[V0_FILE]typescriptreact:file="app/page.tsx" isMerged="true" import { CayleyDickinsonExplorer } from "@/components/cayley-dickinson-explorer" export default function Home() { return (

) } [V0_FILE]json:file="package.json" isMerged="true" { "name": "my-v0-project", "version": "0.1.0", "private": true, "scripts": { "dev": "next dev", "build": "next build", "start": "next start", "lint": "eslint ." }, "dependencies": { "@hookform/resolvers": "^3.10.0", "@radix-ui/react-accordion": "1.2.2", "@radix-ui/react-alert-dialog": "1.1.4", "@radix-ui/react-aspect-ratio": "1.1.1", "@radix-ui/react-avatar": "1.1.2", "@radix-ui/react-checkbox": "1.1.3", "@radix-ui/react-collapsible": "1.1.2", "@radix-ui/react-context-menu": "2.2.4", "@radix-ui/react-dialog": "1.1.4", "@radix-ui/react-dropdown-menu": "2.1.4", "@radix-ui/react-hover-card": "1.1.4", "@radix-ui/react-label": "2.1.1", "@radix-ui/react-menubar": "1.1.4", "@radix-ui/react-navigation-menu": "1.2.3", "@radix-ui/react-popover": "1.1.4", "@radix-ui/react-progress": "1.1.1", "@radix-ui/react-radio-group": "1.2.2", "@radix-ui/react-scroll-area": "1.2.2", "@radix-ui/react-select": "2.1.4", "@radix-ui/react-separator": "1.1.1", "@radix-ui/react-slider": "1.2.2", "@radix-ui/react-slot": "1.1.1", "@radix-ui/react-switch": "1.1.2", "@radix-ui/react-tabs": "1.1.2", "@radix-ui/react-toast": "1.2.4", "@radix-ui/react-toggle": "1.1.1", "@radix-ui/react-toggle-group": "1.1.1", "@radix-ui/react-tooltip": "1.1.6", "@vercel/analytics": "1.3.1", "autoprefixer": "^10.4.20", "class-variance-authority": "^0.7.1", "clsx": "^2.1.1", "cmdk": "1.0.4", "date-fns": "4.1.0", "embla-carousel-react": "8.5.1", "input-otp": "1.4.1", "lucide-react": "^0.454.0", "next": "16.0.0", "next-themes": "^0.4.6", "react": "19.2.0", "react-day-picker": "9.8.0", "react-dom": "19.2.0", "react-hook-form": "^7.60.0", "react-resizable-panels": "^2.1.7", "recharts": "2.15.4", "sonner": "^1.7.4", "tailwind-merge": "^3.3.1", "tailwindcss-animate": "^1.0.7", "vaul": "^0.9.9", "zod": "3.25.76" }, "devDependencies": { "@tailwindcss/postcss": "^4.1.9", "@types/node": "^22", "@types/react": "^19", "@types/react-dom": "^19", "postcss": "^8.5", "tailwindcss": "^4.1.9", "tw-animate-css": "1.3.3", "typescript": "^5" } } [V0_FILE]css:file="app/globals.css" isEdit="true" isMerged="true" @import "tailwindcss"; @import "tw-animate-css"; @custom-variant dark (&:is(.dark *)); /* Linear-inspired design system: monochromatic with sharp edges, no purple */ :root { --background: oklch(1 0 0); --foreground: oklch(0.15 0 0); --card: oklch(0.98 0 0); --card-foreground: oklch(0.15 0 0); --popover: oklch(0.98 0 0); --popover-foreground: oklch(0.15 0 0); --primary: oklch(0.2 0 0); --primary-foreground: oklch(0.98 0 0); --secondary: oklch(0.94 0 0); --secondary-foreground: oklch(0.2 0 0); --muted: oklch(0.96 0 0); --muted-foreground: oklch(0.5 0 0); --accent: oklch(0.45 0.15 25); --accent-foreground: oklch(0.98 0 0); --border: oklch(0.88 0 0); --input: oklch(0.94 0 0); --ring: oklch(0.45 0.15 25); --destructive: oklch(0.55 0.22 25); --destructive-foreground: oklch(0.98 0 0); --chart-1: oklch(0.45 0.15 25); --chart-2: oklch(0.35 0 0); --chart-3: oklch(0.55 0 0); --chart-4: oklch(0.65 0 0); --chart-5: oklch(0.75 0 0); --radius: 0rem; --sidebar: oklch(0.985 0 0); --sidebar-foreground: oklch(0.145 0 0); --sidebar-primary: oklch(0.205 0 0); --sidebar-primary-foreground: oklch(0.985 0 0); --sidebar-accent: oklch(0.97 0 0); --sidebar-accent-foreground: oklch(0.205 0 0); --sidebar-border: oklch(0.922 0 0); --sidebar-ring: oklch(0.708 0 0); } .dark { --background: oklch(0.08 0 0); --foreground: oklch(0.95 0 0); --card: oklch(0.11 0 0); --card-foreground: oklch(0.95 0 0); --popover: oklch(0.11 0 0); --popover-foreground: oklch(0.95 0 0); --primary: oklch(0.85 0 0); --primary-foreground: oklch(0.08 0 0); --secondary: oklch(0.18 0 0); --secondary-foreground: oklch(0.95 0 0); --muted: oklch(0.15 0 0); --muted-foreground: oklch(0.55 0 0); --accent: oklch(0.65 0.18 25); --accent-foreground: oklch(0.08 0 0); --border: oklch(0.22 0 0); --input: oklch(0.18 0 0); --ring: oklch(0.65 0.18 25); --destructive: oklch(0.65 0.22 25); --destructive-foreground: oklch(0.08 0 0); --chart-1: oklch(0.65 0.18 25); --chart-2: oklch(0.55 0 0); --chart-3: oklch(0.65 0 0); --chart-4: oklch(0.75 0 0); --chart-5: oklch(0.85 0 0); --sidebar: oklch(0.205 0 0); --sidebar-foreground: oklch(0.985 0 0); --sidebar-primary: oklch(0.488 0.243 264.376); --sidebar-primary-foreground: oklch(0.985 0 0); --sidebar-accent: oklch(0.269 0 0); --sidebar-accent-foreground: oklch(0.985 0 0); --sidebar-border: oklch(0.269 0 0); --sidebar-ring: oklch(0.439 0 0); } @theme inline { --font-sans: "Geist", "Geist Fallback"; --font-mono: "Geist Mono", "Geist Mono Fallback"; --color-background: var(--background); --color-foreground: var(--foreground); --color-card: var(--card); --color-card-foreground: var(--card-foreground); --color-popover: var(--popover); --color-popover-foreground: var(--popover-foreground); --color-primary: var(--primary); --color-primary-foreground: var(--primary-foreground); --color-secondary: var(--secondary); --color-secondary-foreground: var(--secondary-foreground); --color-muted: var(--muted); --color-muted-foreground: var(--muted-foreground); --color-accent: var(--accent); --color-accent-foreground: var(--accent-foreground); --color-destructive: var(--destructive); --color-destructive-foreground: var(--destructive-foreground); --color-border: var(--border); --color-input: var(--input); --color-ring: var(--ring); --color-chart-1: var(--chart-1); --color-chart-2: var(--chart-2); --color-chart-3: var(--chart-3); --color-chart-4: var(--chart-4); --color-chart-5: var(--chart-5); --radius-sm: calc(var(--radius) - 4px); --radius-md: calc(var(--radius) - 2px); --radius-lg: var(--radius); --radius-xl: calc(var(--radius) + 4px); --color-sidebar: var(--sidebar); --color-sidebar-foreground: var(--sidebar-foreground); --color-sidebar-primary: var(--sidebar-primary); --color-sidebar-primary-foreground: var(--sidebar-primary-foreground); --color-sidebar-accent: var(--sidebar-accent); --color-sidebar-accent-foreground: var(--sidebar-accent-foreground); --color-sidebar-border: var(--sidebar-border); --color-sidebar-ring: var(--sidebar-ring); } @layer base { * { @apply border-border outline-ring/50; } body { @apply bg-background text-foreground font-sans antialiased; } } /* Linear-style precise animations */ @keyframes slide-in { from { opacity: 0; transform: translateY(8px); } to { opacity: 1; transform: translateY(0); } } @keyframes fade-in { from { opacity: 0; } to { opacity: 1; } } @keyframes pulse-border { 0%, 100% { border-color: var(--border); } 50% { border-color: var(--accent); } } .animate-slide-in { animation: slide-in 0.3s ease-out; } .animate-fade-in { animation: fade-in 0.2s ease-out; } .animate-pulse-border { animation: pulse-border 2s ease-in-out infinite; } /* Technical grid pattern */ .technical-grid { background-image: linear-gradient(to right, var(--border) 1px, transparent 1px), linear-gradient(to bottom, var(--border) 1px, transparent 1px); background-size: 20px 20px; } [V0_FILE]typescriptreact:file="components/construction-formula.tsx" isMerged="true" "use client" import { Card } from "@/components/ui/card" import { Code2 } from "lucide-react" export function ConstructionFormula({ level }: { level: number }) { const getFormula = () => { return (

// Cayley-Dickinson Construction

Given algebra A, construct A' = A ⊕ A

// Doubles dimension: dim(A') = 2·dim(A)

(a, b) + (c, d) = (a + c, b + d)

(a, b) × (c, d) = (ac - d̄b, da + bc̄)

DIMENSION

{Math.pow(2, level)}

ITERATION

{level}

Each application of the construction doubles the dimension but loses one algebraic property. The conjugation operation (ā) and multiplication rules preserve the algebraic structure.

) } return (

Construction Formula

{getFormula()} ) } [V0_FILE]typescriptreact:file="components/multiplication-table.tsx" isMerged="true" "use client" import { Card } from "@/components/ui/card" import { Table } from "lucide-react" interface NumberSystem { name: string dimensions: number level: number } export function MultiplicationTable({ system }: { system: NumberSystem }) { const getTable = () => { if (system.level === 0) { return (

Real numbers follow standard multiplication rules

) } if (system.level === 1) { // Complex numbers const basis = ["1", "i"] return (

{basis.map((b) => ( ))}

×	{b}
1	1	i
i	i	-1

) } if (system.level === 2) { // Quaternions const basis = ["1", "i", "j", "k"] const table = [ ["1", "i", "j", "k"], ["i", "-1", "k", "-j"], ["j", "-k", "-1", "i"], ["k", "j", "-i", "-1"], ] return (

{basis.map((b) => ( ))} {table.map((row, i) => ( {row.map((cell, j) => ( ))} ))}

×	{b}
{basis[i]}	{cell}

Note: ij = k, but ji = -k (non-commutative)

) } return null } return (

Multiplication Table

{getTable()} ) } [V0_FILE]typescriptreact:file="components/number-system-visualizer.tsx" isEdit="true" isMerged="true" "use client" import { Card } from "@/components/ui/card" import { useEffect, useRef } from "react" import { Eye } from "lucide-react" interface NumberSystem { name: string symbol: string dimensions: number level: number } export function NumberSystemVisualizer({ system }: { system: NumberSystem }) { const canvasRef = useRef(null) useEffect(() => { const canvas = canvasRef.current if (!canvas) return const ctx = canvas.getContext("2d") if (!ctx) return const rect = canvas.getBoundingClientRect() canvas.width = rect.width * window.devicePixelRatio canvas.height = rect.height * window.devicePixelRatio ctx.scale(window.devicePixelRatio, window.devicePixelRatio) const width = rect.width const height = rect.height const centerX = width / 2 const centerY = height / 2 let animationFrame: number let rotation = 0 const accentColor = "rgb(200, 100, 40)" const primaryColor = "rgb(30, 30, 30)" const secondaryColor = "rgb(100, 100, 100)" const animate = () => { ctx.clearRect(0, 0, width, height) // Technical grid background ctx.strokeStyle = "rgba(200, 200, 200, 0.1)" ctx.lineWidth = 1 for (let i = 0; i < width; i += 20) { ctx.beginPath() ctx.moveTo(i, 0) ctx.lineTo(i, height) ctx.stroke() } for (let i = 0; i < height; i += 20) { ctx.beginPath() ctx.moveTo(0, i) ctx.lineTo(width, i) ctx.stroke() } if (system.dimensions === 1) { // Real numbers - precise line with moving point ctx.strokeStyle = primaryColor ctx.lineWidth = 2 ctx.beginPath() ctx.moveTo(40, centerY) ctx.lineTo(width - 40, centerY) ctx.stroke() // Tick marks for (let i = -3; i <= 3; i++) { const x = centerX + i * 40 ctx.beginPath() ctx.moveTo(x, centerY - 5) ctx.lineTo(x, centerY + 5) ctx.stroke() } const x = centerX + Math.sin(rotation) * 120 ctx.fillStyle = accentColor ctx.fillRect(x - 4, centerY - 4, 8, 8) } else if (system.dimensions === 2) { // Complex plane - axes and rotating vector ctx.strokeStyle = secondaryColor ctx.lineWidth = 1.5 // Axes ctx.beginPath() ctx.moveTo(40, centerY) ctx.lineTo(width - 40, centerY) ctx.moveTo(centerX, 40) ctx.lineTo(centerX, height - 40) ctx.stroke() // Grid lines ctx.strokeStyle = "rgba(100, 100, 100, 0.2)" for (let i = -2; i <= 2; i++) { if (i === 0) continue ctx.beginPath() ctx.moveTo(centerX + i * 50, 40) ctx.lineTo(centerX + i * 50, height - 40) ctx.stroke() ctx.beginPath() ctx.moveTo(40, centerY + i * 50) ctx.lineTo(width - 40, centerY + i * 50) ctx.stroke() } // Rotating vector const vecX = Math.cos(rotation) * 100 const vecY = Math.sin(rotation) * 100 ctx.strokeStyle = accentColor ctx.lineWidth = 2 ctx.beginPath() ctx.moveTo(centerX, centerY) ctx.lineTo(centerX + vecX, centerY + vecY) ctx.stroke() ctx.fillStyle = accentColor ctx.fillRect(centerX + vecX - 4, centerY + vecY - 4, 8, 8) } else if (system.dimensions === 4) { // Quaternions - 4 basis vectors const radius = 80 for (let i = 0; i < 4; i++) { const angle = (i * Math.PI) / 2 + rotation const x = centerX + Math.cos(angle) * radius const y = centerY + Math.sin(angle) * radius // Connection to center ctx.strokeStyle = i === 0 ? accentColor : secondaryColor ctx.lineWidth = i === 0 ? 2 : 1 ctx.globalAlpha = 0.3 ctx.beginPath() ctx.moveTo(centerX, centerY) ctx.lineTo(x, y) ctx.stroke() ctx.globalAlpha = 1 // Point ctx.fillStyle = i === 0 ? accentColor : primaryColor ctx.fillRect(x - 5, y - 5, 10, 10) // Label ctx.fillStyle = secondaryColor ctx.font = "12px monospace" ctx.textAlign = "center" const labels = ["1", "i", "j", "k"] ctx.fillText(labels[i], x, y - 15) } } else if (system.dimensions === 8) { // Octonions - 8 points const radius = 90 for (let i = 0; i < 8; i++) { const angle = (i * Math.PI * 2) / 8 + rotation const x = centerX + Math.cos(angle) * radius const y = centerY + Math.sin(angle) * radius // Connections const nextAngle = ((i + 1) * Math.PI * 2) / 8 + rotation const nextX = centerX + Math.cos(nextAngle) * radius const nextY = centerY + Math.sin(nextAngle) * radius ctx.strokeStyle = secondaryColor ctx.lineWidth = 1 ctx.globalAlpha = 0.2 ctx.beginPath() ctx.moveTo(x, y) ctx.lineTo(nextX, nextY) ctx.stroke() ctx.globalAlpha = 1 ctx.fillStyle = i === 0 ? accentColor : primaryColor ctx.fillRect(x - 4, y - 4, 8, 8) } } else if (system.dimensions === 16) { // Sedenions - nested structure for (let ring = 0; ring < 3; ring++) { const radius = 35 + ring * 35 const points = 4 + ring * 4 for (let i = 0; i < points; i++) { const angle = (i * Math.PI * 2) / points + rotation * (1 + ring * 0.3) const x = centerX + Math.cos(angle) * radius const y = centerY + Math.sin(angle) * radius const size = 7 - ring * 2 ctx.fillStyle = ring === 0 ? accentColor : primaryColor ctx.fillRect(x - size / 2, y - size / 2, size, size) } } } else { // Higher dimensions - complex pattern const numPoints = Math.min(system.dimensions, 32) for (let i = 0; i < numPoints; i++) { const angle = (i * Math.PI * 2) / numPoints + rotation const radius = 60 + Math.sin(rotation * 2 + i) * 30 const x = centerX + Math.cos(angle) * radius const y = centerY + Math.sin(angle) * radius ctx.fillStyle = i % 8 === 0 ? accentColor : primaryColor ctx.fillRect(x - 3, y - 3, 6, 6) } } rotation += 0.015 animationFrame = requestAnimationFrame(animate) } animate() return () => { cancelAnimationFrame(animationFrame) } }, [system]) return (

Visual Representation

{system.dimensions === 1 && "A single dimension - the number line with real values"} {system.dimensions === 2 && "Two dimensions - the complex plane with real and imaginary axes"} {system.dimensions === 4 && "Four dimensions - quaternion space with basis {1, i, j, k}"} {system.dimensions === 8 && "Eight dimensions - octonion structure with 7 imaginary units"} {system.dimensions === 16 && "Sixteen dimensions - sedenion complexity with zero divisors"} {system.dimensions >= 32 && `${system.dimensions} dimensions - beyond conventional visualization`}

) } [V0_FILE]typescriptreact:file="components/operation-playground.tsx" isEdit="true" isMerged="true" "use client" import { Card } from "@/components/ui/card" import { Button } from "@/components/ui/button" import { Input } from "@/components/ui/input" import { useState } from "react" import { Calculator } from "lucide-react" interface NumberSystem { name: string dimensions: number level: number notation: string example: string } export function OperationPlayground({ system }: { system: NumberSystem }) { const [a, setA] = useState("1") const [b, setB] = useState("1") const [operation, setOperation] = useState<"add" | "multiply">("multiply") const demonstrateProperty = () => { if (system.level === 0 || system.level === 1) { return (

COMMUTATIVE PROPERTY

a × b = b × a

Order doesn't matter in multiplication

) } if (system.level === 2) { return (

NON-COMMUTATIVE

i×j = k, but j×i = -k

Order matters! Multiplication is no longer commutative

) } if (system.level === 3) { return (

NON-ASSOCIATIVE

(a×b)×c ≠ a×(b×c)

Grouping matters! Lost associativity

) } if (system.level >= 4) { return (

ZERO DIVISORS EXIST

a×b = 0, but a,b ≠ 0

Non-zero numbers can multiply to zero!

) } } return (

Operation Playground

FIRST NUMBER (a) setA(e.target.value)} placeholder={system.example} className="font-mono" />

SECOND NUMBER (b) setB(e.target.value)} placeholder={system.example} className="font-mono" />

{demonstrateProperty()}

DIMENSIONS

{system.dimensions}

COMPONENTS

{system.dimensions}

) } [V0_FILE]typescriptreact:file="components/property-loss-indicator.tsx" isEdit="true" isMerged="true" "use client" import { Card } from "@/components/ui/card" const properties = [ { name: "Commutative", lostAt: 3, description: "a × b = b × a", detail: "Order independence" }, { name: "Associative", lostAt: 4, description: "(a × b) × c = a × (b × c)", detail: "Grouping independence" }, { name: "Division Algebra", lostAt: 5, description: "Every non-zero element has an inverse", detail: "Invertibility", }, { name: "Alternative", lostAt: 6, description: "Weaker form of associativity", detail: "Partial associativity" }, ] export function PropertyLossIndicator({ level }: { level: number }) { return (

Property Evolution

{properties.map((prop) => { const isLost = level >= prop.lostAt return (

{isLost ? (

) : (

)}

{prop.name}

{prop.detail}

{prop.description}

{isLost &&

LOST AT LEVEL {prop.lostAt - 1}

}

) })}

) } [V0_FILE]typescriptreact:file="components/cayley-dickinson-explorer.tsx" isEdit="true" "use client" import { useState, useEffect } from "react" import { Card } from "@/components/ui/card" import { Button } from "@/components/ui/button" import { Badge } from "@/components/ui/badge" import { ChevronRight, Info, Layers, Binary, GitBranch, Keyboard } from "lucide-react" import { NumberSystemVisualizer } from "@/components/number-system-visualizer" import { OperationPlayground } from "@/components/operation-playground" import { PropertyLossIndicator } from "@/components/property-loss-indicator" import { ConstructionFormula } from "@/components/construction-formula" import { MultiplicationTable } from "@/components/multiplication-table" import { BasisElementPlayground } from "@/components/basis-element-playground" import { RealWorldApplications } from "@/components/real-world-applications" import { HistoricalTimeline } from "@/components/historical-timeline" import { GeometricInterpretation } from "@/components/geometric-interpretation" import { KeyboardShortcuts } from "@/components/keyboard-shortcuts" const numberSystems = [ { name: "Real Numbers", symbol: "ℝ", dimensions: 1, description: "The foundation - just regular numbers on a line", properties: ["Commutative", "Associative", "Division Algebra"], level: 0, notation: "a", example: "5", year: "Ancient", discoverer: "Various civilizations", }, { name: "Complex Numbers", symbol: "ℂ", dimensions: 2, description: "Add the imaginary unit i where i² = -1", properties: ["Commutative", "Associative", "Division Algebra"], level: 1, notation: "a + bi", example: "3 + 4i", year: "1545", discoverer: "Gerolamo Cardano", }, { name: "Quaternions", symbol: "ℍ", dimensions: 4, description: "Three imaginary units: i, j, k with ijk = -1", properties: ["Associative", "Division Algebra"], lostProperty: "Commutativity", level: 2, notation: "a + bi + cj + dk", example: "1 + 2i + 3j + 4k", year: "1843", discoverer: "William Rowan Hamilton", }, { name: "Octonions", symbol: "𝕆", dimensions: 8, description: "Seven imaginary units - things get weird", properties: ["Division Algebra", "Alternative"], lostProperty: "Associativity", level: 3, notation: "a + Σ(bᵢeᵢ) for i=1..7", example: "1 + 2e₁ + 3e₂ + ...", year: "1843", discoverer: "John T. Graves", }, { name: "Sedenions", symbol: "𝕊", dimensions: 16, description: "Fifteen imaginary units - zero divisors appear!", properties: ["Power Associative"], lostProperty: "Division Algebra", level: 4, notation: "a + Σ(bᵢeᵢ) for i=1..15", example: "Complex 16D structure", year: "1845", discoverer: "Various mathematicians", }, { name: "Trigintaduonions", symbol: "𝕋₃₂", dimensions: 32, description: "The pattern continues into mathematical chaos", properties: ["Flexible"], lostProperty: "Alternative", level: 5, notation: "a + Σ(bᵢeᵢ) for i=1..31", example: "Complex 32D structure", year: "Modern", discoverer: "Cayley-Dickinson construction", }, ] export function CayleyDickinsonExplorer() { const [selectedLevel, setSelectedLevel] = useState(0) const [showShortcuts, setShowShortcuts] = useState(false) const currentSystem = numberSystems[selectedLevel] useEffect(() => { const handleKeyPress = (e: KeyboardEvent) => { // Number keys 1-6 to jump to levels if (e.key >= "1" && e.key <= "6") { const level = Number.parseInt(e.key) - 1 if (level < numberSystems.length) { setSelectedLevel(level) } } // Arrow keys for navigation if (e.key === "ArrowRight" && selectedLevel < numberSystems.length - 1) { setSelectedLevel(selectedLevel + 1) } if (e.key === "ArrowLeft" && selectedLevel > 0) { setSelectedLevel(selectedLevel - 1) } // ? to show shortcuts if (e.key === "?") { setShowShortcuts(!showShortcuts) } } window.addEventListener("keydown", handleKeyPress) return () => window.removeEventListener("keydown", handleKeyPress) }, [selectedLevel, showShortcuts]) return (

MATHEMATICAL CONSTRUCTION

Cayley-Dickinson Construction

A recursive process that generates increasingly complex number systems. Each iteration doubles the dimension while systematically losing algebraic properties.

{showShortcuts && setShowShortcuts(false)} />}

LEVEL

{numberSystems.map((system, index) => ( ))}

{/* Left Column - System Info */}

{currentSystem.symbol}

{currentSystem.name}

LEVEL {currentSystem.level} • {currentSystem.dimensions} DIMENSIONS

{currentSystem.description}

NOTATION

{currentSystem.notation}

Example: {currentSystem.example}

DISCOVERED

{currentSystem.year}

{currentSystem.discoverer}

{currentSystem.lostProperty && (

PROPERTY LOST

{currentSystem.lostProperty}

)}

REMAINING PROPERTIES

{currentSystem.properties.map((prop) => (

{prop}

))}

{/* Right Column - Visualizations */}

{currentSystem.level <= 2 && } {currentSystem.level >= 1 && currentSystem.level <= 3 && } {currentSystem.level <= 2 && }

{selectedLevel < numberSystems.length - 1 && (

)}

) } [V0_FILE]typescriptreact:file="components/basis-element-playground.tsx" isFixed="true" isQuickEdit="true" "use client" import { useState } from "react" import { Card } from "@/components/ui/card" import { Button } from "@/components/ui/button" import { Sparkles, X } from "lucide-react" import { Badge } from "@/components/ui/badge" interface NumberSystem { name: string dimensions: number level: number } const basisElements = { 1: [{ symbol: "1", name: "real" }], 2: [ { symbol: "1", name: "real" }, { symbol: "i", name: "imaginary" }, ], 4: [ { symbol: "1", name: "real" }, { symbol: "i", name: "i" }, { symbol: "j", name: "j" }, { symbol: "k", name: "k" }, ], 8: [ { symbol: "1", name: "real" }, { symbol: "e1", name: "e1" }, { symbol: "e2", name: "e2" }, { symbol: "e3", name: "e3" }, { symbol: "e4", name: "e4" }, { symbol: "e5", name: "e5" }, { symbol: "e6", name: "e6" }, { symbol: "e7", name: "e7" }, ], } // Multiplication tables const multiplicationRules: Record>> = { 2: { "1": { "1": "1", i: "i" }, i: { "1": "i", i: "-1" }, }, 4: { "1": { "1": "1", i: "i", j: "j", k: "k" }, i: { "1": "i", i: "-1", j: "k", k: "-j" }, j: { "1": "j", i: "-k", j: "-1", k: "i" }, k: { "1": "k", i: "j", j: "-i", k: "-1" }, }, 8: { "1": { "1": "1", e1: "e1", e2: "e2", e3: "e3", e4: "e4", e5: "e5", e6: "e6", e7: "e7" }, e1: { "1": "e1", e1: "-1", e2: "e4", e3: "e7", e4: "-e2", e5: "e6", e6: "-e5", e7: "-e3" }, e2: { "1": "e2", e1: "-e4", e2: "-1", e3: "e5", e4: "e1", e5: "-e3", e6: "e7", e7: "-e6" }, e3: { "1": "e3", e1: "-e7", e2: "-e5", e3: "-1", e4: "e6", e5: "e2", e6: "-e4", e7: "e1" }, e4: { "1": "e4", e1: "e2", e2: "-e1", e3: "-e6", e4: "-1", e5: "e7", e6: "e3", e7: "-e5" }, e5: { "1": "e5", e1: "-e6", e2: "e3", e3: "-e2", e4: "-e7", e5: "-1", e6: "e1", e7: "e4" }, e6: { "1": "e6", e1: "e5", e2: "-e7", e3: "e4", e4: "-e3", e5: "-e1", e6: "-1", e7: "e2" }, e7: { "1": "e7", e1: "e3", e2: "e6", e3: "-e1", e4: "e5", e5: "-e4", e6: "-e2", e7: "-1" }, }, } export function BasisElementPlayground({ system }: { system: NumberSystem }) { const [selectedA, setSelectedA] = useState(null) const [selectedB, setSelectedB] = useState(null) const [result, setResult] = useState(null) const elements = basisElements[system.dimensions as keyof typeof basisElements] || [] const rules = multiplicationRules[system.dimensions] const handleMultiply = (a: string, b: string) => { if (!rules || !rules[a] || !rules[a][b]) { setResult("undefined") return } setResult(rules[a][b]) } const handleElementClick = (symbol: string) => { if (!selectedA) { setSelectedA(symbol) } else if (!selectedB) { setSelectedB(symbol) handleMultiply(selectedA, symbol) } else { // Reset setSelectedA(symbol) setSelectedB(null) setResult(null) } } const reset = () => { setSelectedA(null) setSelectedB(null) setResult(null) } return (

Basis Element Multiplication

{(selectedA || selectedB) && ( )}

Click two basis elements to see their product. Discover non-commutative pairs where a×b ≠ b×a.

{elements.map((element) => ( ))}

{selectedA && (

{selectedA} {selectedB && ( <> × {selectedB} = {result} )} {!selectedB && × ?}

{selectedB && result && system.level >= 2 && (

CHECK COMMUTATIVITY

{selectedB} × {selectedA} ={" "} {rules && rules[selectedB] && rules[selectedB][selectedA] ? rules[selectedB][selectedA] : "undefined"} {rules && rules[selectedA] && rules[selectedB] && rules[selectedA][selectedB] !== rules[selectedB][selectedA] && ( NON-COMMUTATIVE )}

)}

)} {!selectedA && (

Select a basis element to begin

)} ) } [V0_FILE]typescriptreact:file="components/real-world-applications.tsx" "use client" import { Card } from "@/components/ui/card" import { Badge } from "@/components/ui/badge" import { Rocket, Cpu, Atom, Code, Zap } from "lucide-react" interface NumberSystem { name: string level: number } const applications = { 0: [ { icon: Zap, title: "Everyday Mathematics", description: "Counting, measurement, basic arithmetic", examples: ["Finance", "Engineering", "Physics"], }, ], 1: [ { icon: Zap, title: "Electrical Engineering", description: "AC circuit analysis using phasors", examples: ["Power systems", "Signal processing", "Control theory"], }, { icon: Atom, title: "Quantum Mechanics", description: "Wave functions and probability amplitudes", examples: ["Schrödinger equation", "Quantum computing", "Particle physics"], }, { icon: Code, title: "Signal Processing", description: "Fourier transforms and frequency analysis", examples: ["Audio processing", "Image compression", "Communications"], }, ], 2: [ { icon: Rocket, title: "3D Graphics & Gaming", description: "Rotation and orientation without gimbal lock", examples: ["Unity/Unreal Engine", "Computer graphics", "Animation"], }, { icon: Cpu, title: "Robotics & Aerospace", description: "Spacecraft attitude control and robot kinematics", examples: ["NASA missions", "Drone navigation", "Robot arms"], }, { icon: Code, title: "Computer Vision", description: "3D transformations and camera rotations", examples: ["AR/VR", "SLAM algorithms", "3D reconstruction"], }, ], 3: [ { icon: Atom, title: "Particle Physics", description: "Exceptional Lie groups and symmetries", examples: ["Standard Model", "String theory", "Grand unification"], }, { icon: Zap, title: "Theoretical Physics", description: "Supersymmetry and M-theory", examples: ["10D/11D physics", "Exceptional structures", "Quantum field theory"], }, ], 4: [ { icon: Atom, title: "Pure Mathematics", description: "Algebraic structures and zero divisors", examples: ["Abstract algebra", "Non-associative algebras", "Research"], }, ], 5: [ { icon: Code, title: "Theoretical Research", description: "Exploring mathematical boundaries", examples: ["Pure mathematics", "Algebraic theory", "Academic research"], }, ], } export function RealWorldApplications({ system }: { system: NumberSystem }) { const apps = applications[system.level as keyof typeof applications] || [] if (apps.length === 0) return null return (

Real-World Applications

{apps.map((app, index) => (

{app.title}

{app.description}

{app.examples.map((example) => ( {example} ))}

))}

) } [V0_FILE]typescriptreact:file="components/historical-timeline.tsx" "use client" import { Card } from "@/components/ui/card" import { Clock } from "lucide-react" interface System { name: string year: string discoverer: string level: number } export function HistoricalTimeline({ currentLevel, systems }: { currentLevel: number; systems: System[] }) { return (

Historical Timeline

{/* Timeline line */}

{systems.map((system, index) => (

{/* Timeline dot */}

{system.name} {system.level > 0 && {system.symbol}}

{system.discoverer}

{system.year}

{index === 2 && (

Hamilton carved the quaternion formula into Brougham Bridge in Dublin on October 16, 1843

)} {index === 3 && (

Discovered the same year as quaternions, showing the rapid development of hypercomplex numbers

)}

))}

) } [V0_FILE]typescriptreact:file="components/geometric-interpretation.tsx" "use client" import { Card } from "@/components/ui/card" import { useEffect, useRef } from "react" import { Compass } from "lucide-react" interface NumberSystem { name: string dimensions: number level: number } export function GeometricInterpretation({ system }: { system: NumberSystem }) { const canvasRef = useRef(null) useEffect(() => { const canvas = canvasRef.current if (!canvas) return const ctx = canvas.getContext("2d") if (!ctx) return const rect = canvas.getBoundingClientRect() canvas.width = rect.width * window.devicePixelRatio canvas.height = rect.height * window.devicePixelRatio ctx.scale(window.devicePixelRatio, window.devicePixelRatio) const width = rect.width const height = rect.height const centerX = width / 2 const centerY = height / 2 let animationFrame: number let angle = 0 const accentColor = "rgb(200, 100, 40)" const primaryColor = "rgb(30, 30, 30)" const secondaryColor = "rgb(150, 150, 150)" const animate = () => { ctx.clearRect(0, 0, width, height) if (system.dimensions === 2) { // Complex: rotation in 2D plane ctx.strokeStyle = secondaryColor ctx.lineWidth = 1 // Draw circle ctx.beginPath() ctx.arc(centerX, centerY, 80, 0, Math.PI * 2) ctx.stroke() // Draw rotating vector const x = Math.cos(angle) * 80 const y = Math.sin(angle) * 80 ctx.strokeStyle = accentColor ctx.lineWidth = 2 ctx.beginPath() ctx.moveTo(centerX, centerY) ctx.lineTo(centerX + x, centerY + y) ctx.stroke() // Draw angle arc ctx.strokeStyle = accentColor ctx.lineWidth = 1 ctx.beginPath() ctx.arc(centerX, centerY, 30, 0, angle) ctx.stroke() // Labels ctx.fillStyle = primaryColor ctx.font = "12px monospace" ctx.fillText("Rotation", 10, 20) ctx.fillText(`θ = ${(angle * (180 / Math.PI)).toFixed(0)}°`, 10, 40) } else if (system.dimensions === 4) { // Quaternions: 3D rotation visualization (projected to 2D) const drawSphere = () => { // Draw wireframe sphere ctx.strokeStyle = secondaryColor ctx.lineWidth = 0.5 // Latitude lines for (let lat = -60; lat <= 60; lat += 30) { ctx.beginPath() const r = Math.cos((lat * Math.PI) / 180) * 80 const y = Math.sin((lat * Math.PI) / 180) * 80 ctx.ellipse(centerX, centerY + y, r, r * 0.3, 0, 0, Math.PI * 2) ctx.stroke() } // Longitude lines for (let lon = 0; lon < 180; lon += 30) { ctx.beginPath() for (let t = 0; t <= Math.PI * 2; t += 0.1) { const x = Math.cos(t) * Math.cos((lon * Math.PI) / 180) * 80 const y = Math.sin(t) * 80 const z = Math.cos(t) * Math.sin((lon * Math.PI) / 180) * 80 const projX = centerX + x * Math.cos(angle) - z * Math.sin(angle) const projY = centerY + y if (t === 0) { ctx.moveTo(projX, projY) } else { ctx.lineTo(projX, projY) } } ctx.stroke() } } drawSphere() // Draw rotation axis ctx.strokeStyle = accentColor ctx.lineWidth = 2 ctx.beginPath() ctx.moveTo(centerX, centerY - 100) ctx.lineTo(centerX, centerY + 100) ctx.stroke() // Labels ctx.fillStyle = primaryColor ctx.font = "12px monospace" ctx.fillText("3D Rotation", 10, 20) ctx.fillText("No Gimbal Lock", 10, 40) } angle += 0.02 animationFrame = requestAnimationFrame(animate) } animate() return () => { cancelAnimationFrame(animationFrame) } }, [system]) if (system.dimensions > 4) return null return (

Geometric Interpretation

{system.dimensions === 2 && "Complex multiplication = rotation in the plane. Multiplying by e^(iθ) rotates by angle θ."} {system.dimensions === 4 && "Quaternions represent 3D rotations elegantly. Used in aerospace and game engines to avoid gimbal lock."}

) } [V0_FILE]typescriptreact:file="components/keyboard-shortcuts.tsx" "use client" import { Card } from "@/components/ui/card" import { Button } from "@/components/ui/button" import { X } from "lucide-react" export function KeyboardShortcuts({ onClose }: { onClose: () => void }) { return (

Keyboard Shortcuts

Jump to level 1-6

Next level →

Previous level ←

Toggle shortcuts ?

Linear-style keyboard navigation for power users

) }

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