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      "# This notebook is a semi top-down explanation. This cell needs to be\n",
      "# executed first so that the operators and helper functions are defined\n",
      "# All of this is explained in the later half of the notebook\n",
      "\n",
      "using Compose, Interact\n",
      "Compose.set_default_graphic_size(2inch, 2inch)\n",
      "\n",
      "points_f = [\n",
      "    (.1, .1),\n",
      "    (.9, .1),\n",
      "    (.9, .2),\n",
      "    (.2, .2),\n",
      "    (.2, .4),\n",
      "    (.6, .4),\n",
      "    (.6, .5),\n",
      "    (.2, .5),\n",
      "    (.2, .9),\n",
      "    (.1, .9),\n",
      "    (.1, .1)\n",
      "]\n",
      "\n",
      "f = compose(context(), stroke(\"black\"), line(points_f))\n",
      "\n",
      "rot(pic) = compose(context(rotation=Rotation(-deg2rad(90))), pic)\n",
      "flip(pic) = compose(context(mirror=Mirror(deg2rad(90), 0.5w, 0.5h)), pic)\n",
      "above(m, n, p, q) =\n",
      "    compose(context(),\n",
      "            (context(0, 0, 1, m/(m+n)), p),\n",
      "            (context(0, m/(m+n), 1, n/(m+n)), q))\n",
      "\n",
      "above(p, q) = above(1, 1, p, q)\n",
      "\n",
      "beside(m, n, p, q) =\n",
      "    compose(context(),\n",
      "            (context(0, 0, m/(m+n), 1), p),\n",
      "            (context(m/(m+n), 0, n/(m+n), 1), q))\n",
      "\n",
      "beside(p, q) = beside(1, 1, p, q)\n",
      "\n",
      "over(p, q) = compose(context(),\n",
      "                (context(), p), (context(), q))\n",
      "\n",
      "rot45(pic) =\n",
      "    compose(context(0, 0, 1/sqrt(2), 1/sqrt(2),\n",
      "        rotation=Rotation(-deg2rad(45), 0w, 0h)), pic)\n",
      "\n",
      "# Utility function to zoom out and look at the context\n",
      "zoomout(pic) = compose(context(),\n",
      "                (context(0.2, 0.2, 0.6, 0.6), pic),\n",
      "                (context(0.2, 0.2, 0.6, 0.6), fill(nothing), stroke(\"black\"), strokedash([0.5mm, 0.5mm]),\n",
      "                    polygon([(0, 0), (1, 0), (1, 1), (0, 1)])))\n",
      "\n",
      "function read_path(p_str)\n",
      "    tokens = [try parsefloat(x) catch symbol(x) end for x in split(p_str, r\"[\\s,]+\")]\n",
      "    path(tokens)\n",
      "end\n",
      "\n",
      "fish = compose(context(units=UnitBox(260, 260)), stroke(\"black\"),\n",
      "            read_path(strip(readall(\"fish.path\"))))\n",
      "\n",
      "rotatable(pic) = @manipulate for \u03b8=0:0.001:2\u03c0\n",
      "    compose(context(rotation=Rotation(\u03b8)), pic)\n",
      "end\n",
      "\n",
      "blank = compose(context())\n",
      "\n",
      "fliprot45(pic) = rot45(compose(context(mirror=Mirror(deg2rad(-45))),pic))\n",
      "\n",
      "# Hide this cell.\n",
      "display(MIME(\"text/html\"), \"\"\"<script>\n",
      "var cell = \\$(\".container .cell\").eq(0), ia = cell.find(\".input_area\")\n",
      "if (cell.find(\".toggle-button\").length == 0) {\n",
      "ia.after(\n",
      "    \\$('<button class=\"toggle-button\">Toggle hidden code</button>').click(\n",
      "        function (){ ia.toggle() }\n",
      "        )\n",
      "    )\n",
      "ia.hide()\n",
      "}\n",
      "</script>\"\"\")"
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        "\t    return elem.checked;\n",
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        "    var Textarea = function (typ, id, init) {\n",
        "\tvar attr = { placeholder: init.label },\n",
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        "\tthis.getState = function () {\n",
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        "\tthis.elem = elem;\n",
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        "\n",
        "\tInputWidgets.commInitializer(this);\n",
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        "    Textarea.prototype = Widget;\n",
        "\n",
        "    // RadioButtons\n",
        "    // Dropdown\n",
        "    // HTML\n",
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        "    var InputWidgets = {\n",
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        "\tCheckbox: Checkbox,\n",
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        "\tText: Text,\n",
        "\tTextarea: Textarea,\n",
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        "\tcommInitializer: function (widget) {\n",
        "\t    widget.sendUpdate = function () {};\n",
        "\t}\n",
        "    };\n",
        "\n",
        "    window.InputWidgets = InputWidgets;\n",
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        "\tswitch (type) {\n",
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        "\t\t\ttoinsert.data(\"type\", type);\n",
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     "source": [
      "# Functional Geometry\n",
      "*Functional Geometry* is a paper by Peter Henderson ([original (1982)](users.ecs.soton.ac.uk/peter/funcgeo.pdf), [revisited (2002)](https://cs.au.dk/~hosc/local/HOSC-15-4-pp349-365.pdf)) which deconstructs the MC Escher woodcut *Square Limit*\n",
      "\n",
      "![Square Limit](http://i.imgur.com/LjRzmNM.png)"
     ]
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    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
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     "source": [
      "> A picture is an example of a complex object that can be described in terms of its parts.\n",
      "Yet a picture needs to be rendered on a printer or a screen by a device that expects to\n",
      "be given a sequence of commands. Programming that sequence of commands directly is\n",
      "much harder than having an application generate the commands automatically from the\n",
      "simpler, denotational description."
     ]
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    {
     "cell_type": "markdown",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
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     "source": [
      "A `picture` is a *denotation* of something to draw.\n",
      "\n",
      "e.g. The value of f here denotes the picture of the letter F"
     ]
    },
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     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f"
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      "## Basic Operations on Pictures\n",
      "\n",
      "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:\n",
      "\n",
      "* `rot    : picture \u2192 picture`\n",
      "* `flip   : picture \u2192 picture`\n",
      "* `rot45  : picture \u2192 picture`\n",
      "* `above  : picture \u00d7 picture \u2192 picture`\n",
      "* `above  : int \u00d7 int \u00d7 picture \u00d7 picture \u2192 picture`\n",
      "* `beside : picture \u00d7 picture \u2192 picture`\n",
      "* `beside : int \u00d7 int \u00d7 picture \u00d7 picture \u2192 picture`\n",
      "* `over   : picture \u2192 picture`"
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      "## Rotate and flip\n",
      "\n",
      "#### `rot : picture \u2192 picture`\n",
      "\n",
      "Rotate a picture anti-clockwise by 90\u00b0"
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     "cell_type": "markdown",
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      }
     },
     "source": [
      "### `flip : picture \u2192 picture`\n",
      "\n",
      "Flip a picture along its virtical center axis"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "flip(f)"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
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      "### fliprot45 : picture \u2192 picture\n",
      "\n",
      "rotate the picture anti-clockwise by 45\u00b0, 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."
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      "### Juxtaposition\n",
      "#### `above  : picture \u00d7 picture \u2192 picture`\n",
      "\n",
      "place a picture above another."
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      "\n",
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     },
     "source": [
      "### Superposition\n",
      "\n",
      "#### `over : picture \u2192 picture`\n",
      "\n",
      "place a picture upon another"
     ]
    },
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     "cell_type": "code",
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     "input": [
      "over(f, flip(f))"
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     "language": "python",
     "metadata": {
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      "# Square Limit\n",
      "\n",
      "## The Fish\n",
      "\n",
      "We will now study some of the properties of the fish."
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      "fish |> zoomout"
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      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Tiles"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "There is a certain kind of arrangement that is used to tile parts of the image. We call it `t`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fish2 = fliprot45(fish)\n",
      "fish3 = rot(rot(rot(fish2)))\n",
      "\n",
      "t = over(fish, over(fish2, fish3))\n",
      "\n",
      "t |> zoomout"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "outputs": [
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     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "There is another kind of arrangement of fish which lies at the very center and regions on the diagonals. We will call it `u`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "u = over(over(fish2, rot(fish2)),\n",
      "         over(rot(rot(fish2)), rot(rot(rot(fish2)))))\n",
      "\n",
      "u |> zoomout"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
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 \u2026  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      "## A few useful higher level functions"
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      "`quartet` tiles 4 images in a 2x2 grid"
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      "quartet(p, q, r, s) =\n",
      "    above(beside(p, q), beside(r, s))\n",
      "\n",
      "quartet(f,flip(f),rot(f),f)"
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      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# 2inch x 2inch canvas is no more sufficient, so let's blow it up a bit\n",
      "Compose.set_default_graphic_size(5inch, 5inch)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "quartet(u, u, u, u)     |> zoomout"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "html": [
        "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n",
        "<svg xmlns=\"http://www.w3.org/2000/svg\"\n",
        "     xmlns:xlink=\"http://www.w3.org/1999/xlink\"\n",
        "     version=\"1.2\"\n",
        "     width=\"127mm\" height=\"127mm\" viewBox=\"0 0 127 127\"\n",
        "     stroke=\"none\"\n",
        "     fill=\"#000000\"\n",
        "     stroke-width=\"0.3\"\n",
        "     font-size=\"3.88\"\n",
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     "cell_type": "markdown",
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     "source": [
      "Notice how the fish interlock without leaving out any space in between them. Escher FTW."
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "`cycle` is a quartet of the same picture with each successive tile rotated by 90\u00b0 anti-clockwise"
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      "cycle(p) =\n",
      "    quartet(p, rot(p), rot(rot(p)), rot(rot(rot(p))))\n",
      "\n",
      "cycle(f)"
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     "language": "python",
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     "prompt_number": 23
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    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## More parts"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note: `blank` denotes a blank `picture`\n",
      "\n",
      "There is a certain pattern which makes up the mid region of each of the four edges of the image. We will call this arrangement `side`\n",
      "\n",
      "the 1 in `side1` represents 1 level of recursion. This is the simplest side."
     ]
    },
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     "cell_type": "code",
     "collapsed": false,
     "input": [
      "side1 = quartet(blank, blank, rot(t), t)\n",
      "\n",
      "side1 |> zoomout"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "outputs": [
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     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A side that is 2 levels deep."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "side2 = quartet(side1,side1,rot(t),t)\n",
      "\n",
      "side2 |> zoomout"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "outputs": [
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     "cell_type": "markdown",
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     "source": [
      "A side that is n level deep."
     ]
    },
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     "cell_type": "code",
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     "input": [
      "side(n) =\n",
      "    if n == 1 side1 # basis\n",
      "    else quartet(side(n-1),side(n-1),rot(t),t) # induction\n",
      "    end\n",
      "\n",
      "side(3) |> zoomout"
     ],
     "language": "python",
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    },
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     "input": [
      "\n",
      "# @manipulate lets us watch what happens as the levels increase\n",
      "@manipulate for level=slider(1:4, value=1)\n",
      "    side(level) |> zoomout\n",
      "end\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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       "html": [],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "Slider{Int64}([Input{Int64}] 1,\"\",1,1:4)"
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        "     fill=\"#000000\"\n",
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     "prompt_number": 27
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    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Similarly, there is a certain kind of arrangement which makes up the corners of the artwork.\n",
      "\n",
      "A `corner` level 1 deep is simply"
     ]
    },
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     "cell_type": "code",
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      "corner1 = quartet(blank,blank,blank,u)\n",
      "\n",
      "corner1 |> zoomout"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
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     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A corner 2 levels deep, it is built using corner1, side1 and u."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "corner2 = quartet(corner1,side1,rot(side1),u)\n",
      "\n",
      "corner2 |> zoomout"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "outputs": [
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 \u2026  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      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "An n level deep corner."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "corner(n) =\n",
      "    n == 1 ? corner1 :\n",
      "             quartet(corner(n-1), side(n-1), rot(side(n-1)), u)\n",
      "\n",
      "corner(3) |> zoomout"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "slide"
      }
     },
     "outputs": [
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 \u2026  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      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Touring the corners with a slider:\n",
      "\n",
      "@manipulate for level=slider(1:4, value=1)\n",
      "    corner(level) |> zoomout\n",
      "end"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "Slider{Int64}([Input{Int64}] 1,\"\",1,1:3)"
       ]
      },
      {
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        "     fill=\"#000000\"\n",
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     "cell_type": "markdown",
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     },
     "source": [
      "## Square Limit"
     ]
    },
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We are almost there.\n",
      "\n",
      "Square limit is a nonet of various rotations of `corner` at the corners, `side` at the sides and `u` in the center.\n",
      "\n",
      "The following equation puts this precisely:"
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     "input": [
      "squarelimit(n) =\n",
      "    nonet(corner(n), side(n), rot(rot(rot(corner(n)))),\n",
      "          rot(side(n)), u, rot(rot(rot(side(n)))),\n",
      "          rot(corner(n)), rot(rot(side(n))), rot(rot(corner(n))))"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 32,
       "text": [
        "squarelimit (generic function with 1 method)"
       ]
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     ],
     "prompt_number": 32
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     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# We render a level-3 square limit on a 10inch x 10inch SVG for maximum awesome.\n",
      "draw(SVG(10inch, 10inch), squarelimit(3))"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "fragment"
      }
     },
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    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Implementing the primitives with Compose"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We will now implement the basic operators `rot`, `flip`, `fliprot45`, `above`, `below` and `over` with [Compose.jl](composejl.org). (this explanation is taken mostly from the Compose website)\n",
      "\n",
      "Compose graphics are created as a tree data structure. There are 3 types of objects that make up the nodes of the tree:\n",
      "\n",
      "1. `Context`: An internal node, defines the transformation matrix to be applied for its children\n",
      "2. `Form`: A leaf node that defines some geometry, like a line or a polygon\n",
      "3. `Property`: A leaf node that modifies how its parent's subtree is drawn, like fill color, font family, or line width."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The all-important function in Compose, is called, not surprisingly, `compose`. Calling `compose(a, b)` will return a new tree rooted at `a` and with `b` attached as a child.\n",
      "\n",
      "That's enough to start drawing some simple shapes."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Compose.set_default_graphic_size(2inch, 2inch) # switch back to smaller output\n",
      "\n",
      "compose(context(), rectangle(), fill(\"tomato\"))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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      "The circles represent contexts, the triangles represent properties (fill color in our case) and the square represents a Form (one is a circle another is a rectangle)\n",
      "\n",
      "When a picture needs to be drawn, Compose walks the tree, setting up the correct transformation matrices and resolving the forms into absolute coordinate system depending on the size of the canvas being drawn on.\n",
      "\n",
      "These trees can be really big, but that's OK. Compose will draw them just fine. Let's introspect the tile `.`"
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      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "Let us now describe the picture of the letter F.\n",
      "\n",
      "Notice that we use relative coordinates, where the width of the image is 1 width unit, and the height is 1 height unit"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "points_f = [\n",
      "    (.1, .1),\n",
      "    (.9, .1),\n",
      "    (.9, .2),\n",
      "    (.2, .2),\n",
      "    (.2, .4),\n",
      "    (.6, .4),\n",
      "    (.6, .5),\n",
      "    (.2, .5),\n",
      "    (.2, .9),\n",
      "    (.1, .9),\n",
      "    (.1, .1)\n",
      "]\n",
      "\n",
      "# line function takes a list of points and returns a Line form\n",
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      "Rotating a picture involves wrapping in a context with a rotation."
     ]
    },
    {
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     "input": [
      "\n",
      "# rotate anti-clockwise by 90deg\n",
      "rot(pic) =\n",
      "    compose(context(rotation=Rotation(-\u03c0/2)), pic)\n",
      "\n",
      "rot(f)"
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     "language": "python",
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      "\n",
      "fliprot45(f) |> zoomout"
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      "### The Fish\n",
      "\n",
      "The fish is stored as an [SVG path](https://developer.mozilla.org/en/docs/Web/SVG/Tutorial/Paths) in the `fish.path` file. We read it, tokenize it and create a path `Form` from it."
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      "function read_path(p_str)\n",
      "    tokens = [try parsefloat(x) catch symbol(x) end for x in split(p_str, r\"[\\s,]+\")]\n",
      "    path(tokens)\n",
      "end\n",
      "\n",
      "# The fish was drawn on a canvas of 260x260 units, we will specify this\n",
      "# by wrapping the path in a Unit box of this dimension\n",
      "\n",
      "fish = compose(context(units=UnitBox(260, 260)), stroke(\"black\"),\n",
      "            read_path(strip(readall(\"fish.path\"))))\n"
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     "source": [
      "The zoomout function draws its argument in a smaller context in the middle of the canvas, and marks the extents of the smaller context with a dotted rectangle."
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    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Utility function to zoom out and look at the context\n",
      "zoomout(pic) = compose(context(),\n",
      "                (context(0.2, 0.2, 0.6, 0.6), pic),\n",
      "                (context(0.2, 0.2, 0.6, 0.6), fill(nothing), stroke(\"black\"), strokedash([0.5mm, 0.5mm]),\n",
      "                    polygon([(0, 0), (1, 0), (1, 1), (0, 1)])))\n",
      "\n",
      "zoomout(fish)"
     ],
     "language": "python",
     "metadata": {
      "slideshow": {
       "slide_type": "skip"
      }
     },
     "outputs": [
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      "## In conclusion\n",
      "\n",
      "We described Escher's *Square Limit* from the description of its smaller parts, which in turn were described in terms of their smaller parts.\n",
      "\n",
      "This seemed simple because we chose to talk in terms of an *algebra* to describe pictures. The primitives `rot`, `flip`, `fliprot45`, `above`, `beside` and `over` fit the job perfectly.\n",
      "\n",
      "We were able to describe these primitves in terms of `compose` `contexts`, which the Compose library knows how to render.\n",
      "\n",
      "Denotation can be an easy way to describe a system as well as a practical implementation method.\n",
      "\n",
      "[Abstraction barriers](https://mitpress.mit.edu/sicp/full-text/sicp/book/node29.html) are useful tools that can reduce the cognitive overhead on the programmer. It entails creating layers consisting of functions which only use functions in the same layer or layers below in their own implementation. The layers in our language were:\n",
      "\n",
      "    ------------------[ squarelimit ]------------------\n",
      "    -------------[ quartet, cycle, nonet ]-------------\n",
      "    ---[ rot, flip, fliprot45, above, beside, over ]---\n",
      "    -------[ compose, context, line, path,... ]--------\n",
      "    \n",
      "Drawing this diagram out is a useful way to begin building any library.\n"
     ]
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