Provincial atlasFIFTY-ONE PLATES

The twenty largest and twenty smallest county-level units, each column scaled on its own, so read the rank rather than the bar length. Shenmu City is the largest at 7,451 km² and Beilin District the smallest at 22 km² — a span of about 334 times.

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31 municipal districts cover 42,545 km² (20.8%), 69 counties 144,961 km² (70.8%) and 7 county-level cities 17,267 km² (8.4%). Counts and area do not move together: districts are numerous but compact.

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The same figures as the ten city pages, set down once in a single table — area, span and adjacency all summed from county-boundary geometry. How many units a city governs and how large it is are two different things.

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266 adjacent county pairs across the province, and how the three regions share them. The busiest are Taibai and Ningshan with 9 neighbours each; Changwu, Fugu, Zhenping and Baihe have just one.

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Sort the 107 units by area and add them up: the top 10 hold 24.0%, the top 20 hold 40.3%, and it takes 27 to pass half. At the other end, 16 units are under 500 km² and together hold 1.8%.

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Straight-line distance from each unit's centre point to the centre of Xi'an: median 148 km, with 34 units within 100 km and only 16 at 300 km or more. Guanzhong, southern Shaanxi and northern Shaanxi barely overlap on this axis; the farthest, Fugu County, is 565 km out.

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Changing the centre to each unit's own prefecture seat: median 55.8 km, farthest 201.9 km.

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Each unit's nearest neighbour: median 34 km. Guanzhong is the tightest, southern Shaanxi the loosest.

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How fully the boundary fills its bounding box: median fill 0.61, and the three regions are almost identical.

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Distance from the government seat to the geometric centre of the county boundary: median 10.3 km, and 63 seats lie south of centre.

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The farthest point of a county from its own seat: median 44.2 km, longest in Chencang District at 101.4 km.

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The nearest point of the county boundary measured from the seat: median 8.0 km, and 9 units are under 1 km.

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206 characters in 140 distinct shapes: 阳 is the most frequent, and 101 shapes appear only once.

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107 boundaries, median perimeter ratio 1.58: closest to a circle is Baihe County at 1.25, most folded is Yintai District at 2.81.

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Convexity — boundary area divided by hull area — has a median of 0.795. Baihe County is the fullest at 0.903, Yintai District the most indented at 0.487.

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The 107 boundaries laid out in fourteen columns, every cell the same size and the same normalisation: with scale taken away, only shape is left.

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Counting outward from Xi'an, Fugu County is 10 county boundaries away; the median unit sits 3 rings out.

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46 units share part of their boundary with the provincial line, 43.0% of all 107; for 6 of them more than half of their own boundary is the provincial line.

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The 266 adjacent pairs share 11,281 km in all, median 37.6 km: the longest pair 215.5 km, the shortest only 2.4 km.

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What lies on the other side of each boundary segment: provincial line 13.8%, cross-city line 21.8%, same-city line 62.2%, with 2.3% unaligned.

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Median distance from one county seat to its nearest other seat: 24.7 km — Guanzhong 17.6 km, southern Shaanxi 35.5 km, northern Shaanxi 40.0 km.

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Of the 45 possible city pairs, 18 actually touch, sharing 2,930.4 km — 26.0% of every county boundary in the province.

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107 seats joined by 106 edges, 3,426.5 km in all; 19 edges cross a city line and 5 a regional line, and the two ends of the tree are 1,500.3 km apart.

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107 units, 266 pairs: 4 units touch only one other county, and 4 cut-vertices keep them attached to the rest of the map.

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All-pairs shortest paths: radius 8 boundaries, held by Yintai District and Xunyi County; diameter 16, between Fugu County and Zhenping County.

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Three mutually adjacent counties close a triangle in the boundary graph: 166 in all, with a mean local clustering of 0.50.

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Walking inward from the 46 border units at once: 82 units (76.6%) are one boundary in, and the deepest — Beilin and Lianhu — are 4 in.

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83 of the 266 pairs cross a city line, 31.2% of them; on the map, 38 units have no outside-city neighbour at all, while Zhouzhi and Ningshan have 6 each.

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Every one of the 5,671 seat pairs has a shortest path through the boundary graph; 23.08% of them pass through Huangling County, and 13 units are never on one.

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Lay each unit's neighbours around the compass: the median largest gap is 122.6°. For 19 units every neighbour falls within one half-circle, and 17 of those sit on the provincial border.

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The 266 boundaries folded to 0–180°: each quadrant holds about a quarter, and the best single direction explains only 0.26% of the variation. The one exception is the 26 cross-region boundaries, of which 31.0% run east–west.

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Each boundary squeezed into an equal-area ellipse: median elongation 1.61, with 60 units above 1.5 — yet the long axes point every which way, combining to an R of only 0.051.

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One inscribed circle per unit, centred on the point farthest from the boundary: median radius 15.3 km, deepest Shenmu City at 34.9 km and shallowest Beilin District at 1.5 km.

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The two farthest points on a boundary joined by a straight line: median 64.7 km, longest Shenmu City at 142.3 km, shortest Beilin District at 9.1 km.

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Ten prefecture seats on one ruler: 19 county seats are closer to a neighbouring city's seat than to their own, and Tongchuan alone draws 7 of them.

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The same boundary measured with rulers from 2.8 km down to 90 m: the median perimeter grows 18.5%, and the most broken, Yangling District, grows 47.6%.

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All 5,671 seat pairs: the farthest pair is Ningqiang County–Fugu County at 813.8 km, and a median unit's mean spacing is 216.0 km.

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For each of the 266 adjacent pairs, the distance between the two seats: median 41.3 km, with a correlation of only 0.14 against boundary length — how long two counties share a border says almost nothing about how far apart their seats are.

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Assigning every patch of land to its nearest county seat would move 25.1% of the province to a neighbouring county; each county would keep 75.2% of its area on average.

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Each unit's single longest boundary segment as a share of its whole perimeter: median 30.4%. 25 pairs are each other's longest, and the remaining 57 units choose one-way.

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The province as a 0.01° grid, each cell centre asked only for its distance to the nearest county seat: median 21.0 km, with 30.2% of the area within 15 km and 25.3% beyond 30 km.

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Two parallel lines clamping a county, the narrowest span that holds it: median 42.5 km. Beilin District is 4.1 km across, Yuyang District 105.4 km.

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Wobble under 200 m smoothed away, what is left: median 119 corners. Chencang District has the most at 391, Lianhu District the fewest at 17.

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Pick two people at random inside a county and average their straight-line distance: median 22.7 km — Beilin District 3.0 km, Shenmu City 47.2 km.

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A county plus every county it touches: median 10,190 km². Jingbian County covers the most at 28,886 km², Beilin District the least at 245 km².

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The smallest circle that covers the whole boundary: median radius 33.2 km — Shenmu City needs 71.1 km, Beilin District only 4.6 km. A county fills a median 0.460 of that circle.

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The point farthest from the provincial boundary lies inside Ningshan County, 140.1 km deep — the largest inscribed circle the province can hold. County seats sit a median 50.4 km from the provincial line: Wubao County just 0.21 km, Foping County 121.6 km.

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26 boundary segments totalling 975 km, 8.6% of all shared county boundary: Guanzhong–southern Shaanxi 639 km, Guanzhong–northern Shaanxi 336 km, and northern–southern Shaanxi not a single metre.

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107 units, 266 boundaries: Moran's I is 0.648 for area (log), 0.569 for county depth, 0.275 for seat-to-boundary distance, and only 0.029 for boundary convexity (p = 0.548). Same map, same weights — the gap between the four numbers is the whole point.

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Eight rulers side by side: area (log), county depth, seat-to-boundary distance, boundary convexity, neighbour count, distance to Xi'an, seat offset, and distance to the nearest other seat — each standardised across the 107 units. Specialness is the mean |z| over the eight. Dingbian tops it at 1.91, past the ±1.5 line on seven of the eight rulers; Luochuan sits closest to the middle at 0.19, crossing none.

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Turn the usual question around: if a county seat could sit anywhere inside its county, where would it leave the average spot closest? On an equal-area grid of about 1,600 cell centres per county, that point is the area-uniform 1-median (Fermat–Weber point), found with a coarse 400-point search and four rounds of local refinement. The "penalty" is the mean straight-line distance from the actual seat minus the mean from the optimum. It is surprisingly small: the median county saves 2.14 km and the mean is 2.96 km (3.50 km weighted by area); 83 counties clear 1 km, 40 clear 3 km, 17 clear 5 km, and only 3 clear 10 km, while 13 save less than 0.5 km. Chencang tops the list at 15.41 km — its seat is also the furthest from its own optimum, 47.4 km to the west — followed by Huangling at 12.18 km and Fu xian at 10.70 km; Yanta is almost there already, at 0.03 km. What drives the penalty is how far the seat sits from the area centroid (r = 0.94), not how elongated the county is (r = 0.05 against the length-to-width ratio, 0.34 against the diameter): Wubao saves the largest share, 48.2%, because its small county seat hugs one edge. Distances are straight-line and cells are weighted by land, so this measures territory rather than people, says nothing about road mileage, and is not a proposal to move any seat.

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