31:T54e4,{ class Polyhedron{constructor(n,v,f){this.name=n;this.vertices=v.map(x=>new THREE.Vector3(...x));this.internalFacesList=Object.values(f);const c=new THREE.Vector3();this.vertices.forEach(x=>c.add(x));c.divideScalar(this.vertices.length);this.faceNormals=[];this.faceCenters=[];this.internalFacesList.forEach(F=>{const p0=this.vertices[F[0]],p1=this.vertices[F[1]],p2=this.vertices[F[2]],N=new THREE.Plane().setFromCoplanarPoints(p0,p1,p2).normal,fc=new THREE.Vector3().add(p0).add(p1).add(p2).divideScalar(3);if(N.dot(new THREE.Vector3().subVectors(fc,c))<0)N.negate();this.faceNormals.push(N);this.faceCenters.push(fc);});}} const vertexNames=["A","B","C","D"],vertices=[[1,0,0],[0,1,0],[0,0,1],[1,1,1]],faces=Object.fromEntries(vertexNames.map((n,i)=>[n,vertexNames.map((_,j)=>j).filter(j=>j!==i)])),poly=new Polyhedron("tetrahedron",vertices,faces); const expandSci=x=>{let s=String(x);if(!/[eE]/.test(s))return s;let[m,e]=s.toLowerCase().split("e");e=Number(e);let[a,b=""]=m.split(".");if(e>=b.length)return a+b+"0".repeat(e-b.length);return a+b.slice(0,e)+"."+b.slice(e);}, BI=x=>{if(typeof x==="bigint")return x;if(typeof x==="number"){if(!Number.isFinite(x)||!Number.isInteger(x))throw new Error(`integer must be saved as decimal string: ${x}`);return BigInt(expandSci(x));}return BigInt(String(x));}, gcd=(a,b)=>b?gcd(b,a%b):(a<0n?-a:a), faceOfInts=I=>vertexNames[I.findIndex(x=>x===0n)]??"?", approxRatio=(a,b)=>{a=BI(a);b=BI(b);if(a===0n)return 0;const neg=(a<0n)!==(b<0n);let sa=(a<0n?-a:a).toString(),sb=(b<0n?-b:b).toString();const k=16,ma=Number(sa.slice(0,k))/10**Math.min(k,sa.length),mb=Number(sb.slice(0,k))/10**Math.min(k,sb.length),e=sa.length-sb.length,r=(ma/mb)*10**e;return neg?-r:r;}, ratio=(a,b)=>approxRatio(a,b), baryPointInTet=(row,den,V)=>{const w=row.map(x=>ratio(BI(x),den));return new THREE.Vector3(w[0]*V[0].x+w[1]*V[1].x+w[2]*V[2].x+w[3]*V[3].x,w[0]*V[0].y+w[1]*V[1].y+w[2]*V[2].y+w[3]*V[3].y,w[0]*V[0].z+w[1]*V[1].z+w[2]*V[2].z+w[3]*V[3].z);}, baryPointToVec=(row,den)=>baryPointInTet(row,den,vertices.map(v=>new THREE.Vector3(...v))), latexToPlain=s=>{ s=String(s??""); const group=(str,i)=>{let d=0,j=i+1;for(;j{s=latexToPlain(s);if(!s)return"";const m=String(s).match(/^(\d+)\*sqrt$(\d+)$\/(\d+)$/);if(!m)return s;const a=BI(m[1]),n=BI(m[2]),b=BI(m[3]),num=a*n,g=gcd(num,b),p=num/g,q=b/g;return q===1n?`${p}/sqrt(${n})`:`${p}/(${q}*sqrt(${n}))`;}, heightOf=o=>o.barycentric_points.reduce((m,row)=>{const s=row.map(BI).reduce((a,b)=>a+b,0n);return s>m?s:m;},0n), isSingular=o=>!!(o?.singular||o?.kind==="singular_normal_cone"), displayWordOf=o=>isSingular(o)?(Array.isArray(o.stratum_word)?o.stratum_word.join(" "):String(o.word??o.id)):(o.word??(o.id.includes("_")?o.id.slice(o.id.indexOf("_")+1):o.id)), faceWordOf=o=>isSingular(o)?(o.face_word??o.unfold_word??""):(o.word??""), closedFormOf=o=>lengthClosedForm(o.length_exact??o.normalized_length_exact??o.length_latex??""), copyCountOf=o=>copyCounts.get(o.id)??o.copyCounts??1, familyDimOf=o=>Number(o.family_dimension??o.family?.dimension??0), familyCopyCountOf=o=>Number(o.family_copy_count??o.family?.copy_count??copyCountOf(o)), degeneracyTextOf=o=>{const d=familyDimOf(o);if(d<=0)return copyCountOf(o);const c=familyCopyCountOf(o);return c===1?`∞ (${d}D)`:`${c}∞ (${d}D)`;}, pointStyle=(o,i,ints)=>{ if(isSingular(o)){ const st=o.strata?.[i]??{},lab=st.label??o.stratum_word?.[i]??"singular"; if(st.type==="face"&&st.face)return{label:lab,face:st.face,type:"face",color:faceColors[st.face],css:faceCss[st.face]}; if(st.type==="edge")return{label:lab,face:"edge",type:"edge",color:0xff7a00,css:"#ff7a00"}; if(st.type==="vertex")return{label:lab,face:"vertex",type:"vertex",color:0x8b5cf6,css:"#8b5cf6"}; return{label:lab,face:"?",type:"singular",color:0xff7a00,css:"#ff7a00"}; } const f=faceOfInts(ints);return{label:f,face:f,type:"face",color:faceColors[f],css:faceCss[f]}; }; const ordinaryOrbits=(billiardsData?.orbits??[]).map(o=>({...o,singular:false,kind:o.kind??"ordinary"})), singularOrbits=(singularBilliardsData?.orbits??[]).map(o=>({...o,singular:true,kind:o.kind??"singular_normal_cone",length_numeric:o.length_numeric??o.length,copyCounts:o.copyCounts??1})), allOrbits=[...ordinaryOrbits,...singularOrbits]; let sortKey="length"; const pathColWidth=Math.ceil(Math.max(8,...allOrbits.map(o=>displayWordOf(o).length))/2)+"ch", closedColWidth=Math.ceil(Math.max(10,...allOrbits.map(o=>String(closedFormOf(o)).length))/2)+"ch", lenOf=o=>o.length_numeric??o.length??0, makePathRows=()=>[...allOrbits].map(o=>({o,height:heightOf(o),den:BI(o.barycentric_denominator),deg:degeneracyTextOf(o)})).sort((a,b)=>sortKey==="period"?a.o.period-b.o.period||lenOf(a.o)-lenOf(b.o):sortKey==="height"?(a.heightb.height?1:lenOf(a.o)-lenOf(b.o)):lenOf(a.o)-lenOf(b.o)); let pathRows=makePathRows(),pathData=pathRows[0].o,commonDenom=pathRows[0].den,barycentricRows=[],selectedPointIndex=0,showFactors=false,selectedPathId=pathData.id; const width=600,height=width,renderer=new THREE.WebGLRenderer({antialias:true});renderer.setSize(width,height);renderer.setPixelRatio(window.devicePixelRatio); const scene=new THREE.Scene();scene.background=new THREE.Color(0xfafafa); const camera=new THREE.PerspectiveCamera(50,width/height,.1,100);camera.position.set(1.8,1.8,2.6); const controls=new OrbitControls(camera,renderer.domElement);controls.enableDamping=true;controls.enablePan=true;controls.enableRotate=true;controls.target.set(.5,.5,.5);controls.update(); const chainWidth=1100,chainHeight=140,chainRenderer=new THREE.WebGLRenderer({antialias:true});chainRenderer.setSize(chainWidth,chainHeight);chainRenderer.setPixelRatio(window.devicePixelRatio); const chainScene=new THREE.Scene();chainScene.background=new THREE.Color(0xfafafa); const chainCamera=new THREE.PerspectiveCamera(45,chainWidth/chainHeight,.1,1000);chainCamera.position.set(0,-4,3); const chainControls=new OrbitControls(chainCamera,chainRenderer.domElement); chainControls.enableDamping=true;chainControls.enablePan=true;chainControls.enableRotate=true;chainControls.minPolarAngle=-Infinity;chainControls.maxPolarAngle=Infinity;chainControls.minAzimuthAngle=-Infinity;chainControls.maxAzimuthAngle=Infinity;chainControls.screenSpacePanning=true;chainControls.target.set(0,0,0);chainControls.update(); const solidMaterial=new THREE.LineBasicMaterial({color:0x000000}),dashedMaterial=new THREE.LineDashedMaterial({color:0x888888,dashSize:.04,gapSize:.04}),pathMaterial=new THREE.LineBasicMaterial({color:0xff0000,linewidth:2}),singularPathMaterial=new THREE.LineBasicMaterial({color:0xff7a00,linewidth:4,depthTest:false,depthWrite:false}),cubeMaterial=new THREE.LineBasicMaterial({color:0xcccccc}); const chainSolidMaterial=new THREE.LineBasicMaterial({color:0x111111}),chainPathMaterial=new THREE.LineBasicMaterial({color:0xff0000,linewidth:3,depthTest:false,depthWrite:false}); const faceColors={A:0x4e79a7,B:0xf28e2b,C:0x59a14f,D:0xe15759},faceCss={A:"#4e79a7",B:"#f28e2b",C:"#59a14f",D:"#e15759"}, faceMaterials=Object.fromEntries(vertexNames.map(k=>[k,new THREE.MeshBasicMaterial({color:faceColors[k],transparent:true,opacity:.18,side:THREE.DoubleSide,depthWrite:false})])), chainFaceMaterials=Object.fromEntries(vertexNames.map(k=>[k,new THREE.MeshBasicMaterial({color:faceColors[k],transparent:false,opacity:1,side:THREE.DoubleSide,depthWrite:true})])); const pathLine=new THREE.Line(new THREE.BufferGeometry(),pathMaterial),pathNodesGroup=new THREE.Group(),polyhedronContainer=new THREE.Group(),faceMeshes=[];polyhedronContainer.renderOrder=1;scene.add(pathLine,pathNodesGroup,polyhedronContainer); const chainGroup=new THREE.Group(),chainFaceMeshes=[],chainEdgeLines=[];chainScene.add(chainGroup); const wireframeCube=new THREE.LineSegments(new THREE.EdgesGeometry(new THREE.BoxGeometry(1,1,1)),cubeMaterial);wireframeCube.position.set(.5,.5,.5);wireframeCube.renderOrder=0;scene.add(wireframeCube); const edgeToFaces=new Map();poly.internalFacesList.forEach((f,fi)=>{for(let i=0;i{const geom=new THREE.BufferGeometry().setFromPoints(f.map(i=>poly.vertices[i]));geom.setIndex([0,1,2]);const mesh=new THREE.Mesh(geom,faceMaterials[n]);mesh.userData.faceIdx=fi;mesh.renderOrder=-1;faceMeshes.push(mesh);polyhedronContainer.add(mesh);}); const tableBox=html`

`, table=html`

`, sidePanel=html`

`, chainNote=html`

`; tableBox.appendChild(table); function reflectPointAcrossFace(p,F){const plane=new THREE.Plane().setFromCoplanarPoints(F[0],F[1],F[2]),proj=new THREE.Vector3();plane.projectPoint(p,proj);return proj.multiplyScalar(2).sub(p);} function unfoldedChain(o){ let V=vertices.map(v=>new THREE.Vector3(...v));const tets=[],pts=[],word=faceWordOf(o); if(!word)return{tets,pts}; for(let i=0;iv.clone())); const row=o.barycentric_points[i].map(BI),den=row.reduce((a,b)=>a+b,0n); pts.push(baryPointInTet(row,den,V)); const letter=word[i%word.length],F=faces[letter].map(j=>V[j]); V=V.map(p=>reflectPointAcrossFace(p,F)); } tets.push(V.map(v=>v.clone())); return{tets,pts}; } function addChainFace(V,name,offset){ const idx=faces[name],raw=idx.map(i=>V[i].clone()),geom=new THREE.BufferGeometry().setFromPoints(raw.map(p=>p.clone().add(offset))); geom.setIndex([0,1,2]); const c=V.reduce((s,p)=>s.add(p),new THREE.Vector3()).multiplyScalar(.25),fc=raw.reduce((s,p)=>s.add(p),new THREE.Vector3()).multiplyScalar(1/3); let n=new THREE.Plane().setFromCoplanarPoints(raw[0],raw[1],raw[2]).normal;if(n.dot(new THREE.Vector3().subVectors(fc,c))<0)n.negate(); const mesh=new THREE.Mesh(geom,chainFaceMaterials[name]); mesh.userData={normal:n,center:fc.add(offset.clone()),face:name}; mesh.renderOrder=0;chainFaceMeshes.push(mesh);chainGroup.add(mesh); return mesh; } function addChainEdges(V,offset,faceMap){ for(const[k,fs]of edgeToFaces.entries()){ const adjacent=fs.map(fi=>faceMap.get(vertexNames[fi])).filter(Boolean); if(!adjacent.length)continue; const [i,j]=k.split("-").map(Number),g=new THREE.BufferGeometry().setFromPoints([V[i].clone().add(offset),V[j].clone().add(offset)]),line=new THREE.Line(g,chainSolidMaterial); line.userData.adjacent=adjacent;chainEdgeLines.push(line);chainGroup.add(line); } } function lineTriT(P,D,A,B,C){ const N=new THREE.Vector3().crossVectors(new THREE.Vector3().subVectors(B,A),new THREE.Vector3().subVectors(C,A)); const den=N.dot(D);if(Math.abs(den)<1e-10)return null; const t=N.dot(new THREE.Vector3().subVectors(A,P))/den,Q=P.clone().add(D.clone().multiplyScalar(t)); const v0=new THREE.Vector3().subVectors(B,A),v1=new THREE.Vector3().subVectors(C,A),v2=new THREE.Vector3().subVectors(Q,A); const d00=v0.dot(v0),d01=v0.dot(v1),d11=v1.dot(v1),d20=v2.dot(v0),d21=v2.dot(v1),dd=d00*d11-d01*d01; const v=(d11*d20-d01*d21)/dd,w=(d00*d21-d01*d20)/dd,u=1-v-w,eps=1e-8; return u>=-eps&&v>=-eps&&w>=-eps?t:null; } function extendedLinePts(rpts,rtets,offset,word){ const P=rpts[0],D=rpts.at(-1).clone().sub(P).normalize(),tLast=D.dot(rpts.at(-1).clone().sub(P)); let t0=0,t1=tLast; const check=(T,hidden,backward)=>{ let best=null; vertexNames.forEach(name=>{ if(hidden.has(name))return; const tri=faces[name].map(i=>T[i]); const t=lineTriT(P,D,tri[0],tri[1],tri[2]); if(t==null)return; if(backward&&t<0&&(best==null||t>best))best=t; if(!backward&&t>tLast&&(best==null||tT.map(p=>p.clone().applyQuaternion(q))),rpts=pts.map(p=>p.clone().applyQuaternion(q)); const box=new THREE.Box3();rtets.flat().forEach(p=>box.expandByPoint(p));rpts.forEach(p=>box.expandByPoint(p)); const center=box.getCenter(new THREE.Vector3()),offset=center.clone().multiplyScalar(-1),word=faceWordOf(pathData); rtets.forEach((T,i)=>{ const hidden=new Set(),faceMap=new Map(); if(i>0)hidden.add(word[(i-1)%word.length]); if(i{if(!hidden.has(name))faceMap.set(name,addChainFace(T,name,offset));}); addChainEdges(T,offset,faceMap); }); const line=new THREE.Line(new THREE.BufferGeometry().setFromPoints(extendedLinePts(rpts,rtets,offset,word)),chainPathMaterial); line.renderOrder=10;chainGroup.add(line); const size=box.getSize(new THREE.Vector3()),R=Math.max(size.x,size.y,size.z,1); chainCamera.position.set(0,-1.15*R,.35*R);chainCamera.near=.01;chainCamera.far=10*R;chainCamera.updateProjectionMatrix(); chainControls.target.set(0,0,0);chainControls.update(); } function setPathById(id){ selectedPathId=id;const idx=pathRows.findIndex(r=>r.o.id===id),r=pathRows[idx>=0?idx:0]; pathData=r.o;commonDenom=r.den;selectedPointIndex=0;pathLine.material=isSingular(pathData)?singularPathMaterial:pathMaterial; barycentricRows=pathData.barycentric_points.map((row,i)=>{const ints=row.map(BI);return{ints,...pointStyle(pathData,i,ints)};}); const nodes=pathData.barycentric_points.map(row=>baryPointToVec(row,commonDenom)); pathLine.geometry.dispose();pathLine.geometry=new THREE.BufferGeometry().setFromPoints([...nodes,nodes[0]]); pathNodesGroup.clear();nodes.forEach((p,i)=>{ const r=barycentricRows[i],geom=r.type==="edge"?new THREE.BoxGeometry(1,1,1):r.type==="vertex"?new THREE.OctahedronGeometry(1.15,0):new THREE.SphereGeometry(1,12,12), s=new THREE.Mesh(geom,new THREE.MeshBasicMaterial({color:r.color,depthTest:!isSingular(pathData)})); s.position.copy(p);s.userData.index=i;pathNodesGroup.add(s); }); refreshNodeMaterials();rebuildChain();renderSidePanel();renderTable(); } function refreshNodeMaterials(){pathNodesGroup.children.forEach((s,i)=>{s.material=new THREE.MeshBasicMaterial({color:barycentricRows[i].color,depthTest:!isSingular(pathData)});s.userData.baseScale=i===selectedPointIndex?2.3:(isSingular(pathData)?1.35:1);});} function renderTable(){ const H=[["period","period"],["path",null],["length","length"],["closed form",null],["height","height"],["degeneracy",null]]; table.innerHTML=`${H.map(([h,k])=>`${h}${k===sortKey?" ↑":""}`).join("")}`; table.querySelectorAll("th[data-key]").forEach(th=>{const k=th.dataset.key;if(k)th.onclick=()=>{sortKey=k;pathRows=makePathRows();renderTable();};}); const tb=document.createElement("tbody"); pathRows.forEach(({o,height,deg})=>{ const tr=document.createElement("tr"),sing=isSingular(o),shortPath=displayWordOf(o),cf=closedFormOf(o),sel=o.id===selectedPathId; tr.style.cursor="pointer";tr.style.background=sel?(sing?"#ffe6c7":"#e8f0ff"):(sing?"#fff8ed":"transparent");tr.style.borderLeft=`5px solid ${sing?"#ff7a00":"transparent"}`; tr.innerHTML=`${o.period} ${shortPath} ${lenOf(o).toFixed(6)} ${cf} ${height.toString()} ${deg}`; tr.onclick=()=>setPathById(o.id);tb.appendChild(tr); }); table.appendChild(tb); } function renderSidePanel(){ sidePanel.innerHTML=""; const title=document.createElement("div");title.textContent=isSingular(pathData)?"Barycentric coordinates / singular strata":"Barycentric coordinates";title.style.fontWeight="600";title.style.marginBottom="8px";sidePanel.appendChild(title); if(isSingular(pathData)){const meta=document.createElement("div");meta.textContent=`normal-cone word: ${displayWordOf(pathData)}${familyDimOf(pathData)>0?`\nfamily: ${degeneracyTextOf(pathData)}`:``}`;meta.style.marginBottom="8px";meta.style.color="#7a3b00";sidePanel.appendChild(meta);} const toggle=document.createElement("label");toggle.style.display="block";toggle.style.marginBottom="8px";toggle.style.fontFamily="system-ui,sans-serif";toggle.style.fontSize="12px";toggle.style.whiteSpace="normal"; const cb=document.createElement("input");cb.type="checkbox";cb.checked=showFactors;cb.onchange=()=>{showFactors=cb.checked;renderSidePanel();}; toggle.appendChild(cb);toggle.append(" prime factors");sidePanel.appendChild(toggle); const fmt=x=>showFactors?(factorCache.get(x)??String(x)):String(x),hline=document.createElement("div");hline.textContent=`height: ${fmt(heightOf(pathData))}`;hline.style.marginBottom="8px";sidePanel.appendChild(hline); const displayRows=barycentricRows.map(({ints,label})=>[label,...ints.map(fmt)]),w=[0,1,2,3,4].map(j=>Math.max(...displayRows.map(r=>r[j].length))); displayRows.forEach((entries,i)=>{const row=document.createElement("div"),css=barycentricRows[i].css;row.textContent=`${String(i).padStart(2)} ${entries[0].padStart(w[0])}: [`+entries.slice(1).map((x,j)=>x.padStart(w[j+1])).join(", ")+"]";row.style.cursor="pointer";row.style.padding="2px 6px 2px 4px";row.style.margin="1px 0";row.style.borderRadius="4px";row.style.display="inline-block";row.style.width="fit-content";row.style.borderLeft=`6px solid ${css}`;row.style.background=i===selectedPointIndex?`${css}33`:`${css}14`;if(i===selectedPointIndex){row.style.color="#000";row.style.fontWeight="700";}row.onclick=()=>{selectedPointIndex=i;refreshNodeMaterials();renderSidePanel();};const wrap=document.createElement("div");wrap.style.width="max-content";wrap.appendChild(row);sidePanel.appendChild(wrap);}); } setPathById(selectedPathId); const view=html`

`,bottom=html`

`; bottom.appendChild(renderer.domElement);bottom.appendChild(sidePanel); view.appendChild(tableBox);view.appendChild(bottom);view.appendChild(chainNote);view.appendChild(chainRenderer.domElement); const cameraWorldPos=new THREE.Vector3(); function animate(){ controls.update();chainControls.update(); pathNodesGroup.children.forEach(s=>{const sc=camera.position.distanceTo(s.position)*.005*(s.userData.baseScale??1);s.scale.set(sc,sc,sc);}); camera.getWorldPosition(cameraWorldPos); const back=poly.faceNormals.map((n,i)=>n.dot(new THREE.Vector3().subVectors(poly.faceCenters[i],cameraWorldPos))>0); polyhedronContainer.children.forEach(line=>{const fs=line.userData.faces;if(fs){const isBack=fs.every(fi=>back[fi]);line.material=isBack?dashedMaterial:solidMaterial;if(isBack)line.computeLineDistances();}}); faceMeshes.forEach(m=>m.visible=!back[m.userData.faceIdx]); renderer.render(scene,camera); chainEdgeLines.forEach(e=>{e.visible=e.userData.adjacent.some(m=>m.visible);}); chainRenderer.render(chainScene,chainCamera); } renderer.setAnimationLoop(animate); invalidation.then(()=>renderer.setAnimationLoop(null)); return view; }32:Td8e,{ const maxLength=24,binWidth=2; const expanded=billiardsData.orbits.flatMap(o=>{ const L=+o.length_numeric,mult=copyCounts.get(o.id)??1,out=[]; for(let k=1;k*L<=maxLength;k++)out.push({length:k*L,mult,kind:k===1?"primitive":"nonprimitive"}); return out; }); const bins=d3.range(0,maxLength,binWidth).map(x0=>{ const x1=x0+binWidth,values=expanded.filter(d=>d.length>=x0&&d.lengthd.kind==="primitive"),nonprimitive:values.filter(d=>d.kind==="nonprimitive")}; }); const W=900,H=320,ml=60,mr=20,mt=20,mb=40, x=d3.scaleLinear().domain([0,maxLength]).range([ml,W-mr]), yMax=d3.max(bins,b=>d3.sum(b.primitive,d=>d.mult)+d3.sum(b.nonprimitive,d=>d.mult))??1, y=d3.scaleLinear().domain([0,yMax]).nice().range([H-mb,mt]), color={primitive:"#4e79a7",nonprimitive:"#f28e2b"}; const wrap=html`

`, tip=html``, svg=d3.create("svg").attr("width",W).attr("height",H); const list=vals=>vals.length ? vals.map(e=>`${e.length.toFixed(6)} ${e.mult}`).join("\n") : "(none)"; const showTip=(ev,b,kind)=>{ const p=d3.sum(b.primitive,d=>d.mult),n=d3.sum(b.nonprimitive,d=>d.mult); tip.style.display="block"; tip.style.left=(ev.offsetX+12)+"px"; tip.style.top=(ev.offsetY+12)+"px"; tip.innerHTML=kind==="primitive" ? `

primitive count: ${p}

${list(b.primitive)}

` : `

total: ${p+n}
non-primitive: ${n}
primitive: ${p}

${list(b.nonprimitive)}

${list(b.primitive)}

`; }; svg.append("g").attr("transform",`translate(0,${H-mb})`).call(d3.axisBottom(x)); svg.append("g").attr("transform",`translate(${ml},0)`).call(d3.axisLeft(y)); svg.append("text").attr("x",W/2).attr("y",H-5).attr("text-anchor","middle").text("Euclidean length"); svg.append("text").attr("x",-H/2).attr("y",14).attr("transform","rotate(-90)").attr("text-anchor","middle").text("count with multiplicity"); for(const b of bins){ const xp=x(b.x0)+1,ww=Math.max(0,x(b.x1)-x(b.x0)-2), p=d3.sum(b.primitive,d=>d.mult),n=d3.sum(b.nonprimitive,d=>d.mult),total=p+n; svg.append("rect").attr("x",xp).attr("y",y(p)).attr("width",ww).attr("height",y(0)-y(p)).attr("fill",color.primitive) .on("mousemove",ev=>showTip(ev,b,"primitive")).on("mouseleave",()=>tip.style.display="none"); svg.append("rect").attr("x",xp).attr("y",y(total)).attr("width",ww).attr("height",y(p)-y(total)).attr("fill",color.nonprimitive) .on("mousemove",ev=>showTip(ev,b,"nonprimitive")).on("mouseleave",()=>tip.style.display="none"); } const legend=svg.append("g").attr("transform",`translate(${W-170},20)`); [["primitive","primitive"],["nonprimitive","non-primitive"]].forEach(([k,label],i)=>{ legend.append("rect").attr("x",0).attr("y",i*20).attr("width",12).attr("height",12).attr("fill",color[k]); legend.append("text").attr("x",18).attr("y",i*20+10).text(label); }); wrap.appendChild(svg.node()); wrap.appendChild(tip); return wrap; }33:T6cc,copyCounts = { const expandSci = x => { let s = String(x); if (!/[eE]/.test(s)) return s; let [m, e] = s.toLowerCase().split("e"); e = Number(e); let [a, b = ""] = m.split("."); if (e >= b.length) return a + b + "0".repeat(e - b.length); return a + b.slice(0, e) + "." + b.slice(e); }; const BI = x => { if (typeof x === "bigint") return x; if (typeof x === "number") { if (!Number.isFinite(x) || !Number.isInteger(x)) throw new Error(`integer must be saved as decimal string: ${x}`); return BigInt(expandSci(x)); } return BigInt(String(x)); }; const rowKey = row => row.map(x => BI(x).toString()).join(","); const cyclicKeys = M => { const n = M.length, R = M.map(rowKey), keys = []; for (let s = 0; s < n; s++) keys.push([...Array(n)].map((_, i) => R[(i + s) % n]).join(";")); const RR = [...R].reverse(); for (let s = 0; s < n; s++) keys.push([...Array(n)].map((_, i) => RR[(i + s) % n]).join(";")); return keys; }; const canonicalPathKey = M => cyclicKeys(M).sort()[0]; const perms4 = (() => { const out = []; const rec = (pool, r) => r.length === 4 ? out.push(r) : pool.forEach((x, i) => rec(pool.filter((_, j) => j !== i), [...r, x])); rec([0, 1, 2, 3], []); return out; })(); const copyCount = o => { const seen = new Set(); for (const p of perms4) { const M = o.barycentric_points.map(row => p.map(j => row[j])); seen.add(canonicalPathKey(M)); } return seen.size; }; const allOrbits = [ ...(billiardsData?.orbits ?? []), ...(singularBilliardsData?.orbits ?? []) ]; return new Map(allOrbits.map(o => [o.id, copyCount(o)])); }34:Tae6,## Some legacy data from previous runs based on "physical" simulations: starting with a "near-closed" billiard path, and adjust the reflecting points. | length | closed form | bounce sequence | barycentric coordinates | degeneracy | |:------:|:---------------:|:-----------:|:------------------------:|:----------:| | 3.5777 | ${tex`\dfrac{4}{\sqrt{10}} s`} | abcd | `[[0, 3, 4, 3],`
`[3, 0, 3, 4],`
`[4, 3, 0, 3],`
`[3, 4, 3, 0]]` | 3 | | 4.0567 | ${tex`\dfrac{12}{\sqrt{70}} s`} | adbdcd | `[[0, 16, 16, 3],`
`[9, 9, 17, 0],`
`[16, 0, 16, 3],`
`[17, 9, 9, 0],`
`[16, 16, 0, 3],`
`[9, 17, 9, 0]]`
| 4 | | 4.5255 | ${tex`\dfrac{8}{5} s`} | abcdbc | `[[0, 9, 9, 32],`
`[18, 0, 6, 26],`
`[26, 6, 0, 18],`
`[32, 9, 9, 0],`
`[26, 0, 6, 18],`
`[18, 6, 0, 26]]` | 6 | | 8.3404 | - | adbdcdadcbdc | - | | 8.3415 | - | cabdacadabad | - | | 8.4223 | ${tex`\dfrac{60}{\sqrt{406}} s`} | badbcabcdacd | `[[45, 0, 1280, 96],`
` [0, 63, 1295, 63],`
`[45, 96, 1280, 0],`
`[77, 0, 672, 672],`
`[45, 96, 0, 1280],`
`[0, 63, 63, 1295],`
`[45, 0, 96, 1280],`
`[77, 672, 0, 672],`
`[45, 1280, 96, 0],`
`[0, 1295, 63, 63],`
`[45, 1280, 0, 96],`
`[77, 672, 672, 0]]` | 4 | | 9.0622 | ${tex`\dfrac{72}{\sqrt{505}} s`} | cadbdadcdabada | `[[132, 1292, 0, 596],`
`[0, 1171, 297, 552],`
`[552, 297, 1171, 0],`
`[596, 0, 1292, 132],`
`[288, 594, 1138, 0],`
`[0, 962, 962, 96],`
`[288, 1138, 594, 0],`
`[596, 1292, 0, 132],`
`[552, 1171, 297, 0],`
`[0, 297, 1171, 552],`
`[132, 0, 1292, 596],`
`[0, 594, 1138, 288],`
`[96, 962, 962, 0],`
`[0, 1138, 594, 288]]` | 6 | | 11.0481 | - | bcadbcadbacada | - | | 11.1584 | - | bdabcbdbacbadaca | - | | 11.2331 | ${tex`\dfrac{100}{\sqrt{634}} s`} | abacabdbabdcdbdcacdc | `[[0, 135, 39, 460],`
`[15, 0, 44, 575],`
`[0, 27, 27, 580],`
`[15, 44, 0, 575],`
`[0, 39, 135, 460],`
`[39, 0, 460, 135],`
`[44, 15, 575, 0],`
` [27, 0, 580, 27],`
` [0, 15, 575, 44],`
`[135, 0, 460, 39],`
`[460, 39, 135, 0]]`
`[575, 44, 0, 15],`
`[580, 27, 27, 0],`
`[575, 0, 44, 15],`
`[460, 135, 39, 0],`
`[135, 460, 0, 39],`
` [0, 575, 15, 44],`
`[27, 580, 0, 27],`
`[44, 575, 15, 0],`
`[39, 460, 0, 135]]`
| 3 | | 13.5910 | - | DBABCBADBACBADABACAB | - | | 13.5918 | - | CDBADBCBDBABDCBDADBD | - | | 13.7801 | - | BCADCABDCADCBADCAD | - | | 16.2376 | - | DCBACDCBCACDBCABDBCBAB | - | | 20.2616 | - | ACDBACDABCADBCABDCBADCBD | - | | 20.6587 | - | CDBCACDBCADBCADBCDADBCDBAB | - | | 22.0787 | - | ADCBADBCABDCABDCABDACBADCB | - |35:T10e5,singularBilliardsData = ({ "format": "tetra_singular_normal_cone_condensed_v2", "tetrahedron": { "vertex_order": ["A", "B", "C", "D"], "vertices": [ [1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1] ] }, "notation": { "A": "face hit on F_A", "(AD)": "edge hit on E_AD", "[D]": "vertex hit on V_D" }, "orbits": [ { "id": "s02_AD_BC", "kind": "singular_normal_cone", "period": 2, "word": "(AD)(BC)", "stratum_word": ["(AD)", "(BC)"], "length_exact": "2", "length_numeric": 2.0, "barycentric_denominator": 2, "barycentric_points": [ [1, 0, 0, 1], [0, 1, 1, 0] ] }, { "id": "s02_D_vD", "kind": "singular_normal_cone", "period": 2, "word": "D[D]", "stratum_word": ["D", "[D]"], "length_exact": "4*sqrt(3)/3", "length_numeric": 2.309401076758503, "barycentric_denominator": 3, "barycentric_points": [ [1, 1, 1, 0], [0, 0, 0, 3] ] }, { "id": "s03_AD_D_A", "kind": "singular_normal_cone", "period": 3, "word": "(AD)DA", "stratum_word": ["(AD)", "D", "A"], "length_exact": "4*sqrt(2)/3", "length_numeric": 1.8856180831641267, "barycentric_denominator": 6, "barycentric_points": [ [3, 0, 0, 3], [2, 2, 2, 0], [0, 2, 2, 2] ] }, { "id": "s04_AD_D_B_A", "kind": "singular_normal_cone", "period": 4, "word": "(AD)DBA", "stratum_word": ["(AD)", "D", "B", "A"], "length_exact": "4*sqrt(21)/9", "length_numeric": 2.036700308869266, "barycentric_denominator": 126, "barycentric_points": [ [63, 0, 0, 63], [60, 15, 51, 0], [35, 0, 56, 35], [0, 15, 51, 60] ] }, { "id": "s04_AD_D_AD_A", "kind": "singular_normal_cone", "period": 4, "word": "(AD)D(AD)A", "stratum_word": ["(AD)", "D", "(AD)", "A"], "length_exact": "4*sqrt(3)/3", "length_numeric": 2.3094010767585034, "barycentric_denominator": 6, "barycentric_points": [ [3, 0, 0, 3], [4, 1, 1, 0], [3, 0, 0, 3], [0, 1, 1, 4] ], "family_dimension": 1, "family_copy_count": 6, "representative_of_family": true, "representative_parameter_value": "1/2", "family_parameter": { "name": "t", "range": ["0", "1"], "default": "1/2", "open_interval": true, "description": "edge point [t,0,0,1-t] on E_AD", "length_constant": true, "barycentric_points_formula": [ ["t", "0", "0", "1-t"], ["(1+2*t)/3", "(1-t)/3", "(1-t)/3", "0"], ["t", "0", "0", "1-t"], ["0", "t/3", "t/3", "(3-2*t)/3"] ] } }, { "id": "s05_AD_D_B_C_A", "kind": "singular_normal_cone", "period": 5, "word": "(AD)DBCA", "stratum_word": ["(AD)", "D", "B", "C", "A"], "length_exact": "4*sqrt(26)/9", "length_numeric": 2.2662308949301417, "barycentric_denominator": 234, "barycentric_points": [ [117, 0, 0, 117], [150, 24, 60, 0], [130, 0, 52, 52], [52, 52, 0, 130], [0, 60, 24, 150] ] }, { "id": "s06_AD_BC_AD_D_AD_A", "kind": "singular_normal_cone", "period": 6, "word": "(AD)(BC)(AD)D(AD)A", "stratum_word": ["(AD)", "(BC)", "(AD)", "D", "(AD)", "A"], "length_exact": "2*sqrt(33)/3", "length_numeric": 3.8297084310253537, "barycentric_denominator": 90, "barycentric_points": [ [9, 0, 0, 81], [0, 45, 45, 0], [81, 0, 0, 9], [80, 5, 5, 0], [45, 0, 0, 45], [0, 5, 5, 80] ] }, { "id": "s06_AD_D_BD_B_AB_A", "kind": "singular_normal_cone", "period": 6, "word": "(AD)D(BD)B(AB)A", "stratum_word": ["(AD)", "D", "(BD)", "B", "(AB)", "A"], "length_exact": "sqrt(66)/2", "length_numeric": 4.062019202317982, "barycentric_denominator": 12, "barycentric_points": [ [6, 0, 0, 6], [5, 5, 2, 0], [0, 6, 0, 6], [5, 0, 2, 5], [6, 6, 0, 0], [0, 5, 2, 5] ] } ] 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2f:{"title":"Billiard in Tetrahedra","cells":[{"id":0,"value":"# Billiard in Tetrahedra\n\nLet the *regular* tetrahedron be given by non-adjacent vertices of a cube, e.g.\n\n${tex.block`A = [1, 0, 0] \\\\\nB = [0, 1, 0] \\\\\nC = [0, 0, 1] \\\\\nD = [1, 1, 1]`}\n\nand let the same letters denote the faces opposite them.\n\nA billiard path in any tetrahedron can be unfolded by reflecting the tetrahedron across the faces it hits, so that the trajectory becomes a straight line ${tex`L`} through a chain of reflected copies. For a given face sequence ${tex`w`}, let ${tex`T_w`} be the affine isometry taking the first tetrahedron to the last in this chain. Then a closed billiard path with word ${tex`w`} exists exactly when there is a line ${tex`L`} with ${tex`T_w(L)=L`}, and whose segment through the chain intersects the corresponding faces in their interiors in the prescribed order.","pinned":false,"hidden":false,"mode":"md"},{"id":136,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"## Counts \n\n* without multiplicity or non-primitives\n\n| period | ${[...new Set(billiardsData.orbits.map(o => o.period))].sort((a,b)=>a-b).join(\" | \")} |\n|---|${[...new Set(billiardsData.orbits.map(o => o.period))].map(() => \"---\").join(\"|\")}|\n| count | ${[...new Set(billiardsData.orbits.map(o => o.period))].sort((a,b)=>a-b).map(p => billiardsData.orbits.filter(o => o.period === p).length).join(\" | \")} |","pinned":false,"hidden":false,"mode":"md"},{"id":474,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":394,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":401,"value":"factorCache = {\n const sup = \"⁰¹²³⁴⁵⁶⁷⁸⁹\";\n const toSup = e => String(e).split(\"\").map(d => sup[d]).join(\"\");\n const cache = new Map();\n\n const factor = n => {\n n = BigInt(String(n));\n const s0 = n.toString();\n\n // Very conservative cutoff. Trial division above this can freeze Observable.\n if (s0.replace(\"-\", \"\").length > 12) return s0;\n\n if (n === 0n) return \"0\";\n let x = n < 0n ? -n : n, parts = [];\n\n for (let p = 2n; p * p <= x; p += p === 2n ? 1n : 2n) {\n let e = 0n;\n while (x % p === 0n) x /= p, e++;\n if (e) parts.push(`${p}${toSup(e)}`);\n }\n\n if (x > 1n) parts.push(`${x}¹`);\n return (n < 0n ? \"−\" : \"\") + parts.join(\"\");\n };\n\n return {\n get(x) {\n const k = BigInt(String(x)).toString();\n if (!cache.has(k)) cache.set(k, factor(k));\n return cache.get(k);\n }\n };\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":87,"value":"$34","pinned":false,"hidden":false,"mode":"md"},{"id":246,"value":"## Data","pinned":false,"hidden":false,"mode":"md"},{"id":166,"value":"polyhedronData = FileAttachment(\"observable_data@5.txt\").json()","pinned":true,"hidden":false,"mode":"ojs"},{"id":340,"value":"billiardsData = {\n const txt = await FileAttachment(\"tetrahedron_geometry_first_orbits_cached.json\").text()\n \n const safeTxt = txt.replace(\n /(:|\\[|,)\\s*(-?\\d{16,})(?=\\s*[,}\\]])/g,\n '$1\"$2\"'\n );\n\n return JSON.parse(safeTxt);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":541,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":256,"value":"## Imports","pinned":false,"hidden":false,"mode":"md"},{"id":127,"value":"THREE = require(\"three@0.150.1\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":139,"value":"OrbitControls = (await import(\n \"three@0.150.1/examples/jsm/controls/OrbitControls.js\"\n)).OrbitControls","pinned":true,"hidden":false,"mode":"ojs"},{"id":508,"value":"TrackballControls = (await import( \"three@0.160/examples/jsm/controls/TrackballControls.js\"\n)).TrackballControls","pinned":false,"hidden":false,"mode":"ojs"},{"id":476,"value":"Plot = require(\"@observablehq/plot\")","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W36","stdlib":"1","titling":"auto"}