diff --git a/README.md b/README.md
index 2c3130a..189da0a 100644
--- a/README.md
+++ b/README.md
@@ -13,6 +13,8 @@ Math Visualizer is an interactive collection of mathematical visualizations buil
 >   glyph outlines become one closed path, the path's DFT becomes a chain of rotating
 >   circles, and the chain's tip traces the letters live (pen lifts between glyphs).
 >   Hundreds of circles render in one instanced draw call.
+>   Share a message with `#/fourier?text=YOUR+TEXT` (`&n=` sets the epicycle count) —
+>   the URL updates as you type, and **Copy link** puts it on the clipboard.
 >
 > The midpoint-on-circle and ColorCycle rules remain in the codebase as alternative
 > examples. See [`docs/superpowers/specs/`](docs/superpowers/specs/) for designs and
diff --git a/web/src/lib/__tests__/router.test.ts b/web/src/lib/__tests__/router.test.ts
index 5ecec6a..2536c75 100644
--- a/web/src/lib/__tests__/router.test.ts
+++ b/web/src/lib/__tests__/router.test.ts
@@ -1,5 +1,5 @@
 import { describe, it, expect } from 'vitest';
-import { parseHash, DEFAULT_LAB, LAB_IDS } from '../router';
+import { parseHash, parseHashQuery, buildQuery, DEFAULT_LAB, LAB_IDS } from '../router';
 
 describe('parseHash', () => {
   it('maps #/fourier to fourier', () => {
@@ -22,3 +22,19 @@ describe('parseHash', () => {
     for (const id of LAB_IDS) expect(parseHash(`#/${id}`)).toBe(id);
   });
 });
+
+describe('parseHashQuery / buildQuery', () => {
+  it('extracts the query part of a hash route', () => {
+    expect(parseHashQuery('#/fourier?text=HI&n=5')).toBe('text=HI&n=5');
+    expect(parseHashQuery('#/fourier')).toBe('');
+    expect(parseHashQuery('')).toBe('');
+  });
+  it('route id parsing ignores the query', () => {
+    expect(parseHash('#/fourier?text=HI')).toBe('fourier');
+  });
+  it('buildQuery omits undefined/empty values and encodes the rest', () => {
+    expect(buildQuery({ text: 'HELLO WORLD', n: '12' })).toBe('text=HELLO+WORLD&n=12');
+    expect(buildQuery({ text: undefined, n: '' })).toBe('');
+    expect(new URLSearchParams(buildQuery({ text: 'a&b=c' })).get('text')).toBe('a&b=c');
+  });
+});
diff --git a/web/src/lib/components/__tests__/FourierLab.test.ts b/web/src/lib/components/__tests__/FourierLab.test.ts
index 2b50b2f..2fcfd32 100644
--- a/web/src/lib/components/__tests__/FourierLab.test.ts
+++ b/web/src/lib/components/__tests__/FourierLab.test.ts
@@ -105,3 +105,32 @@ describe('FourierLab.svelte', () => {
     expect(getByText('Sierpinski Pyramid', { selector: 'h2' })).toBeTruthy();
   });
 });
+
+describe('FourierLab.svelte — shareable link params', () => {
+  beforeEach(() => {
+    dispatchSpy.mockClear();
+    updateRuleConfigSpy.mockClear();
+  });
+
+  it('reads text and n from the hash query and uses them for the first push', async () => {
+    navigate('fourier', 'text=HELLO&n=12');
+    const { getByLabelText } = render(App);
+    await vi.waitFor(() => expect(updateRuleConfigSpy).toHaveBeenCalled());
+    const cfg = updateRuleConfigSpy.mock.calls[0][0] as { epicycles: number };
+    expect(cfg.epicycles).toBe(12);
+    expect((getByLabelText('Text to trace') as HTMLInputElement).value).toBe('HELLO');
+  });
+
+  it('keeps the URL in sync as the text changes and offers a copy-link button', async () => {
+    navigate('fourier');
+    const { getByLabelText, getByTitle } = render(App);
+    await vi.waitFor(() => expect(updateRuleConfigSpy).toHaveBeenCalledTimes(1));
+    expect(location.hash).toBe('#/fourier'); // defaults are omitted from the link
+
+    const input = getByLabelText('Text to trace') as HTMLInputElement;
+    await fireEvent.input(input, { target: { value: 'ABC' } });
+    await vi.waitFor(() => expect(updateRuleConfigSpy).toHaveBeenCalledTimes(2));
+    expect(location.hash).toBe('#/fourier?text=ABC');
+    expect(getByTitle('Copy a link to this message')).toBeTruthy();
+  });
+});
diff --git a/web/src/lib/components/labs/FourierLab.svelte b/web/src/lib/components/labs/FourierLab.svelte
index 82437be..da5fd9e 100644
--- a/web/src/lib/components/labs/FourierLab.svelte
+++ b/web/src/lib/components/labs/FourierLab.svelte
@@ -1,19 +1,75 @@
 <script lang="ts">
-  import { onDestroy } from 'svelte';
+  import { onDestroy, untrack } from 'svelte';
   import LabShell from '../LabShell.svelte';
   import FormulaPanel from '../FormulaPanel.svelte';
   import type { LabApi } from '../labApi.svelte';
   import { cmd } from '../../playback/commands';
   import { textToPath } from '../../fourier/textPath';
   import { readSummary, type FourierSummary } from '../../fourier/summary';
+  import { route, replaceQuery } from '../../router.svelte';
+  import { buildQuery } from '../../router';
 
   const DEFAULT_TEXT = 'POIETIC TECH';
   const SAMPLES = 2000;
   const MAX_EPICYCLES = 2000;
   const DEBOUNCE_MS = 150;
+  const DEFAULT_EPICYCLES = 2000;
+  const MAX_TEXT = 40;
 
-  let text = $state(DEFAULT_TEXT);
-  let epicycles = $state(2000);
+  /** Read the shareable-link params from a hash query: `text=…` and `n=…` (epicycles). */
+  function paramsOf(query: string) {
+    const p = new URLSearchParams(query);
+    const t = p.get('text');
+    const n = Math.floor(Number(p.get('n')));
+    return {
+      text: t !== null && t.trim() !== '' ? t.slice(0, MAX_TEXT) : undefined,
+      epicycles: Number.isFinite(n) && n >= 1 ? Math.min(n, MAX_EPICYCLES) : undefined,
+    };
+  }
+  const initialParams = paramsOf(route.query);
+
+  let text = $state(initialParams.text ?? DEFAULT_TEXT);
+  let epicycles = $state(initialParams.epicycles ?? DEFAULT_EPICYCLES);
+  /** Query we last wrote (or consumed), so our own replaceQuery() doesn't re-trigger a push. */
+  let appliedQuery = route.query;
+  let copied = $state(false);
+
+  /** Keep the URL a shareable link to the current message (defaults are omitted). */
+  function syncUrl() {
+    const q = buildQuery({
+      text: text === DEFAULT_TEXT ? undefined : text,
+      n: epicycles === DEFAULT_EPICYCLES ? undefined : String(epicycles),
+    });
+    appliedQuery = q;
+    replaceQuery(q);
+  }
+
+  // A pasted link / back-forward while on this page: adopt its params and redraw.
+  $effect(() => {
+    const q = route.query;
+    untrack(() => {
+      if (q === appliedQuery) return;
+      appliedQuery = q;
+      const p = paramsOf(q);
+      const nextText = p.text ?? DEFAULT_TEXT;
+      const nextN = p.epicycles ?? DEFAULT_EPICYCLES;
+      if (nextText !== text || nextN !== epicycles) {
+        text = nextText;
+        epicycles = nextN;
+        void push();
+      }
+    });
+  });
+
+  async function copyLink() {
+    try {
+      await navigator.clipboard.writeText(location.href);
+      copied = true;
+      window.setTimeout(() => { copied = false; }, 1500);
+    } catch (err) {
+      console.warn('copy link failed:', err);
+    }
+  }
   /** True when the current text produced no drawable path (blank, or no outline). */
   let empty = $state(false);
   /** The DFT terms behind the current trace, read from the engine after each config push (null when nothing is drawn). */
@@ -42,6 +98,7 @@
     }
     if (my !== gen || !api?.engine) return; // superseded, or shell torn down / not ready yet
     empty = path.length === 0;
+    syncUrl();
     if (empty) {
       summary = null;
       return;
@@ -146,6 +203,9 @@
         title="Epicycles"
       />
     </label>
+    <button class="copy" type="button" onclick={copyLink} title="Copy a link to this message">
+      {copied ? 'Copied ✓' : 'Copy link'}
+    </button>
     {#if empty}<span class="hint">Nothing to draw — type some letters.</span>{/if}
   {/snippet}
 </LabShell>
@@ -196,4 +256,15 @@
   }
   span.swatch.pen { background: #fa9959; }
   span.swatch.ink { background: #a6d9f2; }
+  .copy {
+    background: #2a2a2f;
+    color: #eee;
+    border: 1px solid #3a3a40;
+    border-radius: 4px;
+    padding: 0.35rem 0.7rem;
+    font-size: 0.85rem;
+    cursor: pointer;
+    white-space: nowrap;
+  }
+  .copy:hover { background: #34343a; }
 </style>
diff --git a/web/src/lib/router.svelte.ts b/web/src/lib/router.svelte.ts
index e7872ec..385d76a 100644
--- a/web/src/lib/router.svelte.ts
+++ b/web/src/lib/router.svelte.ts
@@ -1,16 +1,37 @@
-// Reactive hash router. `route.id` is a $state field on an exported object
-// (a reassigned `let` export is not allowed for runes) — mutate `route.id`,
-// never reassign `route`.
-import { parseHash, type LabId } from './router';
+// Reactive hash router. `route` is an exported $state object (a reassigned
+// `let` export is not allowed for runes) — mutate its fields, never reassign it.
+//   route.id    — which lab ('#/fourier' → 'fourier')
+//   route.query — the query part of the hash ('#/fourier?text=HI' → 'text=HI')
+import { parseHash, parseHashQuery, type LabId } from './router';
 
 const initialHash = typeof location !== 'undefined' ? location.hash : '';
 
-export const route = $state<{ id: LabId }>({ id: parseHash(initialHash) });
+export const route = $state<{ id: LabId; query: string }>({
+  id: parseHash(initialHash),
+  query: parseHashQuery(initialHash),
+});
 
-/** Navigate programmatically. Updates the hash (so the URL is bookmarkable) and the reactive route. */
-export function navigate(id: LabId) {
-  if (typeof location !== 'undefined') location.hash = `#/${id}`;
+function hashFor(id: LabId, query: string): string {
+  return query ? `#/${id}?${query}` : `#/${id}`;
+}
+
+/** Navigate programmatically (adds a history entry). Updates the hash and the reactive route. */
+export function navigate(id: LabId, query = '') {
+  if (typeof location !== 'undefined') location.hash = hashFor(id, query);
   route.id = id;
+  route.query = query;
+}
+
+/**
+ * Rewrite the current route's query in place — no history entry, and
+ * history.replaceState fires no hashchange — so a lab can keep the URL in
+ * sync with its live state (a shareable link) without re-triggering itself.
+ */
+export function replaceQuery(query: string) {
+  if (typeof history !== 'undefined' && typeof location !== 'undefined') {
+    history.replaceState(history.state, '', `${location.pathname}${location.search}${hashFor(route.id, query)}`);
+  }
+  route.query = query;
 }
 
 let installed = false;
@@ -21,5 +42,6 @@ export function installRouter() {
   installed = true;
   window.addEventListener('hashchange', () => {
     route.id = parseHash(location.hash);
+    route.query = parseHashQuery(location.hash);
   });
 }
diff --git a/web/src/lib/router.ts b/web/src/lib/router.ts
index 64e75f0..073bfde 100644
--- a/web/src/lib/router.ts
+++ b/web/src/lib/router.ts
@@ -11,3 +11,16 @@ export function parseHash(hash: string): LabId {
   const id = hash.replace(/^#\/?/, '').split(/[/?#]/)[0];
   return (LAB_IDS as readonly string[]).includes(id) ? (id as LabId) : DEFAULT_LAB;
 }
+
+/** The query part of a hash route: `#/fourier?text=HI&n=5` → 'text=HI&n=5' ('' when absent). */
+export function parseHashQuery(hash: string): string {
+  const i = hash.indexOf('?');
+  return i === -1 ? '' : hash.slice(i + 1);
+}
+
+/** Serialize params to a query string, omitting undefined/empty values. */
+export function buildQuery(params: Record<string, string | undefined>): string {
+  const q = new URLSearchParams();
+  for (const [k, v] of Object.entries(params)) if (v !== undefined && v !== '') q.set(k, v);
+  return q.toString();
+}