Dial: rotate the ball about the ½–∞ strut; ξ, its odd partner and F(u) pulled back live
Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01FAqjv3n4yABxvJPJHFRhVA
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<!doctype html>
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<html lang="en">
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<head>
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1, viewport-fit=cover">
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<title>The Zeta Dial</title>
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<meta name="description" content="Turn the ball about the ½–∞ strut: s − ½ ↦ e^{iθ}(s − ½). A quarter turn is the Wick rotation, a half turn the functional equation, a full turn is −1 on the odd sheet.">
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<body>
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<main>
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<header>
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<div class="eyebrow">maths · <a href="../">two poles</a> · <a href="../zeta/">zeta ball</a> · <a href="../strut/">strut</a> · the dial</div>
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<h1>The zeta <i>dial</i></h1>
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</header>
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<p class="lede">Turn the ball about the strut through ½ and ∞. On the plane that is a plain rotation about ½: <b>s − ½ ↦ e<sup>iθ</sup>(s − ½)</b>. A quarter turn carries the real line onto the critical line (the Wick rotation); a half turn is the functional equation s ↦ 1 − s; a full turn is the identity, with q = −1. The paint shows the function pulled back through the turn, so you watch cosh become cos.</p>
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<div class="bar">
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<span><label class="k">sheet</label>
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<span class="seg"><button data-fn="0" aria-pressed="true">ξ(s) even</button><button data-fn="1" aria-pressed="false">(s−½)·ξ(s) odd</button><button data-fn="2" aria-pressed="false">F(u), u = (s−½)²</button></span></span>
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<span><label class="k">θ</label><input type="range" id="th" min="0" max="720" step="1" value="0"> <span id="thRead">0°</span></span>
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<span class="seg"><button data-stop="0">0</button><button data-stop="90">¼</button><button data-stop="180">½</button><button data-stop="270">¾</button><button data-stop="360">1</button><button data-stop="720">2</button></span>
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<span><label class="k">height</label><input type="range" id="span" min="6" max="60" step="1" value="24" style="width:120px"> <span id="spanRead">±24</span></span>
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</div>
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<div class="panes">
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<section class="pane">
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<h2 id="title">ξ pulled back through the turn</h2>
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<p class="sub" id="sub">white line: where the real axis has gone · gold: where the critical line has gone · dots: the zeros, carried along</p>
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<div class="view"><canvas id="c"></canvas><canvas id="o"></canvas><span class="busy" id="b">painting</span></div>
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</section>
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<section class="pane">
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<h2>The ball, from above the lamp</h2>
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<p class="sub">looking down the ½–∞ strut: the equator is the rim, the real axis and the imaginary axis are the two diameters; the needle is θ</p>
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<div class="view"><canvas id="d"></canvas></div>
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<div class="read">
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<span class="k">θ</span><span id="r1">0</span>
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<span class="k">on the plane</span><span id="r2">s − ½ ↦ s − ½</span>
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<span class="k">q = e^{iθ/2}</span><span class="q" id="r3">1</span>
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<span class="k">even sheet sees</span><span id="r4">+1</span>
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<span class="k">odd sheet sees</span><span id="r5">+1</span>
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<span class="k">F sees</span><span id="r6">u ↦ u</span>
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</div>
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</section>
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</div>
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<section class="notes">
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<div class="card">
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<h3>Three sheets, one dial</h3>
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<p><b>F(u).</b> Write v = s − ½. ξ(½ + v) = F(v²), and F has positive Taylor coefficients. F lives in the u = v² plane; the dial turns u at double speed, u ↦ e<sup>2iθ</sup>u. A half turn of the ball is a full turn of F. The zeros of ξ, at v = ±iγ, become the single points u = −γ² on F's negative real axis. RH, downstairs: every zero of F is real and negative.</p>
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<p><b>ξ, even.</b> Take the square root: v = √u. Now the half turn exists as its own move, s ↦ 1 − s, and ξ doesn't notice it: ξ(1 − s) = ξ(s). The two roots ±iγ are the zero pair on the equator.</p>
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<p><b>(s − ½)·ξ, odd.</b> Multiply by v. Same zeros, plus a new one at v = 0, the touching point ½. The half turn now shows up as a sign: this sheet sees s ↦ 1 − s as −1. A full turn is +1 again on both sheets; the −1 that a full turn carries in the quaternion reading, q = e<sup>iθ/2</sup>, is what the odd sheet shows at the half.</p>
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<p>So the dial reads: 0 = nothing; ¼ = Wick, real axis onto the critical line, cosh onto cos; ½ = the functional equation (+1 even, −1 odd); 1 = home with q = −1; 2 = home with q = +1.</p>
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</div>
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<div class="card">
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<h3>Reading the paint</h3>
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<p>Hue is the argument of the value, brightness its size. At θ = 0 the real axis is a bright cosh ridge and the critical line a cos stripe with dark zeros. Drag θ: the ridge and the stripe rotate together about ½. At ¼ they have swapped places on the screen. At ½ the picture is back (even sheet) or back with every hue rotated by half a turn (odd sheet, a sign flip).</p>
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<p>Zeros are the first 20 pairs on the critical line; on the F sheet they appear once each, on the negative real u-axis.</p>
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</div>
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</section>
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function zetaEM(sr,si,out){
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let pr=sr,pi=si,sc=1/N;
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if(Math.abs(c)*Math.hypot(tr,ti)<1e-16*Math.hypot(re,im))break;
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const T1=[0,0],T2=[0,0],T3=[0,0],T4=[0,0],T5=[0,0];
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switch(((i%6)+6)%6){case 0:r=v;g=t;b=p;break;case 1:r=q;g=v;b=p;break;case 2:r=p;g=v;b=t;break;case 3:r=p;g=q;b=v;break;case 4:r=t;g=p;b=v;break;default:r=v;g=p;b=q;}
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// paint the chosen sheet at plane point (x,y) = v (sheets 0,1) or u (sheet 2), after the dial's pull-back
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|
||||
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|
||||
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|
||||
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||||
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|
||||
function dial(){
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
g.strokeStyle="rgba(140,140,160,.6)";g.beginPath();g.moveTo(cx,cy-R);g.lineTo(cx,cy+R);g.stroke(); // imaginary axis
|
||||
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|
||||
dot(cx+R,cy,"#000",7);dot(cx+R,cy,"#f0b44c",7,false);dot(cx-R,cy,"#fff",7); // ∞ (z=+1) and ½ (z=−1) on the rim
|
||||
dot(cx,cy,"#4fd1c5",6); // the pole axis 0–∞ seen end-on
|
||||
// needle: θ, and the double-speed hand for q
|
||||
const nx=cx+0.92*R*Math.cos(-th),ny=cy+0.92*R*Math.sin(-th);
|
||||
g.strokeStyle="#e2559a";g.lineWidth=2.5*devicePixelRatio;g.beginPath();g.moveTo(cx,cy);g.lineTo(nx,ny);g.stroke();
|
||||
const qx=cx+0.5*R*Math.cos(-th/2),qy=cy+0.5*R*Math.sin(-th/2);
|
||||
g.strokeStyle="#f0b44c";g.setLineDash([4*devicePixelRatio,3*devicePixelRatio]);g.beginPath();g.moveTo(cx,cy);g.lineTo(qx,qy);g.stroke();g.setLineDash([]);
|
||||
g.fillStyle="#9a93ad";g.font=`${11*devicePixelRatio}px JetBrains Mono`;g.fillText("∞",cx+R+10*devicePixelRatio,cy+4*devicePixelRatio);g.fillText("½",cx-R-22*devicePixelRatio,cy+4*devicePixelRatio);
|
||||
g.fillText("needle: θ · dashed: q = e^{iθ/2}",10*devicePixelRatio,S-10*devicePixelRatio);
|
||||
const d=st.th;$("r1").textContent=d+"°";
|
||||
$("r2").textContent=d%360===0?"s − ½ ↦ s − ½":d%360===180?"s − ½ ↦ −(s − ½) i.e. s ↦ 1 − s":d%360===90?"s − ½ ↦ i(s − ½) (Wick)":d%360===270?"s − ½ ↦ −i(s − ½)":"s − ½ ↦ e^{i·"+d+"°}(s − ½)";
|
||||
const qr=Math.cos(th/2),qi=Math.sin(th/2);$("r3").textContent=(Math.abs(qr)<1e-9?"":qr.toFixed(3))+(Math.abs(qi)<1e-9?"":(qi>0&&Math.abs(qr)>=1e-9?" + ":qi<0?" − ":"")+Math.abs(qi).toFixed(3)+"i");
|
||||
$("r4").textContent=d%360===0?"+1 (home)":d%360===180?"+1 (ξ(1−s) = ξ(s))":"—";
|
||||
$("r5").textContent=d%360===0?"+1 (home)":d%360===180?"−1 (sign flip)":"—";
|
||||
$("r6").textContent="u ↦ e^{i·"+(2*d)%360+"°}u"+((2*d)%360===0?" (home)":"");
|
||||
}
|
||||
function all(){draw(false);clearTimeout(all._t);all._t=setTimeout(()=>draw(true),220);dial();
|
||||
$("title").textContent=st.fn===2?"F pulled back through the turn (u-plane)":st.fn===1?"(s − ½)·ξ pulled back through the turn":"ξ pulled back through the turn";
|
||||
$("sub").textContent=st.fn===2?"white: the real u-axis · gold: the negative real axis, where F's zeros are (once each) · the dial turns u at double speed":"white line: where the real axis has gone · gold: where the critical line has gone · dots: the zeros, carried along"+(st.fn===1?" · pink: the new zero at ½":"");}
|
||||
document.querySelectorAll("[data-fn]").forEach(b=>b.onclick=()=>{st.fn=+b.dataset.fn;document.querySelectorAll("[data-fn]").forEach(x=>x.setAttribute("aria-pressed",x===b));all();});
|
||||
document.querySelectorAll("[data-stop]").forEach(b=>b.onclick=()=>{st.th=+b.dataset.stop;$("th").value=st.th;$("thRead").textContent=st.th+"°";all();});
|
||||
$("th").oninput=e=>{st.th=+e.target.value;$("thRead").textContent=st.th+"°";dial();overlay();clearTimeout($("th")._t);$("th")._t=setTimeout(all,60);};
|
||||
$("span").oninput=e=>{st.span=+e.target.value;$("spanRead").textContent="±"+st.span;clearTimeout($("span")._t);$("span")._t=setTimeout(all,120);};
|
||||
addEventListener("resize",()=>{overlay();dial();});
|
||||
all();
|
||||
})();
|
||||
</script>
|
||||
</body>
|
||||
</html>
|
||||
Reference in new issue
Block a user