Economic Geography & Urban Systems
Christaller K=3, 4, 7 & Beckmann City-Size Formula
Calculate Walter Christaller’s (1933) hexagonal Central Place market-area hierarchy K^(m - 1) and Martin Beckmann’s (1958) recursive urban system population T_m = (R_1 / (1 - k_srv)) * (K / (1 - k_srv))^(m - 1).
Model hexagonal retail, transport, and administrative urban hierarchies!
Walter Christaller (1933) showed that hexagonal market areas tile an isotropic plain without unserved gaps and nest hierarchically under three geometric principles: Marketing Principle K = 3 (each higher-order center sits at vertices shared by 3 lower-order hexagons: 1 + 6*(1/3) = 3), Transport Principle K = 4 (midpoints of 6 sides: 1 + 6*(1/2) = 4), and Administrative Principle K = 7 (full political containment of 6 satellite hexagons: 1 + 6 = 7)! Martin Beckmann (1958) formalized the population size of every tier m >= 1 when a Tier-1 market town serves Rural_Base R_1 and each tier requires Service_Population_Ratio k_srv in (0, 1) per person in its total market system: Tier-1 Town Population is P_1 = (k_srv / (1 - k_srv)) * R_1 (Total System T_1 = R_1 / (1 - k_srv)), and Tier-m Total System Population compounds at geometric factor K / (1 - k_srv): T_m = (R_1 / (1 - k_srv)) * ((K / (1 - k_srv)) ^ (m - 1))!
Verify on Christaller_Factor K = 3 (Marketing Principle, B2), Hierarchy_Order m = 3 (Regional Metropolis, B3), Rural_Hinterland_Base R_1 = 4,000 residents (B4), and Urban_Service_Ratio k_srv = 0.20 (B5): Tier-1 Market Areas Subsumed = 3^(3 - 1) = 9 hexagons; Tier-1 Market-Town Urban Population P_1 = (0.20 / 0.80) * 4,000 = 1,000 (T_1 = 5,000); and Tier-3 Regional System Population T_3 = 5,000 * (3 / 0.80)^2 = 5,000 * 14.0625 = 70,312.50 residents (with Higher-Order Metropolitan Sector k_srv * T_3 = 14,062.50)! Switch B2 to Transport Principle K = 4: Subsumed Hexagons jump to 16 and T_3 expands to 5,000 * (4 / 0.80)^2 = 125,000.00!
| Central Place Metric (K=3 Marketing, Tier m=3, R_1=4,000, k_srv=0.20) | Formula | Verified Output |
|---|---|---|
| Lowest-Order Market Areas Subsumed | K ^ (m - 1) | 9 hexagons (16 at K=4; 49 at K=7) |
| Tier-1 Market Town Population (P_1) | (k_srv / (1 - k_srv)) * R_1 | 1,000.00 (System T_1 = 5,000) |
| Tier-m Total System Population (T_m) | (R_1 / (1 - k_srv)) * (K / (1 - k_srv))^(m - 1) | 70,312.50 residents |
| Christaller Nesting Geometry | K = 3, 4, or 7 | Marketing Principle (K = 3) |
Beckmann (1958) Hierarchical System Population T_m Formula
Validate canonical Christaller nesting K >= 3, integer tier m >= 1, R_1 > 0, and 0 < k_srv < 1!
=LET(k_ch, B2, m_tr, B3, r_bs, B4, ks_r, B5, IF(OR(NOT(OR(k_ch = 3, k_ch = 4, k_ch = 7)), m_tr < 1, INT(m_tr) <> m_tr, r_bs <= 0, ks_r <= 0, ks_r >= 1), "Invalid Central Place Input", ROUND((r_bs / (1 - ks_r)) * ((k_ch / (1 - ks_r)) ^ (m_tr - 1)), 2)))Christaller Principle Classification Formula
Distinguish K = 3 (retail cost minimization), K = 4 (arterial highway alignment), and K = 7 (undivided county/provincial jurisdiction)!
=IF(ISTEXT(E4), E4, IF(B2 = 3, "Marketing Principle (K = 3)", IF(B2 = 4, "Transport Principle (K = 4)", "Administrative Principle (K = 7)")))Try the example
Use the original values or change them to test your understanding.