The ruler land: three ranges, prizes and warps

Round five: the ground itself from the ruler sequence. rulerland.py and rulerland.js are twins checked against one fixture; the Ruler Hills, the Ladder Range and the Dyadic Downs are drawn from them, with a prize on every prize summit and a warp gate on the biggest hill of every region.

The Ruler Hills from the air, region 0. The ground is the ruler sequence itself: column g raises a hill when 2^k divides g+1, its rank k, its height the edge-run ladder at k times three blocks, its radius doubling with the rank. What you see is the rank-12 column of this region (capped to a rank-9 hill by the world's knobs, so it stands 75 blocks tall and 384 cells wide with its rock strata and a snow cap) with rank-4 to rank-7 mounds scattered on their seven lanes. Between them the plain is walkable; the world leaves out ranks 2 and 3, which would be four-cell spikes.
▶ Go there land-1-ruler-hills-from-the-air
The Ruler Hills from the air, region 0. The ground is the ruler sequence itself: column g raises a hill when 2^k divides g+1, its rank k, its height the edge-run ladder at k times three blocks, its radius doubling with the rank. What you see is the rank-12 column of this region (capped to a rank-9 hill by the world's knobs, so it stands 75 blocks tall and 384 cells wide with its rock strata and a snow cap) with rank-4 to rank-7 mounds scattered on their seven lanes. Between them the plain is walkable; the world leaves out ranks 2 and 3, which would be four-cell spikes.
On foot at the foot of the summit hill, lane z = -118. The dome ahead is one column's hill; the bands of rock read in sevens. Everything here is arithmetic before it is ground: pytoolz's rulerland.height_at gives the same number for every cell, and the smoke test checks forty of them against the JavaScript twin.
▶ Go there land-2-ruler-hills-on-foot
On foot at the foot of the summit hill, lane z = -118. The dome ahead is one column's hill; the bands of rock read in sevens. Everything here is arithmetic before it is ground: pytoolz's rulerland.height_at gives the same number for every cell, and the smoke test checks forty of them against the JavaScript twin.
A prize on a summit: a marble plinth of nine cells and a gold block on top, on column 2,601,791 (rank 6, an even square remainder). The prize law is v2(g+1) >= 5 and sqrem(g) even, about one column in sixty-four, 73 a region. Touch the gold and it is yours: a toast, fireworks, a record under its warp address, and the Warp panel on the left counts it.
▶ Go there land-3-ruler-hills-a-prize
A prize on a summit: a marble plinth of nine cells and a gold block on top, on column 2,601,791 (rank 6, an even square remainder). The prize law is v2(g+1) >= 5 and sqrem(g) even, about one column in sixty-four, 73 a region. Touch the gold and it is yours: a toast, fireworks, a record under its warp address, and the Warp panel on the left counts it.
The warp gate of region 0: eight pillars of marble and copper round a glass centre on the region's biggest hill, the rank-12 column 2,600,959. Its warp address is rulerhills-1jqwv-mgzs-j; paste it into the Warp panel from any world and it opens the Ruler Hills exactly here. The panel's Copy link puts a warp= link on the clipboard.
▶ Go there land-4-ruler-hills-the-warp-gate
The warp gate of region 0: eight pillars of marble and copper round a glass centre on the region's biggest hill, the rank-12 column 2,600,959. Its warp address is rulerhills-1jqwv-mgzs-j; paste it into the Warp panel from any world and it opens the Ruler Hills exactly here. The panel's Copy link puts a warp= link on the clipboard.
The Ladder Range from the air. The ruler's ticks as walls: every column block of eight cells stands at a ladder height by its rank (CUM[v2(cx+1)+2] times four, so a rank-3 block is a wall 36 blocks tall, a rank-6 block 96) over a floor of 16 to 24, plus a shelf per lane block, and wherever a step drops a three-cell ramp runs down to the lower neighbour. On foot the big walls are barriers; the monster truck takes the ramps and the helicopter takes everything. Towns stand on the floor by the city law, ranks 7 to 10, with the railway beside them.
▶ Go there land-5-ladder-range-from-the-air
The Ladder Range from the air. The ruler's ticks as walls: every column block of eight cells stands at a ladder height by its rank (CUM[v2(cx+1)+2] times four, so a rank-3 block is a wall 36 blocks tall, a rank-6 block 96) over a floor of 16 to 24, plus a shelf per lane block, and wherever a step drops a three-cell ramp runs down to the lower neighbour. On foot the big walls are barriers; the monster truck takes the ramps and the helicopter takes everything. Towns stand on the floor by the city law, ranks 7 to 10, with the railway beside them.
A town of the Ladder Range with a lake in the cut beside it. The city law places the towns by column rank and terraces the ground under them; the smoke test allows those few cells to leave the ladder law and checks every other sample against it.
▶ Go there land-6-ladder-range-a-town-on-a-plateau
A town of the Ladder Range with a lake in the cut beside it. The city law places the towns by column rank and terraces the ground under them; the smoke test allows those few cells to leave the ladder law and checks every other sample against it.
The Ladder Range's warp gate from above, on the rank-12 column at shelf 60. The ladder's lane rule puts a column's prize on one of seven shelves at ladder spacing north of the towns (60, 124, 252, 508, 1020, 2044, 4092, chosen by the square remainder); the three nearest shelves are left to the towns, which the city law puts on the very columns of the biggest prizes.
▶ Go there land-7-ladder-range-the-gate
The Ladder Range's warp gate from above, on the rank-12 column at shelf 60. The ladder's lane rule puts a column's prize on one of seven shelves at ladder spacing north of the towns (60, 124, 252, 508, 1020, 2044, 4092, chosen by the square remainder); the three nearest shelves are left to the towns, which the city law puts on the very columns of the biggest prizes.
The Dyadic Downs from the air near lattice row 511. A hill at every lattice point (gx, gz) with rank the smaller of the two valuations, the rule that decides a city's tallest block, now applied to whole downs: big domes where both coordinates are highly divisible, water in the low ground between. The world keeps ranks 6 to 9 at half the law's radius, so the domes stand apart.
▶ Go there land-8-dyadic-downs-from-the-air
The Dyadic Downs from the air near lattice row 511. A hill at every lattice point (gx, gz) with rank the smaller of the two valuations, the rule that decides a city's tallest block, now applied to whole downs: big domes where both coordinates are highly divisible, water in the low ground between. The world keeps ranks 6 to 9 at half the law's radius, so the domes stand apart.
On foot on the downs, below row 511. The domes here are rank-6 and rank-7 hills, 45 and 48 blocks tall, with the rank-9 row on the horizon. Every prize of the downs stands on the lattice row that matches its column's rank: row 31 for rank 5, row 63 for rank 6, row 511 for rank 9 and up.
▶ Go there land-9-dyadic-downs-on-foot
On foot on the downs, below row 511. The domes here are rank-6 and rank-7 hills, 45 and 48 blocks tall, with the rank-9 row on the horizon. Every prize of the downs stands on the lattice row that matches its column's rank: row 31 for rank 5, row 63 for rank 6, row 511 for rank 9 and up.
The warp gate of the Dyadic Downs on row 511, the Warp panel open: this spot's address, Add this spot for a bookmark, Copy link, a box for a pasted address, and this region's landmarks (the gate first, then the 73 prizes by distance). A bookmark from another world opens that world at the address.
▶ Go there land-10-dyadic-downs-the-gate
The warp gate of the Dyadic Downs on row 511, the Warp panel open: this spot's address, Add this spot for a bookmark, Copy link, a box for a pasted address, and this region's landmarks (the gate first, then the 73 prizes by distance). A bookmark from another world opens that world at the address.
The prize shelf in the Trophy Room: a row per range with the count, the best rank and the last prize as a warp link, above the new Prize hunter badge (tiers at one, ten and fifty prizes). Every record is read from wander:prizes:v1 by badges.js, the same file the Node test checks with sample and broken records.
▶ Go there land-11-the-prize-shelf
The prize shelf in the Trophy Room: a row per range with the count, the best rank and the last prize as a warp link, above the new Prize hunter badge (tiers at one, ten and fifty prizes). Every record is read from wander:prizes:v1 by badges.js, the same file the Node test checks with sample and broken records.