31:T40f,# 3D PGA Length Quandry ## Bullet Hole in Foot Fixed! I'm exploring Algebraic Geometry by generating Dodecahedron stuff. The shape looks great, and the vertices seem to be in the right place, but the computed distance seems wacko. This makes all the sorting of length between vertices fail. Obviously I have blown a hole in my own foot again. Why is the length from Vertex 5 to 6 (V56 below) only 0.1725? -- 0.8689 would seem right. ## AHA! The key is that points function best when "Normalized," an incantation that must always be uttered before a Length is calculated. By making all the elements of the vertices array normalized upon insertion guarantees that computation on two vertices will yield valid results. ## The change: a single line in function addVertex: ``` var addVertex= (v,idx)=>{ // create a table of all computed vertex points for the Dodecahedron v=v.Normalized; //The Man of Tao choses the least effort with the most effect ``` Makes all the computed lengths consistent (and not too surprisingly NOT 0.8689)32:T1636,// 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, 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,Inscriber=Phi+.035; const RotationPoint=point(0,Inscriber,0),RotateePoint=point(0,Inscriber,0.5); 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 showVertices = ()=>{ var color=0x000c0c0c, hue=0x0002050a; return vertices.flatMap((e,i)=>[hue=color+hue,e,"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,"Face"+i]); } var showEdges = ()=>{ return edges.flatMap((e,i)=>[[vertices[e[0]],vertices[e[1]]],"edge"+i]); } // this routine fails because the Length metric and I are not on computing terms, yet. var addEdge= (v,idx,vertices)=>{ // Create table of known Edges. var shortList = Array(vertices.length).fill(null).map((u,i)=>{ var ii = i, uu= (v&vertices[i]).Length; return {index:ii, metric:uu}}); 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 edges.push([idx,shortList[1].index]); edges.push([idx,shortList[2].index]); // Not working?!???! // edges.push([idx,shortList[3].index]); return true; } var addVertex= (v,idx)=>{ // create a table of all computed vertex points for the Dodecahedron v=v.Normalized; //The Man of Tao choses the least effort with the most effect if(vertices.length == 0){vertices.push(v);v.vertex=0;return v}; var tooMuch=1000; var shortest = vertices.reduce((idx,test,testIdx)=>{ var thisLength=(v&test).Length; if (thisLength .1 ) { // above "same point" threshold? vertices.push(v); v.vertex = (vertices.length-1); return true; } if (l > 0.003) debugger; // are we sure we want to throw this one away? return true; } var makeFace = (point,rotor=R)=>{ // five rotations of 72 degrees, at least we hope var xx=[point]; xx.unshift((rotor>>>xx[0])); xx.unshift((rotor>>>xx[0])); xx.unshift(rotor>>>xx[0]); xx.unshift(rotor>>>xx[0]); xx.every(addVertex); misMatch =(point&(rotor>>>xx[0])).Length; //Last point needs to equal first point for 5 point symmetry return xx; } var flipFace=(face,factor)=>{ var schmo =face.map((nf)=>factor*nf*factor); schmo.every(addVertex); return schmo; } 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,aFace[i1]&aFace[i2]&origin)); return i2; } aFace.reduce(farkle,aFace.length-1); return aFace; } faces.push(flipFlap(makeFace(RotateePoint))); faces.push(flipFlap(makeFace(point(0,-Inscriber,-0.5)))); vertices.every(addEdge); const segment=(i,j)=>[0x00554455,[vertices[i],vertices[j]],`V${i}${j} `+((vertices[i]&vertices[j]).Length).toPrecision(4)]; // 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: V4 "+(vertices[4]&vertices[0]).Length+ " V7 "+(vertices[7]&vertices[0]).Length, "Mismatch"+misMatch, ...segment(0,1), ...segment(1,2), ...segment(0,4), ...segment(0,5), ...segment(5,6), 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}); });33: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})}34: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);35:[] 2f:{"title":"3D PGA Length Quandry","cells":[{"id":0,"value":"$31","pinned":true,"hidden":false,"mode":"md"},{"id":595,"value":"$32","pinned":true,"hidden":false,"mode":"ojs"},{"id":322,"value":"## Plot Definition\n","pinned":false,"hidden":false,"mode":"md"},{"id":4,"value":"Algebra = require(\"ganja.js\")","pinned":true,"hidden":false,"mode":"ojs"},{"id":36,"value":"$33","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":"$34","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W35","stdlib":"1","titling":"auto"}