Equal edges: (4,n,2n) + (3,4,4,n) Infinite family, works for any n, including spherical cases n = 3, 4, 5 (and Euclidean n = 6, but that is par for the course). Corresponds to diminishing operation on cantellated regular tiling (3,4,n,4). Edge: 0.3644287600570477 (for (4,7,14)/(3,4,4,7)) to the limit of 0.7389979438517389 {n,n/2} + {3,n} Infinite family corresponding to diminishing of {3,n} by replacing vertices with n-gons. Both types of vertex allow for mixing under the rule [n] <-> [3,3]; for example, for n = 7 (first hyperbolic case), the allowed vertices are (3,3,3,3,3,3,3), (3,3,3,3,3,7), (3,3,3,7,7), and (3,7,7,7). Edge: 1.0905496635070864 for {7,7/2}/{3,7}, no upper limit (3,3,10,10)/(4,4,5,5) Edge: 0.7671972182513196 Allows mixed vertex (3,4,5,10). Hybrid tilings known. (3,4,8,40)/(3,5,8,8) Edge: 1.037723275211423 (3,5,6,18)/(3,6,6,9) Edge: 1.0531436003163206 {5,4} group Edge: 1.0612750619050353 Contains vertex (3,4,10,20), vertex (3,3,3,3,4,4), and mixed vertex (3,3,4,5,5) ([3,3,4] <-> [5,5]) (3,5,12,12) + (4,5,5,12) Edge: 1.155521135599718 Hybrid tilings known. (4,4,10,10) + (5,5,6,6) Edge: 1.198913757654163 Allows mixed vertex (4,5,6,10). Hybrid tilings known. {4,5} group Edge: 1.2537393258123553 Contains vertex (4,4,10,30). {18,4} group Edge: 1.7191071206150517 Contains vertex (3,3,4,4,6,6) and mixed vertex (3,4,6,18,18) ([3,4,6] <-> [18,18]) (3,4,4,5,5,5) group Edge: 1.8365197862728897 There is a mixing rule [4,5] <-> [3,20]. Through its application, we get vertices (3,3,4,5,5,20) and (3,3,3,5,20,20). {10,5}/{5,6}/{3,10} group Edge: 2.1225501238100706 {10,5} is a part of the infinite family {n,n/2}, so it automatically meshes with {3,10} and mixed vertices: (3,3,10,10,10,10), (3,3,3,3,10,10,10), (3,3,3,3,3,3,10,10), and (3,3,3,3,3,3,3,3,10). However, the same edge length is also shared by {5,6}, which allows for three more mixed vertices: (3,3,3,3,3,5,5,5), (3,3,3,5,5,5,10), and (3,5,5,5,10,10). In addition, the edge length of this group is double of that of the {5,4} group, allowing for mixing with polygons from there. {4,7} group Edge: 2.1408097231178886 There is a mixing rule [4,4] <-> [3,7]. This gives us vertices (3,4,4,4,4,4,7), (3,3,4,4,4,7,7), and (3,3,3,4,7,7,7) {8,6}/{4,8} group Edge: 2.4484524476780756 Allows for a mixed vertex (4,4,4,4,8,8,8). {30,6}/{5,15/2} group Edge: 2.6212181954545266 While {5,15/2} doesn't have an actual legal vertex, it still allows for mixed vertex (5,5,5,5,5,30,30). (3,4)^6/(3^5,4,24^4) Edge: 2.9830177447176047 (3,4)^6/(3,24)^(24/5) [3,4,4,4,4,4] <-> [24,24,24,24] (4,12)^5/(5,6)^5 group Edge: 3.308694702372257 Allows for 4 types of mixed vertices according to the mixing rule [5,6] <-> [4,12]. (3,9)^6 group Edge: 3.3712456612042487 Mixing rule [9,9,9,9,9] <-> [3,4,4,4,4,4,4], leading to vertex (3^7,4^6,9) {12,10}/{5,12} group Edge: 3.6124183254371904 Allows for a mixed vertex (5,5,5,5,5,5,12,12,12,12,12). {4,15}/(5,10)^6 group Edge: 3.7897307817314614 Mixing rule [4,4,4,4,4] <-> [5,5,10,10], allowing for two mixed vertices. (6,15)^6/(3^5,6^2,15^7) Edge: 3.8869800088001836 (6,10)^10/(4,5)^12 group Edge: 4.8897648328518954 Allows for 1 mixed vertex (6,10)^5+(4,5)^6. (3,3n)^5n family Mixing rule is [3,3,3,3,3n] <-> [5n,5n,5n]. For n divisible by 4, the replacing will eventually lead to (3n,5n)^(15n/4). Edge: 2.1225501238100706 (for {3,10}), no upper limit Apeirogons: (3,3,5,oo)/(3,5,5,5) Edge: 0.6196614774944214 (4,oo,oo)/(3,4,4,oo)/(3,3,4,4,4) Edge: 0.7389979438517389 (10,10,oo)/(3,3,oo,oo)/(3,4,6,oo)/(4,4,6,6) Edge: 0.962423650119207 {oo,3}/(4,4,5,oo) Edge: 1.0986122886681093 (4,4,oo,oo)/{6,4} Edge: 1.3169578969248164 Includes hybrid vertex (4,6,6,oo) (3,8,24,oo)/(3,3,3,oo,oo) Edge: 1.3424540464535255 {8,4}/{3,8} includes (4,16,16,oo) (5,5,oo,oo)/(6,6,10,10) Edge: 1.4793877412100649 Includes hybrid vertex (5,6,10,oo) (4,16^2,oo)/{8,4}/{3,8} Edge: 1.5285709194809984 {12,4}/(3,3,3,3,oo,oo) Edge: 1.6628858910586215 Includes hybrid vertex (3,3,12,12,oo) {oo,4}/{4,6} group Edge: 1.7627471740390868 Includes hybrid vertex (4,4,4,oo,oo); Vertex (4,4,4,4,oo,oo) belongs to the {10,5} group. Edge: 2.1225501238100706 Also (4,4,5,5,5,oo), (3,4,4,10,10,oo), (3,3,3,4,4,10,oo), and (3^5,4^2,oo). (3,3,6,6,6,oo,oo)/(3,3,3,4,4,6,6,oo)/(3,3,3,3,4,4,4,4,6) Edge: 2.4578507681281123 [6,oo] <-> [3,4,4] {oo,10}/(5,6)^6 Edge: 3.685460069402227 Includes hybrid vertex (5,5,5,6,6,6,oo,oo,oo,oo,oo) Random results (most likely just coincidences that fit within the stated precision): 5, 5, 10, 10, 13, 17 + 5, 6, 6, 12, 17, 18 (1:1): 2.421014466950248 4, 7, 8, 12, 19, 19 + 5, 6, 7, 10, 12, 17 (1:1): 2.4210618605909233 7, 10, 17, 18, 20, 20 + 11, 11, 11, 12, 13, 16 (1:1): 2.5536322668571807 (4,6,7,9,9,9,11,11,11)+(3,3,4,7,7,7,7,11,11,11) (3,5,5,7,7,7,11,11,12,12)+(3,6,6,6,6,7,9,10,10,12) (3,4,6,6,7,8,9,11,11,12)+(4,5,5,5,5,5,10,11,12,12) Double edges (first vertex is the one with doubled edges): (n,n,2n,2n,2n,2n) + (4,4,2n,2n) Infinite family. Case n = 5 is special because the smaller edge is the "(4,4,10,10) + (5,5,6,6)" case. Edge: 1.9248473002384139 (for (3,3,6,6,6,6)) to the limit of 2.633915793849633 {12,4} + (3,3,12,12) Edge: 1.6628858910586215 (3,3,4,4,12,12) + (3,3,24,24) Edge: 1.8626040551394825 {10,5}/{5,6}/{3,10} group + {5,4} group Edge: 2.1225501238100706 {3,14}/{14,7} + {7,4} Edge: 2.8981494453551733 {8,8} + {8,4}/{3,8} Edge: 3.057141838961996 {6,9} + {9,4} Edge: 3.1613962757130283 {5,10} + {10,4} Edge: 3.2338433350237734 {4,12} + {12,4} Edge: 3.3257717821172426 And {12,4} has double the edge of (3,3,12,12), too! {3,18}/{18,9} + {18,4} and its group Edge: 3.438214241230103 (3,4)^12 + (3,3,3,3,3,6,6,6) Edge: 4.425532044087407 {9,18} + (4,4,18,18,18,18) Edge: 4.746040408980548 Apeirogons: (10,10,oo) and its group + (5,5,oo) Edge: 0.962423650119207 {oo,3} and its group + (4,12,12) Edge: 1.0986122886681093 (7,7,oo,oo) + (7,7,oo) Edge: 1.6191738320894251 {12,4} gets two more vertices for its groups. ((3,3,12,12,oo) and (3,3,3,3,oo,oo)) Edge: 1.6628858910586215 {oo,4}/{4,6} + (8,8,oo) Edge: 1.7627471740390859 (3,3,6,6,6,6) + (10,10,oo) group Edge: 1.9248473002384139 (3,3,3,oo,oo,oo) + (3,3,3,6,6) Edge: 1.9732939220896686 Vertex (4,4,4,4,oo,oo) belongs to the {10,5} group. Edge: 2.1225501238100706 Also (4,4,5,5,5,oo), (3,4,4,10,10,oo), (3,3,3,4,4,10,oo), and (3^5,4^2,oo). {oo,6} + (4,4,oo,oo)/{6,4} group Edge: 2.633915793849633 Limit of the (n,n,2n,2n,2n,2n) + (4,4,2n,2n) family. (3^6,oo^4) + (3,3,4,4,4,4) Edge: 2.931430703894581 (3^12,oo^6) + (3^6,6^2) Edge: 4.04638017467887 (10,oo)^10 + {10,6} Edge: 5.034011024655859 Random results: (8,8,8,9,9,11,12,12,12,12) + (3,5,9,10,10) Triple edges (apeirogons only so far): (4,4,4,4,oo,oo,oo,oo) + (4,4,6,6) group Edge: 2.8872709503576206 Also 1:6 and 2:3 relations through that.