Cell sizes and distances

[Fulgur14]

I decided to separate this to a new discussion -- distances and areas in various types of tilings. The convention is that the absolute unit in HyperRogue world is 3 m.

The {3,7}-based tilings:

{3,7} - (3,3,3,3,3,3,3)
edge length: 3.272 m
center distance (3-3): 1.699 m
area (3): 4.039 m2

r{3,7} - (3,7,3,7) - triheptagonal
edge length: 1.476 m
center distance (3-7): 1.862 m
area (3): 0.915 m2
area (7): 7.255 m2

{7,3} - (7,7,7) - pure heptagonal
edge length: 1.699 m
center distance (7-7): 3.272 m
area (7): 9.425 m2

t{3,7} - (6,6,7) - basic HyperRogue tiling
edge length: 1.003 m
center distance (6-6): 1.699 m
center distance (6-7): 1.862 m
area (6): 2.534 m2
area (7): 3.514 m2

c{3,7} - (3,4,7,4)
edge length: 1.093 m
center distance (3-4): 0.850 m
center distance (4-7): 1.636 m
area (3): 0.509 m2
area (4): 1.169 m2
area (7): 4.144 m2

t{7,3} - (3,14,14)
edge length: 0.789 m
center distance (3-14): 1.862 m
center distance (14-14): 3.272 m
area (3): 0.267 m2
area (14): 8.801 m2

o{3,7} - (4,6,14)
edge length: 0.627 m
center distance (4-6): 0.850 m
center distance (4-14): 1.636 m
center distance (6-14): 1.862 m
area (4): 0.390 m2
area (6): 1.007 m2
area (14): 5.711 m2

s{3,7} - (3,3,3,3,7)
edge length: 1.127 m
center distance (3-3): 0.641 m
center distance (3-7): 1.450 m
area (3): 0.540 m2
area (7): 4.386 m2

[Fulgur14]

Now, let's compare these numbers to two more tiling families: {3,8} and {4,5}:

{3,8} - (3,3,3,3,3,3,3,3)
edge length: 4.586 m
center distance (3-3): 2.181 m
area (3): 7.069 m2

r{3,8} - (3,8,3,8)
edge length: 1.899 m
center distance (3-8): 2.582 m
area (3): 1.487 m2
area (8): 14.884 m2

{8,3} - (8,8,8)
edge length: 2.181 m
center distance (8-8): 4.586 m
area (8): 18.850 m2

t{3,8} - (6,6,8)
edge length: 1.307 m
center distance (6-6): 2.181 m
center distance (6-8): 2.582 m
area (6): 4.214 m2
area (8): 7.612 m2

c{3,8} - (3,4,8,4)
edge length: 1.418 m
center distance (3-4): 1.091 m
center distance (4-8): 2.293 m
area (3): 0.847 m2
area (4): 1.939 m2
area (8): 8.837 m2

t{8,3} - (3,16,16)
edge length: 1.013 m
center distance (3-16): 2.582 m
center distance (16-16): 4.586 m
area (3): 0.438 m2
area (16): 17.681 m2

o{3,8} - (4,6,16)
edge length: 0.809 m
center distance (4-6): 1.091 m
center distance (4-16): 2.293 m
center distance (6-16): 2.582 m
area (4): 0.646 m2
area (6): 1.663 m2
area (16): 11.830 m2

s{3,8} - (3,3,3,3,8)
edge length: 1.459 m
center distance (3-3): 0.823 m
center distance (3-8): 2.055 m
area (3): 0.895 m2
area (8): 9.305 m2

[Fulgur14]

{4,5} - (4,4,4,4,4)
edge length: 3.761 m
center distance (4-4): 3.184 m
area (4): 11.309 m2

r{4,5} - (4,5,4,5)
edge length: 2.302 m
center distance (4-5): 2.528 m
area (4): 4.832 m2
area (5): 8.096 m2

{5,4} - (5,5,5,5)
edge length: 3.184 m
center distance (5-5): 3.761 m
area (5): 14.137 m2

t{4,5} - (5,8,8)
edge length: 1.407 m
center distance (5-8): 2.527 m
center distance (8-8): 3.184 m
area (5): 3.249 m2
area (8): 8.711 m2

c{4,5} - (4,4,4,5)
edge length: 1.651 m
center distance (4-4): 1.592 m
center distance (4-5): 1.881 m
area (4): 2.597 m2
area (5): 4.400 m2

t{5,4} - (4,10,10)
edge length: 1.325 m
center distance (4-10): 2.528 m
center distance (10-10): 3.762 m
area (4): 1.700 m2
area (10): 12.011 m2

o{4,5} - (4,8,10)
edge length: 0.957 m
center distance (4-8): 1.592 m
center distance (4-10): 1.881 m
center distance (8-10): 2.528 m
area (4): 0.901 m2
area (8): 4.226 m2
area (10): 6.602 m2

s{4,5} - (3,3,4,3,5)
edge length: 1.703 m
center distance (3-3): 0.953 m
center distance (3-4): 1.296 m
center distance (3-5): 1.592 m
area (3): 1.207 m2
area (4): 2.753 m2
area (5): 4.662 m2

[Fulgur14]

And we can also look at the domain based on Schwarz triangle (3,3,4). Some patterns repeat from (2,3,8), the {3,8} domain, but some are new:

(3,4,3,4,3,4):
edge length: 3.850 m
center distance (3-4): 2.582 m
area (3): 5.332 m2
area (4): 11.741 m2

(3,6,4,6):
edge length: 2.036 m
center distance (3-6): 2.181 m
center distance (4-6): 2.582 m
area (3): 1.697 m2
area (4): 3.852 m2
area (6): 9.552 m2

(3,3,3,3,3,4):
edge length: 2.287 m
center distance (3-3): 1.250 m
center distance (3-4): 1.693 m
area (3): 2.111 m2
area (4): 4.775 m2