31:T2711,{ const DEFAULT=[{id:"c3_8",name:"C3 rational theta: ABDACBDA",words:["ABDACBDA","BCDBACDB","CADCBADC"],P:["7969759/12106519","7969759/12106519","7969759/12106519"],Q:["4733767/12106519","4733767/12106519","4733767/12106519"]},{id:"quad_333",name:"Quadratic theta: ABC / BDA / CDB",words:["ABC","BDA","CDB"],P:["2902/4825-162*sqrt(14089)/352225","331/4825+864*sqrt(14089)/352225","2626/4825+594*sqrt(14089)/352225"],Q:["2626/4825+594*sqrt(14089)/352225","331/4825+864*sqrt(14089)/352225","2902/4825-162*sqrt(14089)/352225"]}],RAW=typeof thetaNetsJSON==="undefined"?DEFAULT:thetaNetsJSON,NETS=typeof RAW==="string"?JSON.parse(RAW):RAW; const W=width,H=Math.min(680,Math.max(480,Math.round(width*.68))),c=document.createElement("div");c.style.cssText=`position:relative;width:${W}px;height:${H}px;background:white;border-radius:10px;overflow:hidden;`; const s=new THREE.Scene();s.background=new THREE.Color(0xffffff);const cam=new THREE.PerspectiveCamera(38,W/H,.01,100);cam.position.set(2.35,-2.75,2.15);cam.lookAt(.5,.5,.5);const r=new THREE.WebGLRenderer({antialias:true});r.setPixelRatio(Math.min(devicePixelRatio||1,2));r.setSize(W,H);r.domElement.style.display="block";c.appendChild(r.domElement);const ctl=new OrbitControls(cam,r.domElement);ctl.target.set(.5,.5,.5);ctl.enableDamping=true;ctl.dampingFactor=.08;s.add(new THREE.AmbientLight(0xffffff,.82));const dl=new THREE.DirectionalLight(0xffffff,.75);dl.position.set(3,-4,5);s.add(dl); const V={A:[1,0,0],B:[0,1,0],C:[0,0,1],D:[1,1,1]},N={A:[-1,1,1],B:[1,-1,1],C:[1,1,-1],D:[1,1,1]},I=[[1,0,0],[0,1,0],[0,0,1]],COL=[0x2563eb,0xf97316,0x16a34a,0x9333ea,0xdc2626,0x0891b2],CSS=["#2563eb","#f97316","#16a34a","#9333ea","#dc2626","#0891b2"],va=(a,b)=>a.map((x,i)=>x+b[i]),vs=(a,b)=>a.map((x,i)=>x-b[i]),sm=(k,a)=>a.map(x=>k*x),dt=(a,b)=>a.reduce((t,x,i)=>t+x*b[i],0),mv=(M,v)=>M.map(row=>dt(row,v)),mm=(A,B)=>A.map(row=>B[0].map((_,j)=>row.reduce((t,x,k)=>t+x*B[k][j],0))),mt=M=>M[0].map((_,j)=>M.map(row=>row[j])),V3=p=>new THREE.Vector3(p[0],p[1],p[2]),uv=a=>V3(a).normalize(); function expr(x){if(typeof x==="number")return x;if(Array.isArray(x))return x.map(expr);if(x&&typeof x==="object"){if("approx"in x)return +x.approx;if("value"in x)return expr(x.value);if("expr"in x)return expr(x.expr);if("poly"in x&&"interval"in x)return algroot(x.poly,x.interval)}let e=String(x).replace(/√\s*([0-9.]+)/g,"sqrt($1)").replace(/\bsqrt\s*\(/g,"Math.sqrt(").replace(/\^/g,"**");if(/[^0-9+\-*/().,\sMathsqrt]/.test(e))throw Error("unsupported exact expression: "+x);return Function(`"use strict";return (${e})`)()} function algroot(poly,iv){let a=expr(iv[0]),b=expr(iv[1]),f=x=>poly.reduce((y,c)=>y*x+Number(c),0),fa=f(a),fb=f(b);for(let i=0;i<90;i++){let m=(a+b)/2,fm=f(m);if(!fm)return m;if(Math.sign(fm)===Math.sign(fa)){a=m;fa=fm}else{b=m;fb=fm}}return(a+b)/2} function fmt(x){return x&&typeof x==="object"&&!Array.isArray(x)?x.exact||x.expr||x.value||("poly"in x?`root(${x.poly.join(",")}; ${x.interval.join(",")})`:String(x)):String(x)} function xyz(net,k){return (net[k]||net.points?.[k]).map(expr)} function xyzfmt(net,k){return (net[k]||net.points?.[k]).map(fmt)} function refl(f){const n=N[f],nn=dt(n,n),R=I.map((row,i)=>row.map((x,j)=>x-2*n[i]*n[j]/nn)),t=n.map(x=>2*x/nn);return{R,t}}const REF=Object.fromEntries("ABCD".split("").map(f=>[f,refl(f)])),step=(T,f)=>({R:mm(T.R,REF[f].R),t:va(mv(T.R,REF[f].t),T.t)}),tword=w=>{let T={R:I,t:[0,0,0]};for(const ch of w)T=step(T,ch);return T},app=(T,x)=>va(mv(T.R,x),T.t),iapp=(T,x)=>mv(mt(T.R),vs(x,T.t)); function poly(w,P,Q){const Tw=tword(w),Qw=app(Tw,Q),VV=vs(Qw,P);let T={R:I,t:[0,0,0]},pts=[P];for(const ch of w){const Pc=iapp(T,P),Vc=mv(mt(T.R),VV),n=N[ch],u=(1-dt(n,Pc))/dt(n,Vc),y=va(Pc,sm(u,Vc));pts.push(y);T=step(T,ch)}pts.push(Q);return pts} function sph(g,p,rad,col,op=1){const ge=new THREE.SphereGeometry(rad,24,16),ma=new THREE.MeshStandardMaterial({color:col,roughness:.45,metalness:0,transparent:op<1,opacity:op}),x=new THREE.Mesh(ge,ma);x.position.copy(V3(p));g.add(x);return x} function seg(g,a,b,col,rad=.0085){const A=V3(a),B=V3(b),mid=A.clone().add(B).multiplyScalar(.5),dir=B.clone().sub(A),len=dir.length(),ge=new THREE.CylinderGeometry(rad,rad,len,14),ma=new THREE.MeshStandardMaterial({color:col,roughness:.35,metalness:0}),x=new THREE.Mesh(ge,ma);x.position.copy(mid);x.quaternion.setFromUnitVectors(new THREE.Vector3(0,1,0),dir.clone().normalize());g.add(x);return x} function path(g,pts,col){for(let i=0;i+1V[k]),tetG=new THREE.BufferGeometry();tetG.setAttribute("position",new THREE.Float32BufferAttribute(tetPos,3));tetG.setIndex([1,2,3,0,3,2,0,1,3,0,2,1]);tetG.computeVertexNormals();s.add(new THREE.Mesh(tetG,new THREE.MeshStandardMaterial({color:0xf8fafc,transparent:true,opacity:.18,side:THREE.DoubleSide,depthWrite:false,roughness:.6,metalness:0})));s.add(new THREE.LineSegments(new THREE.EdgesGeometry(tetG),new THREE.LineBasicMaterial({color:0x222222,transparent:true,opacity:.72}))); const staticG=new THREE.Group();s.add(staticG);for(const k of["A","B","C","D"])label(staticG,k,va(V[k],k==="D"?[.045,.045,.045]:k==="A"?[.05,-.035,-.035]:k==="B"?[-.035,.05,-.035]:[-.035,-.035,.05]),{fontSize:46,scale:.11,color:"#334155",background:"rgba(255,255,255,0.72)"}); let dyn=new THREE.Group();s.add(dyn); const panel=document.createElement("div");panel.style.cssText="position:absolute;left:12px;top:12px;max-width:520px;padding:10px 12px;border-radius:10px;background:rgba(255,255,255,0.88);box-shadow:0 4px 18px rgba(15,23,42,0.12);font:13px/1.35 system-ui,-apple-system,Segoe UI,sans-serif;color:#0f172a;";const sel=document.createElement("select");sel.style.cssText="width:100%;margin-bottom:8px;font:13px system-ui,sans-serif;padding:4px 6px;border:1px solid #cbd5e1;border-radius:7px;background:white;";NETS.forEach((n,i)=>{const o=document.createElement("option");o.value=i;o.textContent=n.name||n.id||`net ${i+1}`;sel.appendChild(o)});const info=document.createElement("div");panel.appendChild(sel);panel.appendChild(info);c.appendChild(panel); function disposeGroup(g){g.traverse(o=>{if(o.geometry)o.geometry.dispose();if(o.material)Array.isArray(o.material)?o.material.forEach(m=>m.dispose()):o.material.dispose()});s.remove(g)} function draw(i){disposeGroup(dyn);dyn=new THREE.Group();s.add(dyn);const net=NETS[i],P=xyz(net,"P"),Q=xyz(net,"Q"),PF=xyzfmt(net,"P"),QF=xyzfmt(net,"Q"),words=net.words||net.branches?.map(b=>b.word)||[];sph(dyn,P,.034,0x111111);sph(dyn,Q,.034,0x111111);label(dyn,"P",va(P,[.045,.045,.045]),{fontSize:54,scale:.13,color:"#111"});label(dyn,"Q",va(Q,[-.05,-.05,-.05]),{fontSize:54,scale:.13,color:"#111"});const pTips=[],qTails=[];words.forEach((w,j)=>{const col=COL[j%COL.length];path(dyn,poly(w,P,Q),col);const T=tword(w),out=vs(app(T,Q),P),inc=mv(mt(T.R),out),u=uv(out),v=uv(inc);arrow(dyn,P,out,col,.24);pTips.push([P[0]+.24*u.x,P[1]+.24*u.y,P[2]+.24*u.z]);const tail=[Q[0]-.24*v.x,Q[1]-.24*v.y,Q[2]-.24*v.z];arrow(dyn,tail,inc,col,.24);qTails.push(tail)});if(pTips.length>=3)loop(dyn,pTips);if(qTails.length>=3)loop(dyn,qTails);info.innerHTML=`${words.map((w,j)=>`

${w}

`).join("")}

P = (${PF.join(", ")})
Q = (${QF.join(", ")})

`} sel.onchange=()=>draw(+sel.value);draw(0); const cred=document.createElement("div");cred.style.cssText="position:absolute;right:10px;bottom:8px;font:12px system-ui,sans-serif;color:#64748b;background:rgba(255,255,255,0.65);padding:4px 7px;border-radius:8px;";cred.textContent="drag to rotate · wheel to zoom";c.appendChild(cred); let f;!function a(){f=requestAnimationFrame(a);ctl.update();r.render(s,cam)}(); invalidation.then(()=>{cancelAnimationFrame(f);ctl.dispose();r.dispose();s.traverse(o=>{if(o.geometry)o.geometry.dispose();if(o.material)Array.isArray(o.material)?o.material.forEach(m=>m.dispose()):o.material.dispose()})}); return c }32:T99a,thetaNetsJSON = [ { "id": "theta_AB_CBDC_DCAD", "name": "theta: AB / CBDC / DCAD", "words": [ "AB", "CBDC", "DCAD" ], "P": [ "8132885/15127647", "25393315/44820391", "14322912/49459777" ], "Q": [ "20651493/44663206", "19427076/44820391", "12862655/44417228" ], "metadata": { "lengths_bounces": [ 2, 4, 4 ], "total_bounces": 10, "lambda_ratios": [ "1", "1.0050006284544317", "1.0050006284544322" ], "branch_lengths_unfolded": [ "1.3340846905556567", "1.3274466231999438", "1.3274466231999433" ], "lambda_times_length": [ "1.3340846905556567", "1.3340846905556567", "1.3340846905556569" ], "norm_residual": "1.6653345369377348e-16", "balance_residual": "5.8878467200641567e-16", "min_validity_margin": "0.022937750768554699" } }, { "id": "theta_ABC_BDA_CDB", "name": "theta: ABC / BDA / CDB", "words": [ "ABC", "BDA", "CDB" ], "P": [ { "approx": "0.5468580335197735", "exact": "2902/4825-162*sqrt(14089)/352225" }, { "approx": "0.3597623359056577", "exact": "331/4825+864*sqrt(14089)/352225" }, { "approx": "0.7444220981627022", "exact": "2626/4825+594*sqrt(14089)/352225" } ], "Q": [ { "approx": "0.7444220981627022", "exact": "2626/4825+594*sqrt(14089)/352225" }, { "approx": "0.3597623359056577", "exact": "331/4825+864*sqrt(14089)/352225" }, { "approx": "0.5468580335197735", "exact": "2902/4825-162*sqrt(14089)/352225" } ], "metadata": { "lengths_bounces": [ 3, 3, 3 ], "total_bounces": 9, "lambda_ratios": [ "1", "0.89912538205848269", "0.89912538205848325" ], "branch_lengths_unfolded": [ "1.1905943372535268", "1.3241694217637885", "1.3241694217637883" ], "lambda_times_length": [ "1.1905943372535268", "1.1905943372535264", "1.190594337253527" ], "norm_residual": "4.0980616289968964e-16", "balance_residual": "7.3276822290214068e-16", "min_validity_margin": "0.045597546587379689" } } ]33:[{"id":"22ecb9c9b69c7f25f97579d66e42ce9e1d732ad7c2d5819c835db535ef0ae0f8af06ddc7d8f695240fd1f1c58d1293c633860d902bf96572d67b4d8f581ae1d6","href":"https://static.observableusercontent.com/files/22ecb9c9b69c7f25f97579d66e42ce9e1d732ad7c2d5819c835db535ef0ae0f8af06ddc7d8f695240fd1f1c58d1293c633860d902bf96572d67b4d8f581ae1d6","name":"observable_data@5.txt","type":"text/plain","lastModified":1777538128355.97,"size":183228},{"id":"094f0a7f3f958c618e82017a05c1f222bb9a683f638a25cce045d95dfe45707b6c42667757364ba71de8aba19021b90e4eef14b0865c96e53eff3265c87972c9","href":"https://static.observableusercontent.com/files/094f0a7f3f958c618e82017a05c1f222bb9a683f638a25cce045d95dfe45707b6c42667757364ba71de8aba19021b90e4eef14b0865c96e53eff3265c87972c9","name":"tetrahedron_geometry_first_orbits_cached.json","type":"application/json","lastModified":1777538128355.97,"size":617852}] 2f:{"title":"Billiard nets in Tetrahedra","cells":[{"id":0,"value":"# Billiard nets 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":554,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":560,"value":"$32","pinned":true,"hidden":false,"mode":"ojs"},{"id":256,"value":"## Imports","pinned":false,"hidden":false,"mode":"md"},{"id":127,"value":"THREE = require(\"three@0.150.1\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":139,"value":"OrbitControls = (await import(\n \"three@0.150.1/examples/jsm/controls/OrbitControls.js\"\n)).OrbitControls","pinned":false,"hidden":false,"mode":"ojs"},{"id":508,"value":"TrackballControls = (await import( \"three@0.160/examples/jsm/controls/TrackballControls.js\"\n)).TrackballControls","pinned":false,"hidden":false,"mode":"ojs"},{"id":476,"value":"Plot = require(\"@observablehq/plot\")","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}