import Data.List import Data.Set (Set) import qualified Data.Set as Set import Helpers.SetHelpers (flatMap) import Helpers.Subsets (choose) type Face = (Int, Int) type Vertex = (Int, Int) type Polyhex = Set Face type Polytrihex = (Polyhex, Set Vertex) polytrihexes :: Int -> Set Polytrihex polytrihexes 0 = Set.singleton (Set.empty, Set.empty) polytrihexes 1 = Set.fromList [(Set.singleton (0,0), Set.empty), (Set.empty, Set.singleton (1,0))] directions = Set.fromList $ [(x-z,y-z) | [x,y,z] <- Data.List.permutations [-1,0,1]] faceNeighbors :: Face -> Set Face faceNeighbors (x1,y1) = Set.map (\(x2,y2) -> (x1+x2,y1+y2)) directions neighboringFaces :: Polytrihex -> Set Face neighboringFaces (polyhex, _) = Set.difference (flatMap faceNeighbors polyhex) polyhex vertices :: Face -> Set Vertex vertices (x,y) = Set.fromList [(x+1,y),(x-1,y),(x,y+1),(x,y-1),(x-1,y-1),(x+1,y+1)] faces :: Vertex -> Set Face faces (x, y) | even ((x+y) `mod` 3) = Set.fromList [(x+1, y), (x,y+1), (x-1,y-1)] | otherwise = Set.fromList [(x-1, y), (x,y-1), (x+1,y+1)] vertexCandidates :: Polytrihex -> Set Vertex vertexCandidates (p, vs) = Set.difference (flatMap vertices p) vs faceCandidates :: Polytrihex -> Set Face faceCandidates (p, vs) = Set.difference (flatMap faces vs) p fixedChildren :: Polytrihex -> Set Polytrihex fixedChildren polytrihex = Set.union faceAdded vertexAdded where faceAdded = Set.map (insertFace polytrihex) availableFaces where availableFaces = faceCandidates polytrihex insertFace (p, vs) f = (Set.insert f p, vs) vertexAdded = Set.map (insertVertex polytrihex) availableVertices where availableVertices = vertexCandidates polytrihex insertVertex (p,vs) v = (p, Set.insert v vs) dihedralActions :: [(Int, Int) -> (Int, Int)] dihedralActions = [p1, p2, p3, p4, p5, p6, p7, p8, p9, pA, pB, pC] where p1 (a, b) = (a, b) p2 (a, b) = (a-b, -b) p3 (a, b) = (b, a) p4 (a, b) = (b-a, -a) p5 (a, b) = (-b, a-b) p6 (a, b) = (-a, b-a) p7 (a, b) = (-a, -b) p8 (a, b) = (b-a, b) p9 (a, b) = (-b, -a) pA (a, b) = (a-b, a) pB (a, b) = (b, b-a) pC (a, b) = (a, a-b) rotationsAboutOrigin :: Polytrihex -> Set Polytrihex rotationsAboutOrigin (polyhex, vs) = Set.fromList $ map applyIsometry dihedralActions where applyIsometry f = (Set.map f polyhex, Set.map f vs) allCenters :: Polytrihex -> Set Polytrihex allCenters (polyhex,vs) = Set.map shift polyhex where shift cell = (Set.map (shiftBy cell) polyhex, Set.map (shiftBy cell) vs) where shiftBy (x1,y1) (x2,y2) = (x2-x1, y2-y1) canonical :: Polytrihex -> Polytrihex canonical polytrihex = Set.findMax $ flatMap rotationsAboutOrigin $ allCenters polytrihex freeChildren :: Polytrihex -> Set Polytrihex freeChildren = Set.map canonical . fixedChildren a343398_list :: [Int] a343398_list = 1 : map Set.size a343398_structures where a343398_structures = iterate (flatMap freeChildren) $ polytrihexes 1