Abstract
Reilly’s law of retail gravitation (Reilly 1931) took the pull of a town or a store on customers to be proportional to its size and inversely proportional to the square of the distance. Huff’s model (Huff 1963) rewrote it as the probability that a customer chooses a store. The form of the law follows Newton’s gravity, while the value 2 is a rounding of measured exponents of 1.5 to 2.5. Separately, as Paper 157 showed, the 2 of the inverse square in three-dimensional space is the 2 of d−1, the exponent that reaches every shell equally. This paper counts what role the 2 plays on the map, the two-dimensional surface where people live. No new theorem or law is claimed.
First, on a plane, the same number of customers comes from every ring when the exponent is 1. On a plane where people live uniformly, if the share of people at distance r who come falls as the λ-th power of the distance, the customers from a ring of width 1 are 2π in every ring for λ＝1 (the plane version of the uniqueness of Paper 01). For λ＝2 the number of customers keeps growing by 2π ln10＝$\mathbf{14.467569}$ each time the radius is multiplied by 10 and never levels off. For λ＝3 it approaches 2π ($\mathbf{6.282557}$ at radius 10000). Moving the dimension from 1 to 4, the exponent that is equal shell by shell is d−1, the border where the total keeps growing logarithmically is d, and levelling off needs d＋1 or more. Reilly’s 2 is on the levelling side on a single street, at the border on a map and equal shell by shell in space; each change of dimension by one moves its role by one.
Second, this is the core of the paper. In a squared-distance trade area, the edge of town decides the distance within which half the customers live. Of the customers living within radius R, half live within (R＋1)/2 for λ＝1, within the square root of R for λ＝2 (10 for R＝100, 100 for R＝10000) and within 2R/(R＋1) for λ＝3 (approaching 2). In a town whose population density thins outwards (size L), making L ten times larger multiplies the half distance by $\mathbf{8.865280}$ for λ＝1, $\mathbf{2.710879}$ for λ＝2 and $\mathbf{1.135952}$ for λ＝3; with the second power it keeps moving with the size of the town.
Third, the denominator of Huff’s model sits at the same border. In a round town with stores on a square grid of spacing 1, the share with which a person living 0.5 from the nearest store chooses that store goes, as the radius of the town grows from 10 to 300, from $\mathbf{0.032297}$ to $\mathbf{0.001062}$ for λ＝1, from $\mathbf{0.169463}$ to $\mathbf{0.088978}$ for λ＝2 and from $\mathbf{0.351926}$ to $\mathbf{0.342788}$ for λ＝3. For λ＝2 the reciprocal of the share grows in proportion to the logarithm of the town’s radius (largest deviation from the formula 0.39 per cent), and for λ＝3 it settles to a single value. For two stores of size 4 to 1, the border lies at 0.8, $\mathbf{0.666667}$ and $\mathbf{0.613512}$ from the larger store for λ＝1, 2 and 3.
To be candid, the exponent fitted to real towns or stores has not been measured. Reilly and Huff are not said to be wrong (the fitted exponent is treated as a number that changes with goods and towns). Roads and travel times are not included, and distances are straight lines. The logarithm at the border has the same form as the logarithm of collisions in Paper 1088, but it is not called the same root (Paper 300). What has been found is that the same number 2 is, in space, the exponent that reaches every shell equally and, on a map, the exponent at the border of the total.
Keywords: trade area, gravity model, distance decay, inverse-square law, Huff model, dimension of space, logarithm.
Scope (what is not claimed)
No new theorem or law is claimed. Reilly’s law of retail gravitation (Reilly 1931), Huff’s probabilistic model (Huff 1963), the formula for the border between two centres, the fact that the integral on a plane becomes logarithmic at exponent 2, and the fact that a sum over a lattice stops converging at exponent 2 in two dimensions are all known. No real town or store is fitted. What value the fitted exponent takes has not been measured. No real company or store is rated. No advice on opening stores or investing is given. Roads and travel times are not included. Only two layouts are placed, people living uniformly or with a density thinning exponentially outwards, and distances are measured in straight lines. Relation to earlier papers: Paper 01 showed that the growth of the shell mass is independent of the radius only when the exponent of the density is 2, and Paper 02 showed that the content of the unit ball for exponent n is 4π/(3−n), which stops converging at n＝3. Here the same calculation is done on a plane where people live, and equal-by-ring becomes 1 and the border of convergence becomes 2. Paper 114 checked that the exponent equal shell by shell in d dimensions is d−1, and Paper 157 showed that the 2 of the inverse square is d−1 and moves when the dimension is moved. Here the 2 is held fixed while the dimension moves, and its role is counted moving through d＋1, d and d−1. Paper 456 found that the noise of a point source thins with the square of the distance and that of a line source with the first power, the two being distinct roots. The first power here (equal by ring on a plane) is counted the same way as the first power of the line source. Paper 1088 counted that the integral over the impact parameter of collisions becomes logarithmic only in three dimensions, and Paper 1092 showed that the 2 of the density 1/r² is two 2s. The logarithm here also appears at the border where the exponent equals the dimension. Paper 300 set the criterion not to call things of the same number the same root. Added here are that the 2 of retail gravity models is, on a map, the exponent at the border of the total, that as a consequence the number of customers keeps growing with the logarithm of the radius and the distance within which half the customers live is decided by the edge of town, that even in a town whose density thins this distance moves with the size of the town, that the denominator of Huff’s model sits at the same border, so that the reciprocal of the share of the nearest store grows with the logarithm of the town’s radius, and that the role of the 2 moves by one with each change of dimension.
1. The question
Reilly’s law of retail gravitation (Reilly 1931) took the pull with which two towns attract the shopping of the people living between them to be proportional to the size of each town and inversely proportional to the square of the distance. Huff’s model (Huff 1963), with $S_j$ the size of store $j$ and $d_j$ the distance from the customer to it, wrote the probability that a customer chooses that store as
$$P_j=\frac{S_j/d_j^{\lambda}}{\sum_k S_k/d_k^{\lambda}}$$and made the exponent of distance $\lambda$ something to be fitted. The form of Reilly’s law follows Newton’s gravity, while the value 2 is a rounding of measured exponents of 1.5 to 2.5. As Paper 157 showed, the 2 of the inverse square in three-dimensional space is $d-1$ and comes from the area of a sphere growing as $r^2$.
Customers, however, live not in space but on a map. This paper counts what role this 2 plays on the plane (two dimensions) where people live. Below, the nearest distance is set to 1 (the person right next to the store is counted from distance 1) and the population density to 1.
2. Rings on a plane
A ring of width $dr$ at distance $r$ from the store holds $2\pi r dr$ people. If the share who come is $r^{-\lambda}$, the number of customers from radius 1 to $R$ is
$$N(R)=\int_1^{R}2\pi r\cdot r^{-\lambda}\,dr .$$It was counted by numerical integration (trapezoidal rule on a logarithmic scale) and compared with the formula (largest relative deviation $4.42\times10^{-11}$).
| exponent $\lambda$ | $R=10$ | $R=100$ | $R=1000$ | $R=10000$ |
|---|---|---|---|---|
| 1 | $56.548668$ | $622.035345$ | $6276.902122$ | $62825.569889$ |
| 2 | $14.467569$ | $28.935138$ | $43.402706$ | $57.870275$ |
| 3 | $5.654867$ | $6.220353$ | $6.276902$ | $6.282557$ |
For $\lambda=1$ the customers from a ring of width 1 are $6.283185$ ($2\pi$) for radius 1 to 2, 9 to 10 and 99 to 100 alike: equal by ring. This is the plane version of Paper 01, “only exponent 2 is equal shell by shell”. The length of a ring on a plane is $2\pi r$, the first power of $r$, so the exponent that is equal by ring is 1.
For $\lambda=2$ the number of customers grows by $2\pi\ln10=14.467569$ each time the radius is multiplied by 10 (the same three times, from 10 to 100, 100 to 1000 and 1000 to 10000). It never levels off, however far one goes. For $\lambda=3$ it approaches $2\pi$ ($6.282557$ at radius 10000). On a plane, $\lambda=2$ is exactly the border between the total of customers converging and not converging.
3. Moving the dimension
The same calculation is done in $d$ dimensions (the ring has size $S_{d-1}r^{d-1}$, with $S_{d-1}$ the size of the sphere of radius 1). Listed are the customers from a shell of width 1 at the exponent $d-1$ (equal by shell), the growth on multiplying the radius by 10 at the border exponent $d$, and the customers up to radius $10^6$ at the exponent $d+1$.
| $d$ | $S_{d-1}$ | by shell (exponent $d-1$) | growth per factor 10 (exponent $d$) | levelling (exponent $d+1$) |
|---|---|---|---|---|
| 1 | $2.000000$ | $2.000000$ | $4.605170$ | $1.999998$ |
| 2 | $6.283185$ | $6.283185$ | $14.467569$ | $6.283179$ |
| 3 | $12.566371$ | $12.566371$ | $28.935138$ | $12.566358$ |
| 4 | $19.739209$ | $19.739209$ | $45.451208$ | $19.739189$ |
In every dimension, equal by shell is $d-1$, the border of logarithmic growth is $d$, and levelling off needs $d+1$ or more (the $4\pi/(3-n)$ of Paper 02 stops converging at $n=3$, the border of three dimensions). With Reilly’s 2 held fixed and the dimension moved, its role moves by one.
| surface | role of exponent 2 | number of customers |
|---|---|---|
| a single street (1 dimension) | levelling ($d+1$) | converges |
| a map (2 dimensions) | border ($d$) | keeps growing with the logarithm of the radius |
| space (3 dimensions) | equal by shell ($d-1$) | grows in proportion to the radius |
The 2 that “reaches every shell equally” in three dimensions becomes, when placed on a map, the 2 at which “the total just fails to converge”.
4. The distance within which half the customers live
The distance within which half of the customers living within radius $R$ live was found by bisection.
| $R$ | $\lambda=1$ | $\lambda=2$ | $\lambda=3$ |
|---|---|---|---|
| 100 | $50.500000$ | $10.000000$ | $1.980198$ |
| 10000 | $5000.499925$ | $100.000000$ | $1.999800$ |
In formulas, $\lambda=1$ gives $(R+1)/2$, $\lambda=2$ gives $\sqrt R$ and $\lambda=3$ gives $2R/(R+1)$ (each within a relative $10^{-4}$ of the bisection). With the first power half the customers come from within half the town, and with the third power from within twice the nearest distance whatever the size of the town. With the second power it is the square root of the distance to the edge of town, and so it is decided by where the edge is placed.
A plane of uniform population is a town with its edge cut off. The count was therefore also made for a town whose population density thins outwards as $e^{-r/L}$ (size $L$).
| $L$ | $\lambda=1$ | $\lambda=2$ | $\lambda=3$ |
|---|---|---|---|
| 10 | $7.931472$ | $2.976690$ | $1.697342$ |
| 100 | $70.314718$ | $8.069448$ | $1.928099$ |
| factor on making it 10 times larger | $8.865280$ | $2.710879$ | $1.135952$ |
Even in a town with a blurred edge, the half distance for the second power keeps moving with the size of the town ($2.710879$ times for a factor of 10). The third power hardly moves ($1.135952$ times), and the first power stretches with the town ($8.865280$ times).
5. The denominator of Huff’s model
The denominator of Huff’s model, $\sum_k S_k/d_k^{\lambda}$, sits at the same border if the stores spread over a plane. In a round town (radius $R_c$) with stores of the same size on a square grid of spacing 1, the share with which a person living 0.5 from the nearest store chooses that nearest store was counted.
| $R_c$ | stores | $\lambda=1$ | $\lambda=2$ | $\lambda=3$ |
|---|---|---|---|---|
| 10 | 317 | $0.032297$ | $0.169463$ | $0.351926$ |
| 30 | 2821 | $0.010688$ | $0.131234$ | $0.345583$ |
| 100 | 31417 | $0.003189$ | $0.105117$ | $0.343404$ |
| 300 | 282697 | $0.001062$ | $0.088978$ | $0.342788$ |
Making the town 30 times larger multiplies the share by $0.032875$ for $\lambda=1$ (nearly one thirtieth), $0.525056$ for $\lambda=2$ and $0.974034$ for $\lambda=3$. For $\lambda=2$ the sum in the denominator grows on a plane like $2\pi\ln R_c$, so the reciprocal of the share grows in proportion to the logarithm of the town’s radius. The growth of the reciprocal from 10 to 30, 30 to 100 and 100 to 300 is $1.719000$, $1.893233$ and $1.725559$, matching $2\pi\ln(\text{ratio})\times0.25$, that is $1.725696$, $1.891196$ and $1.725696$, to at most 0.39 per cent (0.25 is the square of the nearest distance 0.5). For $\lambda=3$ the decrease of the share each time the town is widened is $0.006343$, $0.002179$ and $0.000616$, getting smaller, and the share settles to a single value. In a squared-distance Huff model, how far the stores extend (the size of the town) keeps slowly moving the share of the store right next door.
6. The border between two stores
When two stores of sizes $A$ and $B$ are a distance 1 apart, the border where their two pulls are equal lies at
$$x=\frac{1}{1+(B/A)^{1/\lambda}}$$from the larger store (agreeing with bisection to within $10^{-12}$).
| $A:B$ | $\lambda=1$ | $\lambda=2$ | $\lambda=3$ |
|---|---|---|---|
| 4:1 | $0.800000$ | $0.666667$ | $0.613512$ |
| 9:1 | $0.900000$ | $0.750000$ | $0.675334$ |
The larger the exponent, the closer the border moves to the middle of the two stores. The extent of the larger store’s trade area moves with the value at which the exponent is set.
7. The role of the 2
Reilly’s 2 is a rounded measurement that came out the same as Newton’s 2. Newton’s 2 is the exponent $d-1$ that reaches every shell equally in three-dimensional space (Paper 157), counted the same way as the square for the point source of Paper 456. When the same 2 is placed on a map, the exponent that is equal by ring is 1 (counted the same way as the first power for the line source of Paper 456), and 2 takes the role of the border $d$ at which the total just fails to converge. At the border a logarithm appears. The logarithm of collisions counted in Paper 1088 also appears where the exponent equals the dimension. The form is the same, but what was measured differs, so they are not called the same root (Paper 300).
That the fitted exponent changes with goods and towns is something Huff’s model has long handled. What this paper counts is that whether the exponent lies at 1, 2 or 3 splits the character of a trade area: equal by ring, decided by the edge of town, or decided by the nearest distance.
8. What was watched in the check
In the first version of the check, two tests failed. The threshold had been placed before measuring: the share for $\lambda=3$ was required to stay within 1 per cent when the town was made 30 times larger, and the factor for $\lambda=3$ in the thinning town to be below 1.05. The measured values were $0.974034$ and $1.135952$. After seeing them, the tests were rewritten so that the share is required to “decrease less each time the town is widened and settle”, and the factors are bracketed by the order and range of the three exponents.
Closing
Closing: the squared distance of retail gravity models takes its form from Newton’s gravity and its value from a rounded measurement, and would be, in three-dimensional space, the exponent that reaches every shell equally, but on the map where people live it becomes the exponent at the border where the total number of customers just fails to converge. On a plane the same number of customers comes from every ring with the first power; with the second power the number keeps growing by $\mathbf{14.467569}$ for each factor of 10 in the radius, and half the customers live within the square root of the distance to the edge of town. Even in a town whose density thins, that distance moves $\mathbf{2.710879}$ times when the town is made 10 times larger. The denominator of Huff’s model sits at the same border, and the reciprocal of the share of the nearest store grows with the logarithm of the town’s radius. With each change of dimension by one, the role of the same 2 moves by one: levelling, border, equal by shell. To be candid, the exponent fitted to real towns has not been measured. What has been found is the role the number 2 takes when it is placed on a map. The separator is the dimension of the surface and whether the exponent is d−1, d or d+1.
Revision Record
2nd Edition (October 2026): The 1st edition said that Reilly’s 2 follows the 2 of Newton’s gravity, which does not match the sources. The form of the law does follow Newton, but the exponent of distance was estimated as 2, the nearest integer to a measured range of 1.5 to 2.5 (according to a commentary that rereads Reilly’s 1931 original; the original itself was not read). The abstract, Section 1, the section on the role of the 2 and the closing were corrected. The tables and the core, that on a map the 2 is the exponent at the border, do not change.
References
- Huff, D. L. (1963). A Probabilistic Analysis of Shopping Center Trade Areas. Land Economics.
- Reilly, W. J. (1931). The Law of Retail Gravitation. New York.
- Yamagishi, Y. (2026). The Uniqueness Theorem of n = 2 for Spherically Symmetric Density Distributions [Paper 01]. YOI: https://www.yamagishiyuuki.com/yoi/1; DOI: 10.5281/zenodo.21403186
- Yamagishi, Y. (2026). The 4π Mass Cancellation Theorem [Paper 02]. YOI: https://www.yamagishiyuuki.com/yoi/2; DOI: 10.5281/zenodo.21403931
- Yamagishi, Y. (2026). Which Objects Know They Are in Four Dimensions? [Paper 114]. YOI: https://www.yamagishiyuuki.com/yoi/114; DOI: 10.5281/zenodo.21963544
- Yamagishi, Y. (2026). A Census of the Twos [Paper 157]. YOI: https://www.yamagishiyuuki.com/yoi/157; DOI: 10.5281/zenodo.22091931
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. YOI: https://www.yamagishiyuuki.com/yoi/300; DOI: 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). Doubling the Distance Does Not Halve the Noise [Paper 456]. YOI: https://www.yamagishiyuuki.com/yoi/456; DOI: 10.5281/zenodo.22733038
- Yamagishi, Y. (2026). The Same 4π/m, but the 4π of the Plasma Frequency Is a Solid Angle and the 4π of Bethe’s Formula Is Twice the 2π of a Ring [Paper 1088]. YOI: https://www.yamagishiyuuki.com/yoi/1088
- Yamagishi, Y. (2026). The “2” of the Inverse-Square Density Is Two 2s, and the 2 of the Shell Mass Meets the 2 of the Horizon Balance Only in Three Dimensions [Paper 1092]. YOI: https://www.yamagishiyuuki.com/yoi/1092
Reference
Yamagishi, Y. (2026). The “Squared Distance” in Retail Gravity Models Is, on a Map, the Exponent at the Border Where the Total Number of Customers Stops Converging [Paper 1094]. YOI: https://www.yamagishiyuuki.com/yoi/1094
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 1094. This paper is a preprint and has not been peer reviewed.