Application of Mathematics in Real-World Usage · Guide 32 · BTT Mathematics Hub
A neighbourhood can become denser without becoming more crowded inside each home. Two districts with the same population density can offer very different access to schools or jobs because the street and transport networks differ. A place can be physically close to a station yet slow to reach if the route is indirect. Urban planning Mathematics therefore lives in denominators, spatial scales, networks and travel-time relationships.
This is a neutral Mathematics lesson, not a recommendation about zoning, housing, transport investment or any particular city. Every district, population, land parcel and network below is fictional. Real planning decisions require local law, engineering, environmental, social and economic evidence.
Population density divides people by area
A fictional district contains 60,000 residents across 12km². Gross population density is5,000 residents/km².
If the population rises to66,000 while area is unchanged, density becomes5,500/km², a10% increase.
Gross density and net density use different land denominators
Suppose the same 60,000 residents occupy12km² of total district area, but only8km² is residential land. Gross density is5,000/km² while net residential density is7,500/km².
Neither number is inherently wrong. They answer different spatial questions. The land denominator must travel with the statistic.
Density is not the same as household crowding
Urban density concerns people or floor area per land area. Household crowding concerns people relative to dwelling space or rooms. A dense district can contain spacious apartments; a low-density district can still contain crowded households.
Similar words can represent different denominators, so planning metrics should be named precisely.
Floor-area ratio compares built floor area with site area
Suppose a site covers10,000m² and contains25,000m² of total floor area across all levels. A simple floor-area ratio is25,000/10,000=2.5.
A ratio2.5 does not mean 250% of the ground is physically covered. Floor area can be stacked vertically across several storeys.
Land-use shares must reconcile to the total area
A fictional 20km² district assigns8km² residential,5 commercial,3 parks and4 transport/utilities. Shares are40%,25%,15% and20%, summing to100%.
If categories overlap, simple addition may exceed100%. The classification rule must say whether uses are mutually exclusive or multilabel.
Weighted density across zones uses total population over total area
Zone A has50,000 people over10km², B has30,000 over20km² and C has20,000 over30km².
Total population is100,000 across60km², so combined density is1,666.67/km². The simple average of the three zone densities—5,000,1,500 and666.67—is not the correct area-weighted regional density.
Accessibility counts reachable opportunities under a rule
A cumulative-opportunity accessibility measure might count jobs reachable within30 minutes by a specified mode. If 80,000 jobs satisfy the rule, accessibility is80,000 opportunities under that definition.
Change the time threshold, travel mode, departure time or opportunity type and the result can change. Accessibility is a defined measurement, not a timeless property of a point on a map.
OECD separates proximity from transport performance
OECD/ITF accessibility work separates nearby opportunities from how effectively a transport mode reaches them. Proximity counts opportunities within a spatial radius; transport performance can be represented as accessible opportunities divided by nearby opportunities.
If100,000 jobs lie within a chosen radius and60,000 are reachable within the chosen travel-time threshold, transport-performance ratio is0.60.
Two places can have equal absolute accessibility for different reasons: one may have many nearby opportunities but weak network performance; another may have fewer nearby opportunities but a faster network.
Population-weighted accessibility describes the average resident
Suppose Zone A has1,000 residents with accessibility50, Zone B has3,000 with accessibility80. Population-weighted accessibility is (1000×50+3000×80)/4000=72.5.
The simple zone mean65 incorrectly treats the small and large populations as equally important when the question asks about the average resident.
Network distance differs from straight-line distance
Two points are2km apart as the crow flies but3km apart along available streets. Circuity ratio is3/2=1.5.
A river, expressway, gated parcel or disconnected street pattern can make Euclidean proximity a poor proxy for usable travel distance.
Travel-time catchments are speed-and-network problems
At an ideal constant walking speed5km/h,15 minutes corresponds to1.25km of path distance. It does not imply a perfect circle of radius1.25km because the pedestrian network constrains which paths exist.
Real catchments use network travel times, crossings, gradients and access points rather than unrestricted straight-line movement.
Graphs represent urban networks as nodes and edges
Intersections or stations can be modelled as nodes; street or transit links as edges. An edge can carry distance, time, cost or capacity as a weight.
Shortest path then means the path with minimum total chosen weight, not necessarily the path with the fewest edges.
Degree measures local connectivity
A node joined directly to five other nodes has degree5 in an undirected simple graph. A cul-de-sac endpoint may have degree1.
Degree is local. It does not by itself measure whether the node lies on many important routes across the wider network.
Betweenness captures route concentration
In a four-node chain A–B–C–D, B and C lie on shortest paths between more node pairs than the endpoints. They have higher betweenness under standard unweighted definitions.
A high-betweenness link or node can indicate network dependence, but operational importance also depends on demand, capacity and alternative routes.
Route redundancy is a connectivity property
If two areas are connected by only one bridge edge, removing that edge disconnects the graph. If two independent paths exist, one closure may leave the areas connected.
Network robustness therefore depends on topology, not merely total kilometres of road or track.
Catchment overlap uses inclusion–exclusion
Service A covers6,000 residents and B covers5,000, with2,000 covered by both. Unique population covered by at least one service is6,000+5,000−2,000=9,000.
Adding the two catchments without subtracting overlap would double-count shared residents.
Coverage percentage uses the target population as denominator
If9,000 of12,000 residents fall inside at least one defined service catchment, coverage is75%.
This does not mean75% use the service. Geographic access and observed usage are different quantities.
A gravity-style accessibility model discounts opportunities by distance
A simple teaching index can be A=Σ Jᵢ/dᵢ², where Jᵢ is opportunities at destination i and dᵢ is distance. For100 jobs at1km and400 jobs at2km, A=100/1²+400/2²=200 index units.
The inverse-square decay is an invented teaching choice here, not a universal planning law. Different accessibility models use different impedance functions.
Spatial aggregation can change apparent patterns
Fine-grained blocks may show pockets of high and low density. Combining them into one large zone replaces that variation with one average.
This is related to the modifiable areal unit problem: statistical relationships can change when the boundaries or scale of spatial aggregation change.
Planning optimisation needs an explicit objective
A model might minimise average travel time, maximise population within service distance, minimise infrastructure cost or balance several objectives.
The mathematically optimal solution depends on the objective and constraints. Mathematics cannot decide which public values should be optimised without an external decision process.
A complete urban calculation states geography and network assumptions
State the area boundary, land denominator, population date, travel mode, time threshold, network representation and weighting method. For accessibility, state which opportunities count. For density, distinguish gross from net.
The return path is people and activities → land and network representation → spatial metric → comparison → planning interpretation.
Practice: twenty urban-planning Mathematics questions
- 60,000 residents over12km²: find gross density.
- 60,000 residents over8km² residential land: find net residential density.
- Population rises10% with fixed area. What happens to density?
- Site10,000m² with25,000m² floor area: find floor-area ratio.
- Land areas8,5,3,4km² out of20: find shares.
- Zones50k/10km²,30k/20km²,20k/30km²: find total regional density.
- 100,000 nearby jobs,60,000 accessible under a threshold: find transport-performance ratio.
- 1,000 residents have accessibility50 and3,000 have80. Find population-weighted value.
- Straight-line distance2km and network distance3km. Find circuity.
- At5km/h, how far is15minutes of ideal path travel?
- A node has five direct neighbours. Find degree.
- Catchments cover6,000 and5,000 with2,000 overlap. Find unique coverage.
- If total target population is12,000, find coverage percentage.
- Using A=ΣJ/d², find index for100 jobs at1km and400 at2km.
- A 4-node chain is A–B–C–D. Which internal nodes have higher betweenness than endpoints?
- If one bridge is the only edge joining two subnetworks, what happens when it is removed?
- Why is straight-line distance not always a good estimate of network distance?
- Why can a simple average of zone densities misrepresent regional density?
- Why can changing zone boundaries change an observed spatial correlation?
- Why must an optimisation model state its objective before “best plan” has mathematical meaning?
Worked answers
- 5,000 residents/km².
- 7,500 residents/km².
- It rises10%.
- 2.5.
- 40%,25%,15%,20%.
- 100,000/60≈1,666.67 residents/km².
- 0.60.
- 72.5.
- 1.5.
- 1.25km.
- 5.
- 9,000.
- 75%.
- 200 index units.
- B and C.
- The graph becomes disconnected between those subnetworks.
- Because barriers and permitted links constrain usable paths.
- Because zones with different areas should not receive equal weight when computing people per total area.
- Because aggregation changes which observations are combined and can smooth or rearrange spatial variation.
- Because optimality is always relative to a defined objective and constraints.
Sources and connected applications
For accessibility measures that separate proximity, transport performance and reachable opportunities, see OECD: Delivering Accessible and Sustainable Mobility and OECD/ITF: Improving Transport Planning for Accessible Cities. For integrated land-use and transit planning context, see the World Bank report Planning for Transit-Oriented Development in Emerging Cities.
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