Regions computed: 265 (258 Natural Earth admin-0 countries/territories + 7 reference shapes). Long side 301 cells (equal-area projection per country).
Verification: identity+all-3s stabilises back to all-3s in 265/265 regions; identity+identity stabilises to identity in 265/265.
All countries together: 7,490,488 cells, 16,751,362 grains, mean height 2.236. Height shares: 0=10.7% 1=8.0% 2=28.2% 3=53.1%.
Countries with >=20,000 cells (n=185) used for rankings below.
Most 'empty' (height-0 share): Gibraltar 14.6%, Monaco 14.3%, Coral Sea Is. 13.9%, São Tomé and Principe 13.6%, Saint Helena 13.5%
Least empty: Baikonur 1.9%, Sierra Leone 8.2%, Aruba 8.8%, Belize 8.9%, Slovakia 9.1%
Largest single solid same-height patch as share of the shape: eSwatini 14.6%, Czechia 13.1%, Sierra Leone 9.9%, Baikonur 9.5%, Eq. Guinea 9.2%
Smallest largest-patch share: Denmark 0.60%, Fr. S. Antarctic Lands 0.90%, Morocco 1.14%, Portugal 1.23%, Luxembourg 1.25%
Highest mean height: eSwatini 2.389, Cook Is. 2.350, Montserrat 2.340; lowest: Baikonur 2.056, Greece 2.142, Denmark 2.146
Circle: 70,688 cells, 143,264 grains, heights 0/1/2/3 = 1.3%/2.8%/88.0%/8.0%, largest solid patch 10.2%. central patch height 2, 6780 cells
Diamond: 45,300 cells, 104,624 grains, heights 0/1/2/3 = 11.2%/2.2%/31.2%/55.5%, largest solid patch 4.1%.
Kolam dot grid: 77,603 cells, 171,622 grains, heights 0/1/2/3 = 9.5%/10.3%/29.8%/50.4%, largest solid patch 0.3%.
Square 128: 16,384 cells, 37,656 grains, heights 0/1/2/3 = 10.2%/3.5%/32.8%/53.6%, largest solid patch 17.8%. central solid block of height 2: 54x54 cells (0.422 of side); full D4 symmetry: True
Square 256: 65,536 cells, 151,536 grains, heights 0/1/2/3 = 10.4%/2.6%/32.3%/54.7%, largest solid patch 17.8%. central solid block of height 2: 108x108 cells (0.422 of side); full D4 symmetry: True
Square 300: 90,000 cells, 208,096 grains, heights 0/1/2/3 = 10.5%/2.7%/31.7%/55.0%, largest solid patch 17.6%. central solid block of height 2: 126x126 cells (0.420 of side); full D4 symmetry: True
Square 64: 4,096 cells, 9,320 grains, heights 0/1/2/3 = 9.7%/3.7%/36.0%/50.6%, largest solid patch 19.4%. central solid block of height 2: 30x30 cells (0.469 of side); full D4 symmetry: True
Across all 258 countries height 3 is the most common value in 254 of them.
No country zero is free of empty cells: min height-0 share 1.9% (Baikonur).
Countries where 3 is NOT the most common height: Baikonur (mode 2, 81.4%), Brazilian I. (mode 2, 43.1%), Maldives (mode 2, 40.2%), Marshall Is. (mode 2, 39.7%)
