-- Copyright (c) David Amos, 2009. All rights reserved. module Math.Projects.MiniquaternionGeometry where import qualified Data.List as L import Math.Common.ListSet as LS import Math.Algebra.Field.Base import Math.Combinatorics.FiniteGeometry (pnf, ispnf, orderPGL) import Math.Combinatorics.Graph (combinationsOf) import Math.Combinatorics.GraphAuts import Math.Algebra.Group.PermutationGroup hiding (order) import qualified Math.Algebra.Group.SchreierSims as SS import Math.Algebra.Group.RandomSchreierSims import Math.Combinatorics.Design as D import Math.Algebra.LinearAlgebra -- ( (<.>), (<+>) ) import Math.Projects.ChevalleyGroup.Classical import Test.QuickCheck -- Sources: -- Miniquaternion Geometry, Room & Kirkpatrick -- Survey of Non-Desarguesian Planes, Charles Weibel -- F9, defined by adding sqrt of -1 to F3. (The Conway poly for F9 is not so convenient for us here) data F9 = F9 F3 F3 deriving (Eq,Ord) instance Show F9 where show (F9 0 0) = "0" show (F9 0 1) = "e" show (F9 0 2) = "-e" show (F9 1 0) = "1" show (F9 1 1) = "1+e" show (F9 1 2) = "1-e" show (F9 2 0) = "-1" show (F9 2 1) = "-1+e" show (F9 2 2) = "-1-e" e = F9 0 1 -- sqrt of -1 instance Num F9 where F9 a1 b1 + F9 a2 b2 = F9 (a1+a2) (b1+b2) F9 a1 b1 * F9 a2 b2 = F9 (a1*a2-b1*b2) (a1*b2+a2*b1) negate (F9 a b) = F9 (negate a) (negate b) fromInteger n = F9 (fromInteger n) 0 f9 = [F9 a b | a <- f3, b <- f3] w = 1-e -- a primitive element - generates the multiplicative group conj (F9 a b) = F9 a (-b) -- This is just the Frobenius aut x -> x^3 norm (F9 a b) = a^2 + b^2 -- == x * conj x instance Fractional F9 where recip x@(F9 a b) = F9 (a/n) (-b/n) where n = norm x instance FiniteField F9 where basisFq _ = [1,e] -- J9, or Q, defined by modifying the multiplication in F9 data J9 = J9 F9 deriving (Eq,Ord) instance Show J9 where show (J9 (F9 0 0)) = "0" show (J9 (F9 0 1)) = "-j" show (J9 (F9 0 2)) = "j" show (J9 (F9 1 0)) = "1" show (J9 (F9 1 1)) = "-k" show (J9 (F9 1 2)) = "i" show (J9 (F9 2 0)) = "-1" show (J9 (F9 2 1)) = "-i" show (J9 (F9 2 2)) = "k" squaresF9 = [1,w^2,w^4,w^6] -- and 0, but not needed here instance Num J9 where J9 x + J9 y = J9 (x+y) 0 * _ = 0 _ * 0 = 0 J9 x * J9 y = if y `elem` squaresF9 then J9 (x*y) else J9 (conj x * y) negate (J9 x) = J9 (negate x) fromInteger n = J9 (fromInteger n) i = J9 w j = J9 (w^6) -- == i-1 k = J9 (w^7) -- == i+1 j9 = [J9 x | x <- f9] -- the aut of J9 that sends i to x autJ9 x = fromPairs [ (a+b*i, a+b*x) | a <- [0,1,-1], b <- [1,-1] ] autA = autJ9 (-i) -- sends i -> -i autB = autJ9 (-k) -- sends j -> -j autC = autJ9 (-j) -- sends k -> -k autsJ9 = [autA, autC] -- these two auts generate the group, which is isomorphic to S3 -- indeed, the auts permute the pairs {i,-i}, {j,-j}, {k,-k} conj' (J9 x) = J9 (conj x) -- Note that conj' x == x .^ autB isAut k sigma = and [sigma x + sigma y == sigma (x+y) | x <- k, y <- k] && and [sigma x * sigma y == sigma (x*y) | x <- k, y <- k] isReal x = x `elem` [0,1,-1] isComplex = not . isReal instance Fractional J9 where recip 0 = error "J9.recip: 0" recip x | isReal x = x | otherwise = -x instance FiniteField J9 where basisFq _ = [1,i,j,k] eltsFq _ = j9 -- Near fields prop_NearField (a,b,c) = a+(b+c) == (a+b)+c && -- addition is associative a+b == b+a && -- addition is commutative a+0 == a && -- additive identity a+(-a) == 0 && -- additive inverse a*(b*c) == (a*b)*c && -- multiplication is associative a*1 == a && 1*a == a && -- multiplicative identity (a+b)*c == a*c + b*c && -- right-distributivity a*0 == 0 instance Arbitrary F9 where arbitrary = do x <- arbitrary :: Gen Int return (f9 !! (x `mod` 9)) instance Arbitrary J9 where arbitrary = do x <- arbitrary :: Gen Int return (j9 !! (x `mod` 9)) prop_NearFieldF9 (a,b,c) = prop_NearField (a,b,c) where types = (a,b,c) :: (F9,F9,F9) prop_NearFieldJ9 (a,b,c) = prop_NearField (a,b,c) where types = (a,b,c) :: (J9,J9,J9) -- PROJECTIVE PLANES ptsPG2 r = [ [0,0,1] ] ++ [ [0,1,x] | x <- r ] ++ [ [1,x,y] | x <- r, y <- r ] -- if r is sorted, then so is the result orthogonalLinesPG2 xs = L.sort [ [x | x <- xs, u <.> x == 0] | u <- xs ] rightLinesPG2 r = [ [0,0,1] : [ [0,1,x] | x <- r] ] ++ -- line at infinity [ [0,0,1] : [ [1,x,y] | y <- r] | x <- r ] ++ -- vertical lines [ [0,1,a] : [ [1,x,y] | x <- r, y <- [x*a+b] ] | a <- r, b <- r ] -- slope multiplies on the right -- if r is sorted, then so is the result, and each line in the result leftLinesPG2 r = [ [0,0,1] : [ [0,1,x] | x <- r] ] ++ -- line at infinity [ [0,0,1] : [ [1,x,y] | y <- r] | x <- r ] ++ -- vertical lines [ [0,1,a] : [ [1,x,y] | x <- r, y <- [a*x+b] ] | a <- r, b <- r ] -- slope multiplies on the left -- Projective plane PG2(F9) phi = design (xs,bs) where xs = ptsPG2 f9 bs = orthogonalLinesPG2 xs -- L.sort [ [x | x <- xs, u <.> x == 0] | u <- xs ] -- Then the collineations of phi consist of projective transformations, -- together with a conjugacy collineation induced by the Frobenius aut -- alternative construction of PG2(F9) - gives same result phi' = design (xs,bs) where xs = ptsPG2 f9 bs = rightLinesPG2 f9 collineationsPhi = l 3 f9 ++ [fieldAut] where D xs bs = phi fieldAut = fromPairs [ (x , map conj x) | x <- xs ] -- in general, this would be PSigmaL(n,Fq), whereas we want PGammaL(n,Fq). However, for F9 they coincide. -- order 84913920 liftToGraph (D xs bs) g = fromPairs $ [(Left x, Left (x .^ g)) | x <- xs] ++ [(Right b, Right (b -^ g)) | b <- bs] -- This construction appears to produce a projective plane -- (However, Room & Kirkpatrick point out that it's not really well-defined -- - if we had chosen different quasi-homogeneous coords, we would have got different results) -- However, it's not the same as either omega or omegaD below omega0 = design (xs,bs) where xs = ptsPG2 j9 bs = orthogonalLinesPG2 xs -- L.sort [ [x | x <- xs, u <.> x == 0] | u <- xs ] -- Room & Kirkpatrick, p103 omega = design (xs,bs) where xs = ptsPG2 j9 bs = rightLinesPG2 j9 -- another construction that produces same result (but slower) omega2 = design (xs,bs) where xs = ptsPG2 j9 bs = [ l | [p,q] <- combinationsOf 2 xs, l <- [line p q], [p,q] == take 2 l] line p q = toListSet $ filter ispnf [(a *> p) <+> (b *> q) | a <- j9, b <- j9] -- Room & Kirkpatrick, p107, p114 collineationsOmega = [r] ++ [s rho sigma | rho <- j9 \\ [0], sigma <- j9 \\ [0], rho == 1 || sigma == 1] ++ [t delta epsilon | delta <- j9, epsilon <- j9, delta * epsilon == 0] -- for generators sufficient to have only one non-zero ++ [u] ++ [a lambda | lambda <- autsJ9] where D xs bs = omega fromMatrix m = fromPairs [ (x, pnf (x <*>> m)) | x <- xs] r = fromMatrix [[1,0,0],[0,0,1],[0,1,0]] -- reflect in the line x = y in the affine subplane s rho sigma = fromPairs $ [([1,x,y], [1,x*rho,y*sigma]) | x <- j9, y <- j9] ++ [([0,1,mu],[0,1,(recip rho)*mu*sigma]) | mu <- j9] ++ [([0,0,1],[0,0,1])] -- leaves "Y" fixed -- fromMatrix [[1,0,0],[0,rho,0],[0,0,sigma]] -- scale x,y -> rho x, sigma y t delta epsilon = fromMatrix [[1,delta,epsilon],[0,1,0],[0,0,1]] -- translation x,y -> x+delta, y+epsilon u = fromPairs $ [([1,x,y], [1,x+y,x-y]) | x <- j9, y <- j9] ++ [([0,1,mu],[0,1,-mu]) | mu <- filter isComplex j9] ++ [([0,1,0],[0,1,1]), ([0,1,1],[0,1,0]), ([0,1,-1],[0,0,1]), ([0,0,1],[0,1,-1])] -- fromMatrix [[1,0,0],[0,1,-1],[0,1,1]] a lambda = fromPairs [ (x, map (.^ lambda) x) | x <- xs] -- order 311040 -- (which means this is also the plane constructed in Weibel?) -- dual plane of omega omegaD = design (xs,bs) where xs = ptsPG2 j9 bs = leftLinesPG2 j9 omegaD1 = D.to1n $ dual omega -- need proof omega /~= omegaD omegaD2 = design (xs,bs) where xs = ptsPG2 j9 bs = [ l | [p,q] <- combinationsOf 2 xs, l <- [line p q], [p,q] == take 2 l] line p q = toListSet $ filter ispnf [(p <* a) <+> (q <* b) | a <- j9, b <- j9] us <* x = map (*x) us -- Room and Kirkpatrick p130 psi = design (xs,bs) where xs = ptsPG2 j9 isReal x = all (`elem` [0,1,-1]) x xrs = ptsPG2 [0,1,-1] -- the thirteen real points, a copy of PG2(F3) within psi bs = toListSet [line p q | p <- xrs, q <- xs, q /= p] line p q = L.sort $ p : [pnf ( (p <* a) <+> q) | a <- j9] -- Room & Kirkpatrick p137 psi2 = design (xs,bs) where xs = ptsPG2 j9 bs = L.sort $ [ [0,0,1] : [ [0,1,x] | x <- j9] ] ++ -- line at infinity, z=0 [ [0,0,1] : [ [1,kappa,y] | y <- j9] | kappa <- j9 ] ++ -- vertical lines x = kappa [ [0,1,m] : [ [1,x,m*x+kappa] | x <- j9 ] | m <- [0,1,-1], kappa <- j9 ] ++ -- lines with real slope [ [0,1,kappa] : [ [1,x,kappa*(x-r)+s] | x <- j9 ] | r <- [0,1,-1], s <- [0,1,-1], kappa <- j9 \\ [0,1,-1] ] -- lines with complex slope -- Room & Kirkpatrick p134-6 collineationsPsi = realProjectivities -- real transvections, generating real projectivities ++ [a lambda | lambda <- autsJ9] where D xs bs = psi n = 3 realTransvections = [elemTransvection n (r,c) l | r <- [1..n], c <- [1..n], r /= c, l <- [1]] realProjectivities = [fromPairs $ [(x, pnf (x <*>> m)) | x <- xs] | m <- realTransvections] a lambda = fromPairs [ (x, map (.^ lambda) x) | x <- xs] -- order 33696 -- The order of a projective plane order (D xs bs) = length (head bs) - 1 isProjectivePlane pi = designParams pi == Just (2,(q^2+q+1,q,1)) where q = order pi collinear (D xs bs) ys = (not . null) [b | b <- bs, all (`elem` b) ys] -- assume p1..4 are distinct isQuadrangle plane ps@[p1,p2,p3,p4] = all (not . collinear plane) (combinationsOf 3 ps) concurrent (D xs bs) ls = (not . null) [x | x <- xs, all (x `elem`) ls] isQuadrilateral plane ls@[l1,l2,l3,l4] = all (not . concurrent plane) (combinationsOf 3 ls) isOval pi ps = length ps == order pi+1 && all (not . collinear pi) (combinationsOf 3 ps) findOvals1 pi = findOvals' 0 ([], points pi) where n = order pi findOvals' i (ls,rs) | i == n+1 = [reverse ls] | otherwise = concatMap (findOvals' (i+1)) [ (r:ls, rs') | r:rs' <- L.tails rs, all (not . collinear pi) (map (r:) (combinationsOf 2 ls)) ] -- if we have a function to quickly generate the line through two points, -- then we just need to see whether the third point is on it, which is much faster than testing collinearity findQuadrangles pi = findQuadrangles' 0 ([], points pi) where findQuadrangles' i (ls,rs) | i == 4 = [reverse ls] | otherwise = concatMap (findQuadrangles' (i+1)) [ (r:ls, rs') | r:rs' <- L.tails rs, all (not . collinear pi) (map (r:) (combinationsOf 2 ls)) ] findOvals pi@(D xs bs) = findOvals' 0 ([],xs) bs where n = order pi findOvals' i (ls,rs) bs | i == n+1 = [reverse ls] | otherwise = concat [let rls = reverse (r:ls) (notchords, chords) = L.partition (\b -> length (rls `LS.intersect` b) < 2) bs rs'' = foldl (\\) rs' chords -- if any line is already a chord, remove remaining points on it from further consideration in findOvals' (i+1) (r:ls, rs'') notchords | r:rs' <- L.tails rs] -- Todo: -- Code that shows that phi is Desarguesian, and omega, omegaD and psi are not {- -- !! NOT WORKING -- finds apparent counterexamples in phi too findNonDesarguesian pi@(D xs bs) = [ [p,x,y,z,x',y',z',k,l,m] | p <- xs, x <- xs \\ [p], y <- xs \\ [p,x], z <- xs \\ [p,x,y], (not . collinear pi) [x,y,z], x' <- line p x \\ L.sort [p,x], y' <- line p y \\ L.sort [p,y], z' <- line p z \\ L.sort [p,z], (not . collinear pi) [x',y',z'], k <- line x y `intersect` line x' y', -- will only have one element l <- line x z `intersect` line x' z', m <- line y z `intersect` line y' z', (not . collinear pi) [k,l,m] ] where line p q = head [b | b <- bs, p `elem` b, q `elem` b] -}