The Scarborough projectThe Z-order curve: one number for a place on the map
Four ways to put postcodes in a list, scored on whether neighbours stay together.
Ordnance Survey's free Code-Point Open file gives a grid reference for every postcode. The learner keeps the two districts YO11 and YO12, which cover Scarborough together with places such as Eastfield, Cayton and Seamer: 1,685 postcodes at 1,606 distinct points, in a rectangle 11,163 metres wide and 16,220 metres tall. For each point the program first finds its true nearest neighbour, which is 77.1 metres away on average. Then it builds the Z key. Each coordinate is turned into an 11-bit cell number, using cells eight metres across, and the two are interleaved: one bit of easting, one bit of northing, and so on, giving a single 22-bit number.
Sorting by that number traces a path shaped like a chain of letter Zs across the map, filling one small square before moving to the next. The test is simple. Put the points in some order, then ask how often a point's true nearest neighbour is the very next entry in the list, and how often it is within eight places.
How well four orderings of 1,606 Scarborough postcode points keep nearest neighbours together (our Python run on OS Code-Point Open)| Ordering | Neighbour is adjacent | Neighbour within 8 places | Mean hop to next entry |
|---|
| Shuffled at random | 0.0% | 0.5% | 3,416 m |
| By easting only | 5.3% | 40.8% | 2,216 m |
| Postcode, A to Z | 26.7% | 61.3% | 503 m |
| Z-order key | 57.3% | 85.4% | 218 m |
Alphabetical postcode order already carries some geography, which is why it beats a one-axis sort. The Z key does much better than either: for 85.4% of points, checking eight entries on each side of it in the list finds the true nearest neighbour, which is 16 comparisons where a full scan needs 1,605. The remaining cases are the catch. A Z curve has seams where it jumps from one large square to the next, and two points on opposite sides of a seam share almost none of their leading bits. Fourteen points have their nearest neighbour across the first split of the map, and the worst pair sits 1,503 places apart in a list of 1,606. A short prefix in common means close together; the reverse is not guaranteed.
Ages 8 to 11
Number the squares of a 4 by 4 grid by drawing Zs inside Zs, then find two touching squares with far-apart numbers.
Ages 11 to 15
Write a Python function that interleaves the bits of two small numbers, and check it by hand.
Ages 15 and up
Key all 1,606 points, sort them, and measure how often the nearest neighbour is within eight places.
Postcode data and our measurements
Postcode grid references are from Ordnance Survey Code-Point Open, used under the Open Government Licence. Contains OS data, Crown copyright and database right 2026. Contains Royal Mail data, Royal Mail copyright and database right 2026. The keys, orderings and percentages are our own calculations.
