Build

Make something with it

Models you can pick up and apply, and technical concepts taught from the mechanism up, so you can build, calculate, design, or prove, not just recognize.

214 entries

Acceleration

Acceleration is the rate of change of velocity, and because velocity has both a size and a direction, an object accelerates whenever it speeds up, slows down, or merely changes direction, even at constant speed.

7 min

Accumulation and the integral

The definite integral of a varying rate or density over an interval is the limit of a sum of thin slices, each slice's contribution approximated as constant, and it recovers the exact total quantity, distance, mass, or energy, that the varying rate produced.

9 min

Action-reaction pairs

Every force one object exerts on a second is matched, at the same instant, by an equal and opposite force the second exerts back on the first, and this pairing is what makes pushing off anything, including empty exhaust gas, actually work.

7 min

Activation Energy

Nothing changes until it clears its starting barrier, so the leverage is in lowering the barrier, not in pushing harder.

7 min

Adverse Selection

When one side of a deal knows the quality on offer and the other cannot, the hidden bad drives out the good until the market itself unravels.

8 min

Angles, rotation, and radians

A radian measures rotation as the arc length it sweeps out divided by the radius, a definition that makes the same amount of turning read identically on any size circle.

6 min

Angular momentum

Angular momentum is rotational inertia times angular velocity, and it stays constant whenever no outside torque acts, which is why spin speeds up as mass pulls inward.

7 min

Antifragility

Beyond resilience lies a third state: things that do not merely survive disorder but grow stronger from it, and can be built on purpose.

8 min

Arithmetic operations as transformations

Each arithmetic operation is a transformation performed on a quantity, addition and multiplication combine two quantities into one, subtraction and division reverse a combination to recover a missing part.

7 min

Base Rates

The prior frequency of a thing in the population, which the mind reliably ignores when handed a vivid particular.

8 min

Bayesian Updating

A rule for changing your mind by the right amount: start with a prior belief, see the evidence, and revise in proportion to how strongly that evidence points.

8 min

Beams, columns, and buckling

A long slender member loaded in compression can suddenly bow sideways and collapse at a load far below the load that would simply crush its material, and buckling is the study of when and why.

7 min

Bearings, bushings, and friction management

A bearing or bushing is the joint that lets one part rotate against another while a designer controls, rather than eliminates, the friction between them.

7 min

Blue Ocean Strategy

Stop fighting rivals in a crowded market; create uncontested market space where competition does not yet exist, by delivering differentiation and low cost at once.

8 min

Bottlenecks and the Theory of Constraints

The single slowest step sets the pace of the whole system, so the only improvement that matters is the one made at the constraint.

8 min

Campbell's Law

The more heavily a quantitative measure is used to make high-stakes decisions, the more it corrupts and distorts the very thing it was meant to track.

8 min

Center of mass and balance

The center of mass is the single point where a distributed body's mass can be treated as concentrated for the purpose of predicting how the whole body moves and whether it stays balanced.

7 min

Ceramics and glasses

Ceramics and glasses are stiff and heat-resistant because their bonds are strong and directional, but that same directionality gives them no way to yield, so their real strength is set by the size of their worst flaw rather than by bond strength, and they carry compression far better than tension.

7 min

Chesterton's Fence

Before you tear down a fence, find out why it was put up; the reformer who cannot explain a thing has not yet earned the right to remove it.

7 min

Circles, radius, and pi

Every circle's circumference divided by its diameter equals the same fixed number, pi, a ratio fixed by the geometry of flatness rather than by any circle's particular size.

6 min

Circular motion and inward acceleration

Moving at constant speed along a circle still requires acceleration, because velocity's direction is constantly changing, and that acceleration always points toward the center.

7 min

Cleavage Theory

Party systems are frozen fossils of old social conflicts: the divisions that split a society when mass democracy arrived became the party families that outlived the conflicts themselves.

10 min

Commutative, associative, and distributive structure

The commutative, associative, and distributive laws state precisely which rearrangements of an arithmetic expression are guaranteed to leave its value unchanged, and part of what makes an operation interesting is which of these guarantees it fails.

7 min

Comparative Advantage

Even if one party is better at producing everything, both gain by specializing in what they give up least to make, and trading for the rest.

8 min

Composites and sandwich structures

A composite combines a stiff, strong, brittle reinforcing phase with a tougher matrix that holds it in place and spreads load into it, and a sandwich structure applies the same trick at the scale of a whole panel, using stiff skins to carry bending and a lightweight core to carry the shear between them.

7 min

Compounding

When returns are reinvested, growth builds on itself, producing an exponential curve that human intuition, tuned to straight lines, badly underestimates.

8 min

Conduction, convection, and radiation

Heat travels by only three physical routes, direct molecular contact, bulk fluid motion, or electromagnetic waves, and every insulation or cooling problem is solved by identifying which route dominates and blocking or exploiting it.

8 min

Congruence and similarity

Two figures are congruent when one can be placed exactly on the other by moving it without stretching, and similar when one is an exact scaled copy of the other, same shape, possibly different size.

7 min

Conservation of mechanical energy

When only conservative forces act, the sum of kinetic and potential energy stays fixed, letting you compare two moments of motion without tracing the path between them.

6 min

Constraints, assumptions, and margins

A margin is the deliberate gap you leave between what a design must survive and what you actually believe it will face, sized to the uncertainty in what you believe.

6 min

Continuity and breaks

A function is continuous at a point when you can draw its graph through that point without lifting the pen, which happens exactly when the limit approaching the point exists and equals the function's actual value there.

8 min

Conway's Law

Any organization that designs a system will produce a design whose structure copies the organization's own communication structure.

8 min

Coordinates and reference frames

A location is meaningless until you fix an origin, a set of directions, and a scale, and describing the same point from a different origin or set of directions gives different numbers for an identical place.

7 min

Counting and cardinality

Counting is a procedure carried out one tag at a time; cardinality is the fact that procedure reveals, that the last tag names the size of the whole group regardless of order.

7 min

Critical Mass

The threshold past which a process stops needing a push and starts sustaining itself.

8 min

Cross product as oriented area and rotation

The cross product of two vectors is a third vector, perpendicular to both, whose length equals the area of the parallelogram they span and whose direction, fixed by a right hand convention, records the sense of rotation from the first vector to the second.

7 min

Crossing the Chasm

A discontinuous gap separates the visionaries who first buy a new technology from the pragmatic mainstream, and most products die in it.

8 min

Deliberate Practice

Skill grows not from logging hours but from focused, effortful practice at the edge of your ability, guided by immediate feedback.

8 min

Design for manufacturing and assembly

Every part in an assembly, and every motion needed to put it together, is a chance for cost, delay, and error, so the deliberate discipline of designing for manufacturing and assembly asks, of each part and each step, whether it truly has to exist.

6 min

Differential equations as rules of evolution

A differential equation describes a system not by giving its state directly but by giving the rule for how that state changes right now, based only on what the state is right now.

6 min

Differentiation rules from structure

The rules for differentiating sums, products, quotients, and compositions of functions are not arbitrary shortcuts, they are theorems, each derived once from the limit definition of the derivative and then reused to differentiate any compound model without recomputing a difference quotient from scratch.

9 min

Dimensional analysis in physics

Every physical quantity carries dimensions built from a small set of base quantities, and any equation claiming to describe nature must have the same dimensions on both sides, which lets you test or repair a formula before ever taking data.

6 min

Disruptive Innovation

Incumbents lose not by managing badly but by managing well: they serve their best customers and cede the low end to cheap, inferior entrants who improve upward and displace them.

8 min

Dot product as alignment

The dot product of two vectors is a single number, equal to the product of their lengths and the cosine of the angle between them, that measures how much one points along the other.

7 min

Double-Loop Learning

Errors can be fixed two ways: by adjusting the action within your existing assumptions, or by questioning the assumptions themselves.

8 min

Dunbar's Number

There is a rough cognitive ceiling, around 150, on the number of stable social relationships one person can maintain at once.

8 min

Duverger's Law

Winner-take-all elections in single-member districts push a polity toward two parties; proportional representation lets many parties survive.

8 min

Effectuation

The decision logic expert entrepreneurs actually use: start from the means you already have, cap your downside at what you can afford to lose, and build the goal with whoever commits, rather than predicting a future and chasing it.

7 min

Elastic and inelastic collisions

Momentum is conserved in every collision between isolated bodies, but kinetic energy is conserved only in the special case of a perfectly elastic collision.

7 min

Emergence

Higher-level order arising from many local interactions with no central control, so the whole comes to have properties none of its parts possess.

8 min

Engineering as constrained choice

Engineering is the practice of finding the best available choice once physical law, cost, time, safety, and human capability have each ruled out some of what you originally wanted.

7 min

Entropy

Because disordered arrangements vastly outnumber ordered ones, any isolated system tends to slide from order to disorder, and reversing that slide always costs energy.

8 min

Equality as a balance relation

Equality is a claim that two expressions name the same quantity, checkable and preservable under matched changes to both sides, not a command to compute.

7 min

Equations as constraints

An equation is not a puzzle with one ritual answer; it is a filter that sorts every candidate value into allowed or disallowed, and the solution set can be empty, one value, several, or infinite.

7 min

Estimation and orders of magnitude

An order-of-magnitude estimate breaks an unknown quantity into a chain of roughly known factors, so that the errors in each factor partially cancel and the product lands within a defensible range of the truth.

7 min

Expected Value

The value of an uncertain prospect is each possible outcome multiplied by its probability, all added together: the long-run average of the gamble.

8 min

Failure modes and graceful degradation

A failure mode is one specific way a function can stop doing what it is supposed to do, and graceful degradation is designing so that when a failure mode occurs, the system loses capability in a controlled way rather than becoming dangerous all at once.

6 min

Fasteners and joints

Every joint, bolted, welded, riveted, glued, press fit, or snapped together, is a deliberate choice about how a load path crosses from one part into another, and the choice trades off strength, cost, and whether the joint can be made, inspected, and undone again.

7 min

Feedback Loops

A feedback loop is a circle of cause and effect in which a system's output loops back to change its own input, either damping the change (balancing) or amplifying it (reinforcing).

8 min

First Principles Thinking

Reason up from the few things you are sure are true, rather than down from what everyone already assumes.

6 min

First-order dynamics

A first-order dynamical system is one whose rate of change depends only on its current value, and that single restriction is strong enough to force the state toward exactly one of three long-term fates: unbounded growth, decay to zero, or settling at equilibrium.

6 min

Flow

The state of absorbed, effortless concentration that arises when a hard task and a matched skill meet, so that action and awareness fuse and the activity becomes worth doing for its own sake.

8 min

Force as interaction

A force is not a substance an object carries; it is what one object does to another during an interaction, and every force names a specific pair and a specific cause.

7 min

Fractions as numbers, ratios, and operators

A fraction is a single number, the answer to a division that whole numbers alone cannot finish, and its many everyday faces, a part of a whole, a ratio, a scaling operator, a point on a line, all name that same number.

7 min

Functional decomposition

Functional decomposition is breaking a desired outcome into the smaller functions that must all happen for it to occur, described by what each function does rather than by what part will do it.

7 min

Functions as machines and relationships

A function is a rule that assigns to every element of an allowed set of inputs exactly one element of an output set, regardless of whether the rule is a formula, a table, or a picture.

7 min

Gall's Law

A complex system that works is invariably found to have evolved from a simple system that worked; you cannot build a working complex system from scratch.

7 min

Game Theory

The mathematics of strategic decisions, where the best thing to do depends on what everyone else is deciding to do.

8 min

Gate Control Theory of Pain

Pain is not a direct readout of injury but a signal a spinal gate can amplify or dampen, which is why rubbing a hurt helps and why a lost limb can still ache.

7 min

Gears, belts, chains, and transmissions

A gear, belt, or chain transmission trades rotational speed for torque, or torque for speed, in an exact ratio set by the size of the wheels it connects.

7 min

Goodhart's Law

Once you turn a measurement into a target, people optimize the measurement rather than the thing it was meant to track, and the measurement goes bad.

7 min

Graphs as pictures of relationships

A graph is a picture in which position stands for a fact, so shape, slope, and turning points are frozen statements about how one quantity depends on another.

7 min

Hanlon's Razor

Never attribute to malice that which is adequately explained by stupidity, carelessness, or incompetence.

5 min

Heat as energy transfer

Heat is not a substance an object contains; it is energy in transit across a boundary, moving because of a temperature difference, and it stops being called heat the instant it arrives.

7 min

Impulse and force over time

The same change in momentum can come from a large force acting briefly or a small force acting longer, since impulse, force multiplied by the time it acts, is what equals the change in momentum.

6 min

Incentives

People do what they are rewarded for doing, so the reward you set, not the outcome you wanted, is what you will actually get.

8 min

Inertia and the persistence of motion

Inertia is the tendency of an object to keep whatever velocity it already has, constant speed in a constant direction, unless something acts on it to change that velocity, which means force is needed to change motion, not to maintain it.

7 min

Infinite sequences and convergence

A sequence converges to a value when, no matter how small a tolerance you demand, all but finitely many of its terms fall within that tolerance of the value, and this precise test is what separates an endless procedure that settles from one that merely keeps changing forever.

7 min

Internal energy

Internal energy is the total kinetic and potential energy stored in the random microscopic motion and configuration of the particles making up a substance, as distinct from any energy the object has by moving or sitting high up as a whole.

7 min

Inversion

To solve a hard problem, turn it around: instead of asking how to succeed, ask what would guarantee failure, and then refuse to do it.

7 min

Jobs to Be Done

Customers do not buy products, they hire them to make progress on a job that arises in a specific circumstance.

7 min

Kinetic energy

Kinetic energy is the energy a moving object carries by virtue of its motion, equal to one half its mass times the square of its speed, derived directly from the work needed to accelerate it from rest.

7 min

Kinetic friction and energy loss

Kinetic friction is a roughly constant force opposing sliding motion that converts organized kinetic energy into heat at the contact interface, at a rate set by the normal force and a material coefficient.

6 min

Length, area, and volume

Length, area, and volume are all the same operation, counting how many unit pieces tile a shape, applied in one, two, and three dimensions respectively, and every area or volume formula is a shortcut for that count.

7 min

Leverage

A small, well-placed input that moves a load far larger than itself, magnifying the result in both directions.

8 min

Levers, linkages, cams, and motion conversion

A lever, linkage, or cam uses pure geometry, not motors or electronics, to transform a given force or motion into a different force or a different, deliberately chosen path.

7 min

Limits as controlled approach

A limit is a precise, checkable statement about what value a quantity gets forced arbitrarily close to as an input approaches a point, without ever requiring the input to actually reach that point.

7 min

Loads, load paths, and free-body thinking

A load path is the actual route a force travels, member by member, from where it is applied to the ground or support that finally absorbs it, and free-body thinking is the method of isolating one member at a time to check whether that route can carry what is asked of it.

7 min

Manufacturing by removal, forming, joining, and addition

Every way of turning raw material into a finished shape works by one of exactly four mechanisms: taking material away, pushing it into a new shape, sticking separate pieces together, or building it up layer by layer.

7 min

Margin of Safety

Buy a thing for meaningfully less than it is worth, so that error, bad luck, and the unforeseen cannot ruin you.

8 min

Maslow's Hierarchy of Needs

Human motivation is organized into tiers, with basic bodily and safety needs generally pressing before higher needs for belonging, esteem, and self-actualization.

8 min

Mass as resistance to acceleration

Mass is not how much stuff an object has or how big it looks; it is a measured resistance, how much acceleration a given force fails to produce.

7 min

Material selection by properties

Choosing a material for a part means finding the region of a trade-off space, stiffness against density, strength against cost, temperature limit against toughness, where every requirement is satisfied, then picking the lightest or cheapest way to sit inside it.

7 min

Metals and alloys

A metal bends and does not shatter because its atoms are held together by a nondirectional sea of shared electrons that lets whole planes of atoms slide past each other, and every practical way of strengthening a metal works by making that sliding harder.

7 min

Modernization Theory

The claim that the richer a country becomes, the more likely it is to become and stay a democracy.

8 min

Momentum as quantity of motion

Momentum, mass times velocity, is conserved for an isolated system because every internal force comes paired with an equal and opposite reaction acting for the same time.

6 min

Moral Hazard

When you are shielded from the downside of a risk, you take more of it than you otherwise would, because someone else pays if it goes wrong.

8 min

Motion graphs

Position time, velocity time, and acceleration time graphs are three pictures of the same motion, connected because slope on one graph gives the value plotted on the next, and area under one gives the change plotted on the graph before it.

7 min

Natural Selection

Where things copy imperfectly and copies compete, the versions that survive and reproduce best come to prevail, and adaptation accumulates with no one designing it.

8 min

Negative numbers as direction and debt

A negative number is the answer that must exist if subtraction is to always make sense, and it names an opposite, in direction or in owed amount, rather than a smaller kind of nothing.

7 min

Network Effects

A product grows more valuable to each user as more people use it, which lets the leader pull away and often take most of the market.

8 min

Normal force and contact constraints

The normal force is the perpendicular push a rigid surface supplies, adjusted to whatever value stops an object from passing through it, not a fixed force equal to weight.

7 min

Observation, measurement, and physical quantities

A physical quantity is not a private sensation but whatever a stated, repeatable procedure of comparison assigns a number and a unit to.

5 min

Occam's Razor

Among competing explanations that account for the same facts, prefer the one that assumes the least.

8 min

Opportunity Cost

The true cost of any choice is the value of the best alternative you gave up to make it.

7 min

Optimization and trade-offs

At a smooth interior maximum or minimum, the rate of change of a function must be exactly zero, because an instant before it was rising or falling and an instant after it reverses, so setting the derivative to zero and checking which case you have is the systematic way to find the best point under a real trade-off.

8 min

Parkinson's Law

Work expands to fill the time available for its completion, and bureaucracies grow of their own accord regardless of the work to be done.

7 min

Patterns, sequences, and rules

A pattern is not proven by noticing repetition in a few terms; it is established by stating a rule that generates every term, and a short list is compatible with more than one such rule.

8 min

Phase transitions

A substance changes phase when a rival arrangement of its molecules, not hotter or colder but more stable at the current temperature and pressure, becomes favorable, and every joule spent making that change happens at constant temperature.

7 min

Physical models and idealization

A physical model is a deliberately simplified stand in for a real system, built by naming and discarding certain features on purpose, and it is only useful once you can state what it ignores and where that neglect starts to matter.

6 min

Pirate Metrics (AARRR)

The customer lifecycle broken into five measurable stages, Acquisition, Activation, Retention, Referral, Revenue, so you fix the one leaking stage instead of chasing a flattering total.

7 min

Place value and number bases

Place value lets a fixed, small set of symbols represent any quantity, however large, by letting a digit's position multiply its meaning.

7 min

Points, lines, angles, and planes

Geometry builds everything, triangles, circles, solids, from a small set of undefined primitives, point, line, and plane, plus a short list of relations, incidence, betweenness, and angle, that constrain how those primitives can meet.

7 min

Polymers and elastomers

A polymer is a tangle of long chain molecules whose backbone is strongly bonded but whose chains are held to each other only weakly, so the same chemical family can behave like a rigid solid, a rubber, or a slow-flowing liquid depending on temperature and how tightly the chains are cross-linked.

7 min

Porter's Five Forces

An industry's long-run profitability is decided by five structural forces, and strategy is the search for a position where those forces are weak and defensible.

7 min

Porter's Generic Strategies

Competitive advantage comes from being the low-cost producer or being different, aimed either broadly or at a niche; try to be everything and you end up stuck in the middle.

9 min

Position, distance, and displacement

Position is where something is relative to a chosen origin, distance is the total length of the path travelled, and displacement is the straight, directed change between start and end, and the three are related but never interchangeable.

7 min

Potential energy and configuration

Potential energy is the work a configuration of interacting objects could still do, stored not in one object but in the relationship between them and a reference.

6 min

Power Laws and Fat Tails

In a power law a handful of events account for most of the total, the distribution has no typical size, and the rare extreme, not the average, is what dominates the outcome and the risk.

8 min

Precision, accuracy, and uncertainty

Precision is how repeatable a measurement is, accuracy is how close it is to the true value, and uncertainty is the honestly stated range within which the true value probably lies.

6 min

Pressure from microscopic collisions

Pressure is not a mysterious substance pushing outward, it is the measured average of countless individual molecular collisions transferring momentum to a surface every instant.

8 min

Probabilistic Thinking

Treat beliefs as probabilities to be sized and updated, not as certainties to be defended.

8 min

Prototypes as questions

A prototype is not a small version of the final product, it is the cheapest object you can build that answers one specific uncertain question.

6 min

Quantity before number

Before any numeral exists, a quantity is just how much of something there is, and you can compare, match, and preserve it without ever naming a number.

5 min

Rate of change

A rate of change is the ratio of how much one quantity changed to how much a related quantity changed alongside it, and comparing rates, not totals, is what tells you which process is truly faster.

8 min

Ratios, proportions, and similarity

A ratio is a relationship between two quantities, of the form how many times one contains the other, that can stay exactly fixed while the quantities themselves grow or shrink together.

6 min

Redundancy

Carry more capacity than the average day requires, because failure comes on the days that are not average.

8 min

Reference frames and relative description

Motion has no meaning stated in isolation; every description of position, velocity, or rest is a description relative to a chosen frame of reference, and changing the frame changes the description without changing what physically happens.

6 min

Regression to the Mean

When you measure something extreme, the next measurement tends to be less extreme, because luck does not repeat, and mistaking that drift for cause is how we fool ourselves.

8 min

Requirements and success criteria

A requirement is a single, checkable statement of what a design must achieve, written so that WHAT is fixed while HOW is left entirely open.

6 min

Rotational inertia

Rotational inertia measures how hard it is to change a body's spin, and it depends not just on how much mass there is, but on how far that mass sits from the axis of rotation.

7 min

Safety factors and uncertainty

A safety factor is the ratio of the load a part is designed to survive to the load it is actually expected to carry, sized deliberately to absorb the variability in materials, manufacturing, loading, and analysis that no calculation can fully eliminate.

7 min

Schelling's Segregation Model

Mild individual preferences for similar neighbors produce extreme collective segregation.

3 min

Second-order dynamics

A second-order dynamical system is one whose evolution rule constrains not the rate of change of its state but the rate of change of that rate, which is what allows a system to overshoot, reverse, and oscillate instead of only growing, decaying, or settling.

7 min

Second-Order Thinking

Ask not just what an action does, but what happens after that: the consequences of the consequences.

8 min

Self-Determination Theory

The theory that people are healthiest and most durably motivated when three basic needs, autonomy, competence, and relatedness, are met, and that external rewards can crowd out the internal motivation they aim to boost.

8 min

Series as accumulated sequences

A series is not a mysterious act of adding infinitely many numbers at once but an ordinary sequence of running totals, called partial sums, and the series has a sum precisely when that sequence of totals converges.

7 min

Signal Detection Theory

A framework separating how well you can tell signal from noise from how willing you are to say yes when unsure.

7 min

Signaling

When quality is hidden, a signal is credible only if it costs the impostor more than it costs the genuine article.

8 min

Sine, cosine, and tangent as geometric ratios

Sine, cosine, and tangent are ratios between the sides of a right triangle that depend only on one angle, never on the triangle's size, because similar triangles always share the same side ratios.

7 min

Speed and velocity

Speed is how fast the distance odometer is ticking, a plain number with no direction, while velocity is the rate of change of displacement, a directed quantity that can be zero, positive, or negative even while speed stays constant.

6 min

Springs, dampers, and suspension

A suspension pairs a spring, which stores energy and pushes back proportional to displacement, with a damper, which dissipates energy proportional to speed, so a vehicle can absorb a bump and then stop moving rather than bounce indefinitely.

7 min

Static equilibrium

A rigid body genuinely at rest must satisfy two independent conditions at once: zero net force, so it does not start moving, and zero net torque, so it does not start turning.

7 min

Static friction

Static friction is not a fixed force but a responsive one, matching whatever is needed to keep two touching surfaces from sliding, up to a maximum set by the normal force and a material coefficient.

6 min

Sternberg's Triangular Theory of Love

Love is built from three components (intimacy, passion, and commitment) whose presence or absence yields eight recognizable kinds of love.

8 min

Supply and Demand

Price is the point at which the quantity people want to buy equals the quantity people want to sell.

8 min

Surface finish, wear, and lubrication

No manufactured surface is truly smooth; it is a landscape of microscopic peaks, and whether two such landscapes slide, stick, or wear each other away depends on those peaks, the load pressing them together, and whatever film separates them.

7 min

SWOT Analysis

A planning grid that sorts what you know about a decision into four boxes: internal Strengths and Weaknesses, external Opportunities and Threats.

7 min

Symmetry and invariance

A symmetry is a transformation that leaves an object or a situation looking exactly the same, and whatever stays unchanged under that transformation is called invariant, a fact that can be used to shortcut a calculation rather than merely decorate its answer.

7 min

System boundaries and interfaces

A system boundary is the deliberately chosen line separating what you are designing from everything else, and an interface is every point where matter, energy, or information crosses that line.

7 min

Taylor approximation

A Taylor approximation rebuilds a smooth function near a point using nothing but that point's value and its successive derivatives, one derivative order at a time, and the leftover error after stopping at any order can itself be written down and bounded exactly.

7 min

Temperature as microscopic energy distribution

Temperature is not a substance or a feeling of hot and cold; it is a number that measures the average kinetic energy per particle in the random jostling motion of matter.

7 min

Tension in strings and cables

Tension is the pulling force transmitted along a flexible connector such as a string or cable, uniform along an ideal massless, frictionless length and solved for by applying Newton's second law to each body it connects.

7 min

Tension, compression, bending, torsion, and shear

Every internal force a member carries can be sorted into five basic types, tension, compression, bending, torsion, and shear, and telling them apart is what lets you predict how and where a part will actually fail.

7 min

The Bargaining Model of War

Because war is costly, a settlement both sides prefer should almost always exist, so war is a failure of bargaining, not its natural outcome.

7 min

The BCG Matrix

A 2x2 that sorts a company's businesses by market growth and relative market share to decide which to feed, which to milk, and which to sell.

8 min

The Big Five Personality Traits

Five broad, statistically independent dimensions, openness, conscientiousness, extraversion, agreeableness, and neuroticism, that keep reappearing whenever personality description is factor-analyzed.

9 min

The Black Swan

A rare, unpredictable, high-consequence event that we insist, only afterward, we should have seen coming.

8 min

The Circle of Competence

Know the perimeter of what you truly understand, and operate strictly inside it, because the boundary matters far more than the size.

7 min

The derivative as local sensitivity

The derivative of a function at a point is the limit of its average rate of change as the interval shrinks to nothing, giving the exact sensitivity of the output to a tiny nudge in the input at that single instant.

7 min

The Eisenhower Matrix

Sort your tasks by importance and urgency, and you will find that the work that matters most is rarely the work that is shouting loudest.

6 min

The Elaboration Likelihood Model

Persuasion travels one of two routes, careful thought about the argument or a shortcut around it, depending on whether the audience is able and motivated to think.

8 min

The Feynman Technique

To understand something, explain it in plain words as if to a beginner; where you stall or reach for jargon, you have found what you do not actually know.

8 min

The first law of thermodynamics

The first law of thermodynamics says a system's internal energy can only change by heat crossing its boundary or work being done on or by it, and the change equals exactly the heat added minus the work done by the system: delta-U = Q - W.

8 min

The Flywheel

Momentum is built turn by turn: no single push transforms anything, but consistent effort in one direction compounds into force that becomes nearly unstoppable.

8 min

The fundamental link between change and accumulation

The instantaneous rate at which a quantity changes and the running total of that quantity are two descriptions of one process, and differentiating the total always hands back the rate, while integrating the rate always hands back the total.

7 min

The Globalization Trilemma

Deep economic integration, the nation-state, and democracy cannot all hold at once: pick any two and you must give up the third.

9 min

The ideal gas model

Treat gas molecules as tiny, free, colliding points with no attraction between them, and pressure, volume, temperature, and particle count lock together into one simple equation, PV = nRT, that predicts real gas behavior remarkably well until the gas is squeezed cold enough or dense enough to condense.

8 min

The Impossible Trinity

A country can have a fixed exchange rate, free capital movement, and independent monetary policy, but only two of the three at once.

8 min

The Johari Window

A four-pane map of the self, split by what you know about yourself and what others know about you, used to widen the shared, honest space between people.

7 min

The Lean Startup

Treat a startup as an experiment engine whose real product is validated learning, and test cheap falsifiable hypotheses before you scale.

7 min

The Lindy Effect

For things that do not age, the longer they have already survived, the longer they can be expected to survive: age is evidence of durability.

8 min

The Long Tail

When shelf space is unlimited, the many niche products can collectively rival the few hits.

8 min

The Map Is Not the Territory

A representation of reality is never the reality itself, and mistaking the two is a basic and costly error.

7 min

The Median Voter Theorem

Under majority rule on a single issue dimension, vote-seeking candidates are pulled toward the position of the median voter.

8 min

The Nash Equilibrium

A situation in which no player can do better by unilaterally changing their own strategy, given what everyone else is doing.

8 min

The Norm Life Cycle

A three-stage account of how a moral standard goes from the claim of a few activists to a rule almost no one questions: emergence, cascade, internalization.

7 min

The Normal Distribution

The bell curve: a model of how quantities scatter when they are the sum of many small independent effects, reliable at the center and dangerous when its thin tails are trusted where they do not belong.

8 min

The Pareto Principle

A minority of causes usually accounts for a majority of effects, so effort spent on the vital few beats effort spread across the trivial many.

7 min

The Peter Principle

In a hierarchy, people are promoted until they reach a job they cannot do, and then they stay there.

7 min

The Policy Window

Policy change happens when three separate streams (problems, policies, and politics) are joined by an entrepreneur during a brief window of opportunity.

8 min

The Principal-Agent Problem

Whenever one person acts on another's behalf but wants different things and cannot be fully watched, their interests drift apart, and closing the gap is never free.

8 min

The Pythagorean relationship

In any right triangle, the area of the square on the hypotenuse always equals the combined area of the squares on the other two sides, a fact provable by rearranging area rather than by measuring it.

7 min

The relation between force, mass, and acceleration

Net force, mass, and acceleration are locked together by one relation, net force equals mass times acceleration, and it can be reconstructed, not just memorized, from how force and mass were separately defined.

7 min

The Rubber Band Model

A hard decision about whether to leave is a rubber band stretched between the bonds that hold you and the pull of something new; name both before you decide.

6 min

The Rubicon Model

Action runs through four phases split by a single point of no return, the moment you decide, and each side of that line demands a different frame of mind.

8 min

The second law and irreversibility

Heat, gases, and mixtures all move toward the overwhelmingly more probable arrangement, and that one-way drift, not any extra force, is the second law.

8 min

The Seven Perspectives of Psychology

Psychology explains the same behavior at seven different levels, from brain chemistry to culture, and the levels are complementary answers to different questions rather than rival theories competing for one.

5 min

The Spacing Effect

The same amount of study spread out over time produces far more durable learning than the same study packed into one session.

8 min

The Swiss Cheese Model

Accidents happen not because one defense fails but because the holes in many defenses momentarily line up to let a hazard through.

8 min

The Transtheoretical Model

A map of behavior change as a sequence of five stages, from not intending to change to sustaining a change, with different help suited to each.

9 min

The Working Memory Model

Short-term memory is not one holding pen but a small team: an executive that directs attention, a sound-based loop, a visual sketchpad, and a buffer that binds them into experience.

7 min

The Yerkes-Dodson Law

Performance improves as arousal rises but only up to a peak, and that peak sits lower for hard tasks than for easy ones.

8 min

Tolerance, fit, and clearance

No two manufactured parts are ever exactly the size on the drawing, so every dimension needs a stated range of acceptable variation, and that range, chosen deliberately, is what makes a fit between two parts predictable.

6 min

Torque and rotational effect

Torque is the turning effect of a force, set by how much force you apply, how far from the pivot, and at what angle.

7 min

Trade-off spaces and objective functions

A trade-off space is the set of designs where improving one property necessarily worsens another, and an objective function is the explicit rule you use to decide which point in that space you actually want.

6 min

Transnational Advocacy Networks

When a government blocks its own citizens, they route around it: appealing to allies abroad who pressure the state from outside, so the demand comes back home like a boomerang.

8 min

Triangles and rigidity

A triangle with three fixed side lengths has exactly one possible shape, which is why triangulated frames resist deformation that four sided frames cannot.

7 min

Trusses, frames, and triangulation

A truss carries load through a rigid arrangement of triangles, so that nearly every member works only in tension or compression, letting a structure be strong while using far less material than a solid beam of the same span.

7 min

Two-dimensional motion and projectiles

Motion in two dimensions, including a thrown or launched projectile's curved path, can be built exactly by treating the horizontal and vertical motions as two independent one-dimensional motions happening at the same time, sharing only the clock.

7 min

Two-Level Games

International negotiation is a game played at two tables at once: bargaining with foreign counterparts while selling any deal back home.

7 min

Units and dimensional reasoning

A measured number is incomplete without its unit, because the unit names the standard the number is a multiple of, and mismatched units make an equation meaningless regardless of whether the digits agree.

6 min

Units, standards, and calibration

A unit is an agreed, reproducible reference amount, and calibration is the act of comparing an instrument against that reference so its readings can be trusted by anyone, not just its owner.

6 min

Universal gravitation

Every two masses in the universe pull on each other with a force that grows with both masses and falls off as the square of the distance between them, the same law that governs a falling apple and a moon in orbit.

8 min

Variables and unknowns

A variable is a symbol paired with a domain of allowed values, standing for an unknown, a varying quantity, or a placeholder for generality depending on how it is used.

7 min

Vectors as directed quantities

A vector is a quantity that has both a magnitude and a direction, and that combines with others by placing them tip to tail rather than by ordinary addition of numbers.

7 min

Veto Players

Policy change requires the agreement of every actor whose consent is needed to pass it; the more of them there are, the more ideologically distant, and the less internally unified, the harder change becomes.

9 min

Via Negativa

Improve by subtraction: the surest gains often come from removing the harmful rather than adding the good.

8 min

War Made the State

Modern states were built not by contract or design but as the by-product of rulers scrambling to fund war: to fight, they had to tax, and to tax they had to build bureaucracy.

9 min

Weight and gravitational interaction

Weight is not a property an object owns; it is the specific gravitational force pulling on it right now, which is why the same object weighs differently on the Moon, in an accelerating elevator, and in orbit, while its mass never changes at all.

7 min

Wheels, rolling resistance, and traction

A wheel trades most sliding friction for a much smaller rolling resistance, but the same contact patch must still provide, through static friction, all the traction a vehicle needs to accelerate, brake, and turn.

7 min

Work as force through distance

Work is the transfer of mechanical energy by a force, measured as the component of that force along an object's displacement times the distance moved, and it is zero whenever force and motion are perpendicular.

8 min

World-Systems Theory

There is one capitalist world-economy, not many national economies climbing the same ladder, and it is built as a single division of labor that enriches a core by draining a periphery.

9 min

Zero as number, placeholder, and boundary

Zero is not one idea but four: a placeholder that fixes position, a count of an empty collection, the boundary between positive and negative, and the number that leaves every quantity unchanged under addition.

7 min

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