of built surface, s(x, t), changes over time:

Here m_k is the share of each year’s building that arrives in projects of size S_k. The equation belongs to the family of non-local reaction–diffusion equations, which describe how a population spreads over space as it grows (Skellam, 1951). It is non-local because the utility u_k(x, t) depends on the distance e(x, t) to the built footprint, so the growth at any one point depends on the form of the whole city.

Baseline Parameter Calibration

Three-Dimensional Extension: Volumetric Morphology

In 3D the rule places floor space, f_i, rather than built surface. The city’s floor space is its population times floor space per person, and floor space per person grows at a rate of its own:

Headroom and utility then take account of the ceiling on floor space and the cost of height:

The headroom for floor space, H_i, is what cell i can still take under its floor area ratio, F_i, applied to the roughly 70 percent of the cell, ρ, that lies in plots rather than streets and open space. The weight η counts each floor already standing, k_i, against adding more. A cell covers its ground up to the coverage limit, σ_iA, before it gains any floors. At η = 0 floor space stacks where access is best and the centre rises first; as η rises, height costs more against distance and the city spreads outward instead.

5. Planning Interventions and Regulatory Instruments

The master plan limits how much each cell may hold, through its floor area ratio F_i and its coverage limit σ_i; it does not assign growth to places. The allocation rule decides where growth goes among the cells with room left. Where the limits bind in the centre, growth moves out to the periphery.

Statutory Land-Use Benchmarks

The land-use shares follow the Urban and Regional Development Plans Formulation and Implementation (URDPFI, 2015) Guidelines, rescaled to the net developable urban area.

Land for transport is of two kinds. The trunk lines change travel costs, c_i, across the region, while local streets take up part of every cell, and the plotted share ρ ≈ 0.7 allows for them.

Spatial Delimitation of the Urbanisable Envelope (L)

The master plan divides the container into an urbanisable area, L, and agricultural land. At year 0 it takes in the contiguous footprint and releases enough zoned land around it to hold the growth projected over the plan’s horizon, multiplied by a release factor M (1.5 by default, adjustable from 0.5 to 3.0). Land outside L stays agricultural, with a coverage limit of 0.05. Land outside the 10 km container is not planned at all and may be covered completely (σ_i = 1.0), as on the unregulated peri-urban fringe typical of developing economies.

Morphological Zoning Configurations

Three classic arrangements of land use are available:

• Concentric Monocentric (Burgess, 1925). Uses in rings around one centre, from the commercial core outward through public, residential, and industrial bands. It has no arms by design.
• Axial Sectors (Hoyt, 1939). Wedges of each use running out along the transport lines, with industry along the railway and open space in the outer wedges. The wedges draw growth into arms.
• Polycentric Nodes (Harris & Ullman, 1945). Three satellite nuclei of employment 2.5–3.2 km from the core, which together take 45 percent of the land released for building.

Ground Coverage and Floor Area Ratio Limits

Ground coverage, σ_i, limits how much of a plot buildings may stand on; the floor area ratio, F_i, limits how much floor space the plot may hold altogether. Their ratio, F_i/σ_i, is the average number of floors when both limits bind.

How far the plan binds is set by a compliance parameter ω ∈ [0, 1], from organic growth that ignores it (ω = 0) to full compliance (ω = 1):

At the default of 0.5, a residential cell whose zone allows coverage of 0.50 may in fact be covered to 0.75; the difference stands for the informal extensions built beyond what a plan permits.

6. Spatial Metrics and Evaluative Instruments

The simulator reads the city with instruments matching those of the retrospective analysis.

Morphological Compactness: The Disconnection Index

The disconnection index, D, divides the mean straight-line distance between all pairs of built cells by the mean distance between two points of a disc with the same area. For a disc of radius R that mean distance is 0.905R, so a city of area A has a mean distance between its built cells of:

A disc scores one, and a city scores more the more it scatters, into satellites or along an elongated form. Counting every pair of built cells directly becomes slow as the city grows, so the simulator uses the Wiener–Khinchin theorem instead: the autocorrelation of the footprint, computed with fast Fourier transforms (FFT), is its covariogram, the number of pairs of built cells at every separation. That gives the exact distribution of pairwise distances in O(N log N) time (Matheron, 1975; Averkov & Bianchi, 2007).

Three-Dimensional Volumetric Profile Factor

In 3D, where floor space stands changes how far apart people are. The mean distance between two residents is:

The profile factor P (Batik et al., 2026) is the mean distance between two residents divided by the mean distance between two cells of footprint. Below one, floor space stands nearer the middle, a pyramid with its height at the core; at one it is spread evenly; above one it stands nearer the edge, with the taller building at the periphery.

Tripartite Growth Accounting Decomposition

Over each five years, the growth of floor space splits into three parts that add up exactly when taken in logs:

The three are the footprint growing (Δ ln A), the average cell of the footprint being built over more fully (Δ ln σ), and buildings gaining floors (Δ ln k).

7. Comparative Dynamics and Empirical Hypotheses

Fourteen experiments test the model. Each changes one parameter or planning choice and grows 10 to 30 cities under both settings, with the same random seeds on each side, so that every city is compared with itself. A prediction counts as confirmed when the effect goes the predicted way in at least 80 percent of cities.

Analysis of Model Failures and Boundary Conditions

Three of the results departed from their predictions:

• Land Conversion Charges and Directional Persistence. Raising the conversion charge κ raises densification’s share of new building in all 30 cities, by 6.7 percentage points on average. The arm ratio does not follow: it rises in only 13 of the 30, and its median change is −0.05. Across cities the arm ratio’s interquartile range is about 0.4 wide, so a small effect of the charge is lost among them; detecting one would take a hundred cities or more.
• Polycentric Zoning Without Localised Agglomeration. A polycentric plan fails to produce a polycentric city in five runs of ten. Higher FAR limits and released land around the outer nuclei add room there, but they make the nuclei no easier to reach from the main centre, where the jobs are. The large projects, the only ones that can afford a distant connection, go instead to the well-connected stretches of the trunk road. With no jobs or agglomeration economies of their own, the planned nuclei stay largely empty, which agrees with what Bertaud (2004) and Anas, Arnott, and Small (1998) observe.
• The Dispersion Parameter (θ). As θ → 0, growth spreads in proportion to available space only on flat ground with no plan, where the arm ratio comes out at 0.98. With water it rises to 1.12, because available space counts water as room and the rule cannot build on it. With a master plan it rises to 1.38, because the zoning limits cut the room on some cells far below what available space credits them with.

The Durability Benchmark

To separate what durability does from what the transport lines do, the rule was run on the linear strip: a town four times as long as it is wide, on flat ground, with no trunk lines.

Over 20 runs of 25 years each, with no pull to the centre (τ = 0), the rule gives the pattern seen in the observed data:

• The arms take 2.02 times the growth their share of available space would warrant.
• New ground in the arms takes 0.88 times the arms’ share of the fringe.
• Densification in the arms takes 0.99 times the arms’ share of the headroom on built cells.

The arm ratio is well above one although neither kind of growth leans towards the arms against the room it could use. Buildings last, so the room for densification lies where the stock already stands, which is largely in the arms, and growth that densifies goes there in proportion. Direction appears in the total because of durability, while new ground at the fringe shows no lean towards the arms at all.

Run it: zero transport pull Run it: a pull of 0.25 per km

8. Theoretical Lineage and Related Literature

The simulator draws on three literatures:

• Cellular Automata and Land-Use Allocation. The model belongs to a line of rules for allocating land use on a grid, from Tobler’s (1970) simulation of growth in the Detroit region, through cellular automata (White, Engelen, & Uljee, 1997; Clarke, Hoppen, & Gaydos, 1997), to models that grow patches of land use (Meentemeyer et al., 2013). Cellular automata change a cell’s state by heuristic rules about its neighbours. Here the allocation rests instead on discrete choice (McFadden, 1974), a multinomial logit in which each cell is weighted by its headroom and each project pays the cost of its connection.
• Statistical Physics of Surface Growth Interfaces. The model is related to physical models of how clusters grow: diffusion-limited aggregation (Batty, Longley, & Fotheringham, 1989), correlated percolation (Makse, Havlin, & Stanley, 1995), and Eden growth, in which a cluster adds cells at random along its edge (Eden, 1961; Marquis & Barthelemy, 2026). Recent work finds that physical boundary roughening can occur without directional bias (Hendrick, Trique, & Manoli, 2026). These models treat a cell as empty or filled, and so leave out how tall buildings are, how densely they cover the ground, and the fact that they last. The simulator adds those: its front grows outward as in the physical models, while built cells fill up and, in 3D, rise.
• Spatial Economics and Capital Durability. The monocentric model of urban economics is radially symmetric by construction (Alonso, 1964; Mills, 1967; Muth, 1969). Vintage models add durable capital (Brueckner, 1980; Wheaton, 1982), and Henderson, Regan, and Venables (2021) add the timing of redevelopment, but these usually keep the monocentric symmetry and so abstract from two-dimensional geometry. More recent spatial models study the frictions of land use and their welfare costs (Harari, 2020; Lall et al., 2021) without modelling how the fringe grows outward. The simulator puts a growth rule with no directional term together with durable buildings, to see whether the corridors a city develops over its history can arise with no preference for direction built into the rule.

9. Development Roadmap

The simulator is planned in three versions, from the physical simulation of this one towards a model of the city in general equilibrium, with prices and the location of jobs determined inside it.

Three questions remain open:

• How forward-looking expectations of prices can enter a rule that at present depends only on the current state of the city.
• How to model rationing by regulation rather than by price, such as land titling, the obstacles to public acquisition, and developer licensing.
• Whether equation (3) has travelling-wave solutions, fronts that advance at a steady speed and form, and whether they are stable.

10. References