# Long time, no blog!

Wow, it's been a long time since I posted here. Well, I've been involved with rc stuff and 3d printing things.

The latter is what led to this post. I ran into a 3d printed maze that was generated programmatically. The authors comment that trying to do this in OpenSCAD was impossible because it was a functional language.

I had to disagree. First, that OpenSCAD is a functional language. It might have functional inspiration, and the developers try to stay with those principles, but as a functional language, it sucks. So much so that I wrote a Haskell library to generate OpenSCAD from a proper functional language, with static type checking and much better error messages.

Second, I disagree that a maze generator is impossible in a functional language. There was a time in the past - long enough ago that I was working in BASIC, C and PostScript - that I wrote maze generators. So I decided to write one in Haskell. The generator is purely functional, though the `State`

monad is used to hide some of the plumbing.

# The algorithm

The algorithm I use is a bog-standard recursive walk of the cells in the maze. We start with walls between all the cells, and pick a random cell. Then take each of it's four neighbors in random order. If a neighbor has been visited, we skip it. Otherwise, remove the wall between those two cells, and recursively walk from that neighbor as well.

You can find this algorithm described on rosettacode. This will include implementations in a number of languages, including a more idiomatic Haskell version.

## Representation

A maze is just an array of cells that either have halls or walls between each cell. Each cell needs two walls, one for each direction, and boolean to note that it's been visited. Which gives us the basic data types for a maze:

```
data Wall = Wall | Hall deriving Eq
data Cell = Cell { x, y :: Wall, visited :: Bool }
type Board = Array (Int, Int) Cell
```

If you're wondering why only two walls and not four, it's because each wall is shared by two cells. So each cell will have the wall with the larger coordinate in each direction. In particular, the `x`

wall is the wall that runs in the `y`

direction with the largest `x`

coordinate, and vice versa.

`Data.Array`

To work with the array, we need a few tools from `Data.Array`

:

```
import Data.Array (Array, array, bounds, (//), (!))
```

We need the `Array`

type for our type declarations, which we've already seen. The `array`

function is used to create the initial array. `bounds`

gets the bounds of the array so we don't have to pass those around. `//`

is used to create a new array with a list of changes to an array, and `!`

is used to reference a `Cell`

in the array.

## Generating the board

So our initial maze needs to have every cell with both walls, and initially not visited. That's just a `Cell wall wall false`

.

But our maze is going to have a border around it, for a number of reasons. We can make the walk easier by creating border cells as visited. So the walk won't need to do bounds checking. And the border on the low coordinate sides will have the wall on that side, and no others. That will simplify drawing the maze. And as a final touch, we'll put entry and exit doors in the maze at the diagonally opposite corners on the axis.

So we have a function `makeCell`

that looks convoluted, but each part is straightforward. For a `Board`

of width `w`

and height `h`

, we make the cell at `i, j`

with:

```
makeCell i j = Cell (if j == 0 then Hall else Wall)
(if i == 0 || i == 1 && j == h || i == w && j == 0
then Hall else Wall)
$ i == 0 || i > w || j == 0 || j > h
```

This creates a `Cell`

with an `x`

`Wall`

except for the first element in the `j`

direction, and a `y`

Wall except for the first element in the `i`

direction, as otherwise we'd try and draw those. We also create `Hall`

's for `1, h`

and `w, 0`

, which will be the entry and exit for the maze.

Finally, if either `x`

or `y`

is 0 or `x`

is greater than `w`

or `y`

greater than `h`

, then mark these border `Cell`

's as `Visited`

, so we won't visit them during the walk. All other cells haven't been visited yet.

So now we can create the array with `array`

and `makeCell`

:

```
makeBoard w h = array ((0, 0), (w+1, h+1))
[((i, j), makeCell i j) | i <- [0..w+1], j <- [0..h+1]]
```

`array`

takes list of pairs of indices and values and converts it to an array whose bounds are given as the first argument. In this case, bounds are `(0, 0)`

and `(w+1, h+1)`

. That makes the border the indices that have 0 for either `x`

or `y`

, and `w+1`

for `x`

and `h+1`

for `y`

. A list comprehension generates the indices, and we call `makeCell`

on them to create the `Cell`

for each index.

## The walk

We start with helper functions to remove each wall from a cell. Well, since this is a functional language, we can't actually remove the wall, so instead well have functions that return a cell with the appropriate wall removed. And one to return a visited cell.

```
clearY, clearX, visit :: Cell -> Cell
clearY cell = cell {y = Hall}
clearX cell = cell {x = Hall}
visit cell = cell {visited = True}
```

A step is represented by a tuple indicating the motion in the `x`

and `y`

directions, so we'll want a list of all possible steps, `allSteps`

:

```
allSteps = [(0, 1), (0, -1), (1, 0), (-1, 0)]
```

The body of the `walkMaze`

function is straightforward. Just pick a random cell in the maze, then call the internal helper `walkCell`

for `allSteps`

, that board position and our original `Board`

:

```
i <- state $ randomR (1, (fst . snd $ bounds origBoard) - 1)
j <- state $ randomR (1, (snd . snd $ bounds origBoard) - 1)
walkCell (i, j) allSteps origBoard
```

`walkCell`

implements the "in every direction from each cell" part of the algorithm by calling itself recursively, removing a random move from the list of moves it was passed on each recursion, stopping when there are no more moves. It uses `doStep`

to walk the `Board`

after that step:

```
walkCell _ [] b = return b
walkCell start steps board = do
step <- (steps !!) <$> (state . randomR) (0, length steps - 1)
walkCell start (delete step steps)
=<< doStep start step (board // [(start, visit $ board ! start)])
```

`doStep`

just calls `walkCell`

on `allSteps`

and the cell it steps to, after removing the wall between the `Cell`

it's stepping `from`

and the new `Celll`

. The last bit is the hard part, requiring examining the move in detail:

```
doStep from@(i, j) (dX, dY) board
| visited neighbor = return board
| dY > 0 = walkCell' $ board // [(from, clearY cell)]
| dY < 0 = walkCell' $ board // [(new, clearY neighbor)]
| dX > 0 = walkCell' $ board // [(from, clearX cell)]
| dX < 0 = walkCell' $ board // [(new, clearX neighbor)]
where cell = board ! from
new = (i + dX, j + dY)
neighbor = board ! new
walkCell' = walkCell new allSteps
```

So we can put all that together to get:

```
walkMaze :: Board -> State StdGen Board
walkMaze origBoard = let
clearY cell = cell {y = Hall}
clearX cell = cell {x = Hall}
visit cell = cell {visited = True}
allSteps = [(0, 1), (0, -1), (1, 0), (-1, 0)]
walkCell _ [] b = return b
walkCell start steps board = do
step <- (steps !!) <$> (state . randomR) (0, length steps - 1)
walkCell start (delete step steps)
=<< doStep start step (board // [(start, visit $ board ! start)])
doStep from@(i, j) (dX, dY) board
| visited neighbor = return board
| dY > 0 = walkCell' $ board // [(from, clearY cell)]
| dY < 0 = walkCell' $ board // [(new, clearY neighbor)]
| dX > 0 = walkCell' $ board // [(from, clearX cell)]
| dX < 0 = walkCell' $ board // [(new, clearX neighbor)]
where cell = board ! from
new = (i + dX, j + dY)
neighbor = board ! new
walkCell' = walkCell new allSteps
in do
i <- state $ randomR (1, (fst . snd $ bounds origBoard) - 1)
j <- state $ randomR (1, (snd . snd $ bounds origBoard) - 1)
walkCell (i, j) allSteps origBoard
```

The key to doing this in a functional language is generating a new `Board`

for the various recursive calls, rather than mutating a `Board`

and just using recursion to keep track of the progress of the walk.

This is liable to create a lot of extra state in each recursion. I haven't made any attempts to minimize that, which you would want to do in a solution for production use. Idiomatic Haskell would use the State monad for the `Board`

to hide the extra plumbing, as is done with the random number generator.

# Displaying the board

While the walk above is the meat of this blog entry, I find the display code interesting, so will cover that as well.

It would be nice to be able to plug in various different types of output to display the maze, so that we can debug with ASCII to a terminal or a Diagram before adding code to generate OpenSCAD code. So we'll use a `Board`

drawing function that takes functions that generate the walls and pastes them together. The type for the function is:

```
drawBoard :: (Board -> Int -> Int -> a) -- make X-direction cell walls
-> (Board -> Int -> Int -> a) -- make Y-direction cell walls
-> ([a] -> b) -- combine [walls] into a row
-> ([b] -> IO ()) -- Draw the board from [rows]
-> Board -- Board to draw
-> IO ()
```

As you can see, it takes two functions that create `Wall`

's, one in each direction. Then a function to combine a list of walls into a row, and finally one that takes a list of rows and outputs the final maze. For a larger program, it might be worthwhile to use a `Render`

data type to hold those for functions, but for a simple demo, it's just extra formula.

The wall drawing functions get the `Board`

and indices, as the indices may be needed to calculate where the wall needs to go. However, we are also going to generate the rows by generating the walls for the `Cell`

's in order of increasing `x`

, then do the same to put the rows together in order of increasing `y`

.

So the actual `drawBoard`

code is:

```
drawBoard makeX makeY makeRow makeMaze board =
makeMaze . concat $ [firstWall]:[drawCells j | j <- [1 .. height]]
where height = (snd . snd $ bounds board) - 1
width = (fst . snd $ bounds board) - 1
firstWall = makeRow [makeX board i 0 | i <- [0 .. width]]
drawCells j = [makeRow [makeY board i j | i <- [0 .. width]],
makeRow [makeX board i j | i <- [0 .. width]]]
```

This builds `firstWall`

, which is the `x`

direction walls for the 0'th `y`

row. We don't bother making the `y`

direction walls for that row, since they aren't part of the maze proper. That `firstWall`

is wrapped in a list and consed onto the list output by `drawCells`

, which outputs a list consisting of a row `x`

walls and a row of the `y`

walls for the `Cell`

's in that `y`

direction. We draw the 0 `Cell`

s in each row to generate the `y`

direction `Wall`

that forms the boundary of the maze. There are no `x`

direction `Wall`

s in those `Cell`

s, but either `makeRow`

or `DrawX`

will be responsible for dealing with any other artifacts that these cells might generate.

That result is passed to `concat`

to turn it into a list of rows instead of a list of lists of rows, which are passed to `makeMaze`

to output the maze.

## Drawing in ASCII

For ASCII output, we only need two extra functions:

```
charX, charY :: Board -> Int -> Int -> String
charX board i j = if y (board ! (i, j)) == Wall then "---+" else " +"
charY board i j = if x (board ! (i, j)) == Wall then " |" else " "
```

An `x`

`Wall`

is a horizontal line of dashes, and a `y`

wall is a vertical bar. `Hall`

's are just blank spaces, except for a `+`

at an intersection. Note that an `x`

`Wall`

is the `y`

element of a `Cell`

, as the `Cell`

element is named for the direction you are facing, but the `Wall`

rendering is named for the direction the wall runs.

`makeRow`

is simply `drop 3 . concat`

, to paste the strings together and then remove the extra `Hall`

's `drawCells`

creates for the 0 cells in each row. `makeMaze`

is just `putStr. unlines`

.

At this point, if you load the module (available via the fossil repository link on the right) into ghci, you can print square grids. Just use `:main 16 8`

to print a 16 by 8 maze. Or on a Unix system, you should be able to do `./maze.hs 16 8`

to generate a maze from the shell.

## Graphical output

That works, but it's not very pretty. So let's do a little graphics. Since I'm not much of a graphics designer, it still won't be very pretty.

### Support routines

This is a bit more complicated, so let's start with a couple of support routines.

`diaSpace`

is used to create a spacer. It takes an `R2`

, which is a direction, and a `Double`

indicating how long it is. It outputs a `Diagram B R2`

, which is something we can draw. Given that it's a spacer, it won't draw anything when drawn.

```
diaSpace :: R2 -> Double -> Diagram B R2
diaSpace unit size = phantom (fromOffsets [unit # scale size] :: Diagram B R2)
```

`diaCell`

does all the work. It needs to know which `Wall`

to check in a `Cell`

, which direction to draw in, any spacer needed, and the cell size. Plus the board and the cell's index:

```
diaCell :: (Cell -> Wall) -> R2 -> Diagram B R2 -> Double -> Board -> Int -> Int
-> Diagram B R2
diaCell side unit space cellSize board i j =
space ||| make (side (board ! (i, j)))
where make Wall = strokeT (fromOffsets [unit # scale cellSize])
make Hall = diaSpace unit cellSize
```

`diaCell`

returns the space in front of the result of calling the internal function `make`

on the `Cell`

s' `Wall`

. `make`

is simple - it uses `diaSpace`

to return a blank space for a `Hall`

, and the `Diagram`

primitives to create a line of length `cellSize`

in the given direction.

### Drawing cells.

Given `diaCell`

, the two routines for drawing walls are simple:

```
diaX, diaY :: Double -> Board -> Int -> Int -> Diagram B R2
diaX = diaCell y unitX mempty
diaY cellSize = diaCell x unitY (diaSpace unitX cellSize) cellSize
```

The type of `diaX`

and `diaY`

match the types needed by `drawBoard`

. `diaX`

is just `diaCell`

with the `y`

`Wall`

selector as it's first argument, the `x`

direction and an empty spacer, as the wall spans the entire length of the `Cell`

. `diaY`

needs the `cellSize`

argument as well, since the spacer it passes to `diaCell`

is a `cellSize`

spacer created by `diaSpace`

.

### Drawing the board

The row creator for `drawBoard`

is simply the `Diagram`

function `hcat`

, which accepts a list of diagrams and puts them together horizontally in a new diagram.

The board creator is almost that simply, but is actually long enough to get it's own function:

```
diaBoard :: Double -> [Diagram B R2] -> IO ()
diaBoard ww rows =
renderCairo "maze.png" Absolute $ vcat rows # centerXY # pad 1.1 # lwO ww
```

As with the row creator, the bulk of the work is done by the `Diagram`

function `vcat`

, which stacks the diagrams up vertically instead of horizontally. That image is then centered by `centerXY`

, padded by `pad 1.1`

, and the line weight is set to the wall width with `lw0 ww`

. That diagram is passed to `renderCairo`

along with some extra arguments so that it creates an appropriately scaled output in the file `maze.png`

.

### Seeing the result

The version of `maze.hs`

in the fossil repository has the Diagram (and OpenSCAD) drawing code commented out. Once you install the `diagrams`

package and the `diagrams-cairo`

package, you can change that. Look for three places where a line starts with `{- Comment out`

. The first two will need to be moved down to the next blank line. The last one will need to be moved down to beneath `diaBoard`

. You can now run this in `ghci`

as `:main 16 8 40 2`

, or as `./maze.hs 16 8 40 2`

. The two new arguments are the size of the cell and the width of the walls to draw. The old ASCII invocations will still work as well.

After running it with 4 arguments, the file `maze.png`

will be created in the current directory, and you can display that.

Expanding this to display images from the command line, or to embed it in an app for solving mazes, is left as an exercise for the reader. In which case, it ought to be made pretty as well.

## Printing in 3d

The inspiration was a 3d-printed maze, so let's do that. This is very similar to the Diagrams code, so the commentary will be a bit shorter.

To show what using an encapsulating data type would look like, this uses the `SCADCell`

data type, consisting of the side selector, a routine to construct the appropriate wall, and a `Vector3d`

to move the wall to the appropriate place in the cell:

```
data SCADCell = SCADCell (Cell->Wall) -- Wall extractor
(Double -> Double -> Double -> Model3d) -- Wall drawing
Vector3d -- translation
```

This is the first argument to `scadCell`

. `scadCell`

just creates a the appropriate wall and base, or a 0-sized block if this is a border `Cell`

. It also need the cell size, wall dimensions and base depth to create those models. `scadX`

and `scadY`

just call `scadCell`

with the appropriate `SCADCell`

.

```
scadX, scadY :: Double -> Double -> Double -> Double -> Board -> Int -> Int
-> Model3d
scadX cs = scadCell (SCADCell y (flip box) (0, cs, 0)) cs
scadY cs = scadCell (SCADCell x box (cs, 0, 0)) cs
scadCell :: SCADCell -> Double -> Double -> Double -> Double ->
Board -> Int -> Int -> Model3d
scadCell (SCADCell side box' move) cs ww wh bd board i j =
make (side $ board ! (i, j))
# translate (cs * fromIntegral (i - 1), cs * fromIntegral (j - 1), 0)
where make Wall = box' ww (cs + ww) (bd + wh) # translate move <> base
make Hall = base
base = if i == 0 || j == 0 then box 0 0 0
else box (cs + ww) (cs + ww) bd
```

Again, there's a function in the library that does exactly what we want for turning the output of the cell drawing routines into a row. So we just use `union`

for this. That same function also serves to join the rows into a board, so we just need to compose it with `draw`

in order to print the maze. However, this prints the maze "upside down" compared to the previous two rendering engines, so we use `mirror`

to fix that as well. No real need, but it feels like the right thing.

You'll need to install version 0.2.1.1 or later of my Haskell OpenSCAD library from hackage and uncomment the appropriate code segments to use it. You can then run it as either `:main`

or `./main.hs`

, using arguments like 16 8 20 2 4 10. That's the same four arguments as the Diagram version, with the depth of the base and the height of the walls added.

To see the results, you'll also need the OpenSCAD application. That can generate an STL file, and getting it to a 3d printer from there is up to you.

# Main

Just for completeness, a brief look at the main routine that ties it together. This is really just a kludge to test the others, but it does the job.

The outline is to get the arguments, map them to integers. Sorry, no fractional sizes here. Then convert those to floats for the things that need them. Switch on the length of the argument list to either raise a usage error or create a `drawBoard'`

function that's just the `drawBoard`

invoked with the functions appropriate to the type of output we want.

Then get a random number generator, and run `mazeWalk`

using it on a board of the appropriate size, which we will use the newly created `drawBoard'`

to output.

```
main :: IO ()
main = do
args <- map read <$> getArgs
let floats = map fromIntegral args
drawBoard' =
case length args of
2 -> drawBoard charX charY (drop 3 . concat) (putStr . unlines)
4 -> drawBoard (diaX cs) (diaY cs) hcat (diaBoard ww)
where [_, _, cs, ww] = floats
6 -> drawBoard (scadX cs ww wh bd) (scadY cs ww wh bd) union
(draw . mirror (0, 1, 0) . union)
where ([_, _, cs, ww, bd, wh]) = floats
{- Comment out drawing argument handling
-}
_ -> error "Width Height [CellSize WallWidth | CellSize WallWidth WallHeight BaseDepth]"
gen <- newStdGen
drawBoard' $ evalState (walkMaze $ makeBoard (head args) (args !! 1)) gen
```