Thesis_Appendix_A.ipynb

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    "#  Appendix A : Inverse limit computation\n",
    "\n",
    "In this jupyter notebook we directly compute the inverse limit given in Appendix A of the PhD thesis: *Matroids, Flag Varieties, and Generalized Permutahedra*."
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   "cell_type": "code",
   "execution_count": 224,
   "id": "05627ce0",
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   "source": [
    "using Oscar"
   ]
  },
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    "**Functions**\n",
    "\n",
    "``ind_2_set`` returns set corresponding to an indicator vector.\n",
    "\n",
    "``symm_diff`` returns symmetric difference of two sets.\n",
    "\n",
    "``matroid_matrix`` returns a $d\\times n$ matrix which we may use to compute the associated ideal, semigroup, and ambient ring of the coordinate ring of its matroid stratum."
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      "text/plain": [
       "matroid_matrix (generic function with 1 method)"
      ]
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   "source": [
    "function ind_2_set(e)\n",
    "    \n",
    "    return [i for i in 1:length(e) if !(e[i]==0)]\n",
    "end\n",
    "\n",
    "function symm_diff(A,B)\n",
    "    \n",
    "    return setdiff(union(A,B),intersect(A,B))\n",
    "    \n",
    "end\n",
    "\n",
    "function matroid_matrix(M,F)\n",
    "    \n",
    "    Bs = bases(M)\n",
    "    \n",
    "    d = rank(M)\n",
    "    \n",
    "    dd = [i for i in 1:d]\n",
    "    \n",
    "    @req dd in Bs \"matroid not standardized, d is not a basis\"\n",
    "\n",
    "    n = length(M)\n",
    "    \n",
    "    P,x = polynomial_ring(F, :x=>(1:d,1:n-d))\n",
    "    \n",
    "    X = hcat(identity_matrix(P,d),matrix(P,x))\n",
    "    \n",
    "    \n",
    "    for i in 1:d, j in d+1:n\n",
    "        \n",
    "        if setdiff(union(setdiff(dd,i),j)) in Bs\n",
    "            \n",
    "            X[i,j] = X[i,j]\n",
    "            \n",
    "            \n",
    "        else\n",
    "            \n",
    "            X[i,j] = 0\n",
    "            \n",
    "        end\n",
    "\n",
    "        \n",
    "    end\n",
    "    \n",
    "    return X\n",
    "end"
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   "source": [
    "**1. Compute matroid subdivision of $\\Delta(2,4)$.**"
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   "id": "d46ea174",
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       "6-element SubObjectIterator{PointVector{QQFieldElem}}:\n",
       " [1, 1, 0, 0]\n",
       " [1, 0, 1, 0]\n",
       " [1, 0, 0, 1]\n",
       " [0, 1, 1, 0]\n",
       " [0, 1, 0, 1]\n",
       " [0, 0, 1, 1]"
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   "source": [
    "H24 = hypersimplex(2,4)\n",
    "V24 = vertices(H24)"
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       "Subdivision of points in ambient dimension 4"
      ]
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   "source": [
    "w = [0,1,0,0,0,0];\n",
    "Sw = subdivision_of_points(H24,w)"
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    "Observe that the subdivision $Sw$ is a split of $\\Delta(2,4)$. We visualize $Sw$. \n",
    "\n",
    "Recall $\\Delta(2,4)$ is isomorphic to the octahedron.\n",
    "\n",
    "We visualize the subdivision below."
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       "var scene = new THREE.Scene();\n",
       "var renderer = new THREE.WebGLRenderer( { antialias: true } );\n",
       "var svgRenderer = new THREE.SVGRenderer( { antialias: true } );\n",
       "renderer.setPixelRatio( window.devicePixelRatio );\n",
       "renderer.setClearColor(0xFFFFFF, 1);\n",
       "svgRenderer.setClearColor(0xFFFFFF, 1);\n",
       "three.appendChild(renderer.domElement);\n",
       "\n",
       "var frustumSize = 4;\n",
       "var cameras = [new THREE.PerspectiveCamera(75, 1, 0.1, 1000), new THREE.OrthographicCamera()];\n",
       "cameras.forEach(function(cam) {\n",
       "    cam.position.set(0, 0, 5);\n",
       "    cam.lookAt(0, 0, 0);  \n",
       "    cam.up.set(0, 1, 0);         \n",
       "});\n",
       "var controls = [new THREE.TrackballControls(cameras[0], three), new THREE.OrbitControls(cameras[1], three)];\n",
       "var camera, control;\n",
       "\n",
       "controls[0].zoomSpeed = 0.2;\n",
       "controls[0].rotateSpeed = 4;\n",
       "\n",
       "\n",
       "// class to allow move points together with labels and spheres\n",
       "var PMPoint = function (x,y,z) {\n",
       "   this.vector = new THREE.Vector3(x,y,z);\n",
       "   this.sprite = null;\n",
       "   this.sphere = null;\n",
       "}\n",
       "PMPoint.prototype.addLabel = function(labelsprite) {\n",
       "   this.sprite = labelsprite;\n",
       "   this.sprite.position.copy(this.vector);\n",
       "}\n",
       "PMPoint.prototype.addSphere = function(spheremesh) {\n",
       "   this.sphere = spheremesh;\n",
       "   this.sphere.position.copy(this.vector);\n",
       "}\n",
       "PMPoint.prototype.set = function(x,y,z) {\n",
       "   this.vector.set(x,y,z);\n",
       "   if (this.sprite) {\n",
       "      this.sprite.position.copy(this.vector);\n",
       "   }\n",
       "   if (this.sphere) {\n",
       "      this.sphere.position.copy(this.vector);\n",
       "   }\n",
       "}\n",
       "PMPoint.prototype.radius = function() {\n",
       "   if (this.sphere) {\n",
       "      return this.sphere.geometry.parameters.radius;\n",
       "   } else {\n",
       "      return 0;\n",
       "   }\n",
       "};\n",
       "// select the target node\n",
       "var target = document.querySelector('#model4000438617');\n",
       "\n",
       "// create an observer instance\n",
       "var observer = new MutationObserver(function(mutations) {\n",
       "   mutations.forEach(function(mutation) {\n",
       "      if (mutation.removedNodes && mutation.removedNodes.length > 0) {\n",
       "         cancelAnimationFrame(renderId);\n",
       "         observer.disconnect();\n",
       "         console.log(\"cancelled frame \"+renderId);\n",
       "      }\n",
       "   });\n",
       "});\n",
       "\n",
       "// configuration of the observer:\n",
       "var config = { childList: true, characterData: true }\n",
       "\n",
       "// pass in the target node, as well as the observer options\n",
       "while (target) {\n",
       "   if (target.className==\"output\") {\n",
       "      observer.observe(target, config);\n",
       "      break;\n",
       "   }\n",
       "   target = target.parentNode;\n",
       "}\n",
       "\n",
       "// COMMON_CODE_BLOCK_END\n",
       "\n",
       "var obj0 = new THREE.Object3D();\n",
       "obj0.name = \"unnamed__1\";\n",
       "obj0.userData.explodable = 1;\n",
       "obj0.userData.points = [];\n",
       "obj0.userData.points.push(new PMPoint(-1, 0, 0));\n",
       "obj0.userData.points.push(new PMPoint(0, 1, 0));\n",
       "obj0.userData.points.push(new PMPoint(0, -1, 0));\n",
       "obj0.userData.points.push(new PMPoint(0, 0, 1));\n",
       "obj0.userData.points.push(new PMPoint(0, 0, -1));\n",
       "\n",
       "obj0.userData.pointradii = 0.02;\n",
       "   <!-- Vertex style -->\n",
       "obj0.userData.pointmaterial = [new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } )];\n",
       "obj0.userData.pointlabels = [\"2\", \"3\", \"4\", \"5\", \"6\"];\n",
       "obj0.userData.edgeindices = [0, 1, 0, 2, 0, 3, 1, 3, 2, 3, 0, 4, 1, 4, 2, 4];\n",
       "   <!-- Edge style -->\n",
       "obj0.userData.edgematerial = new THREE.LineBasicMaterial( { color: 0x000000, linewidth: 1.5, transparent: false } );\n",
       "obj0.userData.facets = [[3, 2, 4, 1], [0, 2, 3], [4, 2, 0], [1, 4, 0], [3, 1, 0]];\n",
       "   <!-- Facet style -->\n",
       "obj0.userData.facetmaterial = new THREE.MeshBasicMaterial( { color: 0x77EC9E, depthFunc: THREE.LessDepth, depthWrite: false, opacity: 0.5, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } );\n",
       "init_object(obj0);\n",
       "scene.add(obj0);\n",
       "\n",
       "var obj1 = new THREE.Object3D();\n",
       "obj1.name = \"unnamed__2\";\n",
       "obj1.userData.explodable = 1;\n",
       "obj1.userData.points = [];\n",
       "obj1.userData.points.push(new PMPoint(1, 0, 0));\n",
       "obj1.userData.points.push(new PMPoint(0, 1, 0));\n",
       "obj1.userData.points.push(new PMPoint(0, -1, 0));\n",
       "obj1.userData.points.push(new PMPoint(0, 0, 1));\n",
       "obj1.userData.points.push(new PMPoint(0, 0, -1));\n",
       "\n",
       "obj1.userData.pointradii = 0.02;\n",
       "   <!-- Vertex style -->\n",
       "obj1.userData.pointmaterial = [new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } ),\n",
       "new THREE.MeshBasicMaterial( { color: 0xFF0000, side: THREE.DoubleSide, transparent: false } )];\n",
       "obj1.userData.pointlabels = [\"1\", \"3\", \"4\", \"5\", \"6\"];\n",
       "obj1.userData.edgeindices = [0, 1, 0, 2, 0, 3, 1, 3, 2, 3, 0, 4, 1, 4, 2, 4];\n",
       "   <!-- Edge style -->\n",
       "obj1.userData.edgematerial = new THREE.LineBasicMaterial( { color: 0x000000, linewidth: 1.5, transparent: false } );\n",
       "obj1.userData.facets = [[4, 2, 3, 1], [0, 2, 4], [3, 2, 0], [1, 3, 0], [4, 1, 0]];\n",
       "   <!-- Facet style -->\n",
       "obj1.userData.facetmaterial = new THREE.MeshBasicMaterial( { color: 0x77EC9E, depthFunc: THREE.LessDepth, depthWrite: false, opacity: 0.5, polygonOffset: true, polygonOffsetFactor: 1, polygonOffsetUnits: 0.5, side: THREE.DoubleSide, transparent: true } );\n",
       "init_object(obj1);\n",
       "scene.add(obj1);\n",
       "\n",
       "// COMMON_CODE_BLOCK_BEGIN\n",
       "function textSpriteMaterial(message, parameters) {\n",
       "    if ( parameters === undefined ) parameters = {};\n",
       "    var fontface = \"Helvetica\";\n",
       "    var fontsize = parameters.hasOwnProperty(\"fontsize\") ? parameters[\"fontsize\"] : 15;\n",
       "    fontsize = fontsize*10;\n",
       "    var lines = message.split('\\\\n');\n",
       "    var size = 512;\n",
       "    for(var i = 0; i<lines.length; i++){\n",
       "        var tmp = lines[i].length;\n",
       "        while(tmp*fontsize > size){\n",
       "           fontsize--;\n",
       "        }\n",
       "    }\n",
       "    \n",
       "    var canvas = document.createElement('canvas');\n",
       "    canvas.width = size;\n",
       "    canvas.height = size;\n",
       "    var context = canvas.getContext('2d');\n",
       "    context.fillStyle = \"rgba(255, 255, 255, 0)\";\n",
       "    context.fill();\n",
       "    context.font = fontsize + \"px \" + fontface;\n",
       "    \n",
       "    // text color\n",
       "    context.fillStyle = \"rgba(0, 0, 0, 1.0)\";\n",
       "     for(var i = 0; i<lines.length; i++){\n",
       "        context.fillText(lines[i], size/2, size/2+i*fontsize);\n",
       "     }\n",
       "    \n",
       "    // canvas contents will be used for a texture\n",
       "    var texture = new THREE.Texture(canvas);\n",
       "    texture.needsUpdate = true;\n",
       "    \n",
       "    var spriteMaterial = new THREE.SpriteMaterial({map: texture, depthTest: true, depthWrite: false, polygonOffset: true, polygonOffsetFactor: -1, polygonOffsetUnits: 1 });\n",
       "    return spriteMaterial;\n",
       "}\n",
       "\n",
       "\n",
       "// ---------------------- INITIALIZING OBJECTS--------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "function init_object(obj) {\n",
       "    if (obj.userData.hasOwnProperty(\"pointmaterial\")) {\n",
       "        init_points(obj);\n",
       "        modelContains.points = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"pointlabels\")) {\n",
       "        init_pointlabels(obj);\n",
       "        modelContains.pointlabels = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"edgematerial\")) {\n",
       "        init_lines(obj);\n",
       "        modelContains.lines = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"edgelabels\")) {\n",
       "        init_edgelabels(obj);\n",
       "        modelContains.edgelabels = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"arrowstyle\")) {\n",
       "        init_arrowheads(obj);\n",
       "        modelContains.arrowheads = true;\n",
       "    }\n",
       "    if (obj.userData.hasOwnProperty(\"facetmaterial\")) {\n",
       "        init_faces(obj);\n",
       "        modelContains.faces = true;\n",
       "    }\n",
       "}\n",
       "\n",
       "function init_points(obj) {\n",
       "    var pointgroup = new THREE.Group();\n",
       "    pointgroup.name = \"points\";\n",
       "    var points = obj.userData.points;\n",
       "    var radii = obj.userData.pointradii;\n",
       "    var materials = obj.userData.pointmaterial;\n",
       "    var geometry,material;\n",
       "    if (!Array.isArray(radii)) {\n",
       "        geometry = new THREE.SphereBufferGeometry(radii);  \n",
       "    }\n",
       "    if (!Array.isArray(materials)) {\n",
       "        material = materials;\n",
       "    }\n",
       "    for (var i=0; i<points.length; i++) {\n",
       "        var point = points[i];\n",
       "        if (Array.isArray(radii)) {\n",
       "            if (radii[i] == 0) {\n",
       "                continue;\n",
       "            }\n",
       "            geometry = new THREE.SphereBufferGeometry(radii[i]);  \n",
       "        } \n",
       "        if (Array.isArray(materials)) {\n",
       "            material = materials[i];     \n",
       "        } \n",
       "        var sphere = new THREE.Mesh(geometry, material);\n",
       "        point.addSphere(sphere);\n",
       "        pointgroup.add(sphere);\n",
       "    }\n",
       "    obj.add(pointgroup);\n",
       "}\n",
       "\n",
       "function init_pointlabels(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var labels = obj.userData.pointlabels;\n",
       "    var pointlabels = new THREE.Group();\n",
       "    pointlabels.name = \"pointlabels\";\n",
       "    if (Array.isArray(labels)) {\n",
       "        for (var i=0; i<points.length; i++) {\n",
       "            var point = points[i];\n",
       "            var spriteMaterial = textSpriteMaterial( labels[i] );\n",
       "\t        var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            point.addLabel(sprite);\n",
       "            pointlabels.add(sprite);\n",
       "        }\n",
       "    } else {\n",
       "        var spriteMaterial = textSpriteMaterial( labels );\n",
       "        for (var i=0; i<points.length; i++) {\n",
       "            var point = points[i];\n",
       "\t        var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            point.addLabel(sprite);\n",
       "            pointlabels.add(sprite);\n",
       "        }\n",
       "    }\n",
       "    obj.add(pointlabels);\n",
       "}\n",
       "\n",
       "function init_lines(obj) {\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var points = obj.userData.points;\n",
       "    var materials = obj.userData.edgematerial;\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    var bufarr = new Float32Array( obj.userData.edgeindices.length * 3 );\n",
       "    var bufattr = new THREE.Float32BufferAttribute( bufarr, 3 );\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    geometry.setAttribute('position', bufattr);\n",
       "    if (Array.isArray(materials)) {     \n",
       "        for (var i=0; i<materials.length; i++) {\n",
       "            geometry.addGroup(2*i,2,i);\n",
       "        }\n",
       "    }\n",
       "    var lines = new THREE.LineSegments(geometry, materials);\n",
       "    lines.name = \"lines\";\n",
       "    obj.add(lines);\n",
       "    updateEdgesPosition(obj);\n",
       "}\n",
       "\n",
       "function init_edgelabels(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var labels = obj.userData.edgelabels;\n",
       "    var edgelabels = new THREE.Group();\n",
       "    edgelabels.name = \"edgelabels\";\n",
       "    if (Array.isArray(labels)) {\n",
       "        for (var i=0; i<edgeindices.length/2; i++) {\n",
       "            var spriteMaterial = textSpriteMaterial( labels[i] );\n",
       "            var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            sprite.position.copy(new THREE.Vector3().addVectors(points[edgeindices[2*i]].vector,points[edgeindices[2*i+1]].vector).multiplyScalar(0.5));\n",
       "            edgelabels.add(sprite);\n",
       "        }\n",
       "    } else {\n",
       "        var spriteMaterial = textSpriteMaterial( labels );\n",
       "        for (var i=0; i<edgeindices.length/2; i++) {\n",
       "            var sprite = new THREE.Sprite(spriteMaterial);\n",
       "            sprite.position.copy(new THREE.Vector3().addVectors(points[edgeindices[2*i]].vector,points[edgeindices[2*i+1]].vector).multiplyScalar(0.5));\n",
       "            edgelabels.add(sprite);\n",
       "        }\n",
       "    }\n",
       "    obj.add(edgelabels);\n",
       "}\n",
       "\n",
       "function init_arrowheads(obj) {\n",
       "    var arrowheads = new THREE.Group();\n",
       "    arrowheads.name = \"arrowheads\";\n",
       "    var arrowstyle = obj.userData.arrowstyle;\n",
       "    var edgeindices = obj.userData.edgeindices;\n",
       "    var edgematerials = obj.userData.edgematerial;\n",
       "    var points = obj.userData.points;\n",
       "    var material;\n",
       "    if (!Array.isArray(edgematerials)) {\n",
       "        material = new THREE.MeshBasicMaterial( {color: edgematerials.color} );\n",
       "    }\n",
       "\n",
       "    for (var i=0; i<edgeindices.length; i=i+2) {\n",
       "        var start = points[edgeindices[i]];\n",
       "        var end = points[edgeindices[i+1]];\n",
       "        var dist = start.vector.distanceTo( end.vector ) - start.radius() - end.radius();\n",
       "        if (dist <= 0) {\n",
       "            continue;\n",
       "        }\n",
       "        var dir = new THREE.Vector3().subVectors(end.vector,start.vector);\n",
       "        dir.normalize();\n",
       "        var axis = new THREE.Vector3().set(dir.z,0,-dir.x);\n",
       "        axis.normalize();\n",
       "        var radians = Math.acos( dir.y );\n",
       "        var radius = dist/25;\n",
       "        var height = dist/5;\n",
       "        var geometry = new THREE.ConeBufferGeometry(radius,height);\n",
       "        var position = new THREE.Vector3().addVectors(start.vector,dir.clone().multiplyScalar(start.radius()+dist-height/2));\n",
       "        if (Array.isArray(edgematerials)) {\n",
       "            material = new THREE.MeshBasicMaterial( {color: edgematerials[i].color} );\n",
       "        }\n",
       "        var cone = new THREE.Mesh( geometry, material );\n",
       "        cone.quaternion.setFromAxisAngle(axis,radians);;\n",
       "        cone.position.copy(position);;\n",
       "        arrowheads.add(cone);\n",
       "    }\n",
       "    obj.add(arrowheads);\n",
       "}\n",
       "\n",
       "function init_faces(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var facets = obj.userData.facets;\n",
       "    obj.userData.triangleindices = [];\n",
       "    for (var i=0; i<facets.length; i++) {\n",
       "        facet = facets[i];\n",
       "        for (var t=0; t<facet.length-2; t++) {\n",
       "            obj.userData.triangleindices.push(facet[0],facet[t+1],facet[t+2]);  \n",
       "        }\n",
       "    }\n",
       "    var bufarr = new Float32Array( obj.userData.triangleindices.length * 3 );\n",
       "    var bufattr = new THREE.Float32BufferAttribute(bufarr,3);\n",
       "    \n",
       "    var materials = obj.userData.facetmaterial;\n",
       "    var geometry = new THREE.BufferGeometry();\n",
       "    var frontmaterials = [];\n",
       "    var backmaterials = [];\n",
       "    geometry.setAttribute('position',bufattr);\n",
       "    if (Array.isArray(materials)) {\n",
       "        var tricount = 0;\n",
       "        var facet;\n",
       "        for (var i=0; i<facets.length; i++) {\n",
       "            facet = facets[i];\n",
       "            geometry.addGroup(tricount,(facet.length-2)*3,i);\n",
       "            tricount += (facet.length-2)*3;\n",
       "        }\n",
       "        for (var j=0; j<materials.length; j++) {\n",
       "            var fmat = materials[j].clone()\n",
       "            fmat.side = THREE.FrontSide;\n",
       "            frontmaterials.push(fmat);\n",
       "            var bmat = materials[j].clone()\n",
       "            bmat.side = THREE.BackSide;\n",
       "            backmaterials.push(bmat);\n",
       "            obj.userData.facetmaterial = frontmaterials.concat(backmaterials);\n",
       "        }\n",
       "    } else if (materials instanceof THREE.Material) {\n",
       "        frontmaterials = materials.clone()\n",
       "        frontmaterials.side = THREE.FrontSide;\n",
       "        backmaterials = materials.clone()\n",
       "        backmaterials.side = THREE.BackSide;\n",
       "        obj.userData.facetmaterial = [frontmaterials, backmaterials];\n",
       "    }\n",
       "    // duplicating the object with front and back should avoid transparency issues\n",
       "    var backmesh = new THREE.Mesh(geometry, backmaterials);\n",
       "    // meshname is used to show/hide objects\n",
       "    backmesh.name = \"backfaces\";\n",
       "    obj.add(backmesh);\n",
       "    var frontmesh = new THREE.Mesh(geometry, frontmaterials);\n",
       "    frontmesh.name = \"frontfaces\";\n",
       "    obj.add(frontmesh);\n",
       "    updateFacesPosition(obj);\n",
       "}\n",
       "// //INITIALIZING\n",
       "\n",
       "\n",
       "function updateFacesPosition(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var indices = obj.userData.triangleindices;\n",
       "    var faces = obj.getObjectByName(\"frontfaces\");\n",
       "    var ba = faces.geometry.getAttribute(\"position\");\n",
       "    for (var i=0; i<indices.length; i++) {\n",
       "        ba.setXYZ(i, points[indices[i]].vector.x, points[indices[i]].vector.y ,points[indices[i]].vector.z); \n",
       "    }\n",
       "    faces.geometry.attributes.position.needsUpdate = true;\n",
       "    \n",
       "}\n",
       "\n",
       "function updateEdgesPosition(obj) {\n",
       "    var points = obj.userData.points;\n",
       "    var indices = obj.userData.edgeindices;\n",
       "    var lines = obj.getObjectByName(\"lines\");\n",
       "    var ba = lines.geometry.getAttribute(\"position\"); \n",
       "    for (var i=0; i<indices.length; i++) {\n",
       "        ba.setXYZ(i, points[indices[i]].vector.x, points[indices[i]].vector.y ,points[indices[i]].vector.z); \n",
       "    }\n",
       "    lines.geometry.attributes.position.needsUpdate = true;\n",
       "}\n",
       "\n",
       "function onWindowResize() {\n",
       "    renderer.setSize( three.clientWidth, three.clientHeight );\n",
       "    svgRenderer.setSize( three.clientWidth, three.clientHeight );\n",
       "    updateCamera();\n",
       "}\n",
       "\n",
       "function updateCamera() {\n",
       "    var width = three.clientWidth;\n",
       "    var height = three.clientHeight;\n",
       "    var aspect = width / height;\n",
       "    if (camera.type == \"OrthographicCamera\") {\n",
       "        camera.left = frustumSize * aspect / - 2;\n",
       "        camera.right = frustumSize * aspect / 2;\n",
       "        camera.top = frustumSize / 2;\n",
       "        camera.bottom = - frustumSize / 2;\n",
       "    } else if (camera.type == \"PerspectiveCamera\") {\n",
       "        camera.aspect = aspect;\n",
       "    }\n",
       "    camera.updateProjectionMatrix();\n",
       "}\n",
       "\n",
       "function changeCamera(event) {\n",
       "    var selindex = event.currentTarget.selectedIndex;\n",
       "    camera = cameras[selindex];\n",
       "    control = controls[selindex];\n",
       "    control.enabled = true; \n",
       "    for (var i=0; i<controls.length; i++) {\n",
       "        if (i!=selindex) {\n",
       "            controls[i].enabled = false;\n",
       "        }\n",
       "    }\n",
       "    updateCamera();\n",
       "}\n",
       "\n",
       "var camtypenode = document.getElementById('cameraType_0');\n",
       "camtypenode.onchange = changeCamera;\n",
       "camtypenode.dispatchEvent(new Event('change'));\n",
       "\n",
       "onWindowResize();\n",
       "window.addEventListener('resize', onWindowResize);\t\n",
       "\n",
       "\n",
       "var xRotationEnabled = false;\n",
       "var yRotationEnabled = false;\n",
       "var zRotationEnabled = false;\n",
       "var rotationSpeedFactor = 1;\n",
       "var settingsShown = false;\n",
       "var labelsShown = true;\n",
       "var intervals = [];\n",
       "var timeouts = [];\n",
       "var explodingSpeed = 0.05;\n",
       "var explodeScale = 0.000001;\n",
       "var XMLS = new XMLSerializer();\n",
       "var svgElement;\n",
       "var renderId;\n",
       "\n",
       "var render = function () {\n",
       "\n",
       "\trenderId = requestAnimationFrame(render);\n",
       "\n",
       "//\tcomment in for automatic explosion\n",
       "//\texplode(updateFactor());\n",
       "\n",
       "    var phi = 0.02 * rotationSpeedFactor;\n",
       "\n",
       "    if (xRotationEnabled) {\n",
       "        scene.rotation.x += phi;\n",
       "    }\n",
       "    if (yRotationEnabled) {\n",
       "        scene.rotation.y += phi;\n",
       "    }\n",
       "    if (zRotationEnabled) {\n",
       "        scene.rotation.z += phi;\n",
       "    }\n",
       "\n",
       "    control.update();\n",
       "    renderer.render(scene, camera);\n",
       "};\n",
       "\n",
       "if ( THREE.WEBGL.isWebGLAvailable() ) {\n",
       "\trender();\n",
       "} else {\n",
       "\tvar warning = WEBGL.getWebGLErrorMessage();\n",
       "\tthree.appendChild( warning );\n",
       "}\n",
       "    \n",
       "function changeTransparency() {\n",
       "    var opacity = 1-Number(event.currentTarget.value);\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        child = scene.children[i];\n",
       "        if ( child.userData.hasOwnProperty(\"facetmaterial\") ) {\n",
       "            if (Array.isArray(child.userData.facetmaterial)) {\n",
       "                for (var j=0; j<child.userData.facetmaterial.length; j++) {\n",
       "                    child.userData.facetmaterial[j].opacity = opacity;\n",
       "                }\n",
       "            } else {\n",
       "                child.userData.facetmaterial.opacity = opacity;\n",
       "            }    \n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function toggleDepthWrite(event) {\n",
       "    depthwrite = event.currentTarget.checked;\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        child = scene.children[i];\n",
       "        if ( child.userData.hasOwnProperty(\"facetmaterial\") ) {\n",
       "            if (Array.isArray(child.userData.facetmaterial)) {\n",
       "                for (var j=0; j<child.userData.facetmaterial.length; j++) {\n",
       "                    child.userData.facetmaterial[j].depthWrite = depthwrite;\n",
       "                }\n",
       "            } else {\n",
       "                child.userData.facetmaterial.depthWrite = depthWrite;\n",
       "            }    \n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function changeRotationX(event){\n",
       "    xRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "function changeRotationY(event){\n",
       "    yRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "function changeRotationZ(event){\n",
       "    zRotationEnabled = event.currentTarget.checked;\n",
       "}\t\n",
       "\n",
       "\n",
       "function changeRotationSpeedFactor(event){\n",
       "    rotationSpeedFactor = Number(event.currentTarget.value);\n",
       "}\n",
       "\n",
       "function resetScene(){\n",
       "    scene.rotation.set(0,0,0);\n",
       "    camera.position.set(0,0,5);\n",
       "    camera.up.set(0,1,0);\n",
       "}\n",
       "\n",
       "function showSettings(event){\n",
       "    document.getElementById('settings_0').style.visibility = 'visible';\n",
       "    document.getElementById('showSettingsButton_0').style.visibility = 'hidden';\n",
       "    document.getElementById('hideSettingsButton_0').style.visibility = 'visible';\n",
       "    settingsShown = true;\n",
       "}\n",
       "\n",
       "function hideSettings(event){\n",
       "    document.getElementById('settings_0').style.visibility = 'hidden';\n",
       "    document.getElementById('showSettingsButton_0').style.visibility = 'visible';\n",
       "    document.getElementById('hideSettingsButton_0').style.visibility = 'hidden';\n",
       "    settingsShown = false;\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "var pos = 150* Math.PI;\n",
       "\n",
       "function updateFactor() {\n",
       "    pos++;\n",
       "    return Math.sin(.01*pos)+1;\n",
       "}\n",
       "\n",
       "// ------------------------ FOLDING ------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "// rotate point p around axis defined by points p1 and p2 by given angle\n",
       "function rotate(p, p1, p2, angle ){   \n",
       "    angle = -angle;\n",
       "    var x = p.x, y = p.y, z = p.z, \n",
       "    a = p1.x, b = p1.y, c = p1.z, \n",
       "    u = p2.x-p1.x, v = p2.y-p1.y, w = p2.z-p1.z;\n",
       "    var result = [];\n",
       "    var L = u*u + v*v + w*w;\n",
       "    var sqrt = Math.sqrt;\n",
       "    var cos = Math.cos;\n",
       "    var sin = Math.sin;\n",
       "\n",
       "    result[0] = ((a*(v*v+w*w)-u*(b*v+c*w-u*x-v*y-w*z))*(1-cos(angle))+L*x*cos(angle)+sqrt(L)*(-c*v+b*w-w*y+v*z)*sin(angle))/L;\n",
       "    result[1] = ((b*(u*u+w*w)-v*(a*u+c*w-u*x-v*y-w*z))*(1-cos(angle))+L*y*cos(angle)+sqrt(L)*(c*u-a*w+w*x-u*z)*sin(angle))/L;\n",
       "    result[2] = ((c*(u*u+v*v)-w*(a*u+b*v-u*x-v*y-w*z))*(1-cos(angle))+L*z*cos(angle)+sqrt(L)*(-b*u+a*v-v*x+u*y)*sin(angle))/L;\n",
       "\n",
       "    return result;\n",
       "}\n",
       "\n",
       "var fold = function(event){\n",
       "    var obj = foldables[Number(event.currentTarget.name)];\n",
       "    var foldvalue = Number(event.currentTarget.value);\n",
       "    var scale = foldvalue - obj.userData.oldscale;\n",
       "\n",
       "    for (var j=0; j<obj.userData.axes.length; j++) {\n",
       "        rotateVertices(obj, j, scale);\n",
       "    }\n",
       "    update(obj);\n",
       "    obj.userData.oldscale += scale;\n",
       "    lookAtBarycenter(obj);\n",
       "}\n",
       "\n",
       "function lookAtBarycenter(obj){\n",
       "    control.target = barycenter(obj);\n",
       "}\n",
       "\n",
       "function barycenter(obj) {\n",
       "    var center = new THREE.Vector3(0,0,0);\n",
       "    var points = obj.userData.points;\n",
       "    for (var i=0; i<points.length; i++){\n",
       "        center.add(points[i].vector);\n",
       "    }\n",
       "    center.divideScalar(points.length);\n",
       "    return center;\n",
       "}\n",
       "\n",
       "function rotateVertices(obj, edge, scale) {\n",
       "    var axes = obj.userData.axes;\n",
       "    var subtrees = obj.userData.subtrees;\n",
       "    var points = obj.userData.points;\n",
       "    var angles = obj.userData.angles;\n",
       "    if (edge < axes.length){\n",
       "        for (var j=0; j<subtrees[edge].length; j++){\n",
       "            var rotP = rotate(points[subtrees[edge][j]].vector, points[axes[edge][0]].vector,points[axes[edge][1]].vector, scale * (Math.PI - angles[edge]));\n",
       "            points[subtrees[edge][j]].set(rotP[0],rotP[1],rotP[2]);\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function update(obj) {\n",
       "   updateFacesPosition(obj);\n",
       "   updateEdgesPosition(obj);\n",
       "}\n",
       "\n",
       "if (foldables.length) {\n",
       "    var settings = document.getElementById('settings_0');\n",
       "    var foldDiv = document.createElement('div');\n",
       "    foldDiv.id = 'fold_0';\n",
       "    var title = document.createElement('strong');\n",
       "    title.innerHTML = 'Fold';\n",
       "    foldDiv.appendChild(title);\n",
       "    foldDiv.className = 'group';\n",
       "    for (var i=0; i<foldables.length; i++) {\n",
       "        var range = document.createElement('input');\n",
       "        range.type = 'range';\n",
       "        range.min = 0;\n",
       "        range.max = 1;\n",
       "        range.value = 0;\n",
       "        range.step = 0.001;\n",
       "        range.name = String(i);\n",
       "        range.oninput = fold;\n",
       "        foldDiv.appendChild(range);\n",
       "    }\n",
       "    lookAtBarycenter(foldables[0]);\n",
       "    settings.insertBefore(foldDiv,settings.childNodes[0]);\n",
       "}\n",
       "\n",
       "    \n",
       "// ---------------------- EXPLOSION ------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "if (explodableModel) {\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        obj = scene.children[i];\n",
       "        if ( obj.userData.explodable ) {\n",
       "            computeCentroid(obj);\n",
       "        }\n",
       "    }\n",
       "    document.getElementById('explodeRange_0').oninput = triggerExplode;\n",
       "    document.getElementById('explodeCheckbox_0').onchange = triggerAutomaticExplode;\n",
       "    document.getElementById('explodingSpeedRange_0').oninput = setExplodingSpeed;\n",
       "    explode(0.000001);\n",
       "}\n",
       "\n",
       "function computeCentroid(obj) {\n",
       "    centroid = new THREE.Vector3();\n",
       "    obj.userData.points.forEach(function(pmpoint) {\n",
       "        centroid.add(pmpoint.vector);\t\t\t\n",
       "    });\n",
       "    centroid.divideScalar(obj.userData.points.length);\n",
       "    obj.userData.centroid = centroid;\n",
       "}\n",
       "\n",
       "function explode(factor) {\n",
       "    for (var i=0; i<scene.children.length; i++) {\n",
       "        var obj = scene.children[i];\n",
       "        if (obj.userData.hasOwnProperty(\"centroid\")) { \n",
       "            var c = obj.userData.centroid;\n",
       "            obj.position.set(c.x*factor, c.y*factor, c.z*factor);\n",
       "        }\n",
       "    }\t\n",
       "}\n",
       "\n",
       "function triggerExplode(event){\n",
       "    explodeScale = Number(event.currentTarget.value);\n",
       "    explode(explodeScale);\n",
       "}\n",
       "\n",
       "function setExplodingSpeed(event){\n",
       "    explodingSpeed = Number(event.currentTarget.value);\n",
       "}\n",
       "\n",
       "function triggerAutomaticExplode(event){\n",
       "    if (event.currentTarget.checked){\n",
       "        startExploding();\n",
       "    } else {\n",
       "        clearIntervals();\n",
       "    }\t\n",
       "}\n",
       "\n",
       "function startExploding(){\n",
       "    intervals.push(setInterval(explodingInterval, 25));\n",
       "}\n",
       "\n",
       "\n",
       "function explodingInterval(){\n",
       "    explodeScale += explodingSpeed;\n",
       "    if (explodeScale <= 6){ \n",
       "        explode(explodeScale);\n",
       "    }\n",
       "    else{\n",
       "        explode(6);\n",
       "        explodeScale = 6;\n",
       "        clearIntervals();\n",
       "        timeouts.push(setTimeout(startUnexploding, 3000));\n",
       "    }\n",
       "    document.getElementById('explodeRange_0').value = explodeScale;\n",
       "}\n",
       "\n",
       "\n",
       "function startUnexploding(){\n",
       "    intervals.push(setInterval(unexplodingInterval, 25));\n",
       "}\n",
       "\n",
       "function unexplodingInterval(){\n",
       "    explodeScale -= explodingSpeed;\n",
       "    if (explodeScale >= 0){\t\n",
       "        explode(explodeScale);\n",
       "    }\n",
       "    else {\n",
       "        explode(0);\n",
       "        explodeScale = 0;\n",
       "        clearIntervals();\n",
       "        timeouts.push(setTimeout(startExploding, 3000));\n",
       "    }\n",
       "    document.getElementById('explodeRange_0').value = explodeScale;\n",
       "}\n",
       "\n",
       "function clearIntervals(){\n",
       "    intervals.forEach(function(interval){\n",
       "        clearInterval(interval);\n",
       "    });\n",
       "    intervals = [];\n",
       "    timeouts.forEach(function(timeout){\n",
       "        clearTimeout(timeout);\n",
       "    });\n",
       "    timeouts = [];\n",
       "}\n",
       "\n",
       "// ---------------------- DISPLAY --------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "const objectTypeInnerHTMLs = { points: \"Points\", pointlabels: \"Point labels\", lines: \"Edges\", edgelabels: \"Edge labels\", faces: \"Faces\", arrowheads: \"Arrow heads\" };\n",
       "const objectTypeVisible = {};\n",
       "Object.assign(objectTypeVisible,modelContains);\n",
       "const sortedObjectTypeKeys = Object.keys(objectTypeInnerHTMLs).sort();\n",
       "const shownObjectTypesList = document.getElementById('shownObjectTypesList_0');\n",
       "\n",
       "function setVisibility(bool,objname) {\n",
       "    for (var i=0; i<scene.children.length; i++){\n",
       "        var obj = scene.children[i].getObjectByName(objname);\n",
       "        if (obj) {\n",
       "            obj.visible = bool;\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "function toggleObjectTypeVisibility(event){\n",
       "    var name = event.currentTarget.name;\n",
       "    var checked = event.currentTarget.checked;\n",
       "    objectTypeVisible[name] = checked;\n",
       "    if (name == \"faces\") {\n",
       "        setVisibility(checked,\"frontfaces\");\n",
       "        setVisibility(checked,\"backfaces\");\n",
       "    } else {\n",
       "        setVisibility(checked,name);\n",
       "    }\n",
       "}\n",
       "\n",
       "for (var i=0; i<sortedObjectTypeKeys.length; i++){\n",
       "    var key = sortedObjectTypeKeys[i];\n",
       "    if (modelContains[key]) {\n",
       "        var objTypeNode = document.createElement('span');\n",
       "        objTypeNode.innerHTML = objectTypeInnerHTMLs[key] + '<br>';\n",
       "        var checkbox = document.createElement('input');\n",
       "        checkbox.type = 'checkbox';\n",
       "        checkbox.checked = true;\n",
       "        checkbox.name = key;\n",
       "        checkbox.onchange = toggleObjectTypeVisibility;\n",
       "        shownObjectTypesList.appendChild(checkbox);\n",
       "        shownObjectTypesList.appendChild(objTypeNode);\n",
       "    }\n",
       "}\n",
       "\n",
       "// ------------------------------------------------------\n",
       "\n",
       "function toggleObjectVisibility(event){\n",
       "    var nr = Number(event.currentTarget.name);\n",
       "    scene.children[nr].visible = event.currentTarget.checked;\n",
       "}\n",
       "\n",
       "// append checkboxes for displaying or hiding objects\n",
       "var shownObjectsList = document.getElementById('shownObjectsList_0');\n",
       "for (var i=0; i<scene.children.length; i++){\n",
       "    obj = scene.children[i];\n",
       "    var objNode = document.createElement('span');\n",
       "    objNode.innerHTML = obj.name + '<br>';\n",
       "    var checkbox = document.createElement('input');\n",
       "    checkbox.type = 'checkbox';\n",
       "    checkbox.checked = true;\n",
       "    checkbox.name = String(i);\n",
       "    checkbox.onchange = toggleObjectVisibility;\n",
       "    shownObjectsList.appendChild(checkbox);\n",
       "    shownObjectsList.appendChild(objNode);\n",
       "}\n",
       "\n",
       "// ---------------------- SVG ------------------------------------------------------\n",
       "// ---------------------------------------------------------------------------------\n",
       "\n",
       "function takeSvgScreenshot() {\n",
       "    if (objectTypeVisible[\"pointlabels\"]) {\n",
       "        setVisibility(false,\"pointlabels\");\n",
       "    }\n",
       "    if (objectTypeVisible[\"edgelabels\"]) {\n",
       "        setVisibility(false,\"edgelabels\");\n",
       "    }\n",
       "    svgRenderer.render(scene,camera);\n",
       "    svgElement = XMLS.serializeToString(svgRenderer.domElement);\n",
       "    \n",
       "    if (objectTypeVisible[\"pointlabels\"]) {\n",
       "        setVisibility(true,\"pointlabels\");\n",
       "    }\n",
       "    if (objectTypeVisible[\"edgelabels\"]) {\n",
       "        setVisibility(true,\"edgelabels\");\n",
       "    }\n",
       "\n",
       "    if (document.getElementById('tab_0').checked){\n",
       "        //show in new tab\n",
       "        var myWindow = window.open(\"\",\"\");\n",
       "        myWindow.document.body.innerHTML = svgElement;\n",
       "    } else{\n",
       "        // download svg file \n",
       "        download(\"screenshot.svg\", svgElement);\n",
       "    }\n",
       "}\n",
       "\n",
       "function download(filename, text) {\n",
       "    var element = document.createElement('a');\n",
       "    element.setAttribute('href', 'data:text/plain;charset=utf-8,' + encodeURIComponent(text));\n",
       "    element.setAttribute('download', filename);\n",
       "\n",
       "    element.style.display = 'none';\n",
       "    document.body.appendChild(element);\n",
       "\n",
       "    element.click();\n",
       "\n",
       "    document.body.removeChild(element);\n",
       "}\n",
       "\n",
       "\n",
       "document.getElementById('transparencyRange_0').oninput = changeTransparency;\n",
       "document.getElementById('depthWriteCheckbox_0').onchange = toggleDepthWrite;\n",
       "document.getElementById('changeRotationX_0').onchange = changeRotationX;\n",
       "document.getElementById('changeRotationY_0').onchange = changeRotationY;\n",
       "document.getElementById('changeRotationZ_0').onchange = changeRotationZ;\n",
       "document.getElementById('resetButton_0').onclick = resetScene;\n",
       "document.getElementById('rotationSpeedRange_0').oninput = changeRotationSpeedFactor;\n",
       "document.getElementById('takeScreenshot_0').onclick = takeSvgScreenshot;\n",
       "document.getElementById('showSettingsButton_0').onclick = showSettings;\n",
       "document.getElementById('hideSettingsButton_0').onclick = hideSettings;\n",
       "\n",
       "\n",
       "// ------------------ SHORTCUTS --------------------------------------------\n",
       "// -------------------------------------------------------------------------\n",
       "\n",
       "/**\n",
       " * http://www.openjs.com/scripts/events/keyboard_shortcuts/\n",
       " * Version : 2.01.B\n",
       " * By Binny V A\n",
       " * License : BSD\n",
       " */\n",
       "shortcut = {\n",
       "\t'all_shortcuts':{},//All the shortcuts are stored in this array\n",
       "\t'add': function(shortcut_combination,callback,opt) {\n",
       "\t\t//Provide a set of default options\n",
       "\t\tvar default_options = {\n",
       "\t\t\t'type':'keydown',\n",
       "\t\t\t'propagate':false,\n",
       "\t\t\t'disable_in_input':false,\n",
       "\t\t\t'target':document,\n",
       "\t\t\t'keycode':false\n",
       "\t\t}\n",
       "\t\tif(!opt) opt = default_options;\n",
       "\t\telse {\n",
       "\t\t\tfor(var dfo in default_options) {\n",
       "\t\t\t\tif(typeof opt[dfo] == 'undefined') opt[dfo] = default_options[dfo];\n",
       "\t\t\t}\n",
       "\t\t}\n",
       "\n",
       "\t\tvar ele = opt.target;\n",
       "\t\tif(typeof opt.target == 'string') ele = document.getElementById(opt.target);\n",
       "\t\tvar ths = this;\n",
       "\t\tshortcut_combination = shortcut_combination.toLowerCase();\n",
       "\n",
       "\t\t//The function to be called at keypress\n",
       "\t\tvar func = function(e) {\n",
       "\t\t\te = e || window.event;\n",
       "\t\t\t\n",
       "\t\t\tif(opt['disable_in_input']) { //Don't enable shortcut keys in Input, Textarea fields\n",
       "\t\t\t\tvar element;\n",
       "\t\t\t\tif(e.target) element=e.target;\n",
       "\t\t\t\telse if(e.srcElement) element=e.srcElement;\n",
       "\t\t\t\tif(element.nodeType==3) element=element.parentNode;\n",
       "\n",
       "\t\t\t\tif(element.tagName == 'INPUT' || element.tagName == 'TEXTAREA') return;\n",
       "\t\t\t}\n",
       "\t\n",
       "\t\t\t//Find Which key is pressed\n",
       "\t\t\tif (e.keyCode) code = e.keyCode;\n",
       "\t\t\telse if (e.which) code = e.which;\n",
       "\t\t\tvar character = String.fromCharCode(code).toLowerCase();\n",
       "\t\t\t\n",
       "\t\t\tif(code == 188) character=\",\"; //If the user presses , when the type is onkeydown\n",
       "\t\t\tif(code == 190) character=\".\"; //If the user presses , when the type is onkeydown\n",
       "\n",
       "\t\t\tvar keys = shortcut_combination.split(\"+\");\n",
       "\t\t\t//Key Pressed - counts the number of valid keypresses - if it is same as the number of keys, the shortcut function is invoked\n",
       "\t\t\tvar kp = 0;\n",
       "\t\t\t\n",
       "\t\t\t//Work around for stupid Shift key bug created by using lowercase - as a result the shift+num combination was broken\n",
       "\t\t\tvar shift_nums = {\n",
       "\t\t\t\t\"`\":\"~\",\n",
       "\t\t\t\t\"1\":\"!\",\n",
       "\t\t\t\t\"2\":\"@\",\n",
       "\t\t\t\t\"3\":\"#\",\n",
       "\t\t\t\t\"4\":\"$\",\n",
       "\t\t\t\t\"5\":\"%\",\n",
       "\t\t\t\t\"6\":\"^\",\n",
       "\t\t\t\t\"7\":\"&\",\n",
       "\t\t\t\t\"8\":\"*\",\n",
       "\t\t\t\t\"9\":\"(\",\n",
       "\t\t\t\t\"0\":\")\",\n",
       "\t\t\t\t\"-\":\"_\",\n",
       "\t\t\t\t\"=\":\"+\",\n",
       "\t\t\t\t\";\":\":\",\n",
       "\t\t\t\t\"'\":\"\\\"\",\n",
       "\t\t\t\t\",\":\"<\",\n",
       "\t\t\t\t\".\":\">\",\n",
       "\t\t\t\t\"/\":\"?\",\n",
       "\t\t\t\t\"\\\\\":\"|\"\n",
       "\t\t\t}\n",
       "\t\t\t//Special Keys - and their codes\n",
       "\t\t\tvar special_keys = {\n",
       "\t\t\t\t'esc':27,\n",
       "\t\t\t\t'escape':27,\n",
       "\t\t\t\t'tab':9,\n",
       "\t\t\t\t'space':32,\n",
       "\t\t\t\t'return':13,\n",
       "\t\t\t\t'enter':13,\n",
       "\t\t\t\t'backspace':8,\n",
       "\t\n",
       "\t\t\t\t'scrolllock':145,\n",
       "\t\t\t\t'scroll_lock':145,\n",
       "\t\t\t\t'scroll':145,\n",
       "\t\t\t\t'capslock':20,\n",
       "\t\t\t\t'caps_lock':20,\n",
       "\t\t\t\t'caps':20,\n",
       "\t\t\t\t'numlock':144,\n",
       "\t\t\t\t'num_lock':144,\n",
       "\t\t\t\t'num':144,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'pause':19,\n",
       "\t\t\t\t'break':19,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'insert':45,\n",
       "\t\t\t\t'home':36,\n",
       "\t\t\t\t'delete':46,\n",
       "\t\t\t\t'end':35,\n",
       "\t\t\t\t\n",
       "\t\t\t\t'pageup':33,\n",
       "\t\t\t\t'page_up':33,\n",
       "\t\t\t\t'pu':33,\n",
       "\t\n",
       "\t\t\t\t'pagedown':34,\n",
       "\t\t\t\t'page_down':34,\n",
       "\t\t\t\t'pd':34,\n",
       "\t\n",
       "\t\t\t\t'left':37,\n",
       "\t\t\t\t'up':38,\n",
       "\t\t\t\t'right':39,\n",
       "\t\t\t\t'down':40,\n",
       "\t\n",
       "\t\t\t\t'f1':112,\n",
       "\t\t\t\t'f2':113,\n",
       "\t\t\t\t'f3':114,\n",
       "\t\t\t\t'f4':115,\n",
       "\t\t\t\t'f5':116,\n",
       "\t\t\t\t'f6':117,\n",
       "\t\t\t\t'f7':118,\n",
       "\t\t\t\t'f8':119,\n",
       "\t\t\t\t'f9':120,\n",
       "\t\t\t\t'f10':121,\n",
       "\t\t\t\t'f11':122,\n",
       "\t\t\t\t'f12':123\n",
       "\t\t\t}\n",
       "\t\n",
       "\t\t\tvar modifiers = { \n",
       "\t\t\t\tshift: { wanted:false, pressed:false},\n",
       "\t\t\t\tctrl : { wanted:false, pressed:false},\n",
       "\t\t\t\talt  : { wanted:false, pressed:false},\n",
       "\t\t\t\tmeta : { wanted:false, pressed:false}\t//Meta is Mac specific\n",
       "\t\t\t};\n",
       "                        \n",
       "\t\t\tif(e.ctrlKey)\tmodifiers.ctrl.pressed = true;\n",
       "\t\t\tif(e.shiftKey)\tmodifiers.shift.pressed = true;\n",
       "\t\t\tif(e.altKey)\tmodifiers.alt.pressed = true;\n",
       "\t\t\tif(e.metaKey)   modifiers.meta.pressed = true;\n",
       "                        \n",
       "\t\t\tfor(var i=0; k=keys[i],i<keys.length; i++) {\n",
       "\t\t\t\t//Modifiers\n",
       "\t\t\t\tif(k == 'ctrl' || k == 'control') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.ctrl.wanted = true;\n",
       "\n",
       "\t\t\t\t} else if(k == 'shift') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.shift.wanted = true;\n",
       "\n",
       "\t\t\t\t} else if(k == 'alt') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.alt.wanted = true;\n",
       "\t\t\t\t} else if(k == 'meta') {\n",
       "\t\t\t\t\tkp++;\n",
       "\t\t\t\t\tmodifiers.meta.wanted = true;\n",
       "\t\t\t\t} else if(k.length > 1) { //If it is a special key\n",
       "\t\t\t\t\tif(special_keys[k] == code) kp++;\n",
       "\t\t\t\t\t\n",
       "\t\t\t\t} else if(opt['keycode']) {\n",
       "\t\t\t\t\tif(opt['keycode'] == code) kp++;\n",
       "\n",
       "\t\t\t\t} else { //The special keys did not match\n",
       "\t\t\t\t\tif(character == k) kp++;\n",
       "\t\t\t\t\telse {\n",
       "\t\t\t\t\t\tif(shift_nums[character] && e.shiftKey) { //Stupid Shift key bug created by using lowercase\n",
       "\t\t\t\t\t\t\tcharacter = shift_nums[character]; \n",
       "\t\t\t\t\t\t\tif(character == k) kp++;\n",
       "\t\t\t\t\t\t}\n",
       "\t\t\t\t\t}\n",
       "\t\t\t\t}\n",
       "\t\t\t}\n",
       "\t\t\t\n",
       "\t\t\tif(kp == keys.length && \n",
       "\t\t\t\t\t\tmodifiers.ctrl.pressed == modifiers.ctrl.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.shift.pressed == modifiers.shift.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.alt.pressed == modifiers.alt.wanted &&\n",
       "\t\t\t\t\t\tmodifiers.meta.pressed == modifiers.meta.wanted) {\n",
       "\t\t\t\tcallback(e);\n",
       "\t\n",
       "\t\t\t\tif(!opt['propagate']) { //Stop the event\n",
       "\t\t\t\t\t//e.cancelBubble is supported by IE - this will kill the bubbling process.\n",
       "\t\t\t\t\te.cancelBubble = true;\n",
       "\t\t\t\t\te.returnValue = false;\n",
       "\t\n",
       "\t\t\t\t\t//e.stopPropagation works in Firefox.\n",
       "\t\t\t\t\tif (e.stopPropagation) {\n",
       "\t\t\t\t\t\te.stopPropagation();\n",
       "\t\t\t\t\t\te.preventDefault();\n",
       "\t\t\t\t\t}\n",
       "\t\t\t\t\treturn false;\n",
       "\t\t\t\t}\n",
       "\t\t\t}\n",
       "\t\t}\n",
       "\t\tthis.all_shortcuts[shortcut_combination] = {\n",
       "\t\t\t'callback':func, \n",
       "\t\t\t'target':ele, \n",
       "\t\t\t'event': opt['type']\n",
       "\t\t};\n",
       "\t\t//Attach the function with the event\n",
       "\t\tif(ele.addEventListener) ele.addEventListener(opt['type'], func, false);\n",
       "\t\telse if(ele.attachEvent) ele.attachEvent('on'+opt['type'], func);\n",
       "\t\telse ele['on'+opt['type']] = func;\n",
       "\t},\n",
       "\n",
       "\t//Remove the shortcut - just specify the shortcut and I will remove the binding\n",
       "\t'remove':function(shortcut_combination) {\n",
       "\t\tshortcut_combination = shortcut_combination.toLowerCase();\n",
       "\t\tvar binding = this.all_shortcuts[shortcut_combination];\n",
       "\t\tdelete(this.all_shortcuts[shortcut_combination])\n",
       "\t\tif(!binding) return;\n",
       "\t\tvar type = binding['event'];\n",
       "\t\tvar ele = binding['target'];\n",
       "\t\tvar callback = binding['callback'];\n",
       "\n",
       "\t\tif(ele.detachEvent) ele.detachEvent('on'+type, callback);\n",
       "\t\telse if(ele.removeEventListener) ele.removeEventListener(type, callback, false);\n",
       "\t\telse ele['on'+type] = false;\n",
       "\t}\n",
       "}\n",
       "\n",
       "shortcut.add(\"Alt+Left\",function() {\n",
       "\tvar event = new Event('click');\n",
       "\tif (settingsShown){\n",
       "\t\tdocument.getElementById('hideSettingsButton_0').dispatchEvent(event);\n",
       "\t} else {\n",
       "\t\tdocument.getElementById('showSettingsButton_0').dispatchEvent(event);\n",
       "\t}\n",
       "});\n",
       "\n",
       "\n",
       "// COMMON_CODE_BLOCK_END\n",
       "\n",
       "});});\n",
       "      </script>\n",
       "   </body>\n",
       "</html>\n"
      ],
      "text/plain": []
     },
     "execution_count": 345,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "O = platonic_solid(\"octahedron\")\n",
    "So = subdivision_of_points(O,w)\n",
    "visualize(So)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ec02aa38",
   "metadata": {},
   "source": [
    "**2. Obtain matroids from cells of subdivision**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 348,
   "id": "1fee133c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element SubObjectIterator{Vector{Int64}}:\n",
       " [1, 2, 3, 4, 6]\n",
       " [1, 3, 4, 5, 6]"
      ]
     },
     "execution_count": 348,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ms = maximal_cells(Sw)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 349,
   "id": "0c12dd06",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{PointVector{QQFieldElem}}:\n",
       " [1, 1, 0, 0]\n",
       " [1, 0, 1, 0]\n",
       " [1, 0, 0, 1]\n",
       " [0, 1, 1, 0]\n",
       " [0, 0, 1, 1]"
      ]
     },
     "execution_count": 349,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E1 = [V24[i] for i in Ms[1]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 350,
   "id": "6cec60b1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{PointVector{QQFieldElem}}:\n",
       " [1, 1, 0, 0]\n",
       " [1, 0, 0, 1]\n",
       " [0, 1, 1, 0]\n",
       " [0, 1, 0, 1]\n",
       " [0, 0, 1, 1]"
      ]
     },
     "execution_count": 350,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E2 = [V24[i] for i in Ms[2]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 351,
   "id": "70edab8e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{PointVector{QQFieldElem}}:\n",
       " [1, 1, 0, 0]\n",
       " [1, 0, 0, 1]\n",
       " [0, 1, 1, 0]\n",
       " [0, 0, 1, 1]"
      ]
     },
     "execution_count": 351,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E12 = [V24[i] for i in intersect(Ms[1],Ms[2])]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 352,
   "id": "d27ceefe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matroid of rank 2 on 4 elements"
      ]
     },
     "execution_count": 352,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B1 = [ind_2_set(e) for e in E1]\n",
    "Q1 = matroid_from_bases(B1,4)\n",
    "\n",
    "B2 = [ind_2_set(e) for e in E2]\n",
    "Q2 = matroid_from_bases(B2,4)\n",
    "\n",
    "B12 = [ind_2_set(e) for e in E12]\n",
    "Q12 = matroid_from_bases(B12,4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f8a1975",
   "metadata": {},
   "source": [
    "**Obtain coordinate ring data**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b3e01a99",
   "metadata": {},
   "source": [
    "Compute data corresponding to maximal cell $\\Delta(Q_1)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 353,
   "id": "4e02e4df",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1   0   x[1, 1]         0]\n",
       "[0   1   x[2, 1]   x[2, 2]]"
      ]
     },
     "execution_count": 353,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X1 = matroid_matrix(Q1,QQ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 354,
   "id": "5fa58dd2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ideal generated by\n",
       "  0"
      ]
     },
     "execution_count": 354,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1 = ideal([det(X1[:,b]) for b in nonbases(Q1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 355,
   "id": "56c0ada2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{QQMPolyRingElem}:\n",
       " 1\n",
       " x[2, 1]\n",
       " x[2, 2]\n",
       " -x[1, 1]\n",
       " x[1, 1]*x[2, 2]"
      ]
     },
     "execution_count": 355,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U1 = [det(X1[:,b]) for b in bases(Q1)]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "48c4888e",
   "metadata": {},
   "source": [
    "Then we have $\\mathbb{C}[\\text{Gr}(Q_1,\\mathbb{C})]\\cong\\frac{U^{-1}_{1}\\mathbb{C}[x_{1,1},x_{2,1},x_{2,2}]}{\\langle0\\rangle}\\cong\\mathbb{C}[x^{\\pm}_{1,1},x^{\\pm}_{2,1},x^{\\pm}_{2,2}]$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01a92627",
   "metadata": {},
   "source": [
    "Compute data corresponding to maximal cell $\\Delta(Q_2)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 356,
   "id": "a66fe8c5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1   0   x[1, 1]   x[1, 2]]\n",
       "[0   1         0   x[2, 2]]"
      ]
     },
     "execution_count": 356,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X2 = matroid_matrix(Q2,QQ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 357,
   "id": "52d8071d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ideal generated by\n",
       "  0"
      ]
     },
     "execution_count": 357,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2 = ideal([det(X2[:,b]) for b in nonbases(Q2)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 358,
   "id": "aafca141",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{QQMPolyRingElem}:\n",
       " 1\n",
       " x[2, 2]\n",
       " -x[1, 1]\n",
       " -x[1, 2]\n",
       " x[1, 1]*x[2, 2]"
      ]
     },
     "execution_count": 358,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U2 = [det(X2[:,b]) for b in bases(Q2)]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "912c6a10",
   "metadata": {},
   "source": [
    "Similar to the above, we have $\\mathbb{C}[\\text{Gr}(Q_2,\\mathbb{C})]\\cong \\mathbb{C}[x^{\\pm}_{1,1},x^{\\pm}_{1,2},x^{\\pm}_{2,2}]$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce046aab",
   "metadata": {},
   "source": [
    "Compute data corresponding to common face of $\\Delta(Q_{1})$ and $\\Delta(Q_{2})$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 362,
   "id": "36a22c9f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1   0   x[1, 1]         0]\n",
       "[0   1         0   x[2, 2]]"
      ]
     },
     "execution_count": 362,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X12 = matroid_matrix(Q12,QQ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 363,
   "id": "07bcede9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ideal generated by\n",
       "  0"
      ]
     },
     "execution_count": 363,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I12 = ideal(unique([det(X12[:,b]) for b in nonbases(Q12)]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 364,
   "id": "4a9b79d2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{QQMPolyRingElem}:\n",
       " 1\n",
       " x[2, 2]\n",
       " -x[1, 1]\n",
       " x[1, 1]*x[2, 2]"
      ]
     },
     "execution_count": 364,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U12 = [det(X12[:,b]) for b in bases(Q12)]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "40b51dbb",
   "metadata": {},
   "source": [
    "Then we have $\\mathbb{C}[\\text{Gr}(Q_{12},\\mathbb{C})\\cong\\mathbb{C}[x^{\\pm}_{1,1},x^{\\pm}_{2,2}]$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7de35af",
   "metadata": {},
   "source": [
    "One may then compute the inverse limit $\\text{Gr}(w)$ directly, noting that in this case $\\text{Gr}(w)$ is isomorphic to the fiber product. $\\text{Gr}(w)\\cong \\text{Gr}(Q_{1},\\mathbb{C})\\times_{\\text{Gr}(Q_{12};\\mathbb{C})}\\text{Gr}(Q_{2};\\mathbb{C})$. Then it follows that $\\mathbb{C}[\\text{Gr}(w)]\\cong\\mathbb{C}[x^{\\pm}_{1,1},x^{\\pm}_{1,2},x^{\\pm}_{2,1},x^{\\pm}_{2,2}]$. Since $\\text{Gr}(w)$ is smooth, irreducible, and $4$-dimensional, we have $\\text{in}_{w}\\text{Gr}^{\\circ}(2,4)\\cong\\text{Gr}(w)$ by Corollary 5.2.2."
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Julia 1.10.4",
   "language": "julia",
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