L2: school
Clocks in the rock
BIOOne exponential, read three ways. Forwards it tells you how much is left; backwards it tells you how old something is; and applied to DNA rather than to nuclei it needs a rock to set it by.
Bio, Geo, Chem
Biology looks like a subject made of exceptions until you start counting. Dose, inheritance, population and reaction rate follow rules tight enough to predict, and predicting them is what these tools do.
Newest in this category
- Great Circle Routes The shortest flight from London to Los Angeles leaves heading north-west, crosses southern Greenland and lands heading south-west: 8,760.1 km, against 9,732.5 km on one steady compass heading. Drag two places on a globe and watch the heading change.
- Ground Temperature: Seasons Underground The surface of the ground warms and cools with the year and the day, and the swing sinks into the soil, shrinking and arriving late. See why the shrink and the delay are one number, where the year turns upside down, and why the night barely reaches half a metre down.
- Blood Alcohol: How the Body Burns It Off Four pints in an 80 kg man peak at 1.26 g/L and fall below 0.5 g/L at 04:05 at the average rate, and anywhere from 02:05 to 08:05 across the healthy range. Widmark's equation with Watson's body water.
Chapters
Short sequences rather than single tools. Each step hands the next one something specific, so they are worth doing in order.
L2: school
Priors, and where they come from
STATNobody objects to the arithmetic in Bayes' theorem. They object to the prior — where the first number came from, and who chose it. These three tools take that question in the order it gets harder: a prior you can look up, a prior the apparatus hands you, and a prior nobody has.
L2: school
Geometry on a curved surface
GEOOn a curved surface the rules proved on paper stop holding, and the amount by which each one fails measures the curvature from inside. A straight route changes its compass heading, an arrow carried without turning comes home turned, a ring comes out the wrong length, and in the hyperbolic plane no triangle can hold more than π.
L2: school
Why a reaction goes
CHEMThe heat a reaction gives out does not decide whether it runs; ice melts by taking heat in. These four pages build the bookkeeping that does decide: enthalpy, which adds like equations; standard free energy, which is an equilibrium constant in other units; the difference between moving K and moving Q; and a battery that reads that difference as a voltage.
L2: school
Mixtures and solutions
CHEMA gas in a mixture acts at its own partial pressure, which sets a diver's depth limit. A liquid boils where its vapour pressure meets the pressure above it. Two liquids that mix badly stop a still at about 95.6% ethanol by mass, dissolved particles make one kelvin of frost worth thirteen bar of osmotic pressure, and a salt dissolves only until its ions reach a fixed product.
L2: school
Growth with a limit
BIOEvery population here is held back by something, and the something gets less obliging at each step: first the crowding it causes itself, then a fixed catch taken off the top, then a predator that is a population in its own right, and finally running out of people left to infect.
L3: undergraduate
The heat, wave and Laplace equations
PHYThree equations built on one operator, the Laplacian, that differ only in what time does. Heat forgets, the sharpest detail first. A wave carries a disturbance away, and on a sheet never quite finishes arriving. Laplace's equation has no time in it at all, and it is where heat forgets its way to: every point the average of its neighbours.
L3: undergraduate
How a weather forecast is made
GEOA forecast is a model run forward from a starting state nobody knows exactly, so forecasters run it many times and let the disagreement say how far to trust it. The second page opens the model and finds a way to fail that has nothing to do with chaos. The third goes back to Lorenz's equations of 1963, where the run's position is soon lost and the shape it draws never is.
L4: specialist
Computing with molecules
BIOFive tools that build one argument: DNA is a real storage medium and a real computer, and both run into limits you can calculate rather than guess at. Each step hands the next a specific number.
Problems solved in full
- The discrete model with 80 generations
- The 3:1 ratio in a real family of four
- A Carbon-14 sample sitting for five 5730 years half-life
- A wood sample that retains 62% of its original carbon-14
- A first order reaction at 298 K with E a = 50 kJ/mol
- An infection that has R₀ = 3 spreading past the immune fraction
- Weighing a protein with a water column
- Counting Avogadro's number with a silicon sphere
- A column that makes both ends pure
- Where limestone really turns, from the hot tables
- How much heat a resting body makes from its oxygen
- Naming a compound from its cluster
- Choosing a nitrox for a 30 m dive, and what it buys
- Benzene's enthalpy of formation, from one combustion
- Silver chloride's Ksp from thermochemistry alone
- A pressure canner at Everest base camp
- A reading of 2.49, and a concentration worth ±107
- One pot, one day, and where the 36,120 kWh went
- Forty animals out there in a park that expects 12
- The 4 km model takes 40 minutes. What does 500 m cost?
- Will it freeze on Thursday?
- The 20-city tour that fits in half a gram, and the 39-city one that does not fit on Earth
- What an exabyte weighs in DNA, and what the constraints add to the bill
- A phone battery that lasts 10,000 hours, and the same battery lasting 384,500 years
What you find out here
- The buffer equation only works in the buffer region Acid-Base Titration
- A dry crossing is free; the warming is paid for entirely by the rain Föhn Winds & the Rain Shadow
- A count that stops at the expected number can never report a surplus The Park Census
- A century of argument over one decimal place, worth a factor of two Allometric Scaling Explorer
- The first unit of absorbance costs 90% of the light. The second costs 9%. The Beer–Lambert Law
- Two indices, and they disagree on purpose Biodiversity Index Calculator
Articles written from these tools
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A fire is measured by the oxygen it takes
Per kilogram of fuel, the things we burn differ more than threefold. Per kilogram of oxygen, most of them land within 5% of one number, because the heat is set by what the oxygen becomes, and in a clean flame it becomes one of two things.
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The 475-year earthquake is not geology
Two odd numbers are stamped on seismic design maps worldwide. Neither was measured. Both are a tolerated failure probability run through one line of algebra — and the same line explains why a 100-year flood is a coin-flip over a lifetime.
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Thirty-nine cities need more DNA than the Earth
DNA computing promised to beat exponential search with molecular parallelism. The counting that kills it is the argument Levinthal used to prove that proteins cannot be searching for their shape — and what survives the argument is not speed at all.
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Kill both species at the same rate and you get more of the pest
The predator–prey model has an exact long-run average, and it sits in a strange place: raise the death rate of both species equally and the average prey population goes up. Volterra found it in fish-market ledgers in 1926.
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Below one it dies out. Above one it runs away. There is nothing in between.
The same threshold governs epidemics, nuclear chain reactions, PCR and surnames: if each unit produces on average fewer than one successor, extinction is certain. The knife edge is exactly at 1, and the mathematics was first worked out on Victorian family names.
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The herd immunity threshold is 66.7%, so why does the epidemic infect 94.1%?
The threshold is where an outbreak stops growing, which is nowhere near where it stops. It keeps infecting people well past that point, and the size of that overshoot is the argument for arriving at the threshold by vaccination rather than by illness.
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Fifty generations of perfect selection halves a recessive gene, and that is the best case
Suppose a recessive condition were lethal in every case, in every generation, with no exceptions. After fifty generations the allele has gone from 2% to 1%. The arithmetic explaining why was published in 1908.
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A radioactive atom does not know how old it is
An atom that has already survived a million years is exactly as likely to decay in the next second as one created moments ago. Exponential decay is the only continuous law with that property, and it is why half-life is a meaningful number at all.
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A magnitude 7 is not slightly worse than a magnitude 6
It releases about 32 times the energy. Logarithmic scales compress catastrophes into numbers that look adjacent, which is exactly the problem with reporting them.
Every tool on this shelf
Acid-Base Titration
A 0.1 M weak acid at pKa 4.76 sits at pH 2.88 where a strong acid gives 1.00, and both still need the same 25.00 mL of base. Strength changes the shape of the curve, never where it ends.
Allometric Scaling Explorer
Metabolism rises as mass^0.75 where skin heat loss predicts 0.67. Across a 10,000-fold mass range those two exponents land a factor of 2.09 apart, which is why one decimal was argued over for a century.
Bayesian Biosignature Update
Apply Bayes' theorem to a biosignature detection: set prior life probability, instrument sensitivity, and false-positive rate to see the true posterior.
Biodiversity Index Calculator
Simpson's dominance λ = Σpᵢ² is the sum of squared shares regulators use to spot a monopoly. Put 80 of 100 individuals in one species and richness stays at 5 while evenness falls to 0.48.
Blood Alcohol: How the Body Burns It Off
Four pints in an 80 kg man peak at 1.26 g/L and fall below 0.5 g/L at 04:05 at the average rate, and anywhere from 02:05 to 08:05 across the healthy range. Widmark's equation with Watson's body water.
Body Mass Index (BMI) Calculator
Scale a body up without changing its shape and BMI still moves. The 175 cm, 70 kg reference reads overweight at 191.4 cm and underweight at 141.6 cm, with not a gram of extra fat anywhere.
Chemical Equilibrium & Le Chatelier Lab
Halve the volume on H₂ + I₂ ⇌ 2HI and conversion sits at 78.0% regardless. Squeeze ammonia from 2 L to 0.25 L and it climbs 36.7% to 69.3%. The exponent on volume is −Δn, and nothing else.
Colligative Properties: Freezing, Boiling & Osmosis
The solution that freezes one kelvin lower holds back 13.29 bar of osmotic pressure at 25 °C. All four colligative effects count dissolved particles and nothing else, and the freezing-point constant is water's latent heat of fusion in disguise.
Country Shape & Flag Challenge
Test your geographical and mathematical skills with country shapes, flag symmetries, and capital cities.
Crystal Packing: How Full Is a Metal?
Face-centred cubic and hexagonal packing both fill 74.05% of space, and nothing fills more. Pick a metal and see its packing fraction, atomic radius and density from its lattice parameter, why iron gets denser when heated through 912 °C, and why zinc packs worse than iron.
Dilution Calculator
Calculate stock and solvent volumes for a dilution using C₁V₁ = C₂V₂.
DNA Codon Translator
Translate DNA sequences into amino acids, visualise codons, and explore the genetic code
DNA Computing Scale
Twenty cities of travelling salesman fit in half a gram of DNA; thirty-nine weigh 1.46 Earths. The encoding is only a constant factor, so the wall sits at 39 cities whether you spend 20 bases each or 200.
DNA Data Storage Capacity
The homopolymer rule everyone names costs 0.9% of the two bits a base can carry. Primers and an index take 53 of every 152 bases, 34.9%, because a strand in a tube has no address.
DNA Strand-displacement Gate
A DNA gate takes 38 minutes to reach 90% output, so a day of chemistry buys a circuit 37.5 layers deep. Molecular logic wins on energy per event by about 10⁴ and loses on time by 10¹¹.
Earthquake Magnitude to Energy
One magnitude point is 31.62× the energy and 10× the ground motion; both are correct and they answer different questions. It takes 1000 magnitude-6 quakes to match a single magnitude-8.
Energy per Operation
Erasing a bit at body temperature costs at least 2.967 × 10⁻²¹ J, and a silicon logic event costs 337,077 times that. The same 10 Wh battery lasts 10,000 hours in silicon and 384,500 years at the floor.
Ensemble Weather Forecasting
Fifty forecasts of one toy atmosphere. At sixteen days every one of the fifty is worse than their own average, the spread predicts the error at r = 0.80 over sixty launches, and the average itself is a map of weather that could not happen.
Faraday's Laws of Electrolysis
Charge divided by the Faraday constant is moles of electrons, and the metal follows: m = (Q/F)(M/z). Work out a plating thickness, or the 2,979,980 ampere-hours behind a tonne of aluminium.
Forest Fire Simulator
Walk the density from 0.57 to 0.62 and the burn jumps from 7.6% of the forest to 87%, with no setting in between. At 0.59 the same forest burns 7.6%, 58.8% or 19.3% depending only on which map you drew.
Fractional Distillation & the Ethanol–Water Azeotrope
Raoult's law says twelve stages turn a 10 mol% ethanol wash into 99.954 mol% ethanol. The real mixture never passes its azeotrope, measured at 95.57% ethanol by mass and 78.17 °C, however many stages you add.
Föhn Winds & the Rain Shadow
Lift an airstream over a ridge and back down: cloud base, how much rains out, and the lee-side warming that follows.
Gibbs Free Energy & Spontaneity
Limestone absorbs 178.8 kJ/mol to decompose and does it anyway above 843.1 °C, because only the sign of ΔS decides whether heat can reverse a reaction. At 25 °C every 5.708 kJ/mol of ΔG is one factor of ten in K.
Great Circle Routes
The shortest flight from London to Los Angeles leaves heading north-west, crosses southern Greenland and lands heading south-west: 8,760.1 km, against 9,732.5 km on one steady compass heading. Drag two places on a globe and watch the heading change.
Ground Temperature: Seasons Underground
The surface of the ground warms and cools with the year and the day, and the swing sinks into the soil, shrinking and arriving late. See why the shrink and the delay are one number, where the year turns upside down, and why the night barely reaches half a metre down.
Hardy-Weinberg Equilibrium
At q = 0.01 one person in 10,000 is affected and one in 50 is a carrier, 198 carriers per case. Removing every affected person from breeding moves q only to 0.0091 in ten generations.
Hazard Return Period & Lifetime Risk
A 100-year flood is a 26.03% risk over a 30-year mortgage, and one full return period gives 63.40%, never certainty. The 475-year design earthquake is 10% in 50 years, inverted.
Heat of Combustion, per Fuel and per Oxygen
Methane gives 50.03 MJ per kilogram and glucose 14.10, yet per kilogram of oxygen both give about 13. Compare sixteen fuels by heat per mole, per kg of fuel and per kg or litre of oxygen, and see why fat beats sugar per gram and loses per breath.
Hess's Law & Reaction Enthalpy
Carbon never burns to pure CO, yet its heat of formation is known to 0.1 kJ/mol, by subtraction. Add measured reactions like equations, and benzene comes out 150.4 kJ/mol more stable than the ring of double bonds it is drawn as.
Interactive Periodic Table Explorer
Explore all 118 elements with a live periodic-trend heatmap, a states-of-matter temperature simulator, electron configurations, and Bohr diagrams.
Isostasy & Mountain Roots
Every kilometre of mountain floats on 5.6 km of root, putting 84.85% of the block underground against an iceberg's 89.46%. Airy's 49.55 km root under Everest is where the model starts to fail.
Isotope Patterns in Mass Spectrometry
One chlorine puts the M+2 peak at 32.00% of M, one bromine at 97.28%. Type any formula and see its whole isotope cluster: why dibromomethane's tallest peak is M+2, and why only 2.90% of insulin molecules have insulin's monoisotopic mass.
Lewis Structure & VSEPR Molecular Lab
Build molecular structures in 2D and explore their 3D VSEPR geometry, orbital lobes, hybridization, and bond angles.
Limiting Reagent & Stoichiometry Lab
6 g of hydrogen supplies 2.9762 mol against nitrogen's 0.9995 and still runs out first: divided by the coefficients it is 0.9921 to 0.9995, and the smaller number is the one that limits.
Logistic Growth with Harvesting
Simulate logistic population growth under constant-rate harvesting - from stable equilibrium to population collapse.
Map Projection Distortion Lab
Drag latitude to compare how Mercator and equal-area maps trade area for shape.
Michaelis-Menten Kinetics
Half speed costs one Km of substrate, 90% costs nine, 99% costs ninety-nine, so Vmax is fitted and never measured. A competitive inhibitor moves Km from 5 to 13.33 and leaves Vmax at 100.
Molecular Clock Calculator
Thirteen aligned bases with one mismatch date a split at 8.1 million years. Add a single mismatch and the date becomes 17.2 million, which is what a molecular clock quoted without a width is worth.
Nernst Cell Voltage & Electrochemistry Lab
Concentration barely moves a cell: at 25 °C with n = 2, a millionfold shift in Q costs under 0.18 V. Temperature moves the slope instead, from 59.16 mV per decade to 66.10 mV at 60 °C.
Oxygen–Haemoglobin Dissociation Curve
Hill-equation saturation, the Bohr shift, arterial oxygen content and how much oxygen a litre of blood actually delivers.
Partial Pressure (Dalton's Law): Diving & Altitude
Air reaches 1.4 bar of oxygen at 56.43 m of seawater, nitrox 32 at 33.44 m and pure oxygen at 3.85 m. Dalton's law, and why Everest's summit gives the lungs less than a third of sea level's oxygen.
PCR Amplification Simulator
Thirty perfect cycles turn one molecule into 1.07 × 10⁹. Drop efficiency by 5% and you lose a factor of 2.14, more than half the yield, because the shortfall is multiplied thirty times rather than added once.
pH / Acid-Base Calculator
Diluting a 250 mL glass of stomach acid from pH 1.5 to coffee's pH 5.0 takes 791 litres of water, and pH 7 is unreachable: a strong acid stalls at 6.79 where water's own ions take over.
Plate Motion Rate Converter
Convert tectonic plate displacement over geological time into a motion rate, and project future distance.
Population Growth & Chaos
Logistic growth in continuous and discrete time - from stable equilibrium to chaos
Predator-Prey Dynamics
Lotka-Volterra population dynamics simulator.
Protein Folding: Levinthal's Paradox
A 21-residue chain searches its whole conformational space in a millisecond, fifty residues take 2,300 years and a hundred take 10²⁷ years. The paradox rules out the algorithm, not the folding.
Punnett Square Calculator
Monohybrid and dihybrid crosses with genotypic and phenotypic ratios
Radioactive Decay Calculator
Five half-lives of carbon-14 is 28,650 years and 3.125% left. From a thousand nuclei that is 31.25, and a quarter of a nucleus does not exist: the count is Poisson and scatters by about ±18%.
Radiometric Dating Calculator
Wood holding 62% of its carbon-14 dates to 3.952 kyr, and a 1% error in that fraction becomes 2.1% in the age.
Reaction Kinetics Calculator
A 60 kJ/mol reaction moved from a 4 °C fridge to a 20 °C room speeds up 4.1 times on 16 K. Q10 belongs to a temperature, not a reaction: the same one gives 2.13 in the fridge and 1.52 in boiling water.
SIR Epidemic Model
Simulate disease spread with SIR/SEIR models - R0, herd immunity, and epidemic curves
Solubility Product (Ksp) & the Common-Ion Effect
Silver chromate's Ksp is 158 times smaller than silver chloride's, and it dissolves 4.92 times more. Six of these eleven salts rank differently by Ksp than by solubility, and the Mohr titration works because of it.
The Beer–Lambert Law
A = εcℓ is one multiplication, and absorbance is a base-ten logarithm, so the first unit of absorbance costs 90% of the beam and the second costs 9% more. The reading a spectrophotometer measures most precisely sits at A = 0.4343, and going either side of it costs accuracy.
The Park Census
A wildlife park counts its animals and the total comes out exactly right. Two statistical arguments from the first Jurassic Park novel, both of them real: why a search that stops at the expected number can never report a surplus, and how the shape of a size distribution gives away breeding.
Vapour Pressure & Boiling Point (Clausius–Clapeyron)
Water boils at 71.8 °C on Everest's summit and 115.8 °C in a canner at 10.5 psi. One latent heat predicts both to within a degree, and the steam tables show exactly where that stops being true.
Weather on a Grid
The wind crosses 1 km in one step of 100 s. Make the cells 1 km wide and the scheme is exact; make them 667 m and the same model, same wind, same physics, passes ten times its starting peak at step 44 on its way to 1.8 × 10³⁰.