# Haskell Quiz/Geodesic Dome Faces/Solution Jkramar

This problem seems to be strongly IO-bound, so actually computing the geodesic faces is not too time-sensitive. Hence there is time for computing the faces from the vertices by trying each triple of vertices to check if it's a face. (In the problem statement, the data for each polyhedron includes a list of the faces, not just the vertices.)

Originally this program represented vectors with ordinary tuples, and in some places that was definitely more understandable. But of course it's more fun this way.

```import Prelude hiding (sum, foldr1, foldr, maximum, mapM_)
import Control.Applicative
import Data.Foldable

data V3 a = V !a !a !a deriving (Read,Show)
instance Foldable V3 where foldr f x (V a b c) = f a\$f b\$f c x
instance Functor V3 where fmap f (V a b c) = V (f a) (f b) (f c)
instance Applicative V3 where
pure a = V a a a
V f g h <*> V a b c = V (f a) (g b) (h c)
type Tri a = V3 (V3 a)

(.+) :: (Num a) => V3 a -> V3 a -> V3 a
(.+) = liftA2 (+)

(.-) :: (Num a) => V3 a -> V3 a -> V3 a
(.-) = liftA2 (-)

(.*) :: (Num a) => a -> V3 a -> V3 a
(.*) = fmap.(*)

dot :: (Num a) => V3 a -> V3 a -> a
dot = (sum.).liftA2 (*)

cross :: (Num a) => V3 a -> V3 a -> V3 a
cross (V x y z) (V x' y' z')=V (y*z'-z*y') (z*x'-x*z') (x*y'-y*x')

det :: (Num a) => V3 (V3 a) -> a
det (V a b c) = (a `cross` b) `dot` c

-- matrix multiplication
mmul :: (Num a) => V3 (V3 a) -> V3 (V3 a) -> V3 (V3 a)
mmul t = fmap\$foldr1 (.+).(<*>t).fmap (.*)

-- chooses xs!!n gives the combinations of n elements from xs
chooses :: [a] -> [[[a]]]
chooses = foldr consider\$[[]]:repeat [] where
consider x cs = zipWith (flip (++)) cs\$map (x:)<\$>[]:cs

-- inefficient function to generate all the positively-oriented faces of the
-- triangle-faced polyhedron with vertices at vs
faces :: (Real a) => [V3 a] -> [Tri a]
faces vs = filter isFace \$ orient <\$> tris vs where
orient t@(V a b c) = if (signum\$det t)==1 then t else V a c b
tris xs = (\[a,b,c]->V a b c) <\$> (chooses xs!!3)
farth (V a b c) = maximum.(dot ((a.-b) `cross` (a.-c))<\$>)
isFace t@(V a b c) = farth t [a,b,c] == farth t vs

-- each triangle side is broken into n pieces, unlike in the problem statement,
-- where they use n+1 for some reason
shatter :: (Fractional a) => Int -> Tri a -> [Tri a]
shatter n t = mmul t'<\$>coords where
basis = V (V 1 0 0) (V 0 1 0) (V 0 0 1)
fwd = [(V k j (n-1-k-j).+) <\$> basis|k<-[0..n-1], j<-[0..n-1-k]]
bwd = [(V k j (n+1-k-j).-) <\$> basis|k<-[1..n], j<-[1..n-k]]
t' = ((1/fromIntegral n).*)<\$>t
coords = (fmap fromIntegral<\$>)<\$>fwd++bwd

geode :: (RealFloat a) => [V3 a] -> Int -> [Tri a]
geode vs n = fmap normize<\$>(shatter n=<<faces vs) where
normize v = (1/sqrt (dot v v)).*v

cyc :: V3 a -> [V3 a]
cyc (V a b c) = [V a b c, V b c a, V c a b]

tetrahed :: (Floating a) => [V3 a]
tetrahed = [V (x*sqrt 1.5) (-sqrt 2/3) (-1/3)|x<-[-1,1]]++
[V 0 (2*sqrt 2/3) (-1/3), V 0 0 1]

octahed :: (Num a) => [V3 a]
octahed = cyc =<< [V x 0 0|x<-[-1,1]]

icosahed :: (Floating a) => [V3 a]
icosahed = cyc =<< [V x (y*(1+sqrt 5/2)) 0|x<-[-1, 1], y<-[-1, 1]]

main :: IO ()
main = mapM_ print (geode octahed 51::[Tri Double])```