2a:null 2b:["$","title",null,{"children":"Clifford torus | Torben Jansen | Observable"}] 30:T6b8,md`# Clifford torus A [clifford torus](https://en.wikipedia.org/wiki/Clifford_torus) (named after [William Kingdon Clifford](https://en.wikipedia.org/wiki/William_Kingdon_Clifford)) is the cartesian product of two Circles (${tex`S_{a}^1 \times S_{b}^1`}). It resides in ${tex`\R^4`}, as opposed to in ${tex`\R^3`}. The visualization below shows a [stereographic projection](https://en.wikipedia.org/wiki/Stereographic_projection) of a rotating clifford torus on a canvas. Given a four dimensional vector ${tex`\vec v`}, and *assuming* that it is a [quaternion](https://en.wikipedia.org/wiki/Quaternion), we can rotate the vector ${tex`\vec v`} with a pair of unit quaternions ${tex`\boldsymbol q_1`} and ${tex`\boldsymbol q_2`}. ${ tex.block` f(\vec v) = \boldsymbol q_1 \vec v \boldsymbol q_2 = \begin{pmatrix} a_1 & -b_1 & -c_1 & -d_1 \\ b_1 & a_1 & -d_1 & c_1 \\ c_1 & d_1 & a_1 & -b_1 \\ d_1 & -c_1 & b_1 & a_1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix} \begin{pmatrix} a_2 & -b_2 & -c_2 & -d_2 \\ b_2 & a_2 & d_2 & -c_2 \\ c_2 & -d_2 & a_2 & b_2 \\ d_2 & c_2 & -b_2 & a_2 \end{pmatrix} ` } A pair of unit quaternions can represent any [rotation in 4D space](https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Pairs_of_unit_quaternions_as_rotations_in_4D_space). Arbitrary four dimensional rotations have 6 degrees of freedom, each matrix represents 3 of those 6 degrees of freedom. Drag the mouse over the canvas to manually change the rotation! *See also these notebooks: [Clif4d: A Track-Torus](https://observablehq.com/@vorth/clif4d-a-track-torus) by Scott Vorthmann [Clifford Torus](https://observablehq.com/@timhau/clifford-torus) by Tim Hau* `31:T816,{ const context = DOM.context2d(width, height); let xy = 0, yz = 0, xz = 0; let xw = 0, yw = 0, zw = 0; let mouseDown = false; let lastX = null; let lastY = null; function handleMouseDown(event) { mouseDown = true; lastX = event.clientX; lastY = event.clientY; context.canvas.style.cursor = "grabbing"; } function handleMouseUp(event) { mouseDown = false; context.canvas.style.cursor = "grab"; } function handleMouseMove(event) { if (!mouseDown) { return } const newX = event.clientX const newY = event.clientY const sensitivity = 0.012 let deltaX = (newX - lastX) * sensitivity; let deltaY = -(newY - lastY) * sensitivity; xz += -deltaY; zw += deltaY; yz += deltaX; yw += deltaX; lastX = newX lastY = newY; } function drawCircle(index) { let [x, y] = torus.data[index[0]].project2D(); context.beginPath(); context.moveTo(x, y); for (let j = 1; j < torus._LOD; ++j) { [x, y] = torus.data[index[j]].project2D(); context.lineTo(x, y); } context.closePath(); context.stroke(); } context.translate(width / 2, height / 2); context.scale(scale, scale); context.lineWidth = 1 / scale; context.fillStyle = "white"; context.canvas.onmousedown = handleMouseDown; context.canvas.onmouseup = handleMouseUp; context.canvas.onmousemove = handleMouseMove; context.canvas.style.cursor = "grab"; while (true) { if (!mouseDown) { xy += 0.00; yz += 0.02; xz += 0.00; xw += 0.00; yw += 0.00; zw += 0.00; } torus.rotate(xy, yz, xz, xw, yw, zw); context.fillRect(-width / 2, -height / 2, width, height); for (let i = 0, start; i < torus._LOD; ++i) { context.strokeStyle = colors[i]; drawCircle(torus.indices[0][i]); } for (let i = 0; i < torus._LOD; ++i) { context.strokeStyle = colors[i]; drawCircle(torus.indices[1][i]); } yield context.canvas; } }32:T572,class CliffordTorus { constructor(radius1, radius2, LOD = 50) { this._r1 = radius1; this._r2 = radius2; this._LOD = LOD; this.rotor = new Rotor4(); this.data = new Array(this._LOD ** 2); this.indices = [ Array.from({ length: this._LOD }, () => new Uint16Array(this._LOD)), Array.from({ length: this._LOD }, () => new Uint16Array(this._LOD)) ]; this._createData(); this._createIndices(); } *[Symbol.iterator]() { for(let i = 0; i < this.data.length; ++i) { yield this.data[i]; } } _createData() { for (let i = 0; i < this.data.length; ++i) { let v = (i % this._LOD) * step, u = Math.floor(i / this._LOD) * step; this.data[i] = this.getPoint(u, v).setCache(); } } _createIndices() { for (let i = 0; i < this._LOD; ++i) { for (let j = 0; j < this._LOD; ++j) { this.indices[0][i][j] = i * this._LOD + j; this.indices[1][i][j] = j * this._LOD + i; } } } getPoint(u, v) { return new Point4().set( this._r1 * Math.cos(u) * rad, this._r1 * Math.sin(u) * rad, this._r2 * Math.cos(v) * rad, this._r2 * Math.sin(v) * rad ); } rotate(a, b, c, d, e, f) { this.rotor.angles(a, b, c, d, e, f); for (let i = 0; i < this.data.length; ++i) { this.data[i].rotate(this.rotor); } } }33:T538,class Point4 extends Vector4 { constructor(x, y, z, w) { super(); Object.defineProperty(this, "_cache", { value: Object.defineProperties({}, { data: { value: new Float64Array(4), writeable: true }, vec3: { value: new Float64Array(3), writeable: true }, vec2: { value: new Float64Array(2), writeable: true } }) }); } reset() { return this.set( this._cache.data[0], this._cache.data[1], this._cache.data[2], this._cache.data[3] ); } setCache() { this._cache.data[0] = this[0]; this._cache.data[1] = this[1]; this._cache.data[2] = this[2]; this._cache.data[3] = this[3]; return this; } rotate([m1, m2]) { return this.reset() .applyMatrix(m1) .applyMatrix(m2); } project3D(distance = 2) { let d = distance - this[3]; this._cache.vec3[0] = this[0] / d; this._cache.vec3[1] = this[1] / d; this._cache.vec3[2] = this[2] / d; return this._cache.vec3; } project2D(distance = 2) { let d1 = distance - this[3], d2 = distance - this[2] / d1; this._cache.vec2[0] = this[0] / d1 / d2; this._cache.vec2[1] = this[1] / d1 / d2; return this._cache.vec2; } }34:[] 2e:{"id":"cdf1b33337cd1997","slug":"clifford-torus","title":"Clifford torus","thumbnail":"a6018e2785069b8c1d13f12f215c0517eeabed9272a992808108a74e4ff687c4","access_level":"public","create_time":"$D2020-04-10T10:49:42.115Z","update_time":"$D2021-03-26T17:36:31.411Z","view_time":null,"first_public_time":"$D2020-04-13T18:14:54.634Z","has_unpublished_edits":null,"fork":false,"stars":21,"starred":false,"owner":{"type":"team","id":"03836a98946da550","login":"toja","name":"Torben 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