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point.w);\n \n return {\n x: point.x * factor,\n y: point.y * factor,\n z: point.z * factor\n };\n }\n\n // Stereographic projection from 4D to 3D\n static stereographic4DTo3D(point: Vector4D, radius: number = 1): Vector3D {\n const denominator = radius - point.w;\n if (Math.abs(denominator) < 0.001) {\n // Point at infinity, return a large value\n return { x: point.x * 1000, y: point.y * 1000, z: point.z * 1000 };\n }\n \n const factor = radius / denominator;\n return {\n x: point.x * factor,\n y: point.y * factor,\n z: point.z * factor\n };\n }\n\n // Double perspective projection from 5D to 3D\n static perspective5DTo3D(point: Vector5D, settings: ProjectionSettings): Vector3D {\n const { distance } = settings;\n \n // First projection: 5D to 4D\n const factor1 = distance / (distance - point.v);\n const point4D: Vector4D = {\n x: point.x * factor1,\n y: point.y * factor1,\n z: point.z * factor1,\n w: point.w * factor1\n };\n \n // Second projection: 4D to 3D\n return this.perspective4DTo3D(point4D, settings);\n }\n\n // Orthogonal projection from 5D to 3D (dropping two dimensions)\n static orthogonal5DTo3D(point: Vector5D, dropDimensions: [number, number] = [3, 4]): Vector3D {\n const coords = [point.x, point.y, point.z, point.w, point.v];\n const result = [0, 0, 0];\n let resultIndex = 0;\n \n for (let i = 0; i < 5; i++) {\n if (!dropDimensions.includes(i) && resultIndex < 3) {\n result[resultIndex] = coords[i];\n resultIndex++;\n }\n }\n \n return {\n x: result[0] || 0,\n y: result[1] || 0,\n z: result[2] || 0\n };\n }\n\n // Schlegel projection for 4D polytopes\n static schlegel4DTo3D(point: Vector4D, center: Vector4D, projectionPoint: Vector4D): Vector3D {\n // Vector from projection point to the point\n const direction = {\n x: point.x - projectionPoint.x,\n y: point.y - projectionPoint.y,\n z: point.z - projectionPoint.z,\n w: point.w - projectionPoint.w\n };\n \n // Find intersection with the 3D hyperplane (w = 0)\n if (Math.abs(direction.w) < 0.001) {\n return { x: point.x, y: point.y, z: point.z };\n }\n \n const t = -projectionPoint.w / direction.w;\n return {\n x: projectionPoint.x + t * direction.x,\n y: projectionPoint.y + t * direction.y,\n z: projectionPoint.z + t * direction.z\n };\n }\n\n // Parallel projection with custom direction\n static parallelProjection4D(point: Vector4D, direction: Vector4D, plane: Vector4D): Vector3D {\n // Project point onto the 3D plane perpendicular to the direction\n const dot = point.x * direction.x + point.y * direction.y + \n point.z * direction.z + point.w * direction.w;\n const planeDot = plane.x * direction.x + plane.y * direction.y + \n plane.z * direction.z + plane.w * direction.w;\n \n if (Math.abs(planeDot) < 0.001) {\n return { x: point.x, y: point.y, z: point.z };\n }\n \n const t = dot / planeDot;\n return {\n x: point.x - t * plane.x,\n y: point.y - t * plane.y,\n z: point.z - t * plane.z\n };\n }\n\n // Fisheye projection for wide-angle viewing\n static fisheye4DTo3D(point: Vector4D, fov: number = Math.PI / 2): Vector3D {\n const r = Math.sqrt(point.x * point.x + point.y * point.y + point.z * point.z + point.w * point.w);\n if (r < 0.001) return { x: 0, y: 0, z: 0 };\n \n const theta = Math.acos(Math.abs(point.w) / r);\n const phi = Math.atan2(point.y, point.x);\n const psi = Math.atan2(Math.sqrt(point.x * point.x + point.y * point.y), point.z);\n \n const fisheyeR = (2 * theta / fov) * Math.sin(fov / 2);\n \n return {\n x: fisheyeR * Math.cos(phi) * Math.sin(psi),\n y: fisheyeR * Math.sin(phi) * Math.sin(psi),\n z: fisheyeR * Math.cos(psi)\n };\n }\n\n // Adaptive projection that chooses the best method based on the data\n static adaptiveProjection(points: Vector4D[], settings: ProjectionSettings): Vector3D[] {\n // Analyze the distribution of w-coordinates\n const wValues = points.map(p => p.w);\n const minW = Math.min(...wValues);\n const maxW = Math.max(...wValues);\n const wRange = maxW - minW;\n \n // Choose projection method based on w-distribution\n if (wRange < 0.1) {\n // Points are mostly in the same w-plane, use orthogonal\n return points.map(p => this.orthogonal4DTo3D(p));\n } else if (minW > -settings.distance * 0.9) {\n // No points too close to the projection center, use perspective\n return points.map(p => this.perspective4DTo3D(p, settings));\n } else {\n // Some points might cause issues with perspective, use stereographic\n return points.map(p => this.stereographic4DTo3D(p));\n }\n }\n\n // Cross-section projection: project a 4D cross-section to 2D\n static crossSection4DTo2D(point: Vector4D, normal: Vector4D): { x: number; y: number } {\n // Find two orthogonal vectors in the 3D subspace perpendicular to normal\n const u = this.findOrthogonalVector4D(normal);\n const v = this.crossProduct4D(normal, u);\n \n // Project point onto the 2D plane spanned by u and v\n const uDot = point.x * u.x + point.y * u.y + point.z * u.z + point.w * u.w;\n const vDot = point.x * v.x + point.y * v.y + point.z * v.z + point.w * v.w;\n \n return { x: uDot, y: vDot };\n }\n\n // Helper function to find an orthogonal vector in 4D\n private static findOrthogonalVector4D(v: Vector4D): Vector4D {\n // Find the component with the smallest absolute value\n const abs = [Math.abs(v.x), Math.abs(v.y), Math.abs(v.z), Math.abs(v.w)];\n const minIndex = abs.indexOf(Math.min(...abs));\n \n const result = { x: 0, y: 0, z: 0, w: 0 };\n switch (minIndex) {\n case 0: result.x = 1; break;\n case 1: result.y = 1; break;\n case 2: result.z = 1; break;\n case 3: result.w = 1; break;\n }\n \n // Make it orthogonal using Gram-Schmidt\n const dot = result.x * v.x + result.y * v.y + result.z * v.z + result.w * v.w;\n const vLengthSq = v.x * v.x + v.y * v.y + v.z * v.z + v.w * v.w;\n \n if (vLengthSq > 0.001) {\n const factor = dot / vLengthSq;\n result.x -= factor * v.x;\n result.y -= factor * v.y;\n result.z -= factor * v.z;\n result.w -= factor * v.w;\n }\n \n // Normalize\n const length = Math.sqrt(result.x * result.x + result.y * result.y + \n result.z * result.z + result.w * result.w);\n if (length > 0.001) {\n result.x /= length;\n result.y /= length;\n result.z /= length;\n result.w /= length;\n }\n \n return result;\n }\n\n // Helper function for 4D cross product (returns a vector orthogonal to both inputs)\n private static crossProduct4D(a: Vector4D, b: Vector4D): Vector4D {\n // This is a simplified version - 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The application runs as an HTTP val\n2. Navigate to the main page to start exploring higher-dimensional geometry\n3. Use the control panel to select shapes, adjust transformations, and modify viewing parameters\n4. Interact with the 3D viewport to explore projections and cross-sections\n\n## Mathematical Background\n\nThis application implements advanced concepts from higher-dimensional geometry:\n- Orthogonal projections from nD to 3D space\n- Rotation matrices in 4D and 5D spaces\n- Cross-sectional analysis of higher-dimensional objects\n- Real-time transformation calculations optimized for performance","label",[129],"a83b53f4-59a1-11f0-bf43-f69ea79377d9","shared","6a7dd57b-35c1-426f-9cb3-ba018acef2b3",2,["D",1751721129544],"directory",[139],"pulls","branches","forks","envVars",{"_152":62,"_154":62},"connections",{"_152":62,"_153":62},"count","unfilledCount","emptyCount",{"_42":138,"_31":139,"_29":140,"_131":-5,"_40":41,"_39":141,"_133":58,"_37":199,"_13":62,"_33":143,"_35":-5,"_23":24,"_136":-5,"_163":200,"_26":201},{"_42":195,"_31":178,"_29":174,"_131":140,"_40":41,"_39":190,"_133":58,"_37":196,"_13":62,"_33":143,"_35":-5,"_23":24,"_136":-5,"_163":197,"_26":198},{"_42":187,"_31":188,"_29":189,"_131":174,"_40":41,"_39":190,"_133":24,"_37":191,"_13":62,"_33":34,"_35":192,"_23":24,"_136":-5,"_163":193,"_26":194},{"_42":179,"_31":180,"_29":181,"_131":174,"_40":41,"_39":182,"_133":24,"_37":183,"_13":62,"_33":34,"_35":184,"_23":24,"_136":-5,"_163":185,"_26":186},{"_42":173,"_31":32,"_29":30,"_131":174,"_40":41,"_39":14,"_133":24,"_37":175,"_13":62,"_33":34,"_35":36,"_23":24,"_136":-5,"_163":176,"_26":177},{"_42":166,"_31":167,"_29":168,"_131":140,"_40":41,"_39":141,"_133":58,"_37":169,"_13":62,"_33":34,"_35":170,"_23":24,"_136":-5,"_163":171,"_26":172},{"_42":128,"_31":129,"_29":130,"_131":-5,"_40":41,"_39":132,"_133":24,"_37":162,"_13":62,"_33":20,"_35":135,"_23":24,"_136":-5,"_163":164,"_26":165},["D",1751721114669],"children",[],[129],"a83b47f6-59a1-11f0-bf43-f69ea79377d9","types.ts","a83b48d2-59a1-11f0-bf43-f69ea79377d9",["D",1751721129544],"// Shared types for higher-dimensional geometry visualization\n\nexport interface Vector4D {\n x: number;\n y: number;\n z: number;\n w: number;\n}\n\nexport interface Vector5D {\n x: number;\n y: number;\n z: number;\n w: number;\n v: number;\n}\n\nexport interface Vector3D {\n x: number;\n y: number;\n z: number;\n}\n\nexport interface Matrix4D {\n elements: number[][]; // 4x4 matrix\n}\n\nexport interface Matrix5D {\n elements: number[][]; // 5x5 matrix\n}\n\nexport interface Edge {\n start: number;\n end: number;\n}\n\nexport interface Face {\n vertices: number[];\n}\n\nexport interface Cell {\n faces: number[];\n}\n\nexport interface Shape4D {\n vertices: Vector4D[];\n edges: Edge[];\n faces: Face[];\n cells: Cell[];\n name: string;\n}\n\nexport interface Shape5D {\n vertices: Vector5D[];\n edges: Edge[];\n faces: Face[];\n cells: Cell[];\n hypercells: Cell[];\n name: string;\n}\n\nexport interface ProjectionSettings {\n distance: number;\n perspective: boolean;\n viewAngle: number;\n}\n\nexport interface TransformationState {\n rotation4D: {\n xy: number;\n xz: number;\n xw: number;\n yz: number;\n yw: number;\n zw: number;\n };\n rotation5D?: {\n xy: number;\n xz: number;\n xw: number;\n xv: number;\n yz: number;\n yw: number;\n yv: number;\n zw: number;\n zv: number;\n wv: number;\n };\n scale: number;\n translation: Vector4D | Vector5D;\n}\n\nexport interface CameraState {\n position: Vector3D;\n target: Vector3D;\n up: Vector3D;\n fov: number;\n}\n\nexport interface LightingState {\n ambient: number;\n directional: {\n intensity: number;\n direction: Vector3D;\n };\n point: {\n intensity: number;\n position: Vector3D;\n };\n}\n\nexport interface CrossSectionSettings {\n enabled: boolean;\n dimension: number;\n position: number;\n normal: Vector4D | Vector5D;\n}\n\nexport interface RenderSettings {\n showVertices: boolean;\n showEdges: boolean;\n showFaces: boolean;\n wireframe: boolean;\n opacity: number;\n color: string;\n edgeColor: string;\n vertexColor: string;\n}\n\nexport type ShapeType = \n | 'tesseract' \n | '16-cell' \n | '600-cell' \n | '120-cell'\n | 'penteract'\n | '5-simplex'\n | '5-orthoplex'\n | 'custom';\n\nexport interface AppState {\n currentShape: ShapeType;\n dimension: 4 | 5;\n transformations: TransformationState;\n camera: CameraState;\n lighting: LightingState;\n projection: ProjectionSettings;\n crossSection: CrossSectionSettings;\n rendering: RenderSettings;\n isAnimating: boolean;\n animationSpeed: number;\n}",[],[139,167],"f8e4ba70-59a1-11f0-bc55-f69ea79377d9","10e2870c-b7b7-48c6-9e40-545c61d33aa9",["D",1751721264877],[],[139,178,32],"math","dbfdd892-59a1-11f0-a68e-f69ea79377d9","geometry5d.ts","dbfdd950-59a1-11f0-a68e-f69ea79377d9",4,["D",1751721216383],"// 5D Geometry calculations and shape generation\n\nimport { Vector5D, Vector3D, Shape5D, Edge, Face, Cell, Matrix5D } from '../types.ts';\n\nexport class Geometry5D {\n // Create a 5D rotation matrix for the given plane and angle\n static createRotationMatrix(plane: string, angle: number): Matrix5D {\n const cos = Math.cos(angle);\n const sin = Math.sin(angle);\n const matrix = Array(5).fill(null).map(() => Array(5).fill(0));\n \n // Initialize as identity matrix\n for (let i = 0; i < 5; i++) {\n matrix[i][i] = 1;\n }\n\n switch (plane) {\n case 'xy':\n matrix[0][0] = cos; matrix[0][1] = -sin;\n matrix[1][0] = sin; matrix[1][1] = cos;\n break;\n case 'xz':\n matrix[0][0] = cos; matrix[0][2] = -sin;\n matrix[2][0] = sin; matrix[2][2] = cos;\n break;\n case 'xw':\n matrix[0][0] = cos; matrix[0][3] = -sin;\n matrix[3][0] = sin; matrix[3][3] = cos;\n break;\n case 'xv':\n matrix[0][0] = cos; matrix[0][4] = -sin;\n matrix[4][0] = sin; matrix[4][4] = cos;\n break;\n case 'yz':\n matrix[1][1] = cos; matrix[1][2] = -sin;\n matrix[2][1] = sin; matrix[2][2] = cos;\n break;\n case 'yw':\n matrix[1][1] = cos; matrix[1][3] = -sin;\n matrix[3][1] = sin; matrix[3][3] = cos;\n break;\n case 'yv':\n matrix[1][1] = cos; matrix[1][4] = -sin;\n matrix[4][1] = sin; matrix[4][4] = cos;\n break;\n case 'zw':\n matrix[2][2] = cos; matrix[2][3] = -sin;\n matrix[3][2] = sin; matrix[3][3] = cos;\n break;\n case 'zv':\n matrix[2][2] = cos; matrix[2][4] = -sin;\n matrix[4][2] = sin; matrix[4][4] = cos;\n break;\n case 'wv':\n matrix[3][3] = cos; matrix[3][4] = -sin;\n matrix[4][3] = sin; matrix[4][4] = cos;\n break;\n }\n\n return { elements: matrix };\n }\n\n // Apply transformation matrix to a 5D vector\n static transformVector(vector: Vector5D, matrix: Matrix5D): Vector5D {\n const { x, y, z, w, v } = vector;\n const m = matrix.elements;\n \n return {\n x: m[0][0] * x + m[0][1] * y + m[0][2] * z + m[0][3] * w + m[0][4] * v,\n y: m[1][0] * x + m[1][1] * y + m[1][2] * z + m[1][3] * w + m[1][4] * v,\n z: m[2][0] * x + m[2][1] * y + m[2][2] * z + m[2][3] * w + m[2][4] * v,\n w: m[3][0] * x + m[3][1] * y + m[3][2] * z + m[3][3] * w + m[3][4] * v,\n v: m[4][0] * x + m[4][1] * y + m[4][2] * z + m[4][3] * w + m[4][4] * v\n };\n }\n\n // Project 5D point to 3D using double perspective projection\n static projectTo3D(point: Vector5D, distance1: number = 2, distance2: number = 2): Vector3D {\n // First project from 5D to 4D\n const factor1 = distance1 / (distance1 - point.v);\n const point4D = {\n x: point.x * factor1,\n y: point.y * factor1,\n z: point.z * factor1,\n w: point.w * factor1\n };\n \n // Then project from 4D to 3D\n const factor2 = distance2 / (distance2 - point4D.w);\n return {\n x: point4D.x * factor2,\n y: point4D.y * factor2,\n z: point4D.z * factor2\n };\n }\n\n // Generate penteract (5D cube) vertices and structure\n static createPenteract(): Shape5D {\n const vertices: Vector5D[] = [];\n \n // Generate all 32 vertices of a penteract\n for (let i = 0; i < 32; i++) {\n vertices.push({\n x: (i & 1) ? 1 : -1,\n y: (i & 2) ? 1 : -1,\n z: (i & 4) ? 1 : -1,\n w: (i & 8) ? 1 : -1,\n v: (i & 16) ? 1 : -1\n });\n }\n\n // Generate edges\n const edges: Edge[] = [];\n for (let i = 0; i < 32; i++) {\n for (let j = i + 1; j < 32; j++) {\n // Two vertices are connected if they differ in exactly one coordinate\n const diff = [\n vertices[i].x !== vertices[j].x ? 1 : 0,\n vertices[i].y !== vertices[j].y ? 1 : 0,\n vertices[i].z !== vertices[j].z ? 1 : 0,\n vertices[i].w !== vertices[j].w ? 1 : 0,\n vertices[i].v !== vertices[j].v ? 1 : 0\n ].reduce((sum, val) => sum + val, 0);\n \n if (diff === 1) {\n edges.push({ start: i, end: j });\n }\n }\n }\n\n // Generate faces (simplified)\n const faces: Face[] = [];\n for (let dim1 = 0; dim1 < 5; dim1++) {\n for (let dim2 = dim1 + 1; dim2 < 5; dim2++) {\n for (let val1 = -1; val1 <= 1; val1 += 2) {\n for (let val2 = -1; val2 <= 1; val2 += 2) {\n for (let val3 = -1; val3 <= 1; val3 += 2) {\n const faceVertices: number[] = [];\n for (let i = 0; i < 32; i++) {\n const coords = [vertices[i].x, vertices[i].y, vertices[i].z, vertices[i].w, vertices[i].v];\n let matches = 0;\n if (coords[dim1] === val1) matches++;\n if (coords[dim2] === val2) matches++;\n if (matches === 2) {\n faceVertices.push(i);\n }\n }\n if (faceVertices.length === 4) {\n faces.push({ vertices: faceVertices });\n }\n }\n }\n }\n }\n }\n\n // Generate cells and hypercells (simplified)\n const cells: Cell[] = [];\n const hypercells: Cell[] = [];\n \n // This is a simplified version - full implementation would be much more complex\n for (let i = 0; i < Math.min(faces.length, 40); i += 6) {\n if (i + 5 < faces.length) {\n cells.push({ faces: [i, i + 1, i + 2, i + 3, i + 4, i + 5] });\n }\n }\n\n for (let i = 0; i < Math.min(cells.length, 10); i += 8) {\n if (i + 7 < cells.length) {\n hypercells.push({ faces: [i, i + 1, i + 2, i + 3, i + 4, i + 5, i + 6, i + 7] });\n }\n }\n\n return {\n vertices,\n edges,\n faces,\n cells,\n hypercells,\n name: 'Penteract'\n };\n }\n\n // Generate 5-simplex (5D tetrahedron)\n static create5Simplex(): Shape5D {\n const vertices: Vector5D[] = [\n { x: 1, y: 0, z: 0, w: 0, v: 0 },\n { x: 0, y: 1, z: 0, w: 0, v: 0 },\n { x: 0, y: 0, z: 1, w: 0, v: 0 },\n { x: 0, y: 0, z: 0, w: 1, v: 0 },\n { x: 0, y: 0, z: 0, w: 0, v: 1 },\n { x: -1/5, y: -1/5, z: -1/5, w: -1/5, v: -1/5 }\n ];\n\n const edges: Edge[] = [];\n // Connect all vertices to each other\n for (let i = 0; i < 6; i++) {\n for (let j = i + 1; j < 6; j++) {\n edges.push({ start: i, end: j });\n }\n }\n\n // Generate triangular faces\n const faces: Face[] = [];\n for (let i = 0; i < 6; i++) {\n for (let j = i + 1; j < 6; j++) {\n for (let k = j + 1; k < 6; k++) {\n faces.push({ vertices: [i, j, k] });\n }\n }\n }\n\n // Generate tetrahedral cells\n const cells: Cell[] = [];\n for (let i = 0; i < 6; i++) {\n for (let j = i + 1; j < 6; j++) {\n for (let k = j + 1; k < 6; k++) {\n for (let l = k + 1; l < 6; l++) {\n cells.push({ faces: [i, j, k, l] }); // Simplified\n }\n }\n }\n }\n\n const hypercells: Cell[] = [\n { faces: [0, 1, 2, 3, 4] } // Single 5-cell\n ];\n\n return {\n vertices,\n edges,\n faces,\n cells,\n hypercells,\n name: '5-Simplex'\n };\n }\n\n // Calculate cross-section of 5D shape with a 4D hyperplane\n static calculateCrossSection(shape: Shape5D, vValue: number): Vector3D[] {\n const crossSectionPoints: Vector3D[] = [];\n \n // Find intersections of edges with the hyperplane v = vValue\n for (const edge of shape.edges) {\n const v1 = shape.vertices[edge.start];\n const v2 = shape.vertices[edge.end];\n \n // Check if edge crosses the hyperplane\n if ((v1.v <= vValue && v2.v >= vValue) || (v1.v >= vValue && v2.v <= vValue)) {\n if (Math.abs(v1.v - v2.v) > 0.001) {\n // Calculate intersection point\n const t = (vValue - v1.v) / (v2.v - v1.v);\n const intersection4D = {\n x: v1.x + t * (v2.x - v1.x),\n y: v1.y + t * (v2.y - v1.y),\n z: v1.z + t * (v2.z - v1.z),\n w: v1.w + t * (v2.w - v1.w)\n };\n \n // Project 4D intersection to 3D\n const factor = 2 / (2 - intersection4D.w);\n crossSectionPoints.push({\n x: intersection4D.x * factor,\n y: intersection4D.y * factor,\n z: intersection4D.z * factor\n });\n }\n }\n }\n \n return crossSectionPoints;\n }\n\n // Multiply two 5D matrices\n static multiplyMatrices(a: Matrix5D, b: Matrix5D): Matrix5D {\n const result = Array(5).fill(null).map(() => Array(5).fill(0));\n \n for (let i = 0; i < 5; i++) {\n for (let j = 0; j < 5; j++) {\n for (let k = 0; k < 5; k++) {\n result[i][j] += a.elements[i][k] * b.elements[k][j];\n }\n }\n }\n \n return { elements: result };\n }\n\n // Create composite rotation matrix from multiple rotations\n static createCompositeRotation(rotations: { [key: string]: number }): Matrix5D {\n let result: Matrix5D = { \n elements: Array(5).fill(null).map((_, i) => \n Array(5).fill(null).map((_, j) => i === j ? 1 : 0)\n )\n };\n\n for (const [plane, angle] of Object.entries(rotations)) {\n if (angle !== 0) {\n const rotMatrix = this.createRotationMatrix(plane, angle);\n result = this.multiplyMatrices(result, rotMatrix);\n }\n }\n\n return result;\n }\n}",[],[139,178,180],"c358fc86-59a1-11f0-9b50-f69ea79377d9","geometry4d.ts","c358fd62-59a1-11f0-9b50-f69ea79377d9",3,["D",1751721175037],"// 4D Geometry calculations and shape generation\n\nimport { Vector4D, Vector3D, Shape4D, Edge, Face, Cell, Matrix4D } from '../types.ts';\n\nexport class Geometry4D {\n // Create a 4D rotation matrix for the given plane and angle\n static createRotationMatrix(plane: string, angle: number): Matrix4D {\n const cos = Math.cos(angle);\n const sin = Math.sin(angle);\n const matrix = [\n [1, 0, 0, 0],\n [0, 1, 0, 0],\n [0, 0, 1, 0],\n [0, 0, 0, 1]\n ];\n\n switch (plane) {\n case 'xy':\n matrix[0][0] = cos; matrix[0][1] = -sin;\n matrix[1][0] = sin; matrix[1][1] = cos;\n break;\n case 'xz':\n matrix[0][0] = cos; matrix[0][2] = -sin;\n matrix[2][0] = sin; matrix[2][2] = cos;\n break;\n case 'xw':\n matrix[0][0] = cos; matrix[0][3] = -sin;\n matrix[3][0] = sin; matrix[3][3] = cos;\n break;\n case 'yz':\n matrix[1][1] = cos; matrix[1][2] = -sin;\n matrix[2][1] = sin; matrix[2][2] = cos;\n break;\n case 'yw':\n matrix[1][1] = cos; matrix[1][3] = -sin;\n matrix[3][1] = sin; matrix[3][3] = cos;\n break;\n case 'zw':\n matrix[2][2] = cos; matrix[2][3] = -sin;\n matrix[3][2] = sin; matrix[3][3] = cos;\n break;\n }\n\n return { elements: matrix };\n }\n\n // Apply transformation matrix to a 4D vector\n static transformVector(vector: Vector4D, matrix: Matrix4D): Vector4D {\n const { x, y, z, w } = vector;\n const m = matrix.elements;\n \n return {\n x: m[0][0] * x + m[0][1] * y + m[0][2] * z + m[0][3] * w,\n y: m[1][0] * x + m[1][1] * y + m[1][2] * z + m[1][3] * w,\n z: m[2][0] * x + m[2][1] * y + m[2][2] * z + m[2][3] * w,\n w: m[3][0] * x + m[3][1] * y + m[3][2] * z + m[3][3] * w\n };\n }\n\n // Project 4D point to 3D using perspective projection\n static projectTo3D(point: Vector4D, distance: number = 2): Vector3D {\n const factor = distance / (distance - point.w);\n return {\n x: point.x * factor,\n y: point.y * factor,\n z: point.z * factor\n };\n }\n\n // Generate tesseract (4D cube) vertices and structure\n static createTesseract(): Shape4D {\n const vertices: Vector4D[] = [];\n \n // Generate all 16 vertices of a tesseract\n for (let i = 0; i < 16; i++) {\n vertices.push({\n x: (i & 1) ? 1 : -1,\n y: (i & 2) ? 1 : -1,\n z: (i & 4) ? 1 : -1,\n w: (i & 8) ? 1 : -1\n });\n }\n\n // Generate edges (32 edges total)\n const edges: Edge[] = [];\n for (let i = 0; i < 16; i++) {\n for (let j = i + 1; j < 16; j++) {\n // Two vertices are connected if they differ in exactly one coordinate\n const diff = [\n vertices[i].x !== vertices[j].x ? 1 : 0,\n vertices[i].y !== vertices[j].y ? 1 : 0,\n vertices[i].z !== vertices[j].z ? 1 : 0,\n vertices[i].w !== vertices[j].w ? 1 : 0\n ].reduce((sum, val) => sum + val, 0);\n \n if (diff === 1) {\n edges.push({ start: i, end: j });\n }\n }\n }\n\n // Generate faces (24 square faces)\n const faces: Face[] = [];\n // Each face is defined by 4 vertices that form a square\n // This is a simplified version - full implementation would be more complex\n for (let dim1 = 0; dim1 < 4; dim1++) {\n for (let dim2 = dim1 + 1; dim2 < 4; dim2++) {\n for (let val1 = -1; val1 <= 1; val1 += 2) {\n for (let val2 = -1; val2 <= 1; val2 += 2) {\n const faceVertices: number[] = [];\n for (let i = 0; i < 16; i++) {\n const coords = [vertices[i].x, vertices[i].y, vertices[i].z, vertices[i].w];\n if (coords[dim1] === val1 && coords[dim2] === val2) {\n faceVertices.push(i);\n }\n }\n if (faceVertices.length === 4) {\n faces.push({ vertices: faceVertices });\n }\n }\n }\n }\n }\n\n // Generate cells (8 cubic cells)\n const cells: Cell[] = [];\n for (let w = -1; w <= 1; w += 2) {\n cells.push({ faces: faces.map((_, i) => i).slice(0, 6) }); // Simplified\n }\n\n return {\n vertices,\n edges,\n faces,\n cells,\n name: 'Tesseract'\n };\n }\n\n // Generate 16-cell (4D cross-polytope)\n static create16Cell(): Shape4D {\n const vertices: Vector4D[] = [\n { x: 1, y: 0, z: 0, w: 0 },\n { x: -1, y: 0, z: 0, w: 0 },\n { x: 0, y: 1, z: 0, w: 0 },\n { x: 0, y: -1, z: 0, w: 0 },\n { x: 0, y: 0, z: 1, w: 0 },\n { x: 0, y: 0, z: -1, w: 0 },\n { x: 0, y: 0, z: 0, w: 1 },\n { x: 0, y: 0, z: 0, w: -1 }\n ];\n\n const edges: Edge[] = [];\n // Connect vertices that are not opposite to each other\n for (let i = 0; i < 8; i++) {\n for (let j = i + 1; j < 8; j++) {\n const v1 = vertices[i];\n const v2 = vertices[j];\n // Check if they're not opposite (dot product != -1)\n const dot = v1.x * v2.x + v1.y * v2.y + v1.z * v2.z + v1.w * v2.w;\n if (Math.abs(dot + 1) > 0.001) {\n edges.push({ start: i, end: j });\n }\n }\n }\n\n // Generate triangular faces\n const faces: Face[] = [];\n for (let i = 0; i < 8; i++) {\n for (let j = i + 1; j < 8; j++) {\n for (let k = j + 1; k < 8; k++) {\n // Check if these three vertices form a valid triangular face\n const v1 = vertices[i];\n const v2 = vertices[j];\n const v3 = vertices[k];\n \n // Simplified check - in a real implementation, this would be more sophisticated\n const sum = Math.abs(v1.x + v2.x + v3.x) + \n Math.abs(v1.y + v2.y + v3.y) + \n Math.abs(v1.z + v2.z + v3.z) + \n Math.abs(v1.w + v2.w + v3.w);\n \n if (Math.abs(sum - 1) < 0.001) {\n faces.push({ vertices: [i, j, k] });\n }\n }\n }\n }\n\n const cells: Cell[] = [];\n // Each cell is a tetrahedron\n for (let i = 0; i < faces.length; i += 4) {\n if (i + 3 < faces.length) {\n cells.push({ faces: [i, i + 1, i + 2, i + 3] });\n }\n }\n\n return {\n vertices,\n edges,\n faces,\n cells,\n name: '16-Cell'\n };\n }\n\n // Calculate cross-section of 4D shape with a 3D hyperplane\n static calculateCrossSection(shape: Shape4D, wValue: number): Vector3D[] {\n const crossSectionPoints: Vector3D[] = [];\n \n // Find intersections of edges with the hyperplane w = wValue\n for (const edge of shape.edges) {\n const v1 = shape.vertices[edge.start];\n const v2 = shape.vertices[edge.end];\n \n // Check if edge crosses the hyperplane\n if ((v1.w <= wValue && v2.w >= wValue) || (v1.w >= wValue && v2.w <= wValue)) {\n if (Math.abs(v1.w - v2.w) > 0.001) {\n // Calculate intersection point\n const t = (wValue - v1.w) / (v2.w - v1.w);\n const intersection = {\n x: v1.x + t * (v2.x - v1.x),\n y: v1.y + t * (v2.y - v1.y),\n z: v1.z + t * (v2.z - v1.z)\n };\n crossSectionPoints.push(intersection);\n }\n }\n }\n \n return crossSectionPoints;\n }\n\n // Multiply two 4D matrices\n static multiplyMatrices(a: Matrix4D, b: Matrix4D): Matrix4D {\n const result = Array(4).fill(null).map(() => Array(4).fill(0));\n \n for (let i = 0; i < 4; i++) {\n for (let j = 0; j < 4; j++) {\n for (let k = 0; k < 4; k++) {\n result[i][j] += a.elements[i][k] * b.elements[k][j];\n }\n }\n }\n \n return { elements: result };\n }\n\n // Create composite rotation matrix from multiple rotations\n static createCompositeRotation(rotations: { [key: string]: number }): Matrix4D {\n let result: Matrix4D = { elements: [\n [1, 0, 0, 0],\n [0, 1, 0, 0],\n [0, 0, 1, 0],\n [0, 0, 0, 1]\n ]};\n\n for (const [plane, angle] of Object.entries(rotations)) {\n if (angle !== 0) {\n const rotMatrix = this.createRotationMatrix(plane, angle);\n result = this.multiplyMatrices(result, rotMatrix);\n }\n }\n\n return result;\n }\n}",[],[139,178,188],"c3591a90-59a1-11f0-9b50-f69ea79377d9",["D",1751721175037],[157,158,159],[139,178],["D",1751721129544],[156,160],[139],["D",1751721027052],{"_96":95,"_103":99},{"_96":95,"_97":-5,"_98":99,"_18":54,"_100":101,"_102":24},{"_13":85,"_31":86,"_42":41,"_87":43,"_88":-5,"_89":-5,"_90":-5,"_91":-5,"_92":-5,"_93":44,"_94":95},"ctx",{"_78":-5,"_123":239,"_240":241,"_242":243,"_244":24,"_245":24,"_246":24,"_247":58,"_248":-5,"_249":-7,"_250":-7,"_251":62,"_252":24,"_253":24,"_254":-5,"_255":24,"_256":24,"_257":24,"_258":-5},"ENV",{"_210":211,"_212":213,"_214":86,"_215":216,"_217":218,"_219":220,"_221":222,"_223":224,"_225":226,"_227":228,"_229":230,"_231":232,"_233":234,"_235":236,"_237":238},"REACT_APP_SENTRY_DSN","https://586020d86f05489db63a554690acf1e9@o1398359.ingest.sentry.io/6724499","RENDER_GIT_COMMIT","c73f21da490985bff19c279eb0e09db24d688e4f","RENDER_GIT_BRANCH","POSTHOG_API_TOKEN","phc_nmoyuAkStEZuUp69rYlg1DZgrIt19Gx0Y4Hwf8tGgJP","POSTHOG_API_HOST","https://wart.val.town","WEB_API_TEMPLATE","https://{val}.web.val.run","APP_URL","https://www.val.town","MODULE_URL","https://esm.town","VANITY_WEB_API_TEMPLATE","https://{label}.val.run/","DENOLS_URL","https://lsp.val.town","NODE_ENV","production","CLOUDFLARE_TURNSTILE_SITE_KEY","0x4AAAAAACtaIZJH7s9rBMZ6","GOOGLE_CLIENT_ID","875989696899-jtsqkhlbdv5jrlsk8631p7ipjg3c6o62.apps.googleusercontent.com","STRIPE_PUBLISHABLE_KEY","pk_live_51N2zvWEpZTTLipIkVEwXrUG9AT0yeOOEYxszarZT1jpsZV58Ly6wlg3SltZpHN26CUalOBmUSY7adQRQiSkCZz2500qSXNkf8Y","API_URL","https://api.val.town",[],"preferencesKeybindings","default","preferencesTheme","light","preferencesFixers","preferencesCopilot","preferencesSelectionAi","preferencesCloseBrackets","preferencesChatPanelState","preferencesDprint","preferencesDenofmt","notificationCount","flagDenols","isBusinessOrgMember","businessOrgHandle","isBusinessOrgAdmin","isHigherTownieTier","navCollapsed","currentOrgId"]

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