Press n or j to go to the next uncovered block, b, p or k for the previous block.
| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 | export interface MathConstant { id: string symbol: string name: string value: number /** Decimal places needed to distinguish this constant from nearby integers/fractions */ revealPrecision: number description: string ttsExplanation: string category: 'math' | 'physics' | 'fun' /** MathML markup for proper mathematical typesetting (rendered as DOM overlay) */ mathml: string /** Prompt suffix for generating a whimsical metaphor illustration (flat vector style) */ metaphorPrompt: string /** Prompt suffix for generating a classical geometric math illustration */ mathPrompt: string /** Path to metaphor illustration (populated after generation) */ metaphorImage?: string /** Path to math-style illustration (populated after generation) */ mathImage?: string } /** Shared prompt prefix for metaphor-style illustrations */ export const METAPHOR_PROMPT_PREFIX = 'Clean flat vector illustration, bold colors, simple geometric shapes, no text, no labels, square 1:1 composition, white background.' /** Shared prompt prefix for math-style illustrations (classical geometric, Byrne's Euclid inspired) */ export const MATH_PROMPT_PREFIX = "Classical geometric diagram in the style of Oliver Byrne's illustrated Euclid. Clean precise constructions with bold flat primary colors (red, blue, gold, black) on warm cream parchment background. No text, no labels, no numbers, no letters. Square 1:1 composition. Mathematically accurate and instructional. Simple clean shapes, visible compass and straightedge construction marks where appropriate." /** Theme-specific prompt modifiers appended during generation */ export const THEME_MODIFIERS = { light: { metaphor: 'IMPORTANT: Use a clean white (#ffffff) background. Ensure all colors are vibrant and visible against the light background. Same subject and composition.', math: 'IMPORTANT: Use a clean white (#ffffff) background instead of cream parchment. Keep the same bold primary colors (red, blue, gold) and clean geometric construction style. Same composition.', }, dark: { metaphor: 'IMPORTANT: Use a dark navy background (#1e1e28). Use bright, saturated colors with high contrast against the dark background. Same subject and composition.', math: 'IMPORTANT: Use a dark aged-vellum background (#1a1a2e). Use bright vivid primary colors (vermillion, bright blue, bright gold) with high contrast against the dark ground. Keep the same clean geometric construction style. Same composition.', }, } as const /** Euler-Mascheroni constant */ const EULER_MASCHERONI = 0.5772156649 /** Feigenbaum constant (first) */ const FEIGENBAUM_DELTA = 4.6692016091 /** Fine-structure constant */ const FINE_STRUCTURE = 1 / 137.035999084 export const MATH_CONSTANTS: MathConstant[] = [ { id: 'pi', symbol: 'π', name: 'Pi', value: Math.PI, revealPrecision: 2, description: "Every circle has a diameter — that's the distance all the way across, from one side to the other. If you walk straight across a round pond, that's one diameter. Walking all the way around the edge takes a little more than three of those. No matter how big the pond, the walk around is always exactly pi times the walk across. Pi ties every circle's trip around to its distance across.", ttsExplanation: "This is pi! Every circle has a diameter — that's the distance all the way across, from one side to the other. If you wrap a string around any circle, the string will be exactly pi times as long as that distance across. Watch: this circle is one diameter wide, and when it rolls one full turn, it covers exactly pi on the number line. Pi starts with 3.14159 and goes on forever without repeating.", category: 'math', mathml: '<math><mi>π</mi></math>', metaphorPrompt: 'A tiny person walking straight across a perfectly round pond in one step, then the same person walking all the way around the edge of the pond taking a little more than three steps. Show the diameter path and the circumference path with different colors.', mathPrompt: 'A circle with its diameter drawn as a bold red line. Below the circle, the circumference unrolled into a straight blue line. The diameter length is marked off along the unrolled line three times with a visible remainder, showing the ratio is slightly more than 3. Compass arcs visible at construction points.', metaphorImage: '/images/constants/pi-metaphor.png', mathImage: '/images/constants/pi-math.png', }, { id: 'e', symbol: 'e', name: "Euler's Number", value: Math.E, revealPrecision: 2, description: "Imagine a magic vine where every tiny new leaf immediately starts growing its own baby leaves the very moment it appears. Because every new speck of green adds to the growth instantly without waiting for a turn, the plant pushes to become almost three times as big as it started. This specific size is nature's limit for how much something can expand when every single part of it is growing continuously. It is the universe's secret number for perfectly smooth, non-stop change.", ttsExplanation: "This is e, Euler's number! It's about 2.718. If you put one dollar in a bank that doubles your money continuously, after one year you'd have e dollars. It shows up everywhere in nature — in spirals, in how things grow, and even in music.", category: 'math', mathml: '<math><mi>e</mi></math>', metaphorPrompt: 'A magic vine growing from a single stem, with each tiny leaf sprouting its own baby leaves instantly. The vine is lush and fractal-like, almost tripling from its original size. Warm greens with golden highlights.', mathPrompt: 'The hyperbola y=1/x drawn as a clean black curve. The region under the curve from x=1 to x=e shaded in translucent blue, representing area exactly equal to 1. A red vertical line at x=e. Thin rectangular Riemann-sum slices visible within the shaded region, showing how the area is measured.', metaphorImage: '/images/constants/e-metaphor.png', mathImage: '/images/constants/e-math.png', }, { id: 'phi', symbol: 'φ', name: 'Golden Ratio', value: (1 + Math.sqrt(5)) / 2, revealPrecision: 2, description: 'The golden ratio, found in art, architecture, and nature.', ttsExplanation: "This is phi, the golden ratio! It's about 1.618. Sunflower seeds spiral in patterns based on this number. Artists and architects have used it for thousands of years because shapes built with it look naturally beautiful.", category: 'math', mathml: '<math><mi>φ</mi></math>', metaphorPrompt: 'A sunflower viewed from above with spiral seed patterns radiating from the center. Golden petals arranged in a fibonacci spiral. Warm yellows and oranges.', mathPrompt: 'A golden rectangle subdivided recursively: each subdivision peels off a red square, leaving a smaller blue golden rectangle, repeated four or five times inward. A smooth golden spiral arc traced through the corners of the squares in black. Construction lines showing the proportional division at each step.', metaphorImage: '/images/constants/phi-metaphor.png', mathImage: '/images/constants/phi-math.png', }, { id: 'sqrt2', symbol: '√2', name: 'Square Root of 2', value: Math.SQRT2, revealPrecision: 2, description: 'If you have a square garden tile, walking straight through the middle is a secret shortcut compared to walking along the sides. This path is longer than one side but shorter than two, and it is a special distance that regular counting numbers can never measure perfectly. It teaches us that cutting diagonally across a shape creates a new kind of number that goes on forever. This number is the hidden key that turns a square into two perfect triangles.', ttsExplanation: "This is the square root of 2! It's about 1.414. If you draw a perfect square where each side is 1, the diagonal corner-to-corner distance is exactly this number. Ancient Greeks discovered it can't be written as a simple fraction — that blew their minds!", category: 'math', mathml: '<math><msqrt><mn>2</mn></msqrt></math>', metaphorPrompt: 'A square garden tile viewed from above with a glowing diagonal line cutting across it as a shortcut. Two tiny people — one walking along two sides, one cutting diagonally. The diagonal splits the square into two perfect triangles.', mathPrompt: 'A blue unit square with its diagonal drawn as a bold red line. On each side of the square, a small gold square is constructed outward (area 1 each). On the diagonal, a larger red square is constructed (area 2), visually showing that the diagonal square equals the two side squares combined. Right-angle marks at the corners. Compass arc at the diagonal endpoint.', metaphorImage: '/images/constants/sqrt2-metaphor.png', mathImage: '/images/constants/sqrt2-math.png', }, { id: 'sqrt3', symbol: '√3', name: 'Square Root of 3', value: Math.sqrt(3), revealPrecision: 2, description: 'Imagine you are inside a square box and you stretch a string from the bottom corner all the way through the air to the top opposite corner. This distance is longer than the floor and longer than the wall; it is the longest reach possible inside that space. It shows us exactly how much "deeper" the world is when you move from flat drawings to solid blocks. This number defines the structural shape of honeycomb cells and how bubbles stack together in a pile.', ttsExplanation: "This is the square root of 3! It's about 1.732. If you make a perfect triangle where all three sides are the same length of 2, the height from base to tip is exactly this number.", category: 'math', mathml: '<math><msqrt><mn>3</mn></msqrt></math>', metaphorPrompt: 'A transparent cube/box viewed in 3D perspective with a glowing string stretched from the bottom-front-left corner to the top-back-right corner — the space diagonal. Honeycomb cells and stacked bubbles nearby.', mathPrompt: 'A large equilateral triangle with side length 2, filled in light blue. The altitude drawn from the apex perpendicular to the base as a bold red line, splitting the triangle into two right triangles. The base halves marked in gold. Compass arcs visible at the apex and base vertices showing the construction of the equilateral triangle.', metaphorImage: '/images/constants/sqrt3-metaphor.png', mathImage: '/images/constants/sqrt3-math.png', }, { id: 'tau', symbol: 'τ', name: 'Tau', value: 2 * Math.PI, revealPrecision: 2, description: "Every circle has a radius — that's the distance from the center to the edge, which is half the distance across. One full trip around any circle is exactly tau times the radius. Pi uses the whole distance across, but tau uses just the half, so tau is exactly 2 times pi. Tau is the number of one complete turn.", ttsExplanation: "This is tau! Every circle has a radius — that's the distance from the center to the edge, which is half the distance across. If you wrap a string around any circle, the string will be exactly tau times as long as the radius. Watch: this circle has a radius of one, so it's two wide. When it rolls one full turn, it covers exactly tau on the number line. The pi circle was smaller because pi measures against the whole distance across. Tau measures against just the half, so tau is exactly 2 times pi!", category: 'math', mathml: '<math><mi>τ</mi></math>', metaphorPrompt: 'A spinning wheel or merry-go-round completing one full revolution, with a trail showing the complete circular path. A stick figure riding it all the way around back to the start. Bright, dynamic, sense of motion.', mathPrompt: 'A circle with its radius drawn in red. The full circumference traced as a bold blue arc going all the way around (one complete turn). The radius length is marked off along the circumference six times with tick marks, showing it fits slightly more than six times. A faint dashed semicircle in gold contrasts the full turn (tau) against the half turn (pi).', metaphorImage: '/images/constants/tau-metaphor.png', mathImage: '/images/constants/tau-math.png', }, { id: 'ln2', symbol: 'ln 2', name: 'Natural Log of 2', value: Math.LN2, revealPrecision: 2, description: 'If our magic vine grows smoothly, this number tells you the exact moment on the clock when the vine becomes double its size. It acts as a bridge, converting the smooth, continuous growth of nature into the simple idea of doubling. It marks the specific time when "one" leaf turns into "two" leaves in a world of perfect, non-stop growth. This number is the universal "waiting time" required for nature to reach the next level.', ttsExplanation: "This is the natural logarithm of 2! It's about 0.693. It tells you how long it takes for something to double when it's growing continuously. If bacteria double every hour, this number connects the doubling to the growth rate.", category: 'math', mathml: '<math><mrow><mi>ln</mi><mo>\u2061</mo><mn>2</mn></mrow></math>', metaphorPrompt: 'The same magic vine from e, but now with a clock or hourglass beside it showing the exact moment when one leaf becomes two leaves. The vine at the halfway-point of its growth journey. A bridge connecting "1" to "2".', mathPrompt: 'The hyperbola y=1/x drawn as a clean black curve. The area under the curve from x=1 to x=2 shaded in translucent red. Vertical lines at x=1 (blue) and x=2 (blue) bounding the shaded region. The shaded area represents ln(2). Thin horizontal reference line at y=1 in light gold. Clean axis lines.', metaphorImage: '/images/constants/ln2-metaphor.png', mathImage: '/images/constants/ln2-math.png', }, { id: 'gamma', symbol: 'γ', name: 'Euler-Mascheroni Constant', value: EULER_MASCHERONI, revealPrecision: 2, description: 'If you build a smooth slide out of square blocks, a tiny jagged edge will always stick out. This number is the size of that stubborn gap that refuses to disappear, no matter how small you make the blocks. It is the permanent "error code" nature gets when it tries to turn smooth flows into counting numbers.', ttsExplanation: "This is gamma, the Euler-Mascheroni constant! It's about 0.577. If you add up one half plus one third plus one quarter and keep going, it grows like a logarithm but always a little bit more. That little bit extra is gamma. Nobody knows if it's rational or irrational!", category: 'math', mathml: '<math><mi>γ</mi></math>', metaphorPrompt: 'A smooth curved slide built from tiny square blocks, with a small jagged gap visible between the smooth curve and the stepped blocks. The gap is highlighted in a contrasting color, stubbornly present no matter how small the blocks get.', mathPrompt: 'A smooth ln(x) curve drawn in black. Above it, a descending staircase of rectangles (the harmonic series partial sums: heights 1, 1/2, 1/3, 1/4...) drawn with blue outlines. The persistent gap between the top of the staircase and the smooth curve is shaded in red — this strip represents gamma. The gap stays visually constant even as both curve and staircase rise together.', metaphorImage: '/images/constants/gamma-metaphor.png', mathImage: '/images/constants/gamma-math.png', }, { id: 'feigenbaum', symbol: 'δ', name: 'Feigenbaum Constant', value: FEIGENBAUM_DELTA, revealPrecision: 2, description: 'When a dripping tap speeds up, the rhythm splits into two beats, then four, rushing toward a crash. This number is the stopwatch that tells the drops exactly when to split again as they zoom into chaos. It is the universal speed limit for how fast simple order turns into a storm.', ttsExplanation: "This is delta, the Feigenbaum constant! It's about 4.669. It appears in chaos theory — when simple rules create complex patterns, this number keeps showing up. It's the same whether you're looking at dripping faucets, animal populations, or electronic circuits.", category: 'math', mathml: '<math><mi>δ</mi></math>', metaphorPrompt: 'A dripping faucet where drops split into two streams, then four, then eight, rushing toward chaotic splashing. The rhythm of splitting visualized as branching paths. Blues and silvers.', mathPrompt: 'A clean bifurcation diagram showing the period-doubling cascade. A single branch at left splits into two (red), then four (blue), then eight (gold), with the splits accelerating toward a chaotic region on the right. Brackets between successive fork points show the shrinking intervals whose ratio converges to delta. Simple tree-like branching, clean lines, no scatter.', metaphorImage: '/images/constants/feigenbaum-metaphor.png', mathImage: '/images/constants/feigenbaum-math.png', }, { id: 'alpha', symbol: 'α', name: 'Fine-Structure Constant', value: FINE_STRUCTURE, revealPrecision: 4, description: 'This number tells us how "sticky" light is, acting like the strength of the velcro holding atoms together. If this number were even a tiny bit different, the sun would not shine and the solid ground would crumble apart like dust. It is the specific strength setting of the universe that allows light to grab onto matter. It acts as the cosmic volume knob for electricity.', ttsExplanation: "This is alpha, the fine-structure constant! It's roughly 1 divided by 137. It controls how strongly light interacts with matter. If this number were even slightly different, atoms wouldn't work, chemistry wouldn't exist, and neither would we! Physicist Richard Feynman called it one of the greatest mysteries in physics.", category: 'physics', mathml: '<math><mi>α</mi></math>', metaphorPrompt: 'Beams of light sticking to atoms like velcro, with a cosmic volume/dial knob controlling how strongly light grabs matter. A sun shining, solid ground stable. Cosmic purple and gold tones.', mathPrompt: 'A clean Feynman diagram: two straight red lines (electrons) meeting a wavy blue line (photon) at a single vertex point. The vertex drawn as a bold black dot. The diagram is simple and centered — just the basic QED vertex showing how light couples to matter. Clean straight lines and a smooth sinusoidal photon line.', metaphorImage: '/images/constants/alpha-metaphor.png', mathImage: '/images/constants/alpha-math.png', }, { id: 'ramanujan', symbol: '−1⁄12', name: 'Ramanujan Summation', value: -1 / 12, revealPrecision: 2, description: "If you trap infinite waves between two walls, they don't push them apart\u2014they suck them together. The math shows that adding up all that pushing creates a tiny, negative weight pulling backward. It is the secret suction holding empty space together.", ttsExplanation: 'This is negative one twelfth! The brilliant mathematician Ramanujan showed that if you add up 1 plus 2 plus 3 plus 4 and keep going forever, in a special mathematical sense, you get negative one twelfth. It sounds impossible, but physicists actually use this result in string theory!', category: 'fun', mathml: '<math><mrow><mo>−</mo><mfrac><mn>1</mn><mn>12</mn></mfrac></mrow></math>', metaphorPrompt: 'Two parallel walls with invisible waves trapped between them, the walls being sucked together instead of pushed apart. A tiny vacuum force pulling inward. Dark space between the plates with subtle wave patterns.', mathPrompt: 'Two thick parallel vertical red lines (conducting plates). Between them, three or four standing wave patterns drawn in blue (discrete modes: 1, 2, 3 half-wavelengths that fit between the plates). Outside the plates, denser wave patterns in gold (continuous spectrum). Bold black arrows on each plate pointing inward, showing the net attractive Casimir force.', metaphorImage: '/images/constants/ramanujan-metaphor.png', mathImage: '/images/constants/ramanujan-math.png', }, ] |