In [1]:

```
# This notebook is a semi top-down explanation. This cell needs to be
# executed first so that the operators and helper functions are defined
# All of this is explained in the later half of the notebook
using Compose, Interact
Compose.set_default_graphic_size(2inch, 2inch)
points_f = [
(.1, .1),
(.9, .1),
(.9, .2),
(.2, .2),
(.2, .4),
(.6, .4),
(.6, .5),
(.2, .5),
(.2, .9),
(.1, .9),
(.1, .1)
]
f = compose(context(), stroke("black"), line(points_f))
rot(pic) = compose(context(rotation=Rotation(-deg2rad(90))), pic)
flip(pic) = compose(context(mirror=Mirror(deg2rad(90), 0.5w, 0.5h)), pic)
above(m, n, p, q) =
compose(context(),
(context(0, 0, 1, m/(m+n)), p),
(context(0, m/(m+n), 1, n/(m+n)), q))
above(p, q) = above(1, 1, p, q)
beside(m, n, p, q) =
compose(context(),
(context(0, 0, m/(m+n), 1), p),
(context(m/(m+n), 0, n/(m+n), 1), q))
beside(p, q) = beside(1, 1, p, q)
over(p, q) = compose(context(),
(context(), p), (context(), q))
rot45(pic) =
compose(context(0, 0, 1/sqrt(2), 1/sqrt(2),
rotation=Rotation(-deg2rad(45), 0w, 0h)), pic)
# Utility function to zoom out and look at the context
zoomout(pic) = compose(context(),
(context(0.2, 0.2, 0.6, 0.6), pic),
(context(0.2, 0.2, 0.6, 0.6), fill(nothing), stroke("black"), strokedash([0.5mm, 0.5mm]),
polygon([(0, 0), (1, 0), (1, 1), (0, 1)])))
function read_path(p_str)
tokens = [try parsefloat(x) catch symbol(x) end for x in split(p_str, r"[\s,]+")]
path(tokens)
end
fish = compose(context(units=UnitBox(260, 260)), stroke("black"),
read_path(strip(readall("fish.path"))))
rotatable(pic) = @manipulate for Î¸=0:0.001:2Ï€
compose(context(rotation=Rotation(Î¸)), pic)
end
blank = compose(context())
fliprot45(pic) = rot45(compose(context(mirror=Mirror(deg2rad(-45))),pic))
# Hide this cell.
display(MIME("text/html"), """<script>
var cell = \$(".container .cell").eq(0), ia = cell.find(".input_area")
if (cell.find(".toggle-button").length == 0) {
ia.after(
\$('<button class="toggle-button">Toggle hidden code</button>').click(
function (){ ia.toggle() }
)
)
ia.hide()
}
</script>""")
```

*Functional Geometry* is a paper by Peter Henderson (original (1982), revisited (2002)) which deconstructs the MC Escher woodcut *Square Limit*

A `picture`

is a *denotation* of something to draw.

e.g. The value of f here denotes the picture of the letter F

In [2]:

```
f
```

Out[2]:

We begin specifying the algebra of pictures we will use to describe *Square Limit* with a few operations that operate on pictures to give other pictures, namely:

`rot : picture â†’ picture`

`flip : picture â†’ picture`

`rot45 : picture â†’ picture`

`above : picture Ã— picture â†’ picture`

`above : int Ã— int Ã— picture Ã— picture â†’ picture`

`beside : picture Ã— picture â†’ picture`

`beside : int Ã— int Ã— picture Ã— picture â†’ picture`

`over : picture â†’ picture`

`rot : picture â†’ picture`

Rotate a picture anti-clockwise by 90Â°

In [3]:

```
rot(f)
```

Out[3]:

`flip : picture â†’ picture`

Flip a picture along its virtical center axis

In [4]:

```
flip(f)
```

Out[4]:

In [5]:

```
rot(flip(f))
```

Out[5]:

rotate the picture anti-clockwise by 45Â°, then flip it across the new virtical axis. In the paper this is implemented as $flip(rot45(fish))$. This function is rather specific to the problem at hand.

In [6]:

```
fliprot45(fish) |> zoomout # zoomout shows the bounding box
```

Out[6]:

`above : picture Ã— picture â†’ picture`

place a picture above another.

In [7]:

```
above(f, f)
```

Out[7]:

`above : int Ã— int Ã— picture Ã— picture â†’ picture`

given `m`

, `n`

, `picture1`

and `picture2`

, return a picture where `picture1`

is placed above `picture2`

such that their heights occupy the total height in m:n ratio

In [8]:

```
above(1, 2, f, f)
```

Out[8]:

`beside : picture Ã— picture â†’ picture`

Similar to `above`

but in the left-to-right direction.

In [9]:

```
beside(f, f)
```

Out[9]:

`beside : int Ã— int Ã— picture Ã— picture â†’ picture`

In [10]:

```
beside(1, 2, f, f)
```

Out[10]:

In [11]:

```
above(beside(f, f), f)
```

Out[11]:

`over : picture â†’ picture`

place a picture upon another

In [12]:

```
over(f, flip(f))
```

Out[12]:

We will now study some of the properties of the fish.

In [13]:

```
fish |> zoomout
```

Out[13]: