{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "da7f1bfe-b499-4682-bc76-53abe52dbcfa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<iframe srcdoc=\"&lt;!DOCTYPE html&gt;\n",
       "&lt;html&gt;\n",
       "&lt;head&gt;\n",
       "&lt;title&gt;&lt;/title&gt;\n",
       "&lt;meta charset=&quot;utf-8&quot;&gt;\n",
       "&lt;meta name=viewport content=&quot;width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0&quot;&gt;\n",
       "&lt;style&gt;\n",
       "\n",
       "    body { margin: 0px; overflow: hidden; }\n",
       "\n",
       "    #menu-container { position: absolute; bottom: 30px; right: 40px; cursor: default; }\n",
       "\n",
       "    #menu-message { position: absolute; bottom: 0px; right: 0px; white-space: nowrap;\n",
       "                    display: none; background-color: #F5F5F5; padding: 10px; }\n",
       "\n",
       "    #menu-content { position: absolute; bottom: 0px; right: 0px;\n",
       "                    display: none; background-color: #F5F5F5; border-bottom: 1px solid black;\n",
       "                    border-right: 1px solid black; border-left: 1px solid black; }\n",
       "\n",
       "    #menu-content div { border-top: 1px solid black; padding: 10px; white-space: nowrap; }\n",
       "\n",
       "    #menu-content div:hover { background-color: #FEFEFE; }\n",
       "\n",
       "    .dark-theme #menu-container { color: white; }\n",
       "\n",
       "    .dark-theme #menu-message { background-color: #181818; }\n",
       "\n",
       "    .dark-theme #menu-content { background-color: #181818; border-color: white; }\n",
       "\n",
       "    .dark-theme #menu-content div { border-color: white; }\n",
       "\n",
       "    .dark-theme #menu-content div:hover { background-color: #303030; }\n",
       "\n",
       "&lt;/style&gt;\n",
       "\n",
       "&lt;/head&gt;\n",
       "\n",
       "&lt;body&gt;\n",
       "\n",
       "&lt;script src=&quot;/nbextensions/threejs-sage/r122/three.min.js&quot;&gt;&lt;/script&gt;\n",
       "&lt;script&gt;\n",
       "  if ( !window.THREE ) document.write(&#x27; \\\n",
       "&lt;script src=&quot;https://cdn.jsdelivr.net/gh/sagemath/threejs-sage@r122/build/three.min.js&quot;&gt;&lt;\\/script&gt; \\\n",
       "            &#x27;);\n",
       "&lt;/script&gt;\n",
       "        \n",
       "&lt;script&gt;\n",
       "\n",
       "    var options = {&quot;animate&quot;: false, &quot;animationControls&quot;: true, &quot;aspectRatio&quot;: [1.0, 1.0, 1.0], &quot;autoScaling&quot;: [false, false, false], &quot;autoPlay&quot;: true, &quot;axes&quot;: false, &quot;axesLabels&quot;: false, &quot;axesLabelsStyle&quot;: null, &quot;decimals&quot;: 2, &quot;delay&quot;: 20, &quot;frame&quot;: false, &quot;loop&quot;: true, &quot;projection&quot;: &quot;perspective&quot;, &quot;theme&quot;: &quot;light&quot;, &quot;viewpoint&quot;: false};\n",
       "    var animate = options.animate;\n",
       "\n",
       "    if ( options.theme === &#x27;dark&#x27; )\n",
       "        document.body.className = &#x27;dark-theme&#x27;;\n",
       "\n",
       "    var scene = new THREE.Scene();\n",
       "\n",
       "    var renderer = new THREE.WebGLRenderer( { antialias: true, preserveDrawingBuffer: true } );\n",
       "    renderer.setPixelRatio( window.devicePixelRatio );\n",
       "    renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "    renderer.setClearColor( options.theme === &#x27;dark&#x27; ? 0 : 0xffffff, 1 );\n",
       "    document.body.appendChild( renderer.domElement );\n",
       "\n",
       "    var b = [{&quot;x&quot;:-5.0, &quot;y&quot;:-5.0, &quot;z&quot;:-5.0}, {&quot;x&quot;:5.0, &quot;y&quot;:5.0, &quot;z&quot;:5.0}]; // bounds\n",
       "\n",
       "    if ( b[0].x === b[1].x ) {\n",
       "        b[0].x -= 1;\n",
       "        b[1].x += 1;\n",
       "    }\n",
       "    if ( b[0].y === b[1].y ) {\n",
       "        b[0].y -= 1;\n",
       "        b[1].y += 1;\n",
       "    }\n",
       "    if ( b[0].z === b[1].z ) {\n",
       "        b[0].z -= 1;\n",
       "        b[1].z += 1;\n",
       "    }\n",
       "\n",
       "    var rxRange = Math.sqrt( Math.pow( b[1].z - b[0].z, 2 )\n",
       "                            + Math.pow( b[1].y - b[0].y, 2 ) );\n",
       "    var ryRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].z - b[0].z, 2 ) );\n",
       "    var rzRange = Math.sqrt( Math.pow( b[1].x - b[0].x, 2 )\n",
       "                            + Math.pow( b[1].y - b[0].y, 2 ) );\n",
       "    var xRange = b[1].x - b[0].x;\n",
       "    var yRange = b[1].y - b[0].y;\n",
       "    var zRange = b[1].z - b[0].z;\n",
       "\n",
       "    var ar = options.aspectRatio;\n",
       "    var a = [ ar[0], ar[1], ar[2] ]; // aspect multipliers\n",
       "\n",
       "    var autoScaling = options.autoScaling;\n",
       "    var autoAspect = 2.5;\n",
       "    if ( xRange &gt; autoAspect * rxRange &amp;&amp; autoScaling[0] ) a[0] = autoAspect * rxRange / xRange;\n",
       "    if ( yRange &gt; autoAspect * ryRange &amp;&amp; autoScaling[1] ) a[1] = autoAspect * ryRange / yRange;\n",
       "    if ( zRange &gt; autoAspect * rzRange &amp;&amp; autoScaling[2] ) a[2] = autoAspect * rzRange / zRange;\n",
       "\n",
       "    // Distance from (xMid,yMid,zMid) to any corner of the bounding box, after applying aspectRatio\n",
       "    var midToCorner = Math.sqrt( a[0]*a[0]*xRange*xRange + a[1]*a[1]*yRange*yRange + a[2]*a[2]*zRange*zRange ) / 2;\n",
       "\n",
       "    var xMid = ( b[0].x + b[1].x ) / 2;\n",
       "    var yMid = ( b[0].y + b[1].y ) / 2;\n",
       "    var zMid = ( b[0].z + b[1].z ) / 2;\n",
       "\n",
       "    var box = new THREE.Geometry();\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[0].x, a[1]*b[0].y, a[2]*b[0].z ) );\n",
       "    box.vertices.push( new THREE.Vector3( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) );\n",
       "    var boxMesh = new THREE.Line( box );\n",
       "    var boxColor = options.theme === &#x27;dark&#x27; ? &#x27;white&#x27; : &#x27;black&#x27;;\n",
       "    if ( options.frame ) scene.add( new THREE.BoxHelper( boxMesh, boxColor ) );\n",
       "\n",
       "    if ( options.axesLabels ) {\n",
       "\n",
       "        var d = options.decimals; // decimals\n",
       "        var offsetRatio = 0.1;\n",
       "        var al = options.axesLabels;\n",
       "        var als = options.axesLabelsStyle || [{}, {}, {}];\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var xm = xMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(xm) ) xm = xm.substr(1);\n",
       "        addLabel( al[0] + &#x27;=&#x27; + xm, a[0]*xMid, a[1]*b[1].y+offset, a[2]*b[0].z, als[0] );\n",
       "        addLabel( ( b[0].x ).toFixed(d), a[0]*b[0].x, a[1]*b[1].y+offset, a[2]*b[0].z, als[0] );\n",
       "        addLabel( ( b[1].x ).toFixed(d), a[0]*b[1].x, a[1]*b[1].y+offset, a[2]*b[0].z, als[0] );\n",
       "\n",
       "        var offset = offsetRatio * a[0]*( b[1].x - b[0].x );\n",
       "        var ym = yMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(ym) ) ym = ym.substr(1);\n",
       "        addLabel( al[1] + &#x27;=&#x27; + ym, a[0]*b[1].x+offset, a[1]*yMid, a[2]*b[0].z, als[1] );\n",
       "        addLabel( ( b[0].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[0].y, a[2]*b[0].z, als[1] );\n",
       "        addLabel( ( b[1].y ).toFixed(d), a[0]*b[1].x+offset, a[1]*b[1].y, a[2]*b[0].z, als[1] );\n",
       "\n",
       "        var offset = offsetRatio * a[1]*( b[1].y - b[0].y );\n",
       "        var zm = zMid.toFixed(d);\n",
       "        if ( /^-0.?0*$/.test(zm) ) zm = zm.substr(1);\n",
       "        addLabel( al[2] + &#x27;=&#x27; + zm, a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*zMid, als[2] );\n",
       "        addLabel( ( b[0].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[0].z, als[2] );\n",
       "        addLabel( ( b[1].z ).toFixed(d), a[0]*b[1].x, a[1]*b[0].y-offset, a[2]*b[1].z, als[2] );\n",
       "\n",
       "    }\n",
       "\n",
       "    function addLabel( text, x, y, z, style ) {\n",
       "\n",
       "        var color = style.color || &#x27;black&#x27;;\n",
       "        var fontSize = style.fontSize || 14;\n",
       "        var fontFamily = style.fontFamily || &#x27;monospace&#x27;;\n",
       "        var fontStyle = style.fontStyle || &#x27;normal&#x27;;\n",
       "        var fontWeight = style.fontWeight || &#x27;normal&#x27;;\n",
       "        var opacity = style.opacity || 1;\n",
       "\n",
       "        if ( options.theme === &#x27;dark&#x27; )\n",
       "            if ( color === &#x27;black&#x27; || color === &#x27;#000000&#x27; )\n",
       "                color = &#x27;white&#x27;;\n",
       "\n",
       "        if ( Array.isArray( fontStyle ) ) {\n",
       "            fontFamily = fontFamily.map( function( f ) {\n",
       "                // Need to put quotes around fonts that have whitespace in their names.\n",
       "                return /\\s/.test( f ) ? &#x27;&quot;&#x27; + f + &#x27;&quot;&#x27; : f;\n",
       "            }).join(&#x27;, &#x27;);\n",
       "        }\n",
       "\n",
       "        var canvas = document.createElement( &#x27;canvas&#x27; );\n",
       "        var context = canvas.getContext( &#x27;2d&#x27; );\n",
       "        var pixelRatio = Math.round( window.devicePixelRatio );\n",
       "\n",
       "        // For example: italic bold 20px &quot;Times New Roman&quot;, Georgia, serif\n",
       "        var font = [fontStyle, fontWeight, fontSize + &#x27;px&#x27;, fontFamily].join(&#x27; &#x27;);\n",
       "\n",
       "        context.font = font;\n",
       "        var width = context.measureText( text ).width;\n",
       "        var height = fontSize;\n",
       "\n",
       "        // The dimensions of the canvas&#x27;s underlying image data need to be powers\n",
       "        // of two in order for the resulting texture to support mipmapping.\n",
       "        canvas.width = THREE.MathUtils.ceilPowerOfTwo( width * pixelRatio );\n",
       "        canvas.height = THREE.MathUtils.ceilPowerOfTwo( height * pixelRatio );\n",
       "\n",
       "        // Re-compute the unscaled dimensions after the power of two conversion.\n",
       "        width = canvas.width / pixelRatio;\n",
       "        height = canvas.height / pixelRatio;\n",
       "\n",
       "        canvas.style.width = width + &#x27;px&#x27;;\n",
       "        canvas.style.height = height + &#x27;px&#x27;;\n",
       "\n",
       "        context.scale( pixelRatio, pixelRatio );\n",
       "        context.fillStyle = color;\n",
       "        context.font = font; // Must be set again after measureText.\n",
       "        context.textAlign = &#x27;center&#x27;;\n",
       "        context.textBaseline = &#x27;middle&#x27;;\n",
       "        context.fillText( text, width/2, height/2 );\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var materialOptions = { map: texture, sizeAttenuation: false, depthWrite: false };\n",
       "        if ( opacity &lt; 1 ) {\n",
       "            // Setting opacity=1 would cause the texture&#x27;s alpha component to be\n",
       "            // discarded, giving the text a black background instead of the\n",
       "            // background being transparent.\n",
       "            materialOptions.opacity = opacity;\n",
       "        }\n",
       "        var sprite = new THREE.Sprite( new THREE.SpriteMaterial( materialOptions ) );\n",
       "        sprite.position.set( x, y, z );\n",
       "\n",
       "        // Scaling factor, chosen somewhat arbitrarily so that the size of the text\n",
       "        // is consistent with previously generated plots.\n",
       "        var scale = 1/625;\n",
       "        if ( options.projection === &#x27;orthographic&#x27; ) {\n",
       "            scale = midToCorner/256; // Needs to scale along with the plot itself.\n",
       "        }\n",
       "        sprite.scale.set( scale * width, scale * height, 1 );\n",
       "\n",
       "        scene.add( sprite );\n",
       "\n",
       "        return sprite;\n",
       "\n",
       "    }\n",
       "\n",
       "    if ( options.axes ) scene.add( new THREE.AxesHelper( Math.min( a[0]*b[1].x, a[1]*b[1].y, a[2]*b[1].z ) ) );\n",
       "\n",
       "    var camera = createCamera();\n",
       "    camera.up.set( 0, 0, 1 );\n",
       "    camera.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "\n",
       "    // camera is positioned so that the line from the camera to the center\n",
       "    // of the bounding sphere of the objects makes an angle of 60 degrees with x-axis\n",
       "    // and an angle of 30 degrees with z-axis and the field of view of the camera looking\n",
       "    // at the center has an angle of 45 degrees.\n",
       "    const sin8 = Math.sin(Math.PI / 8);\n",
       "    const sin5 = Math.sin(Math.PI / 5);\n",
       "    const cos5 = Math.cos(Math.PI / 5);\n",
       "    const sin3 = Math.sin(Math.PI / 3);\n",
       "    const cos3 = Math.cos(Math.PI / 3);\n",
       "    var r = midToCorner / sin8;\n",
       "    var offset = new THREE.Vector3( r * sin3 * cos5, r * sin3 * sin5, r * cos3 );\n",
       "\n",
       "    if ( options.viewpoint ) {\n",
       "\n",
       "        var aa = options.viewpoint;\n",
       "        var axis = new THREE.Vector3( aa[0][0], aa[0][1], aa[0][2] ).normalize();\n",
       "        var angle = aa[1] * Math.PI / 180;\n",
       "        var q = new THREE.Quaternion().setFromAxisAngle( axis, angle ).inverse();\n",
       "\n",
       "        offset.set( 0, 0, offset.length() );\n",
       "        offset.applyQuaternion( q );\n",
       "\n",
       "    }\n",
       "\n",
       "    camera.position.add( offset );\n",
       "\n",
       "    function createCamera() {\n",
       "\n",
       "        var aspect = window.innerWidth / window.innerHeight;\n",
       "\n",
       "        // Scale the near and far clipping planes along with the overall plot size.\n",
       "        var nearClip = 0.01 * midToCorner;\n",
       "        var farClip = 100 * midToCorner;\n",
       "\n",
       "        if ( options.projection === &#x27;orthographic&#x27; ) {\n",
       "            var camera = new THREE.OrthographicCamera( -1, 1, 1, -1, -farClip, farClip );\n",
       "            updateCameraAspect( camera, aspect );\n",
       "            return camera;\n",
       "        }\n",
       "\n",
       "        return new THREE.PerspectiveCamera( 45, aspect, nearClip, farClip );\n",
       "\n",
       "    }\n",
       "\n",
       "    function updateCameraAspect( camera, aspect ) {\n",
       "\n",
       "        if ( camera.isPerspectiveCamera ) {\n",
       "            camera.aspect = aspect;\n",
       "        } else if ( camera.isOrthographicCamera ) {\n",
       "            // Fit the camera frustum to the bounding box&#x27;s diagonal so that the entire plot fits\n",
       "            // within at the default zoom level and camera position.\n",
       "            if ( aspect &gt; 1 ) { // Wide window\n",
       "                camera.top = midToCorner;\n",
       "                camera.right = midToCorner * aspect;\n",
       "            } else { // Tall or square window\n",
       "                camera.top = midToCorner / aspect;\n",
       "                camera.right = midToCorner;\n",
       "            }\n",
       "            camera.bottom = -camera.top;\n",
       "            camera.left = -camera.right;\n",
       "        }\n",
       "\n",
       "        camera.updateProjectionMatrix();\n",
       "\n",
       "    }\n",
       "\n",
       "    var lights = [{&quot;x&quot;:-5, &quot;y&quot;:3, &quot;z&quot;:0, &quot;color&quot;:&quot;#7f7f7f&quot;, &quot;parent&quot;:&quot;camera&quot;}];\n",
       "    for ( var i=0 ; i &lt; lights.length ; i++ ) {\n",
       "        var light = new THREE.DirectionalLight( lights[i].color, 1 );\n",
       "        light.position.set( a[0]*lights[i].x, a[1]*lights[i].y, a[2]*lights[i].z );\n",
       "        if ( lights[i].parent === &#x27;camera&#x27; ) {\n",
       "            light.target.position.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "            scene.add( light.target );\n",
       "            camera.add( light );\n",
       "        } else scene.add( light );\n",
       "    }\n",
       "    scene.add( camera );\n",
       "\n",
       "    var ambient = {&quot;color&quot;:&quot;#7f7f7f&quot;};\n",
       "    scene.add( new THREE.AmbientLight( ambient.color, 1 ) );\n",
       "\n",
       "    var controls = new THREE.OrbitControls( camera, renderer.domElement );\n",
       "    controls.target.set( a[0]*xMid, a[1]*yMid, a[2]*zMid );\n",
       "    controls.addEventListener( &#x27;change&#x27;, function() { if ( !animate ) render(); } );\n",
       "\n",
       "    window.addEventListener( &#x27;resize&#x27;, function() {\n",
       "\n",
       "        renderer.setSize( window.innerWidth, window.innerHeight );\n",
       "        updateCameraAspect( camera, window.innerWidth / window.innerHeight );\n",
       "        if ( window.rescaleFatLines ) rescaleFatLines();\n",
       "        if ( !animate ) render();\n",
       "\n",
       "    } );\n",
       "\n",
       "    var texts = [];\n",
       "    for ( var i=0 ; i &lt; texts.length ; i++ ) addText( texts[i] );\n",
       "\n",
       "    function addText( json ) {\n",
       "        var sprite = addLabel( json.text, a[0]*json.x, a[1]*json.y, a[2]*json.z, json );\n",
       "        sprite.userData = json;\n",
       "    }\n",
       "\n",
       "    var points = [];\n",
       "    for ( var i=0 ; i &lt; points.length ; i++ ) addPoint( points[i] );\n",
       "\n",
       "    function addPoint( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        var v = json.point;\n",
       "        geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "\n",
       "        var canvas = document.createElement( &#x27;canvas&#x27; );\n",
       "        canvas.width = 128;\n",
       "        canvas.height = 128;\n",
       "\n",
       "        var context = canvas.getContext( &#x27;2d&#x27; );\n",
       "        context.arc( 64, 64, 64, 0, 2 * Math.PI );\n",
       "        context.fillStyle = json.color;\n",
       "        context.fill();\n",
       "\n",
       "        var texture = new THREE.Texture( canvas );\n",
       "        texture.needsUpdate = true;\n",
       "\n",
       "        var transparent = json.opacity &lt; 1 ? true : false;\n",
       "        var size = camera.isOrthographicCamera ? json.size : json.size/100;\n",
       "        var material = new THREE.PointsMaterial( { size: size, map: texture,\n",
       "                                                   transparent: transparent, opacity: json.opacity,\n",
       "                                                   alphaTest: .1 } );\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var mesh = new THREE.Points( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    var lines = [];\n",
       "    for ( var i=0 ; i &lt; lines.length ; i++ ) addLine( lines[i] );\n",
       "\n",
       "    function addLine( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i &lt; json.points.length ; i++ ) {\n",
       "            var v = json.points[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v[0], a[1]*v[1], a[2]*v[2] ) );\n",
       "        }\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var transparent = json.opacity &lt; 1 ? true : false;\n",
       "        var materialOptions = { color: json.color, linewidth: json.linewidth,\n",
       "                                transparent: transparent, opacity: json.opacity };\n",
       "\n",
       "        var mesh;\n",
       "        if ( json.linewidth &gt; 1 &amp;&amp; window.createFatLineStrip ) {\n",
       "            mesh = createFatLineStrip( geometry, materialOptions );\n",
       "        } else {\n",
       "            var material = new THREE.LineBasicMaterial( materialOptions );\n",
       "            mesh = new THREE.Line( geometry, material );\n",
       "        }\n",
       "\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
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150442, 150443], [150444, 150445, 150446], [150447, 150448, 150449], [150450, 150451, 150452], [150453, 150454, 150455], [150456, 150457, 150458], [150459, 150460, 150461], [150462, 150463, 150464], [150465, 150466, 150467], [150468, 150469, 150470], [150471, 150472, 150473], [150474, 150475, 150476], [150477, 150478, 150479], [150480, 150481, 150482], [150483, 150484, 150485], [150486, 150487, 150488], [150489, 150490, 150491], [150492, 150493, 150494], [150495, 150496, 150497], [150498, 150499, 150500], [150501, 150502, 150503], [150504, 150505, 150506], [150507, 150508, 150509], [150510, 150511, 150512], [150513, 150514, 150515], [150516, 150517, 150518], [150519, 150520, 150521], [150522, 150523, 150524], [150525, 150526, 150527], [150528, 150529, 150530], [150531, 150532, 150533], [150534, 150535, 150536], [150537, 150538, 150539], [150540, 150541, 150542], [150543, 150544, 150545], [150546, 150547, 150548], [150549, 150550, 150551], [150552, 150553, 150554], [150555, 150556, 150557], [150558, 150559, 150560], [150561, 150562, 150563], [150564, 150565, 150566], [150567, 150568, 150569], [150570, 150571, 150572], [150573, 150574, 150575], [150576, 150577, 150578], [150579, 150580, 150581], [150582, 150583, 150584], [150585, 150586, 150587], [150588, 150589, 150590], [150591, 150592, 150593], [150594, 150595, 150596], [150597, 150598, 150599], [150600, 150601, 150602], [150603, 150604, 150605], [150606, 150607, 150608], [150609, 150610, 150611], [150612, 150613, 150614], [150615, 150616, 150617], [150618, 150619, 150620], [150621, 150622, 150623], [150624, 150625, 150626], [150627, 150628, 150629], [150630, 150631, 150632], [150633, 150634, 150635], [150636, 150637, 150638], [150639, 150640, 150641], [150642, 150643, 150644], [150645, 150646, 150647], [150648, 150649, 150650], [150651, 150652, 150653], [150654, 150655, 150656], [150657, 150658, 150659], [150660, 150661, 150662], [150663, 150664, 150665], [150666, 150667, 150668], [150669, 150670, 150671], [150672, 150673, 150674], [150675, 150676, 150677], [150678, 150679, 150680], [150681, 150682, 150683], [150684, 150685, 150686], [150687, 150688, 150689], [150690, 150691, 150692], [150693, 150694, 150695], [150696, 150697, 150698], [150699, 150700, 150701], [150702, 150703, 150704], [150705, 150706, 150707], [150708, 150709, 150710], [150711, 150712, 150713], [150714, 150715, 150716], [150717, 150718, 150719], [150720, 150721, 150722], [150723, 150724, 150725], [150726, 150727, 150728], [150729, 150730, 150731], [150732, 150733, 150734], [150735, 150736, 150737], [150738, 150739, 150740], [150741, 150742, 150743], [150744, 150745, 150746], [150747, 150748, 150749], [150750, 150751, 150752], [150753, 150754, 150755], [150756, 150757, 150758], [150759, 150760, 150761], [150762, 150763, 150764], [150765, 150766, 150767], [150768, 150769, 150770], [150771, 150772, 150773], [150774, 150775, 150776], [150777, 150778, 150779], [150780, 150781, 150782], [150783, 150784, 150785], [150786, 150787, 150788], [150789, 150790, 150791], [150792, 150793, 150794], [150795, 150796, 150797], [150798, 150799, 150800], [150801, 150802, 150803], [150804, 150805, 150806], [150807, 150808, 150809], [150810, 150811, 150812], [150813, 150814, 150815], [150816, 150817, 150818], [150819, 150820, 150821], [150822, 150823, 150824], [150825, 150826, 150827], [150828, 150829, 150830], [150831, 150832, 150833], [150834, 150835, 150836], [150837, 150838, 150839], [150840, 150841, 150842], [150843, 150844, 150845], [150846, 150847, 150848], [150849, 150850, 150851], [150852, 150853, 150854], [150855, 150856, 150857], [150858, 150859, 150860], [150861, 150862, 150863], [150864, 150865, 150866], [150867, 150868, 150869], [150870, 150871, 150872], [150873, 150874, 150875], [150876, 150877, 150878], [150879, 150880, 150881], [150882, 150883, 150884], [150885, 150886, 150887], [150888, 150889, 150890], [150891, 150892, 150893], [150894, 150895, 150896], [150897, 150898, 150899], [150900, 150901, 150902], [150903, 150904, 150905], [150906, 150907, 150908], [150909, 150910, 150911], [150912, 150913, 150914], [150915, 150916, 150917], [150918, 150919, 150920], [150921, 150922, 150923], [150924, 150925, 150926], [150927, 150928, 150929], [150930, 150931, 150932], [150933, 150934, 150935], [150936, 150937, 150938], [150939, 150940, 150941], [150942, 150943, 150944], [150945, 150946, 150947], [150948, 150949, 150950], [150951, 150952, 150953], [150954, 150955, 150956], [150957, 150958, 150959], [150960, 150961, 150962], [150963, 150964, 150965], [150966, 150967, 150968], [150969, 150970, 150971], [150972, 150973, 150974], [150975, 150976, 150977], [150978, 150979, 150980], [150981, 150982, 150983], [150984, 150985, 150986], [150987, 150988, 150989], [150990, 150991, 150992], [150993, 150994, 150995], [150996, 150997, 150998], [150999, 151000, 151001], [151002, 151003, 151004], [151005, 151006, 151007], [151008, 151009, 151010], [151011, 151012, 151013], [151014, 151015, 151016], [151017, 151018, 151019], [151020, 151021, 151022], [151023, 151024, 151025], [151026, 151027, 151028], [151029, 151030, 151031], [151032, 151033, 151034], [151035, 151036, 151037], [151038, 151039, 151040], [151041, 151042, 151043], [151044, 151045, 151046], [151047, 151048, 151049], [151050, 151051, 151052], [151053, 151054, 151055], [151056, 151057, 151058], [151059, 151060, 151061], [151062, 151063, 151064], [151065, 151066, 151067], [151068, 151069, 151070], [151071, 151072, 151073], [151074, 151075, 151076], [151077, 151078, 151079], [151080, 151081, 151082], [151083, 151084, 151085], [151086, 151087, 151088], [151089, 151090, 151091], [151092, 151093, 151094], [151095, 151096, 151097], [151098, 151099, 151100], [151101, 151102, 151103], [151104, 151105, 151106], [151107, 151108, 151109], [151110, 151111, 151112], [151113, 151114, 151115], [151116, 151117, 151118], [151119, 151120, 151121], [151122, 151123, 151124], [151125, 151126, 151127], [151128, 151129, 151130], [151131, 151132, 151133], [151134, 151135, 151136], [151137, 151138, 151139], [151140, 151141, 151142], [151143, 151144, 151145], [151146, 151147, 151148], [151149, 151150, 151151], [151152, 151153, 151154], [151155, 151156, 151157], [151158, 151159, 151160], [151161, 151162, 151163], [151164, 151165, 151166], [151167, 151168, 151169], [151170, 151171, 151172], [151173, 151174, 151175], [151176, 151177, 151178], [151179, 151180, 151181], [151182, 151183, 151184], [151185, 151186, 151187], [151188, 151189, 151190], [151191, 151192, 151193], [151194, 151195, 151196], [151197, 151198, 151199], [151200, 151201, 151202], [151203, 151204, 151205], [151206, 151207, 151208], [151209, 151210, 151211], [151212, 151213, 151214], [151215, 151216, 151217], [151218, 151219, 151220], [151221, 151222, 151223], [151224, 151225, 151226], [151227, 151228, 151229], [151230, 151231, 151232], [151233, 151234, 151235], [151236, 151237, 151238], [151239, 151240, 151241], [151242, 151243, 151244], [151245, 151246, 151247], [151248, 151249, 151250], [151251, 151252, 151253], [151254, 151255, 151256], [151257, 151258, 151259], [151260, 151261, 151262], [151263, 151264, 151265], [151266, 151267, 151268], [151269, 151270, 151271], [151272, 151273, 151274], [151275, 151276, 151277], [151278, 151279, 151280], [151281, 151282, 151283], [151284, 151285, 151286], [151287, 151288, 151289], [151290, 151291, 151292], [151293, 151294, 151295], [151296, 151297, 151298], [151299, 151300, 151301], [151302, 151303, 151304], [151305, 151306, 151307], [151308, 151309, 151310], [151311, 151312, 151313], [151314, 151315, 151316], [151317, 151318, 151319], [151320, 151321, 151322], [151323, 151324, 151325], [151326, 151327, 151328], [151329, 151330, 151331], [151332, 151333, 151334], [151335, 151336, 151337], [151338, 151339, 151340], [151341, 151342, 151343], [151344, 151345, 151346], [151347, 151348, 151349], [151350, 151351, 151352], [151353, 151354, 151355], [151356, 151357, 151358], [151359, 151360, 151361], [151362, 151363, 151364], [151365, 151366, 151367], [151368, 151369, 151370], [151371, 151372, 151373], [151374, 151375, 151376], [151377, 151378, 151379], [151380, 151381, 151382], [151383, 151384, 151385], [151386, 151387, 151388], [151389, 151390, 151391], [151392, 151393, 151394], [151395, 151396, 151397], [151398, 151399, 151400], [151401, 151402, 151403], [151404, 151405, 151406], [151407, 151408, 151409], [151410, 151411, 151412], [151413, 151414, 151415], [151416, 151417, 151418], [151419, 151420, 151421], [151422, 151423, 151424], [151425, 151426, 151427], [151428, 151429, 151430], [151431, 151432, 151433], [151434, 151435, 151436], [151437, 151438, 151439], [151440, 151441, 151442], [151443, 151444, 151445], [151446, 151447, 151448], [151449, 151450, 151451], [151452, 151453, 151454], [151455, 151456, 151457], [151458, 151459, 151460], [151461, 151462, 151463], [151464, 151465, 151466], [151467, 151468, 151469], [151470, 151471, 151472], [151473, 151474, 151475], [151476, 151477, 151478], [151479, 151480, 151481], [151482, 151483, 151484], [151485, 151486, 151487], [151488, 151489, 151490], [151491, 151492, 151493], [151494, 151495, 151496], [151497, 151498, 151499], [151500, 151501, 151502], [151503, 151504, 151505], [151506, 151507, 151508], [151509, 151510, 151511], [151512, 151513, 151514], [151515, 151516, 151517], [151518, 151519, 151520], [151521, 151522, 151523], [151524, 151525, 151526], [151527, 151528, 151529], [151530, 151531, 151532], [151533, 151534, 151535], [151536, 151537, 151538], [151539, 151540, 151541], [151542, 151543, 151544], [151545, 151546, 151547], [151548, 151549, 151550], [151551, 151552, 151553], [151554, 151555, 151556], [151557, 151558, 151559], [151560, 151561, 151562]], &quot;color&quot;: &quot;#7fffd4&quot;, &quot;opacity&quot;: 1.0}];\n",
       "    for ( var i=0 ; i &lt; surfaces.length ; i++ ) addSurface( surfaces[i] );\n",
       "\n",
       "    function addSurface( json ) {\n",
       "\n",
       "        var useFaceColors = &#x27;faceColors&#x27; in json ? true : false;\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "        for ( var i=0 ; i &lt; json.vertices.length ; i++ ) {\n",
       "            var v = json.vertices[i];\n",
       "            geometry.vertices.push( new THREE.Vector3( a[0]*v.x, a[1]*v.y, a[2]*v.z ) );\n",
       "        }\n",
       "        for ( var i=0 ; i &lt; json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j &lt; f.length - 2 ; j++ ) {\n",
       "                var face = new THREE.Face3( f[0], f[j+1], f[j+2] );\n",
       "                if ( useFaceColors ) face.color.set( json.faceColors[i] );\n",
       "                geometry.faces.push( face );\n",
       "            }\n",
       "        }\n",
       "        geometry.computeVertexNormals();\n",
       "\n",
       "        var side = json.singleSide ? THREE.FrontSide : THREE.DoubleSide;\n",
       "        var transparent = json.opacity &lt; 1 ? true : false;\n",
       "        var depthWrite = &#x27;depthWrite&#x27; in json ? json.depthWrite : !transparent;\n",
       "        var flatShading = json.useFlatShading ? json.useFlatShading : false;\n",
       "\n",
       "        var material = new THREE.MeshPhongMaterial( { side: side,\n",
       "                                     color: useFaceColors ? &#x27;white&#x27; : json.color,\n",
       "                                     vertexColors: useFaceColors ? THREE.FaceColors : THREE.NoColors,\n",
       "                                     transparent: transparent, opacity: json.opacity,\n",
       "                                     shininess: 20, flatShading: flatShading,\n",
       "                                     depthWrite: depthWrite } );\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var mesh = new THREE.Mesh( geometry, material );\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        if ( transparent &amp;&amp; json.renderOrder ) mesh.renderOrder = json.renderOrder;\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "        if ( json.showMeshGrid ) addSurfaceMeshGrid( json );\n",
       "\n",
       "    }\n",
       "\n",
       "    function addSurfaceMeshGrid( json ) {\n",
       "\n",
       "        var geometry = new THREE.Geometry();\n",
       "\n",
       "        for ( var i=0 ; i &lt; json.faces.length ; i++ ) {\n",
       "            var f = json.faces[i];\n",
       "            for ( var j=0 ; j &lt; f.length ; j++ ) {\n",
       "                var k = j === f.length-1 ? 0 : j+1;\n",
       "                var v1 = json.vertices[f[j]];\n",
       "                var v2 = json.vertices[f[k]];\n",
       "                // vertices in opposite directions on neighboring faces\n",
       "                var nudge = f[j] &lt; f[k] ? .0005*zRange : -.0005*zRange;\n",
       "                geometry.vertices.push( new THREE.Vector3( a[0]*v1.x, a[1]*v1.y, a[2]*(v1.z+nudge) ) );\n",
       "                geometry.vertices.push( new THREE.Vector3( a[0]*v2.x, a[1]*v2.y, a[2]*(v2.z+nudge) ) );\n",
       "            }\n",
       "        }\n",
       "\n",
       "        var c = new THREE.Vector3();\n",
       "        geometry.computeBoundingBox();\n",
       "        geometry.boundingBox.getCenter( c );\n",
       "        geometry.translate( -c.x, -c.y, -c.z );\n",
       "\n",
       "        var gridColor = options.theme === &#x27;dark&#x27; ? &#x27;white&#x27; : &#x27;black&#x27;;\n",
       "        var linewidth = json.linewidth || 1;\n",
       "        var materialOptions = { color: gridColor, linewidth: linewidth };\n",
       "\n",
       "        var mesh;\n",
       "        if ( linewidth &gt; 1 &amp;&amp; window.createFatLineSegments ) {\n",
       "            mesh = createFatLineSegments( geometry, materialOptions );\n",
       "        } else {\n",
       "            var material = new THREE.LineBasicMaterial( materialOptions );\n",
       "            mesh = new THREE.LineSegments( geometry, material );\n",
       "        }\n",
       "\n",
       "        mesh.position.set( c.x, c.y, c.z );\n",
       "        mesh.userData = json;\n",
       "        scene.add( mesh );\n",
       "\n",
       "    }\n",
       "\n",
       "    function render() {\n",
       "\n",
       "        if ( window.updateAnimation ) animate = updateAnimation();\n",
       "        if ( animate ) requestAnimationFrame( render );\n",
       "\n",
       "        renderer.render( scene, camera );\n",
       "\n",
       "    }\n",
       "\n",
       "    render();\n",
       "    controls.update();\n",
       "    if ( !animate ) render();\n",
       "\n",
       "\n",
       "    // menu functions\n",
       "\n",
       "    function toggleMenu() {\n",
       "\n",
       "        var m = document.getElementById( &#x27;menu-content&#x27; );\n",
       "        if ( m.style.display === &#x27;block&#x27; ) m.style.display = &#x27;none&#x27;\n",
       "        else m.style.display = &#x27;block&#x27;;\n",
       "\n",
       "    }\n",
       "\n",
       "\n",
       "    function saveAsPNG() {\n",
       "\n",
       "        var a = document.body.appendChild( document.createElement( &#x27;a&#x27; ) );\n",
       "        a.href = renderer.domElement.toDataURL( &#x27;image/png&#x27; );\n",
       "        a.download = &#x27;screenshot&#x27;;\n",
       "        a.click();\n",
       "\n",
       "    }\n",
       "\n",
       "    function saveAsHTML() {\n",
       "\n",
       "        toggleMenu(); // otherwise visible in output\n",
       "        event.stopPropagation();\n",
       "\n",
       "        var blob = new Blob( [ &#x27;&lt;!DOCTYPE html&gt;\\n&#x27; + document.documentElement.outerHTML ] );\n",
       "        var a = document.body.appendChild( document.createElement( &#x27;a&#x27; ) );\n",
       "        a.href = window.URL.createObjectURL( blob );\n",
       "        a.download = suggestFilename();\n",
       "        a.click();\n",
       "\n",
       "        function suggestFilename() {\n",
       "            if ( !document.title ) {\n",
       "                return &#x27;graphic.html&#x27;;\n",
       "            } else if ( /\\.html?$/i.test( document.title ) ) {\n",
       "                return document.title; // already ends in .htm or .html\n",
       "            } else {\n",
       "                return document.title + &#x27;.html&#x27;;\n",
       "            }\n",
       "        }\n",
       "\n",
       "    }\n",
       "\n",
       "    function getViewpoint() {\n",
       "\n",
       "        function roundTo( x, n ) { return +x.toFixed(n); }\n",
       "\n",
       "        var v = camera.quaternion.inverse();\n",
       "        var r = Math.sqrt( v.x*v.x + v.y*v.y + v.z*v.z );\n",
       "        var axis = [ roundTo( v.x / r, 4 ), roundTo( v.y / r, 4 ), roundTo( v.z / r, 4 ) ];\n",
       "        var angle = roundTo( 2 * Math.atan2( r, v.w ) * 180 / Math.PI, 2 );\n",
       "\n",
       "        var textArea = document.createElement( &#x27;textarea&#x27; );\n",
       "        textArea.textContent = JSON.stringify( axis ) + &#x27;,&#x27; + angle;\n",
       "        textArea.style.csstext = &#x27;position: absolute; top: -100%&#x27;;\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( &#x27;copy&#x27; );\n",
       "\n",
       "        var m = document.getElementById( &#x27;menu-message&#x27; );\n",
       "        m.innerHTML = &#x27;Viewpoint copied to clipboard&#x27;;\n",
       "        m.style.display = &#x27;block&#x27;;\n",
       "        setTimeout( function() { m.style.display = &#x27;none&#x27;; }, 2000 );\n",
       "\n",
       "    }\n",
       "\n",
       "    function getCamera() {\n",
       "\n",
       "        function roundTo( x, n ) { return +x.toFixed(n); }\n",
       "\n",
       "        var pos = camera.position;\n",
       "        var pos_r = [ roundTo( pos.x, 4 ), roundTo( pos.y, 4 ), roundTo( pos.z, 4 ) ];\n",
       "   //     var up = camera.up; // up is always (0,0,1)\n",
       "        var textArea = document.createElement(&#x27;textarea&#x27;);\n",
       "        var cam_position = JSON.stringify(pos_r);\n",
       "        textArea.textContent = &#x27;,camera_position=&#x27; + cam_position;\n",
       "        textArea.style.csstext = &#x27;position: absolute; top: -100%&#x27;;\n",
       "        document.body.append( textArea );\n",
       "        textArea.select();\n",
       "        document.execCommand( &#x27;copy&#x27; );\n",
       "\n",
       "        var m = document.getElementById( &#x27;menu-message&#x27; );\n",
       "        m.innerHTML = &#x27;Camera position &#x27;+ cam_position+&#x27; copied to clipboard&#x27;;\n",
       "        m.style.display = &#x27;block&#x27;;\n",
       "        setTimeout( function() { m.style.display = &#x27;none&#x27;; }, 2000 );\n",
       "\n",
       "    }\n",
       "\n",
       "&lt;/script&gt;\n",
       "\n",
       "&lt;div id=&quot;menu-container&quot; onclick=&quot;toggleMenu()&quot;&gt;&amp;#x24d8;\n",
       "&lt;div id=&quot;menu-message&quot;&gt;&lt;/div&gt;\n",
       "&lt;div id=&quot;menu-content&quot;&gt;\n",
       "&lt;div onclick=&quot;saveAsPNG()&quot;&gt;Save as PNG&lt;/div&gt;\n",
       "&lt;div onclick=&quot;saveAsHTML()&quot;&gt;Save as HTML&lt;/div&gt;\n",
       "&lt;div onclick=&quot;getCamera()&quot;&gt;Get camera&lt;/div&gt;\n",
       "&lt;div onclick=&quot;getViewpoint()&quot;&gt;Get viewpoint&lt;/div&gt;\n",
       "&lt;div&gt;Close Menu&lt;/div&gt;\n",
       "&lt;/div&gt;&lt;/div&gt;\n",
       "\n",
       "\n",
       "&lt;/body&gt;\n",
       "&lt;/html&gt;\n",
       "\"\n",
       "        width=\"100%\"\n",
       "        height=\"400\"\n",
       "        style=\"border: 0;\">\n",
       "</iframe>\n"
      ],
      "text/plain": [
       "Graphics3d Object"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var('x,y,z')\n",
    "p=x^3+y^3+z^3+1-0.25*(x+y+z+1)^3\n",
    "r=5\n",
    "color='aquamarine'\n",
    "s = implicit_plot3d(p==0, (x,-r,r), (y,-r,r), (z,-r,r), plot_points=100, region=lambda x,y,z: x**2+y**2+z**2<=r*r, color=color)\n",
    "s.show(frame=False, viewer='threejs')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ebffd5a8-c58d-4552-b578-e0171f83ff8e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^3 - (x + y + z + 1)^2*(0.250000000000000*x + 0.250000000000000*y + 0.250000000000000*z + 0.250000000000000) + y^3 + z^3 + 1"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x*x*x+y*y*y+z*z*z+1-0.25*(x+y+z+1)*(x+y+z+1)*(x+y+z+1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b86da0c6-c6ca-4857-bac0-312101edd7ca",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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