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
- Hazard Return Period & Lifetime Risk Calculate cumulative hazard risk over exposure windows and invert risk tolerance to find required design return periods.
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Map Projection Distortion Lab
Drag latitude to compare how Mercator and equal-area maps trade area for shape.
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Limiting Reagent & Stoichiometry Lab
Find the limiting reagent, theoretical yield, leftover excess and percent yield from balanced reactions.
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
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.
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
- An enzyme with V max = 100 and K m = 5
- Two sequences of 13 aligned bases differing at a single site
- A patient with haemoglobin of 9 g/dL and arterial PO₂ 100 mmHg
- Thirty PCR cycles at perfect efficiency turning one molecule into a billion
- Carbon at 12.011 u rebuilt from 12 C and 13 C isotopes
- Pure water at pH 7 and a small pH change
- An ocean 3000 km wide opened over 100 million years
- 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
- Forty animals out there in a park that expects 12
- 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
- Why squeezing does nothing to hydrogen iodide and everything to ammonia
- Pacific air at 8 °C crossing a 3,000 m crest to Calgary
- Erosion strips a kilometre off the top of Everest
- Energy per kilogram of the metals a Daniell cell consumes
- Lotka–Volterra equilibrium with α = 1.1 and the measured averages
- A 100-residue protein, three shapes per residue, and a search that outlasts the universe
- How deep a DNA circuit can get in one day, at 38 minutes a gate
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
- Two indices, and they disagree on purpose Biodiversity Index Calculator
- A better telescope cannot lower the false-positive rate Bayesian Biosignature Update
Articles written from these tools
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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
Simulate strong and weak acid titration curves — watch the pH jump at the equivalence point and explore the Henderson-Hasselbalch buffer region.
Allometric Scaling Explorer
Explore Kleiber's law and metabolic scaling relationships across species
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
Calculate Shannon, Simpson, and evenness indices from species abundance data
Body Mass Index (BMI) Calculator
Calculate BMI from height and weight, estimate body fat with the Deurenberg equation, and see why age and athletic build change interpretation but not the BMI formula itself.
Chemical Equilibrium & Le Chatelier Lab
Explore chemical equilibrium shifts under volume, temperature, and concentration changes across four reactions.
Country Shape & Flag Challenge
Test your geographical and mathematical skills with country shapes, flag symmetries, and capital cities.
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
Calculate the DNA mass required to represent every candidate route.
DNA Data Storage Capacity
Calculate real DNA storage capacity after biochemical and oligo overhead.
DNA Strand-displacement Gate
Explore Boolean DNA logic with a reduced kinetic model.
Earthquake Magnitude to Energy
Convert earthquake magnitude into joules and TNT equivalent, and compare nonlinear energy scaling.
Energy per Operation
Compare event energies with Landauer's thermodynamic bound.
Forest Fire Simulator
Simulate fire spread across a tree lattice and explore non-linear percolation threshold transitions.
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.
Hardy-Weinberg Equilibrium
Model allele frequencies under selection, test genotype counts for Hardy-Weinberg equilibrium with χ², and derive carrier frequencies.
Hazard Return Period & Lifetime Risk
Calculate cumulative hazard risk over exposure windows and invert risk tolerance to find required design return periods.
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
Calculate Airy, Pratt, and iceberg flotation compensation, root depths, and lithosphere buoyancy.
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
Find the limiting reagent, theoretical yield, leftover excess and percent yield from balanced reactions.
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
Explore enzyme kinetics: V vs [S], Lineweaver-Burk plots, and inhibition types
Molecular Clock Calculator
Estimate divergence time from sequence differences using the Jukes-Cantor model
Nernst Cell Voltage & Electrochemistry Lab
Calculate standard and non-standard galvanic cell potentials, Nernst slopes, free energy changes (ΔG), and equilibrium constants (K).
Oxygen–Haemoglobin Dissociation Curve
Hill-equation saturation, the Bohr shift, arterial oxygen content and how much oxygen a litre of blood actually delivers.
PCR Amplification Simulator
Simulate exponential DNA amplification: copy number, efficiency, and Ct values
pH / Acid-Base Calculator
Convert between pH, pOH, [H+], and [OH-] and classify acidic or basic solutions.
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
Compare exhaustive conformational search with real folding times.
Punnett Square Calculator
Monohybrid and dihybrid crosses with genotypic and phenotypic ratios
Radioactive Decay Calculator
Model exponential radioactive decay with half-life, activity, and decay curves
Radiometric Dating Calculator
Estimate geological sample age from isotope half-life using parent fraction or parent-daughter ratio.
Reaction Kinetics Calculator
Explore reaction rates, Arrhenius equation, and concentration-time profiles for different reaction orders
SIR Epidemic Model
Simulate disease spread with SIR/SEIR models - R0, herd immunity, and epidemic curves
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.