31:T19ce,// Create a Clifford Algebra with 3,0,1 metric. Algebra(3,0,1,()=>{ var camera=1+0e0, time=.20; // The geometric elements of 3D PGA. // Grade-1 elements (reflections) represent planes. // Grade-2 elements (bireflections or line reflections) represent lines. // Grade-3 elements (trireflections or point reflections) represent points. var plane = (a,b,c,d)=>(a*1e1 + b*1e2 + c*1e3 + d*1e0).Normalized, line = (...plucker)=>(1e0+plucker*[1e01,1e02,1e03,1e12,1e13,1e23]).Normalized, point = (x,y,z)=>!(1e0 + x*1e1 + y*1e2 + z*1e3).Normalized, origin=1e123, EX=-1e023, EY=1e013, EZ=-1e012; // Transformations in PGA are represented by versors, which // are compositions (geometric product) of reflections (vectors). // Both rotations and translations can be generated by exponentiating // the element they leave invariant. (the line you rotate around) var motor = (line,angle_or_distance)=>Math.E**(angle_or_distance/2.0 * line); // As in 2D finding transforms between elements only requires the square root. var sqrt = motor => ((Math.sign(motor.s) + motor)*(1 + 0.5*(motor.s + motor.Grade(4)))).Normalized; // motors can be interpolated (similar to quaternions in two ways). var lerp = (motor, x) => (Math.sign(motor().s)*(1-x) + x*motor).Normalized; const Phi = (Math.sqrt(5.0)+1.0)*0.5, Phee = Phi-1.0, Root3=Math.sqrt(3); var sideLength=0.85, outerSphere = sideLength * Phi * Root3 * 0.5, innerSphere = sideLength * Phi * Phi * 0.5 * (1/Math.sqrt(3-Phi)), opposite=Math.sqrt( outerSphere*outerSphere - innerSphere*innerSphere); const RotationPoint=point(0,innerSphere,0),RotateePoint=point(0,innerSphere,opposite); var R = motor(RotationPoint&origin,Math.PI/2.5); // A rotation around the y-axis. var faces=[],vertices=[],edges=[]; var misMatch=1.0; // after 5 rotations, we should be this close to the actual beginning var vertexLimit=0; var showVertex = (i)=>[vertices[i],vertices[i].name||i]; var showVertices = ()=>{ var color=0x000c0c0c, hue=0x0002050a; return vertices.flatMap((e,i)=>[hue=color+hue,e,e.name||"V"+e.vertex]); } var colorIze = (vector,startColor=0x02c04030,boostColor=0x0003050a)=>{ return vector.flatMap((e,i)=>[startColor += boostColor,e]); } var showFaces = ()=>{ return faces.flatMap((e,i)=>[e.flatMap((v)=>vertices[v]),"Face"+i]); } var showEdges = ()=>{ return edges.flatMap((e,i)=>[[vertices[e[0]],vertices[e[1]]],"edge"+i]); } var addEdge= (v,idx,vertices)=>{ // Create table of known Edges. var createMetrics = (u,i)=>{return {index:i, metric:(v.n&vertices[i].n).Length}}; var shortList = Array(vertices.length).fill(null).map(createMetrics); shortList.sort((v0,v1)=>{ return (v0.metric - v1.metric)} ) // If the Length metric returned consistent results, shortlist 1,2,3 would be edge points for this vertex shortList.slice(1,4).every((item)=>{ if(idx{ // create a table of all computed vertex points for the Dodecahedron v.n=v.Normalized; v.name=name; if(vertices.length == 0){vertices.push(v);v.vertex=0;return v.vertex;}; var tooMuch=1000; var shortest = vertices.reduce((idx,test,testIdx)=>{ var thisLength=(v.n&test.n).Length; if (thisLength .1 ) { // above "same point" threshold? vertices.push(v); //No push neww vertex v.vertex = (vertices.length-1); return v.vertex; } if (l>vertexLimit) // keep track of how wobbly are the wobblies vertexLimit=l; return shortest; // return existing vertex number } var makeFace = (point,rotor=R)=>{ // five rotations of 72 degrees, at least we hope var result=[]; for (var i=0; i<5; i++){ result.push(addVertex(point)); point = rotor>>>point; } misMatch =(vertices[result[0]]&point.Normalized).Length; //Last point needs to equal first point for 5 point symmetry return result; } var flipFace=(face,factor)=>{ var schmo =face.map((nf)=>factor*vertices[nf]*factor); //schmo.every(addVertex); return schmo.map(addVertex); } var flipFlap=(aFace)=>{ //flip a pentagonal face along one of it's edges and the origin var farkle =function(i1,v,i2){ faces.push(flipFace(aFace,vertices[aFace[i1]]&vertices[aFace[i2]]&origin)); return i2; } aFace.reduce(farkle,aFace.length-1); // push the five flipped faces return aFace; // return the original face } // Create a pentagonal face and flip it five tiimes. then do it again from opposite the origin faces.push(flipFlap(makeFace(point(0,innerSphere,opposite)))); faces.push(flipFlap(makeFace(point(0,-innerSphere,-opposite)))); vertices.every(addEdge); const Opposite = (h,a)=>Math.sqrt(h*h-a*a); const GnomonHeight=Opposite(1.0,Phi/2.0),PointA=point(GnomonHeight,0,0); const segment=(i,j)=>[0x00554455,[vertices[i],vertices[j]],`V${vertices[i].name||i}-${vertices[j].name||j} `+((vertices[i].n&vertices[j].n).Length).toPrecision(4)]; var GT1=addVertex(point(0,Phi/2,0),"GT1"), GT2=addVertex(point(0,-Phi/2,0),"GT2"), GT3=addVertex(point(GnomonHeight,0,0),"GT3"), BigO=addVertex(origin,"OO"); var GT=[GT1,GT2,GT3]; // Graph it return this.graph(()=>{ // Animation time. var t = 0.5+0.5*Math.sin(performance.now()/1000); var PA=line(0,0,0,Phi,0,Phi),PC=line(0,0,0,Phi,0,-1/Phi); debugger; t = t+ 0.001; camera.set(Math.cos(t)+Math.sin(t)*1e23+Math.sin(t)*0.5*1e13); return [ "Dodecahedron Verticies: Rotator Mismatch"+misMatch,"\nFaceFlip Mismatch="+vertexLimit, ...showVertex(BigO), //...segment(GT1,GT3), //...segment(GT1,GT2), //...segment(GT2,GT3), //...segment(BigO,GT3), //...segment(BigO,GT1), //...segment(BigO,GT2), //...segment(BigO,GT1), //0x00402080,GT,GW,PointA,"size = "+GnomonHeight, //0x000f0f00,1e0+RotationPoint&origin,"Vertex Line", 0x00000000,...colorIze(showVertices()), 0x08827384,...colorIze(showFaces()), 0x002f2f2f,...showEdges(), //0x04000000,point(0,0,0),"origin", [origin,RotateePoint],[RotateePoint,RotationPoint] ]; },{grid:false, labels:true,lineWidth:3, h:-0.4, p:0.4, scale:1.4, animate:false/*,camera*/}); });32:T1224, plot=PGA3d.inline`()=>{ // In dual projectivized space, the origin is represented by the e12 bivector. // in 3D PGA, grade-1 elements or vectors (e0,e1,e2) represent // reflections AND planes (the invariant of a reflection) var plane = (a,b,c,d)=>a*1e1 + b*1e2 + c*1e3 + d*1e0; // grade-2 elements or bivectors (e01,e02,e03,e12,e13,e23) represent // rotations/translations AND lines/infinite lines (the invariant // of a rotation/translation). var line = (...plucker)=>plucker*[1e01,1e02,1e03,1e12,1e13,1e23]; // grade-3 elements or trivectors (e012,e013,e023,e123) represent // point/glide/roto reflections AND points (the invariants ..) // Again, we use dualisation for basis independent definition. var point = (x,y,z)=>!(1e0 + x*1e1 + y*1e2 + z*1e3); // Lets define some points to demonstrate basic incidence and rendering. var phi=(1.0+Math.sqrt(5.0))/2.0, phee=phi-1.0; var A=point(0,0,0), B=point(phi,0,phee), C=point(-1,-1,-1), D=point(.8,-1,.8), E=point(-.8,-1,.8); var bc = ()=>B & C, p = ()=>A & B & C; var localPlot =[]; var Dof = ${DegreesOfFreedom}; var dist_points = (P1,P2)=>(P1.Normalized&P2.Normalized).Length; var myPeeps={}; //var rot1=Algebra.rotor(phi,0,phee); debugger; for (var key in Dof) {var val=Dof[key]; myPeeps[key]=[]; for (var peep in Dof) { if (val == Dof[peep]) continue var dist=dist_points(val,Dof[peep]); if (dist < 1.5) myPeeps[key].push(peep); } } var color=0x303030ff; var colorIncrement=0x00082010; var i = 0; var faceId=0; var faces=[]; debugger; for (var me in Dof) { var meVal=Dof[me],uno,unoVal=Dof[uno=myPeeps[me][0]],tother,totherVal=Dof[tother=myPeeps[me][1]]; var t1,t2,zeePlane=totherVal&meVal&unoVal; var onPlane = (nicname)=>{ if (nicname == me) return false; return (zeePlane&Dof[nicname]).Length < 0.1; } //if(i++ ) continue //check if uno's peeps are coplaner -- only one will be if (onPlane(myPeeps[uno][0])) t1=myPeeps[uno][0]; else if (onPlane(myPeeps[uno][1])) t1=myPeeps[uno][1]; else if (onPlane(myPeeps[uno][2])) t1=myPeeps[uno][2]; if (onPlane(myPeeps[tother][0])) t2=myPeeps[tother][0]; else if (onPlane(myPeeps[tother][1])) t2=myPeeps[tother][1]; else if (onPlane(myPeeps[tother][2])) t2=myPeeps[tother][2]; faces[key]=[Dof[t2],Dof[t1],Dof[uno],Dof[me],Dof[tother]] /*localPlot.push(color,faces[key]);*/ /* localPlot.push(0x00602222,unoVal,'Uno '+uno); localPlot.push(0x00602222,totherVal,'Tother '+tother); localPlot.push(0x00602222,meVal,'ME '+me); localPlot.push(0x00602222,Dof[t2],'T2 '+t2); localPlot.push(0x00602222,Dof[t1],'T1 '+t1); */ color = color + colorIncrement; } //localPlot.push(0x00202020, val, key); //localPlot.push(0xF0f5f2f2,[Dof.FumFunFum, Dof.FohFinFee, Dof.FohFinFen, Dof.FumFunFun, Dof.FieFenFoh],"S0"); //localPlot.push(0xF0f2f5f2,[Dof.FunFunFum, Dof.FohFinFee, Dof.FohFinFen, Dof.FunFunFun, Dof.FinFenFoh],"S1"); //localPlot.push(0xc0f2f2f5,[Dof.FunFunFum, Dof.FenFohFie, Dof.FeeFohFie, Dof.FumFunFum ,Dof.FohFinFee],"S2"); //localPlot.push(0x20fcf2f2,[Dof.FunFunFum, Dof.FenFohFie, Dof.FunFumFum, Dof.FinFeeFoh, Dof.FinFenFoh],"S3"); //localPlot.push(0xc0f2f2f5,[Dof.FunFunFum, Dof.FenFohFie, Dof.FeeFohFie, Dof.FumFunFum ,Dof.FohFinFee],"S2"); //localPlot.push(0x20c02020,[Dof.FohFieFee, Dof.FumFumFun, Dof.FeeFohFie/*, Dof.FohFieFen*/]); // Sum points to get the average position. var avg = A+B+E, bc = B+C; // Join (&) points into lines var l = ()=>avg & bc; var axis0 = A&point(0,-1,0); // Intersect a line and a plane into a point. var intersect = ()=>l ^ (A & E & D); // Sum lines to get average lines. var l2 = ()=>l.Normalized + bc.Normalized; return [ "Drag A,B,C,D,E", // title. // 0xD0FFE1, [A,B,D], // polygons 0x00AA88, [A,B],[A,C],[B,C],//[A,D],[A,E],[C,E],[E,D],[D,B], // edges 0x224488, A,"A",B,"B",C,"C",//D,"D",E,"E", // points 0x900000, axis0, "AXIS - 0", 0x884488, bc,"B&C", ()=>p*0.1,"A&B&C", // join of points 0x884488, ()=>B+C, "B+C", ()=>A+B+E, "A+B+E", l, // sum of points 0x00AA88, intersect, "line ^ plane", // meets 0xFF8844, l2, "sum of lines", // sum of lines. ] return localPlot; }`(); 33:Tf8b,## Ganja starterkit : PGA3D P(R*_3,0,1) This example implements the table on page 15 of [Gunn's Geometric Algebra for Computer Graphics](http://page.math.tu-berlin.de/~gunn/Documents/Papers/GAforCGTRaw.pdf). We apply the same strategy from above and start from a Clifford Algebra in R_3,0,1. ```javascript // Create a Clifford Algebra with 3 positive and one zero generator. Algebra(3,0,1,()=>{ // Output algebra info to the console. this.describe(); // The default basis is 1,e0,e1,e2,e3,e01,e02,e03,e12,e13,e23,e012,e013,e023,e123,e0123 // The metric for vectors is 0, 1, 1, 1 - Vectors will represent planes. // The metric for bivectors is 0, 0, 0, -1, -1, -1 - Bivectors will represent lines. // The metric for trivectors is 0, 0, 0, -1 - Trivectors will represent points. // The pseudoscalar is degenerate, so use the built-in duality operator instead. // The bivectors consist of three motor elements and three rotation element - // exactly what is needed to represent translations and rotations in euclidean space. // In dual projectivized space, the origin is represented by the e12 bivector. var origin=1e123, EX=-1e023, EY=1e013, EZ=-1e012; // Points and planes can be specified directly. var point = (x,y,z)=>origin+x*EX+y*EY+z*EZ, plane = (a,b,c,d)=>a*1e1+b*1e2+c*1e3+d*1e0; // Table from "Geometric Algebra for Computer Graphics" p.15 var LineFromPoints = (P,Q)=>P&Q, LineFromPlanes = (a,b)=>a^b, PointFromPlanes = (a,b,c)=>a^b^c, PlaneFromPoints = (P,Q,R)=>P&Q&R, DistPointToPlane = (a,P)=>a&P, DistPoints = (P,Q)=>(P&Q).Length, AnglePlanes = (a,b)=>Math.acos((a<P<P<P<PI^a, PlaneThroughPointPerpLine = (PI,P)=>P<P<P<(P<PI&EP, AngleLines = (PI,EP)=>Math.acos((PI<a*X*a, Rotor = (PI,alpha)=>Math.cos(alpha/2) + Math.sin(alpha/2)*PI, RotationAroundLine = (X,PI,alpha)=>Rotor(PI,alpha)*X*~Rotor(PI,alpha), Translator = (x,y,z)=>1+0.5*(x*EX+y*EY+z*EZ); // To demonstrate graphing, we create some points and lines. // Users can drag points in the graph, lambda expressions can be // used to create dynamic updating items. var A = point(-1,-1, 0), B = point(1,-1, 0), C = point(-1,1,0); // Ganja.js can directly graph 3D PGA elements. Pass in an array of // items to render. (numbers are colors, strings are labels, PGA points // and lines are rendered automatically and arrays can be used for line // segments and polygons). The graph function returns a HTML SVG element. document.body.appendChild(this.graph([ // use numbers to set the current color. 0x444444, // strings label the items they follow, first string is a title. "title", // render points (user can drag these) A, B, C, "Label for point", // render lines ()=>(B&C&A)<<(B&C)<<-A,"Label for line", // line segments ()=>[A,B], "Label for segment", // polygons 0xffeeee, ()=>[A,B,C], 0xff7777, "Label for polygon" ],{grid:false, animate:false})); // When using the animation mode, all lambda's will be evaluated every frame. // Use Date.now() or similar. (many examples in the coffeeshop.) }); ```34:Tb04,PGA3D = notTheThumb&&quick_sheet({ title : "3D Euclidean Projective GA", titlesym : "", altname : "mv", sample : "1e0+1e1+1e2+1e3", sig : [3,0,1], evaluate : "1e1*1e12", cards : [{ description : "Plane Based Geometry", origin : "e_{123}", points : "e_{123} + xe_{012} + ye_{013} + ze_{023}", "ideal points" : "xe_{012} + ye_{013} + ze_{023}", lines : "p_1 \\vee p_2", planes : "p_1 \\vee p_2 \\vee p_3", join : "p_1 \\vee p_2", meet : "l_1 \\wedge l_2", rotors : "e^{\\frac\\alpha 2B}", motors : "e^{\\frac\\delta 2B}" },{ description:"Sub-Algebras", text:md`

Scalars	1
Complex numbers	e.g. 1,e₁₂
Hyperbolic numbers	e.g. 1,e₁
Dual numbers	e.g. 1,e₀
Quaternions	1,e₁₂, e₁₃, e₂₃
Dual Quaternions	1,e₀₁, e₀₂, e₀₃,e₁₂, e₁₃, e₂₃, e₀₁₂₃

` },{ description : "Matrices", text : md`\`\`\`javascript var M = [[1e1,1e2],[0,1]]; // 2x2 PGA matrix. var V = [1,1e123]; // 1x2 PGA vector. var R = M*V; // matrix-vector product. var N = M*M; // matrix-matrix product. var s = V*V; // vector-vector dot product. var O = !M; // conjugate-transpose matrix. \`\`\` ` },{ description : "Graphing", text : md`SVG and webGL output can be used for 3D PGA with support for points, lines, line segments, polygons, labels and colors. Shown below is a euclidean screw motion. ${ Algebra(3,0,1,()=>{ var bv = 1e02+1e13, screw=[]; for (var j=0; j<10; j++) screw = screw.concat( [...Array(40)] .map((x,i)=>Math.E**((i/16-1.2)*bv)* (Math.cos(j/10*Math.PI)+Math.sin(j/10*Math.PI)*1e13)>>> (!(1e0+1e1))) .map((x,i,a)=>[x,i?a[i-1]:x]) ); return this.graph(screw,{grid:1,width:window.innerWidth-100}); }) } ${md` \`\`\`javascript Algebra(3,0,1,()=>{ var bv = 1e02+1e13, screw=[]; for (var j=0; j<10; j++) screw = screw.concat( [...Array(40)] .map((x,i)=>Math.E**((i/16-1.2)*bv)>>> (Math.cos(j/10*Math.PI)+Math.sin(j/10*Math.PI)*1e13)>>> (!(1e0+1e1))) .map((x,i,a)=>[x,i?a[i-1]:x]) ); return this.graph(screw); }); \`\`\``} ` }] })35:Tfba,quick_sheet = (data)=>{ var al = Algebra(data.sig[0],data.sig[1],data.sig[2]); var len = data.sig[0]+data.sig[1]+data.sig[2]; if (len > 6) debugger var a=al.describe(); if (len<6) { var A=new al(), B=new al(); for (var i=0; i2?"A":("("+A+")").replace(/_(\d+)/g,"_{$1}"); var BS = len>2?"B":("("+B+")").replace(/_(\d+)/g,"_{$1}"); var evalex = (x)=>{ try { return al.inline(new Function("return "+x))().toString().replace(/_(\d+)/g,"_{$1}"); } catch (e) { return e.message; } } var updateex = (x,el)=>{ while (el.firstChild) el.removeChild(el.firstChild); el.appendChild(md`= ${tex`${evalex(x)}`}`) }} console.log('here'); return Object.assign(md`# ${data.title} ${tex`\mathbb {${data.titlesym}} \cong \mathbb{R}_{${data.sig}}`}

create

${md`\`\`\`javascript var ${data.titlesym||"A"} = Algebra(${data.sig.join(',')}); var ${data.altname||"x"} = Algebra(${data.sig.join(',')},()=>${data.sample}); \`\`\``}

basis and metric

${(len<6?a.basis:a.basis[1]).map(x=>"${(len<6?a.metric:a.metric[1]).map((x,i)=>("

"+(""+x).replace(/e(\d+)/,"e_$1")).join('')}

")+((x==1)?"+1":x)).join('')}

${a.mulTable&&md`

Cayley Table

${a.mulTable.map(x=>x.map(x=>'')}

'+x.replace(/e(\d+)/,"e_$1")).join('')).join('

`||""} ${data.cards&&data.cards.map(x=>md`

${x.description}

${ Object.keys(x).length>2? html`${Object.keys(x).slice(1).map((y,i)=>html`

${y}	${tex`${x[y]}`}`)}

` :md`${x.text}`}

`)||""} ${len in{"1":1,"2":1,"3":1}&&md`

Operators

${len>2?"

	A	":""}${len>2?md`${tex`${("("+A+")").replace(/_(\d+)/g,"_{$1}")}`}`:""} ${len>2?"
	B	":""}${len>2?md`${tex`${("("+B+")").replace(/_(\d+)/g,"_{$1}")}`}`:""}
Addition	+	${tex`${AS}+${BS}=${(""+A.Add(B)).replace(/_(\d+)/g,"_{$1}")}`}
Subtraction	-	${tex`${AS}-${BS}=${(""+A.Sub(B)).replace(/_(\d+)/g,"_{$1}")}`}
Geometric Product	*	${tex`${AS}*${BS}=${(""+A.Mul(B)).replace(/_(\d+)/g,"_{$1}")}`}
Outer Product	^	${tex`${AS} \wedge ${BS}=${(""+A.Wedge(B)).replace(/_(\d+)/g,"_{$1}")}`}
Left Contraction	<<	${tex`${AS}\cdot ${BS}=${(""+A.Dot(B)).replace(/_(\d+)/g,"_{$1}")}`}
Regressive Product	&	${tex`${AS} \vee ${BS}=${(""+A.Vee(B)).replace(/_(\d+)/g,"_{$1}")}`}
Conjugation	~	${tex`\overline{${AS}} =${(""+A.Conjugate).replace(/_(\d+)/g,"_{$1}")}`}
Dual	!	${tex`!{${AS}} =${(""+A.Dual).replace(/_(\d+)/g,"_{$1}")}`}

`||""} ${data.evaluate&&((()=>{ var r=md`

Expression Evaluator

= ${tex`${evalex(data.evaluate)}`}

`; var i = r.querySelector('input'); var j = r.querySelector('div.out') i.onkeyup = ()=>updateex(i.value,j); return r; })())||""} `,{className:"sheet",id:data.title})}36:T6bb,// Generated by CoffeeScript 1.12.7 DegreesOfFreedom= (function() { var origin=1e123, EX=-1e023, EY=1e013, EZ=-1e012; // Points and planes can be specified directly. var point = PGA3d.inline((x,y,z)=>(1e123-x*1e023+y*1e013-z*1e012)), plane = PGA3d.inline((a,b,c,d)=>a*1e1+b*1e2+c*1e3+d*1e0); var Basis0, Basis1, Basis2, Basis3, Dof, Names, Phee, Phi, i, j, k, l, len, len1, len2, len3, m, makeName, makePoint, val; Phi = (1.0 + Math.sqrt(5.0)) / 2.0; Phee = Phi - 1.0; Basis0 = [[1, 1, 1], [1, 1, -1], [1, -1, 1], [1, -1, -1], [-1, 1, 1], [-1, 1, -1], [-1, -1, 1], [-1, -1, -1]]; Basis1 = [[0, Phi, Phee], [0, Phi, -Phee], [0, -Phi, Phee], [0, -Phi, -Phee]]; Basis2 = [[Phee, 0, Phi], [-Phee, 0, Phi], [Phee, 0, -Phi], [-Phee, 0, -Phi]]; Basis3 = [[Phi, Phee, 0], [Phi, -Phee, 0], [-Phi, Phee, 0], [-Phi, -Phee, 0]]; Names = {}; makeName = function(val, name) { var index; index = "val" + val; return Names["" + val] = name; }; makeName(0, 'Foh'); makeName(1, 'Fum'); makeName(-1, 'Fun'); makeName(Phi, 'Fie'); makeName(-Phi, 'Fin'); makeName(Phee, 'Fee'); makeName(-Phee, 'Fen'); Dof = {}; makePoint = function(triple) { debugger; return Dof[Names[triple[0]] + Names[triple[1]] + Names[triple[2]]] = point(...triple); }; for (i = j = 0, len = Basis0.length; j < len; i = ++j) { val = Basis0[i]; makePoint(val); } for (i = k = 0, len1 = Basis1.length; k < len1; i = ++k) { val = Basis1[i]; makePoint(val); } for (i = l = 0, len2 = Basis2.length; l < len2; i = ++l) { val = Basis2[i]; makePoint(val); } for (i = m = 0, len3 = Basis3.length; m < len3; i = ++m) { val = Basis3[i]; makePoint(val); } return Dof; }).call(this);37:[] 2f:{"title":"3D PGA icosahedral basis","cells":[{"id":0,"value":"# 3D PGA icosahedral basis\n## PGA3D -- taken from https://ganja.js/examples/example_pga3d_points_and_lines.html and https://observablehq.com/@enkimute/ganja-js-cheat-sheets\n\nCode is broken up in three parts -\n- initialization (bottom of page)\n- display plot (adjacent below)\n- scene definition just below the plot. N.B. the code in backticks is GANJA, not JS\n- referance of PGA3d follows","pinned":true,"hidden":false,"mode":"md"},{"id":794,"value":"\nr_{u}=a\\,{\\frac {\\sqrt {3}}{2}}\\phi \nr\ni\n=\na\nϕ\n2\n2\n3\n−\nϕ\n r_{i}=a\\,{\\frac {\\phi ^{2}}{2{\\sqrt {3-\\phi }}}}\nr\nm\n=\na\nϕ\n2\n2\nr_{m}=a\\,{\\frac {\\phi ^{2}}{2}}","pinned":false,"hidden":false,"mode":"tex"},{"id":595,"value":"$31","pinned":true,"hidden":false,"mode":"ojs"},{"id":322,"value":"## Plot Definition\n","pinned":false,"hidden":false,"mode":"md"},{"id":44,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":91,"value":"$33","pinned":false,"hidden":false,"mode":"md"},{"id":32,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":4,"value":"Algebra = require(\"ganja.js\")","pinned":true,"hidden":false,"mode":"ojs"},{"id":36,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":38,"value":"notTheThumb=true","pinned":true,"hidden":false,"mode":"ojs"},{"id":81,"value":"// Create a Clifford Algebra with 3 positive and one zero generator.\nPGA3d=Algebra(3,0,1);","pinned":false,"hidden":false,"mode":"ojs"},{"id":347,"value":"$36","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W37","stdlib":"1","titling":"auto"}