diff --git a/engine-proto/ConstructionViewer.jl b/engine-proto/ConstructionViewer.jl
new file mode 100644
index 0000000..7cb0450
--- /dev/null
+++ b/engine-proto/ConstructionViewer.jl
@@ -0,0 +1,203 @@
+module Viewer
+
+using Blink
+using Colors
+using Printf
+
+export ConstructionViewer, display!, opentools!, closetools!
+
+# === Blink utilities ===
+
+append_to_head!(w, type, content) = @js w begin
+  @var element = document.createElement($type)
+  element.appendChild(document.createTextNode($content))
+  document.head.appendChild(element)
+end
+
+style!(w, stylesheet) = append_to_head!(w, "style", stylesheet)
+
+script!(w, code) = append_to_head!(w, "script", code)
+
+# === construction viewer ===
+
+mutable struct ConstructionViewer
+  win::Window
+  
+  function ConstructionViewer()
+    # create window and open developer console
+    win = Window(Blink.Dict(:width => 620, :height => 830))
+    
+    # set stylesheet
+    style!(win, """
+      body {
+        background-color: #ccc;
+      }
+      
+      /* the maximum dimensions keep Ganja from blowing up the canvas */
+      #view {
+        display: block;
+        width: 600px;
+        height: 600px;
+        margin-top: 10px;
+        margin-left: 10px;
+        border-radius: 10px;
+        background-color: #f0f0f0;
+      }
+      
+      #control-panel {
+        width: 600px;
+        height: 200px;
+        box-sizing: border-box;
+        padding: 5px 10px 5px 10px;
+        margin-top: 10px;
+        margin-left: 10px;
+        overflow-y: scroll;
+        border-radius: 10px;
+        background-color: #f0f0f0;
+      }
+      
+      #control-panel > div {
+        margin-top: 5px;
+        padding: 4px;
+        border-radius: 5px;
+        border: solid;
+        font-family: monospace;
+      }
+    """)
+    
+    # load Ganja.js
+    loadjs!(win, "https://unpkg.com/ganja.js")
+    
+    # create global functions and variables
+    script!(win, """
+      // create algebra
+      var CGA3 = Algebra(4, 1);
+      
+      // initialize element list and palette
+      var elements = [];
+      var palette = [];
+      
+      // declare handles for the view and its options
+      var view;
+      var viewOpt;
+      
+      // declare handles for the controls
+      var controlPanel;
+      var visToggles;
+      
+      // create scene function
+      function scene() {
+        commands = [];
+        for (let n = 0; n < elements.length; ++n) {
+          if (visToggles[n].checked) {
+            commands.push(palette[n], elements[n]);
+          }
+        }
+        return commands;
+      }
+      
+      function updateView() {
+        requestAnimationFrame(view.update.bind(view, scene));
+      }
+    """)
+    
+    @js win begin
+      # create view
+      viewOpt = Dict(
+        :conformal => true,
+        :gl => true,
+        :devicePixelRatio => window.devicePixelRatio
+      )
+      view = CGA3.graph(scene, viewOpt)
+      view.setAttribute(:id, "view")
+      view.removeAttribute(:style)
+      document.body.replaceChildren(view)
+      
+      # create control panel
+      controlPanel = document.createElement(:div)
+      controlPanel.setAttribute(:id, "control-panel")
+      document.body.appendChild(controlPanel)
+    end
+    
+    new(win)
+  end
+end
+
+function display!(viewer::ConstructionViewer, elements::Matrix)
+  # load elements
+  elements_full = [
+    [0; elt; fill(0, 26)]
+    for elt in eachcol(elements)
+  ]
+  @js viewer.win elements = $elements_full.map((elt) -> @new CGA3(elt))
+  
+  # generate palette. this is Gadfly's `default_discrete_colors` palette,
+  # available under the MIT license
+  palette = distinguishable_colors(
+    length(elements_full),
+    [LCHab(70, 60, 240)],
+    transform = c -> deuteranopic(c, 0.5),
+    lchoices = Float64[65, 70, 75, 80],
+    cchoices = Float64[0, 50, 60, 70],
+    hchoices = range(0, stop=330, length=24)
+  )
+  palette_packed = [RGB24(c).color for c in palette]
+  @js viewer.win palette = $palette_packed
+  
+  # create visibility toggles
+  @js viewer.win begin
+    controlPanel.replaceChildren()
+    visToggles = []
+  end
+  for (elt, c) in zip(eachcol(elements), palette)
+    vec_str = join(map(t -> @sprintf("%.3f", t), elt), ", ")
+    color_str = "#$(hex(c))"
+    style_str = "background-color: $color_str; border-color: $color_str;"
+    @js viewer.win begin
+      @var toggle = document.createElement(:div)
+      toggle.setAttribute(:style, $style_str)
+      toggle.checked = true
+      toggle.addEventListener(
+        "click",
+        () -> begin
+          toggle.checked = !toggle.checked
+          toggle.style.backgroundColor = toggle.checked ? $color_str : "inherit";
+          updateView()
+        end
+      )
+      toggle.appendChild(document.createTextNode($vec_str))
+      visToggles.push(toggle);
+      controlPanel.appendChild(toggle);
+    end
+  end
+  
+  # update view
+  @js viewer.win updateView()
+end
+
+function opentools!(viewer::ConstructionViewer)
+  size(viewer.win, 1240, 830)
+  opentools(viewer.win)
+end
+
+function closetools!(viewer::ConstructionViewer)
+  closetools(viewer.win)
+  size(viewer.win, 620, 830)
+end
+
+end
+
+# ~~~ sandbox setup ~~~
+
+# in the default view, e4 + e5 is the point at infinity
+elements = sqrt(0.5) * BigFloat[
+  1  1 -1 -1  0;
+  1 -1  1 -1  0;
+  1 -1 -1  1  0;
+  0  0  0  0 -sqrt(6);
+  1  1  1  1  2
+]
+
+# show construction
+viewer = Viewer.ConstructionViewer()
+Viewer.display!(viewer, elements)
\ No newline at end of file
diff --git a/engine-proto/ganja-test/ganja-test.html b/engine-proto/ganja-test/ganja-test.html
new file mode 100644
index 0000000..d6efc90
--- /dev/null
+++ b/engine-proto/ganja-test/ganja-test.html
@@ -0,0 +1,76 @@
+<!DOCTYPE html>
+<html>
+<head>
+  <style>
+    body {
+      background-color: #ffe0f0;
+    }
+    
+    /* needed to keep Ganja canvas from blowing up */
+    canvas {
+      min-width: 600px;
+      max-width: 600px;
+      min-height: 600px;
+      max-height: 600px;
+    }
+  </style>
+  <script src="https://unpkg.com/ganja.js"></script>
+</head>
+<body>
+  <p><button onclick="flip()">Flip</button></p>
+  <script>
+  // in the default view, e4 + e5 is the point at infinity
+  let CGA3 = Algebra(4, 1);
+  let elements = [
+    CGA3.inline(() => Math.sqrt(0.5)*( 1e1 + 1e2 + 1e3 + 1e5))(),
+    CGA3.inline(() => Math.sqrt(0.5)*( 1e1 - 1e2 - 1e3 + 1e5))(),
+    CGA3.inline(() => Math.sqrt(0.5)*(-1e1 + 1e2 - 1e3 + 1e5))(),
+    CGA3.inline(() => Math.sqrt(0.5)*(-1e1 - 1e2 + 1e3 + 1e5))(),
+    CGA3.inline(() => -Math.sqrt(3)*1e4 + Math.sqrt(2)*1e5)()
+  ];
+  
+  // set up palette
+  var colorIndex;
+  var palette = [0xff00b0, 0x00ffb0, 0x00b0ff, 0x8040ff, 0xc0c0c0];
+  function nextColor() {
+    colorIndex = (colorIndex + 1) % palette.length;
+    return palette[colorIndex];
+  }
+  function resetColorCycle() {
+    colorIndex = palette.length - 1;
+  }
+  resetColorCycle();
+  
+  // create scene function
+  function scene() {
+    commands = [];
+    resetColorCycle();
+    elements.forEach((elt) => commands.push(nextColor(), elt));
+    return commands;
+  }
+  
+  // initialize graph
+  let graph = CGA3.graph(
+    scene,
+    {
+      conformal: true, gl: true, grid: true
+    }
+  )
+  document.body.appendChild(graph);
+  
+  function flip() {
+    let last = elements.length - 1;
+    for (let n = 0; n < last; ++n) {
+      // reflect
+      elements[n] = CGA3.Mul(CGA3.Mul(elements[last], elements[n]), elements[last]);
+      
+      // de-noise
+      for (let k = 6; k < elements[n].length; ++k) {
+        elements[n][k] = 0;
+      }
+    }
+    requestAnimationFrame(graph.update.bind(graph, scene));
+  }
+  </script>
+</body>
+</html>
diff --git a/engine-proto/ganja-test/ganja-test.jl b/engine-proto/ganja-test/ganja-test.jl
new file mode 100644
index 0000000..6c55061
--- /dev/null
+++ b/engine-proto/ganja-test/ganja-test.jl
@@ -0,0 +1,127 @@
+using Blink
+using Colors
+
+# === utilities ===
+
+append_to_head!(w, type, content) = @js w begin
+  @var element = document.createElement($type)
+  element.appendChild(document.createTextNode($content))
+  document.head.appendChild(element)
+end
+
+style!(w, stylesheet) = append_to_head!(w, "style", stylesheet)
+
+script!(w, code) = append_to_head!(w, "script", code)
+
+function add_element!(vec)
+  # add element
+  full_vec = [0; vec; fill(0, 26)]
+  n = @js win elements.push(@new CGA3($full_vec))
+  
+  # generate palette. this is Gadfly's `default_discrete_colors` palette,
+  # available under the MIT license
+  palette = distinguishable_colors(
+    n,
+    [LCHab(70, 60, 240)],
+    transform = c -> deuteranopic(c, 0.5),
+    lchoices = Float64[65, 70, 75, 80],
+    cchoices = Float64[0, 50, 60, 70],
+    hchoices = range(0, stop=330, length=24)
+  )
+  palette_packed = [RGB24(c).color for c in palette]
+  @js win palette = $palette_packed
+end
+
+# === build page ===
+
+# create window and open developer console
+win = Window()
+opentools(win)
+
+# set stylesheet
+style!(win, """
+  body {
+    background-color: #ffe0f0;
+  }
+
+  /* needed to keep Ganja canvas from blowing up */
+  canvas {
+    min-width: 600px;
+    max-width: 600px;
+    min-height: 600px;
+    max-height: 600px;
+  }
+""")
+
+# load Ganja.js
+loadjs!(win, "https://unpkg.com/ganja.js")
+
+# create global functions and variables
+script!(win, """
+  // create algebra
+  var CGA3 = Algebra(4, 1);
+  
+  // initialize element list and palette
+  var elements = [];
+  var palette = [];
+  
+  // declare visualization handle
+  var graph;
+  
+  // create scene function
+  function scene() {
+    commands = [];
+    for (let n = 0; n < elements.length; ++n) {
+      commands.push(palette[n], elements[n]);
+    }
+    return commands;
+  }
+  
+  function flip() {
+    let last = elements.length - 1;
+    for (let n = 0; n < last; ++n) {
+      // reflect
+      elements[n] = CGA3.Mul(CGA3.Mul(elements[last], elements[n]), elements[last]);
+      
+      // de-noise
+      for (let k = 6; k < elements[n].length; ++k) {
+        elements[n][k] = 0;
+      }
+    }
+    requestAnimationFrame(graph.update.bind(graph, scene));
+  }
+""")
+
+# set up controls
+body!(win, """
+  <p><button id="flip-button" onclick="flip()">Flip</button></p>
+""", async = false)
+
+# === set up visualization ===
+
+# list elements. in the default view, e4 + e5 is the point at infinity
+elements = sqrt(0.5) * BigFloat[
+  1  1 -1 -1  0;
+  1 -1  1 -1  0;
+  1 -1 -1  1  0;
+  0  0  0  0 -sqrt(6);
+  1  1  1  1  2
+]
+
+# load elements
+for vec in eachcol(elements)
+  add_element!(vec)
+end
+
+# initialize visualization
+@js win begin
+  graph = CGA3.graph(
+    scene,
+    Dict(
+      "conformal" => true,
+      "gl" => true,
+      "grid" => true
+    )
+  )
+  document.body.appendChild(graph)
+end
\ No newline at end of file
diff --git a/engine-proto/gram-test/descent-test.jl b/engine-proto/gram-test/descent-test.jl
index 9b6fb7a..2f20143 100644
--- a/engine-proto/gram-test/descent-test.jl
+++ b/engine-proto/gram-test/descent-test.jl
@@ -10,10 +10,10 @@ using PolynomialRoots
 # the difference between the matrices `target` and `attempt`, projected onto the
 # subspace of matrices whose entries vanish at each empty index of `target`
 function proj_diff(target, attempt)
-  I, J, values = findnz(target)
+  J, K, values = findnz(target)
   result = zeros(size(target)...)
-  for (i, j, val) in zip(I, J, values)
-    result[i, j] = val - attempt[i, j]
+  for (j, k, val) in zip(J, K, values)
+    result[j, k] = val - attempt[j, k]
   end
   result
 end
@@ -21,24 +21,24 @@ end
 # === example ===
 
 # the Lorentz form
-Q = diagm([-1, 1, 1, 1, 1])
+Q = diagm([1, 1, 1, 1, -1])
 
 # initialize the partial gram matrix for an arrangement of seven spheres in
 # which spheres 1 through 5 are mutually tangent, and spheres 3 through 7 are
 # also mutually tangent
-I = Int64[]
 J = Int64[]
+K = Int64[]
 values = BigFloat[]
-for i in 1:7
-  for j in 1:7
-    if (i <= 5 && j <= 5) || (i >= 3 && j >= 3)
-      push!(I, i)
+for j in 1:7
+  for k in 1:7
+    if (j <= 5 && k <= 5) || (j >= 3 && k >= 3)
       push!(J, j)
-      push!(values, i == j ? 1 : -1)
+      push!(K, k)
+      push!(values, j == k ? 1 : -1)
     end
   end
 end
-gram = sparse(I, J, values)
+gram = sparse(J, K, values)
 
 # set the independent variable
 #
@@ -54,11 +54,11 @@ gram[1, 6] = gram[6, 1]
 # in this initial guess, the mutual tangency condition is satisfied for spheres
 # 1 through 5
 guess = sqrt(0.5) * BigFloat[
-  1  1  1  1  2        0.2  0.1;
-  0  0  0  0 -sqrt(6)  0.3 -0.2;
   1  1 -1 -1  0       -0.1  0.3;
   1 -1  1 -1  0       -0.5  0.4;
-  1 -1 -1  1  0        0.1 -0.2
+  1 -1 -1  1  0        0.1 -0.2;
+  0  0  0  0 -sqrt(6)  0.3 -0.2;
+  1  1  1  1  2        0.2  0.1;
 ]
 
 # search parameters
@@ -108,11 +108,11 @@ println("\nLoss: ", loss, "\n")
 
 # === algebraic check ===
 
-R, gens = polynomial_ring(Generic.Rationals{BigInt}(), ["x", "t₁", "t₂", "t₃"])
+R, gens = polynomial_ring(AbstractAlgebra.Rationals{BigInt}(), ["x", "t₁", "t₂", "t₃"])
 x = gens[1]
 t = gens[2:4]
 
-S, u = polynomial_ring(Generic.Rationals{BigInt}(), "u")
+S, u = polynomial_ring(AbstractAlgebra.Rationals{BigInt}(), "u")
 
 M = matrix_space(R, 7, 7)
 gram_symb = M(R[
diff --git a/engine-proto/gram-test/gram-test.jl b/engine-proto/gram-test/gram-test.jl
index 4b2f859..0e88ff4 100644
--- a/engine-proto/gram-test/gram-test.jl
+++ b/engine-proto/gram-test/gram-test.jl
@@ -11,7 +11,7 @@ function printbad(msg)
   println(" ", msg)
 end
 
-F, gens = rational_function_field(Generic.Rationals{BigInt}(), ["a₁", "a₂", "b₁", "b₂", "c₁", "c₂"])
+F, gens = rational_function_field(AbstractAlgebra.Rationals{BigInt}(), ["a₁", "a₂", "b₁", "b₂", "c₁", "c₂"])
 a = gens[1:2]
 b = gens[3:4]
 c = gens[5:6]
diff --git a/engine-proto/gram-test/overlap-test.jl b/engine-proto/gram-test/overlap-test.jl
index beeaeca..e75531a 100644
--- a/engine-proto/gram-test/overlap-test.jl
+++ b/engine-proto/gram-test/overlap-test.jl
@@ -11,7 +11,7 @@ function printbad(msg)
   println(" ", msg)
 end
 
-F, gens = rational_function_field(Generic.Rationals{BigInt}(), ["x", "t₁", "t₂", "t₃"])
+F, gens = rational_function_field(AbstractAlgebra.Rationals{BigInt}(), ["x", "t₁", "t₂", "t₃"])
 x = gens[1]
 t = gens[2:4]