What's new
Essays arrive in groups rather than one at a time, and a group usually opens up a subject not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.
28 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
Algebra
The biggest box built from signs
Fill a square table with plus and minus ones and ask how large its determinant can be. The columns all have the same length, so the answer is a box with fixed edges — largest when every corner is square, which is possible only when the size is a multiple of four.
Algebra
The count a fold cannot change
A curved map can fold the plane over itself, so that one point has three preimages and its neighbour has one. Count each preimage with the sign of the determinant there and the jump disappears — the signed count is the same everywhere, and it is a whole number.
Analysis
The error that keeps coming back to its worst
Judge a polynomial by its largest error on an interval and there is exactly one best one of each degree. It is recognised without comparing it to anything else — its error rises to the same largest size, alternately above and below, one more time than there are coefficients.
Analysis
An ellipse, not a disc
A Taylor series converges on a disc, and the disc's radius is the distance to the nearest singularity. Ask instead how well polynomials can follow a function on an interval, and the answer is an ellipse with the interval's ends as its foci — the largest one the function is smooth inside.
Applied
Prices at every corner
The duality theorem says a linear program's best value equals its dual's, and says nothing about how to find either. The simplex method finds both at once — it walks from corner to corner, and at each one asks the constraints that meet there for prices. A negative price names an edge that climbs; when none is negative, the prices are the proof.
Applied
The cube that takes every corner
The simplex method is fast on every program anybody meets in practice. In 1972 Victor Klee and George Minty squashed a cube so that the method, choosing the steepest edge each time, visits all of its corners — 2ⁿ − 1 moves in n variables, with the optimum one edge from the start.
Computation
The centre a straightedge cannot find
Give a straightedge one circle and its centre, and it can do everything a compass can. Take the centre away and it cannot even find it again — because to a straightedge a circle has no centre. The maps that keep a circle and its straight lines are the motions of the hyperbolic plane, and in that plane the centre is a point like any other.
Computation
The lengths dividers cannot reach
A pair of dividers carries a length from one place to another and draws nothing. With a straightedge it finds midpoints, parallels and right angles, and it draws the regular 17-gon. It cannot draw a segment of length √(1 + √2) — and the reason is not on the page at all, but in the other root of the equation that number solves.
Discrete
Every crowd holds a bowl or a dome
Among enough points in the plane, some k of them always bend upward like a bowl or some l bend downward like a dome. The number that forces it is a binomial coefficient, it is exactly right, and it proves that every large enough crowd contains a convex polygon — with a bound nobody could lower to the true answer for eighty years.
Discrete
The longest climb of a shuffle
Erdős and Szekeres guarantee that any n distinct numbers hold a climb or a fall of √n, and there are orders that allow nothing more. Shuffle a deck at random instead and the longest climb is almost exactly 2√n — with fluctuations of size n to the power one-sixth, distributed exactly as the largest eigenvalue of a large random matrix.
Dynamics
Where the time goes on an attractor
A chaotic orbit's position is unpredictable within a few dozen steps. How it divides its time is not — start anywhere, follow long enough, and the fraction of time spent in each region comes out the same. That distribution lives on a set of no area, and among the infinitely many ways an orbit could spend its time, typical starts pick exactly one.
Dynamics
A region the orbit cannot leave
That the Hénon map and the Lorenz flow have attractors takes two lines of arithmetic each — a region mapped strictly inside itself, a quantity that falls outside an ellipsoid. That the attractors are strange took a computer and twenty years for Lorenz, and for Hénon's classical parameters it has never been done. The difference between the two questions is the difference between a region and what lives in it.
Geometry
Every point has a partner across the bisectors
Draw the three lines from the corners of a triangle through any point, reflect each in the bisector of its own angle, and the three reflections meet again. The pairing this makes swaps the centroid with the symmedian point and the orthocentre with the circumcentre, bends every straight line into a conic through the corners, and sends the circumcircle to infinity.
Geometry
A triangle that fits once fits everywhere
Put one circle inside another and try to fit a triangle between them, its corners on the outer circle and its sides touching the inner. Usually no triangle fits. But if one does, then one fits starting from every point of the outer circle — and whether it does is decided by a single equation in the two radii and the distance between the centres.
Logic
Two lists that are one order
The rationals and the fractions with a power of two below are different sets of numbers, and as orders they are exactly the same — any two countable orders that are dense and have no ends can be matched, point for point, keeping every comparison. The proof is a zigzag, and it settles every question the language of order can ask.
Logic
The graph that coin tosses always make
Take infinitely many vertices and toss a coin for every pair to decide whether they are joined. The result is random in every detail — and, with probability one, it is always the same graph. The same graph can be written down without any coins, by joining two numbers when one binary digit of the larger is a one.
Number
How evenly the fractions spread
List every fraction between nought and one with denominator at most n, in order. They spread across the interval almost evenly, and how fast the unevenness shrinks as n grows is — exactly, provably — the Riemann hypothesis. The link runs through a second fact: set the fractions round a circle and add them as arrows, and what is left is a whole number.
Number
Counting the classes that break factorisation
The class number measures how badly unique factorisation fails in a field, and defined through ideals it looks impossible to compute. Gauss computed it by hand, for every field he wanted, by counting quadratic forms — and every form can be squeezed, by changes of variable that keep its values, into exactly one small standard shape.
Probability
The narrowest door sets the pace
How fast a chain forgets is an eigenvalue, and nobody can compute the eigenvalues of a chain worth studying. Cheeger's inequality trades the eigenvalue for a picture — the narrowest door in the state space — and pins the one between the square of the other and twice it. Both ends of that range are reached, on graphs small enough to search completely.
Probability
The forgetting that happens all at once
A single small chain forgets its start gradually, a little more with every step. A family of large ones can do something different — stay almost perfectly informed about where it began, and then lose all of it inside a window far shorter than the wait. That cliff is the cutoff phenomenon, and it is why "seven shuffles" is an answer rather than a convention.
Topology
Every surface is sewn from pants
A sphere with three holes — a pair of pants — is the smallest piece a surface with two or more handles can be cut into. Every such surface falls apart into them, and the Euler characteristic alone says how many pieces and how many cuts, however the cutting is done. What it does not say is the pattern, and the patterns are counted by drawing each one as a graph: two for two handles, five for three, seventeen for four.
Topology
The fewest corners a surface needs
Build a closed surface from triangles, any two meeting along a whole edge, at a single corner or not at all, and ask for the fewest corners. Two lines of counting give a floor for every surface, in terms of its Euler characteristic alone. The torus meets it with seven, the projective plane with six — and the Klein bottle, which the counting says could be built from seven, cannot, as a search of every possible arrangement shows.
Before that
Everything published earlier, newest first. Titles only — the cards are on the full listing.
27 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Thirty-one moves from solved — algebra
- Which graphs let the tokens go anywhere — algebra
- Two numbers decide the flow — analysis
- The matrix that has no logarithm — analysis
- Prices the bidders raise — applied
- The prices nobody can break away from — applied
- A rectangle made only of squares — computation
- No odd number of equal triangles — computation
- The order decides the colours — discrete
- How many colours the plane needs — discrete
- Where the Collatz map is a coin — dynamics
- Fourteen fractions that list the primes — dynamics
- The skeleton inside a shape — geometry
- Three mirrors make every solid — geometry
- The program that prints itself — logic
- No algorithm can read what a program does — logic
- Two halves a sieve cannot tell apart — number
- The patterns primes are allowed to make — number
- Points on a lattice that see almost nothing — probability
- An error bar for points that are not random — probability
- Every count a solid can have — topology
- Counting targets by their holes — topology
26 September 2026
22 essays on geometry, logic, number, topology, dynamics, probability, discrete, applied, algebra, analysis and computation
- At least as many lines as points — geometry
- Rows of three, planted on a cubic — geometry
- A contradiction that is only a sum — logic
- Two terms made equal, and no more — logic
- The wait for the next sum of two squares — number
- A fraction on the circle forces a whole point — number
- A whole number split into two that are not — topology
- Six points in space and a pair that must link — topology
- Counting in a base that is not a whole number — dynamics
- Where a base-β orbit spends its time — dynamics
- Evidence measured in decibans — probability
- The envelope that always looks better — probability
- Enough partners in every finite group — discrete
- Matching as they arrive — discrete
- Envy that any single item would cure — applied
- A rent nobody envies — applied
- How a polynomial breaks modulo the primes — algebra
- Units that form a lattice — algebra
- A geometric series whose ratio is a matrix — analysis
- A series that converges to minus one — analysis
- A curve that divides any angle — computation
- Two weak sources make one fair bit — computation
25 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, number, probability, logic and topology
- What is left when the middle is taken out — algebra
- The polygon a lattice becomes from far away — algebra
- A coin in front of every power — analysis
- Two sign patterns that land together — analysis
- One extra person on one side — applied
- The matching in the middle — applied
- Twenty cards with no set among them — computation
- The polynomial that bounds the caps — computation
- Three places cut apart — discrete
- The cheapest way to send — discrete
- A whole interval of speeds — dynamics
- The last circle to break — dynamics
- The least wall for equal rooms — geometry
- Every chord slid to the middle — geometry
- Divisors that add to three times the number — number
- A ratio nobody else has — number
- A coin that lets the first player win — probability
- Three patterns in a circle — probability
- A proof that says which half — logic
- No table of truth values is enough — logic
- How many changes undo a knot — topology
- Several colours on every vertex — topology
24 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Discs that fence in the eigenvalues — algebra
- The highest point on the sphere is an eigenvalue — algebra
- A plucked string keeps its corners — analysis
- The ripples that make a series run away — analysis
- How often the majority goes in a circle — applied
- A majority wiser than its members — applied
- A Latin square with boxes — computation
- Orthogonal squares are a code — computation
- Multiplying every number on the dial at once — discrete
- A root lifted one digit at a time — discrete
- The ball that stays outside the table — dynamics
- The word a straight line spells — dynamics
- The diagonal no unit measures — geometry
- Tiles that never repeat — geometry
- The word that cannot describe itself — logic
- Arithmetic with addition alone — logic
- The shape a random partition takes — number
- Two counts that agree for no visible reason — number
- The median of many small averages — probability
- A sphere that is nearly all equator — probability
- A polynomial that tells left from right — topology
- The crossings an alternating knot cannot lose — topology
23 September 2026
22 essays on number, geometry, logic, algebra, analysis, dynamics, probability, discrete, computation, applied and topology
- Two families of solutions, and a box that holds both — number
- Sixty needs two digits and sixty-one needs ten — number
- The player who meets the first long run — geometry
- The remainders that count the roots — geometry
- One thing in each region is enough — logic
- The conclusion is what survives the erasing — logic
- The sums of roots of unity that add to nothing — algebra
- Three ways to pair four roots — algebra
- The length the derivative never sees — analysis
- A length counted by the lines that cross it — analysis
- A double root halves the error instead of squaring it — dynamics
- A cubic method that is Newton's in disguise — dynamics
- The ground a walk covers — probability
- A walk that may not step where it has been — probability
- Cars that park, and trees that grow — discrete
- A random tree is one part in e leaves — discrete
- No set with a line in every direction is small — computation
- A sum of two sets modulo a prime cannot be small — computation
- What a missing input is worth — applied
- Cutting a link costs both of its ends the same — applied
- The loops on a torus that never cross themselves — topology
- A twist that carries one loop to another — topology
22 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Twenty-four ways to set a cube down — algebra
- Necklaces made of symmetries — algebra
- A limit can jump at every fraction — analysis
- Uniform, except on a small set — analysis
- The nearest consistent verdict — applied
- Agendas that cannot contradict themselves — applied
- The rate a noisy channel allows — computation
- Erasures a code can see — computation
- Every place changes back — discrete
- The walk through the middle levels — discrete
- Sensitivity comes free — dynamics
- Chaos on a set nobody lands on — dynamics
- Seven pieces and an equilateral middle — geometry
- Eighteen equilateral triangles — geometry
- A plane through the cube — logic
- Half the cube and √n neighbours — logic
- An order that proves a prime — number
- Two primes where Fermat holds twice — number
- Fourteen people within a day — probability
- A room where nobody is alone — probability
- Twelve pentagons, whatever the hexagons — topology
- Zero in four dimensions — topology
21 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Every power sum, from the coefficients alone — algebra
- The roots of the slope stay inside — algebra
- The sum that steps over every whole number — analysis
- A coin in front of every term — analysis
- When one of the two numbers is missing — applied
- The lines the optimum lies under — applied
- The price of a construction — computation
- One cell short of a transversal — computation
- A walk that splices in its own detours — discrete
- The streets a postman walks twice — discrete
- Almost every orbit is fair — dynamics
- A solvable chaos of every degree — dynamics
- One number for every chord through a point — geometry
- Every side measured by one diameter — geometry
- Six sentences from two quantifiers — logic
- A quantifier is a shadow — logic
- A factorisation that hides its primes — number
- Factoring uniquely with no way to divide — number
- The cells a permutation must miss — probability
- A round table with no couple together — probability
- Opposite labels that have to meet — topology
- The colours a circle forces — topology
18 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Where two roots run into each other — algebra
- One number under every bell — algebra
- An endless region with a finite area — analysis
- A rectangle cut by a curve — analysis
- The people every stable answer leaves out — applied
- A ring that no pairing can break — applied
- Fair bits from an unfair coin — computation
- A filter that changes only the spread — computation
- Moves that only ever add edges — discrete
- The densest graph without a square — discrete
- A road where nobody overtakes — dynamics
- No local rule can count the votes — dynamics
- Three ordinary lines from a count — geometry
- The fewest ordinary lines a polygon allows — geometry
- A function that adds and is nowhere a line — logic
- Infinitely many guessers, finitely many wrong — logic
- The fraction Lambert built for the tangent — number
- The pattern in e's continued fraction — number
- Two children and the sentence about one of them — probability
- One coin, counted by runs and by wakings — probability
- Every way to pair a polygon's edges — topology
- The surface a random gluing makes — topology
16 September 2026
44 essays on discrete, applied, analysis, number, probability, topology, geometry, algebra, computation, dynamics and logic
- What the search has when it fails — discrete
- The piece that cannot pair off — discrete
- Patience instead of a contract — applied
- The reading that is almost right — applied
- The series everything else is measured against — analysis
- The repair at the boundary — analysis
- The two squares actually produced — number
- Almost no number is one — number
- The one that hardly ever comes up — probability
- Two patterns, one chance, different waits — probability
- A count that can say zero — topology
- A map that offers a choice — topology
- Pinned between two sequences — geometry
- The slice that has to match — geometry
- Counted across and counted down — algebra
- The cycles and the cuts — algebra
- More blocks than points — computation
- A plane in a list of numbers — computation
- An area that never finishes — dynamics
- Covering rather than avoiding — dynamics
- One gadget defeats every refinement — logic
- The boundary at three variables — logic
- Any unevenness brings the match sooner — probability
- A collision that finds a factor — probability
- The sum of the parts, taken again — number
- Why a quarter of numbers overshoot — number
- A polynomial behind the colourings — topology
- The surface a knot bounds — topology
- The dark lines are one point's orbit — dynamics
- The window that opens with a stutter — dynamics
- The angle that is really an area — analysis
- When two circular motions come home — analysis
- Sums of powers, read off a staircase — geometry
- Three triangular numbers, and no fewer — geometry
- A number larger than every number — logic
- A countable field that passes for the line — logic
- How many cuts a fair share costs — applied
- The product that makes a division fair — applied
- Seven powers in a space of six — algebra
- The integers a field contains — algebra
- A plane no field built — computation
- The curve that no three points in line define — computation
- Five spokes squeezed into K5 — discrete
- The few points that cut a flat graph — discrete
14 September 2026
22 essays on discrete, number, applied, analysis, algebra, dynamics, probability, topology, geometry, logic and computation
- The product that deals the labels — discrete
- The coefficient that is a polynomial — discrete
- One residue whose powers are all of them — number
- The exponent that is smaller than Euler's — number
- Five weighings and the question is closed — applied
- An objection one player makes to another — applied
- Which curves have a length at all — analysis
- The length belongs to the journey — analysis
- How fast the ball fills — algebra
- The edge that is as big as the ball — algebra
- How a lock comes apart — dynamics
- A rotation in different coordinates — dynamics
- The bound is the answer to a search — probability
- No single input can move it far — probability
- Every word driven to a normal form — topology
- The third number a surface needs — topology
- One circle touching four — geometry
- A centre is three weights — geometry
- Not one step but a continuum — logic
- Refutable in something small — logic
- Two instruments with one reach — computation
- A quintic a sliding mark reaches — computation
13 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Every third coefficient — algebra
- On the circle and never home — algebra
- The subsequence that has to exist — analysis
- A sum read from inside — analysis
- The table inside every quota — applied
- Where the rounding runs out — applied
- Solutions that come in multiples of p — computation
- Give or take twice the square root — computation
- When several pairs share the roads — discrete
- One tree for every cut — discrete
- A matrix that counts the returns — dynamics
- The folds that measure chaos — dynamics
- Half a circle against a wall — geometry
- Nearly the most means nearly round — geometry
- Two literals make an arrow — logic
- A failed search is a proof — logic
- A method that is allowed to miss — number
- Why the expansion has to repeat — number
- A sum stopped early still says something — probability
- How many get their own hat — probability
- Two opposite points that agree twice — topology
- As many cuts as colours — topology
12 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Moving a map across a product — algebra
- A denominator that reaches past the radius — analysis
- A series that converges nowhere — analysis
- A plane disguised as an arrow — algebra
- A mixture that is a population — applied
- Worth more for being seen first — applied
- The obstruction that was the only one — computation
- Equal area on a sphere, without a rectangle — computation
- A remainder read two digits at a time — discrete
- Averaging down the triangle — discrete
- A dimension for every rate of crowding — dynamics
- The room a jagged graph takes up — dynamics
- Six points on a conic, and the line they share — geometry
- Equal diagonals in a curved quadrilateral — geometry
- Every derivation is a term — logic
- The sentence between a premise and its consequence — logic
- The arcs a line crosses on its way to a number — number
- The function that sends fractions to binary — number
- The thresholds that nest — probability
- Add the odds from the end — probability
- Covering a surface multiplies its count — topology
- What a branch point subtracts — topology
9 September 2026
33 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- The square that cannot be negative — algebra
- One subtraction clears a direction — algebra
- The nearest point of a flat thing — algebra
- An error with an unknown in it — analysis
- The centre is a choice — analysis
- The points that ruin the fit — analysis
- A signal both can see — applied
- The landscape nobody is looking at — applied
- Two equilibria and no way to choose — applied
- A dissection that never comes apart — computation
- Slid, but never turned — computation
- Finitely many, and nobody says how many — computation
- The run that lands one place along — discrete
- Every entry counts the routes to it — discrete
- The carries decide the divisibility — discrete
- Infinite on one side and nought on the other — dynamics
- A carpet with two dimensions — dynamics
- A dimension from the stretching rates — dynamics
- Aimed at one focus, turned towards the other — geometry
- Two families that cross at right angles — geometry
- One sign decides which curve — geometry
- The assumption a proof pays back — logic
- A lemma, and the proof that never mentions one — logic
- The instance that has to be guessed — logic
- Two matrices that generate the tree — number
- Every rational in one sequence — number
- The fractions that beat every smaller one — number
- When the numbers are shown — probability
- Giving up on the best — probability
- Half of what an oracle takes — probability
- The symmetries a cover has of its own — topology
- A covering is a permutation — topology
- Folding a graph until it decides — topology
8 September 2026
21 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- The puzzle that is exactly half solvable — algebra
- When the label may be a matrix — algebra
- Three points, however many there are — analysis
- A wall between two bodies — analysis
- Four ways out, and what each costs — applied
- The best a code can be — computation
- Past half the distance — computation
- Sixteen polygons with one dot inside — discrete
- The dots a circle catches — discrete
- How short a cycle could be — dynamics
- Every pattern happens exactly once — dynamics
- The least area a width can hold — geometry
- The same question in space — geometry
- The game the algorithm was playing — logic
- A language that can name a set — logic
- Every class, and in equal shares — number
- The sieve that cannot finish — number
- The window where the giant is born — probability
- Sharp, or merely a threshold — probability
- One chart is never enough — topology
- The circles that fill a three-sphere — topology
5 September 2026
50 essays on geometry, discrete, topology, dynamics, computation, applied, probability, analysis, logic, number and algebra
- One dimension up, and the circles disappear — geometry
- Every site in the middle of its own cell — geometry
- When the sites are not the same size — geometry
- The tree inside the triangulation — geometry
- The shape described from outside — geometry
- Counting the paths that go wrong — discrete
- One word, and four objects — discrete
- The equation a sequence satisfies — discrete
- The solid whose corners are triangulations — discrete
- The theorem that has no version in space — discrete
- The group a space has at a point — topology
- Every cover is a subgroup — topology
- Cutting a space to find its group — topology
- Why the second group commutes — topology
- Angles survive and areas do not — topology
- The sphere that complex numbers live on — topology
- The flow that is really a map — dynamics
- A closer start buys only time — dynamics
- Stretch, fold, and what is left — dynamics
- Neither a surface nor a solid — dynamics
- Almost every number comes down — dynamics
- The heuristic that cannot be a proof — dynamics
- The test that ranks the generators — computation
- Four numbers and the rule is yours — computation
- Randomness that has to be earned — computation
- Nineteen thousand bits of state — computation
- One circle, and a straightedge — computation
- The compass that will not open — computation
- Seats to parties and places at once — applied
- The corners are whole assignments — applied
- A price for every person and task — applied
- One table, two lotteries — applied
- Where the corners stop being whole — applied
- No rule escapes the doctrinal paradox — applied
- Deciding the premises or the conclusion — applied
- How long until it forgets — probability
- The moment a giant appears — probability
- Two thresholds, not one — probability
- Finding a threshold with two moments — probability
- A staircase with no steps — analysis
- A line under every point — analysis
- The function seen from its tangents — analysis
- Where the guarantee stops — analysis
- How many worlds a formula can need — logic
- Nearly always, or nearly never — logic
- The distance a sentence can see — logic
- Infinitely many of one kind — number
- Which infinitudes are proved — number
- Two hundred and forty directions — algebra
- The only bit that survives — algebra
3 September 2026
20 essays on number, algebra, analysis, probability and logic
- The two supplements, and where the eight comes from — number
- The symbol is the sign of a shuffle — number
- One sum, squared two ways — number
- Which primes a form takes — number
- A rotation of four-space takes two of them — algebra
- The identity that multiplies sums of squares — algebra
- The integers among the quaternions — algebra
- What is lost at eight — algebra
- No interval in it, and length to spare — analysis
- Covering a set from outside — analysis
- Which functions can be added up — analysis
- A set that has no size at all — analysis
- The chain that stops — probability
- The chain that runs the same backwards — probability
- The time spent and the share held — probability
- Where the shares have nowhere to go — probability
- Two diagrams the language cannot tell apart — logic
- Worlds built out of sentences — logic
- The axiom with no property of the arrows — logic
- Necessity that means provable — logic
1 September 2026
20 essays on topology, computation, applied, geometry and discrete
- A curve that has area — topology
- Every loop is a circle in disguise — topology
- Two pieces, in every dimension — topology
- A ball whose outside is not one — topology
- Every necklace, in order — computation
- A memory of four bits — computation
- A page that knows where it is — computation
- A cycle for every pair — computation
- Five rules and one dial — applied
- The rule with no favourites — applied
- Two out of three, and never all three — applied
- Choosing what unfair means — applied
- Eight circles touching three — geometry
- Curvatures that stay whole — geometry
- The number four points agree on — geometry
- Every flat graph is a pile of circles — geometry
- Everybody's share of the chains — discrete
- The cube cut into chains — discrete
- The largest family that always meets — discrete
- How many ways to sort it — discrete
30 August 2026
20 essays on computation, logic, algebra, number and dynamics
- A field's worth of squares — computation
- The plane hiding in the squares — computation
- Nine thousand four hundred and eight — computation
- Sixteen of five hundred and seventy-six — computation
- One step in front of infinitely many — logic
- Every ordinal in base omega — logic
- Reached from below, or not at all — logic
- An ordinal as a growth rate — logic
- The only function that behaves like a volume — algebra
- One point in every big enough shape — algebra
- A determinant that counts trees — algebra
- The same sum without its minus signs — algebra
- Every partition, hidden in a product — number
- The terms that cancel almost everything — number
- The size of a number with no formula — number
- Every fifth one divides — number
- A table folded into a surface — dynamics
- A room that cannot be lit — dynamics
- The obstacle that makes a table chaotic — dynamics
- The triangle nobody can settle — dynamics
29 August 2026
20 essays on topology, number, logic, dynamics and applied
- The bottle that needs a fourth dimension — topology
- A disc sewn to a Möbius band — topology
- Two sheets over a one-sided surface — topology
- Orientation is a sign — topology
- Which roots refuse to be fractions — number
- A tail too small to be a whole number — number
- An integral that cannot be a whole number — number
- Approached too fast to be algebraic — number
- The arithmetic that loses subtraction — logic
- A line with as many points as a square — logic
- Countable, and everywhere — logic
- The size that cannot be pinned down — logic
- How fast the staircase arrives — dynamics
- The same map in different coordinates — dynamics
- The histogram an orbit leaves — dynamics
- The orbit a computer draws — dynamics
- None of the four conditions is spare — applied
- A share of the votes is not a share of the power — applied
- Too many orders to list — applied
- Sharing a cost that is not the sum of its parts — applied
28 August 2026
20 essays on algebra, analysis, geometry, probability and discrete
- The polynomial whose roots are the stretches — algebra
- The same map in a better basis — algebra
- Symmetry forces a right angle — algebra
- What a map does to a circle — algebra
- The area that names the number — analysis
- The equation with only one answer — analysis
- The exponential of a square — analysis
- The constant that counts what does not happen — analysis
- Thirteen more when one word is dropped — geometry
- The four that are allowed to cross themselves — geometry
- Six in four dimensions, and three forever after — geometry
- The five solids as three groups — geometry
- The error that does not care how many dimensions — probability
- Sampling where the answer lives — probability
- A walk that samples a distribution — probability
- Points too even to be random — probability
- Eighteen people, and the seventeen that escape — discrete
- The colouring nobody has ever seen — discrete
- Three in a row on the number line — discrete
- The sequence that cannot avoid a staircase — discrete
26 August 2026
15 essays on geometry, analysis, discrete, probability, dynamics, number, logic, computation, topology, algebra and applied
- The line with only two points on it — geometry
- The curve of the average, and the average of the curve — analysis
- A limit that forgets to be continuous — analysis
- The edge that forces a triangle — discrete
- The bottleneck is the whole story — discrete
- The widest layer and the longest chain — discrete
- The moment everything joins up — probability
- A dimension that is not a whole number — dynamics
- The orbit written as a word — dynamics
- One solution that makes all the others — number
- The choice nobody can write down — logic
- The mark that changes what is reachable — computation
- Linked, and no two of them are — topology
- The group that will not come apart — algebra
- The objection nobody can make louder — applied
25 August 2026
15 essays on topology, probability, algebra, applied, computation, logic and dynamics
- Zero can mean two different things — topology
- The subgroup that is freer than the group — topology
- When the whole histogram deviates — probability
- A shared root, found without finding it — algebra
- The crossings that will not come out even — algebra
- The lattice that runs the other way — algebra
- A multiplication that remembers the order — algebra
- The order everybody arrives in — applied
- A split nobody can walk away from — applied
- The court that contradicts itself — applied
- Equal area is enough, and equal volume is not — computation
- The straightedge buys nothing — computation
- The planes a recurrence cannot leave — computation
- The axiom is the shape of the graph — logic
- The staircase that is flat almost everywhere — dynamics
23 August 2026
15 essays on geometry, discrete, topology, dynamics, probability, logic, computation, applied and algebra
- The map that trades circles for lines — geometry
- One bottleneck and nothing else — discrete
- A polynomial that counts — discrete
- Two loops and one number — topology
- The same loop, unrolled — topology
- A bounce is a fold of the table — dynamics
- A twist that cannot avoid two points — dynamics
- The tail is not a bell — probability
- A game that decides what can be said — logic
- A proof with one rule — logic
- Every word once, around a cycle — computation
- A lottery over whole assignments — applied
- The polygon an equation forces — algebra
- The group drawn as a map — algebra
- A tower whose degrees multiply — algebra
22 August 2026
15 essays on number, geometry, probability, discrete, analysis and topology
- A tree that holds every triple — number
- Circles that are diamonds and squares — geometry
- The walk that becomes a curve — probability
- Three colours force a triangle — discrete
- A map that shrinks everything — analysis
- Where the fixed point escapes — topology
- Counting what has no formula — number
- Always one before the double — number
- The sieve written as a product — number
- The shape that averaging leaves alone — probability
- An average that never settles — probability
- How fast the bell arrives — probability
- A rectangle grown on two sides — analysis
- The flat map that fits closest — analysis
- The slope of the mirror image — analysis
21 August 2026
15 essays on geometry, topology, probability, analysis and discrete
- The rope that squares a corner — geometry
- Two right angles and the diagonal of a box — geometry
- The triangle that a globe gets wrong — geometry
- Two trees, and every edge in exactly one of them — topology
- Seven hundred and twenty degrees of gap — topology
- The solid where the answer is not two — topology
- The path folded at its first touch — probability
- Half the time is the rarest answer — probability
- Two barriers and a fair game — probability
- Where the coefficients come from — analysis
- When the period grows without bound — analysis
- The corners go first — analysis
- Five colours, and a chain that can be followed — discrete
- Counting the colourings — discrete
- Seven regions on a doughnut — discrete
20 August 2026
15 essays on topology, algebra, probability, discrete, analysis and geometry
- One line that halves them both — topology
- A loop that cannot be pulled tight — topology
- Colours that count more than three — topology
- The blocks a subgroup cuts out — algebra
- What a map throws away — algebra
- What the coefficients already know — algebra
- How far from the average a thing can be — probability
- The average settles and the wobble does not — probability
- When to stop looking — probability
- Two graphs that will not lie flat — discrete
- Sixteen trees on four points — discrete
- A curve with a corner at every point — analysis
- Almost none of it left, and still uncountably many — analysis
- Every ray comes back to the other focus — geometry
- Three trisectors and a triangle nobody expected — geometry
19 August 2026
15 essays on algebra, probability, topology, discrete, analysis and geometry
- Eight ways to leave a square alone — algebra
- Colourings nobody can tell apart — algebra
- The number that says how much room is left — algebra
- A loop that cannot miss the middle — algebra
- Nobody gets their own hat — probability
- How long until every one turns up — probability
- The rule that forgets where it came from — probability
- Which side of the line is inside — topology
- Every surface is a sphere with handles — topology
- A walk that changes one thing at a time — discrete
- Area by counting dots — discrete
- The sum that fits in one square — analysis
- The staircase that is not the diagonal — analysis
- The most area a fence can hold — geometry
- Nine points on one circle — geometry
17 August 2026
15 essays on applied
- The majority that goes in a circle — applied
- Five rules and five winners — applied
- Four conditions, and no rule that has all of them — applied
- A lie that pays — applied
- The seat that vanishes when the house grows — applied
- One cuts and the other chooses — applied
- Three people and a trimmed piece — applied
- Envy-free, up to one item — applied
- Nobody has a reason to run away — applied
- The side that proposes wins — applied
- No stable rule is safe from a lie — applied
- Two numbers that have to meet — applied
- What a constraint is worth — applied
- The value from both sides — applied
- The road that makes everyone later — applied
15 August 2026
15 essays on computation
- What two points can build — computation
- Every step is a square root — computation
- The cube that will not double — computation
- The angle that will not divide by three — computation
- Which polygons can be drawn — computation
- The circle that will not square — computation
- Distance is a picture — computation
- Sixteen spheres that fill a cube — computation
- Finding the error without reading the message — computation
- A polynomial through the gaps — computation
- The field with four elements — computation
- Every element is a power of one of them — computation
- Seven points, seven lines — computation
- A schedule where every pair meets once — computation
- The thirty-six officers — computation
14 August 2026
15 essays on logic
- A formula is a corner of a cube — logic
- One connective is enough — logic
- The map that puts neighbours side by side — logic
- Four circles cannot do it — logic
- Twenty-four out of two hundred and fifty-six — logic
- Every row, or one column — logic
- The tree that closes — logic
- Two worlds that both obey the rules — logic
- An infinite tree has an infinite path — logic
- The row that is not on the list — logic
- A list that cannot contain itself — logic
- The sentence that says it has no proof — logic
- Two injections make a bijection — logic
- A sequence that explodes and still stops — logic
- The middle that is not excluded — logic
12 August 2026
15 essays on dynamics
- The staircase that shows the whole orbit — dynamics
- A point that pulls, and a point that pushes — dynamics
- The road paved with doublings — dynamics
- A constant that does not care which map — dynamics
- A difference too small to draw — dynamics
- How fast two orbits part — dynamics
- The shape in every picture of itself — dynamics
- One c, one picture — dynamics
- Where Newton's method goes instead — dynamics
- Eight rules and a triangle — dynamics
- The rule that computes — dynamics
- Three gaps and no more — dynamics
- The orbit that must come back — dynamics
- Two lobes and no cycle — dynamics
- The question nobody can answer — dynamics
11 August 2026
15 essays on number
- The primes are what is left over — number
- There is no last prime — number
- A fraction that never closes — number
- Every fraction, exactly once — number
- How close a fraction can get — number
- One way to factor, and no other — number
- The shape of a number's divisors — number
- Numbers that are their own parts — number
- Two squares, and a lattice — number
- Every triple, on one circle — number
- Necklaces that prove a theorem — number
- Two dials at once — number
- Counting one rectangle, twice — number
- The square that cannot shrink — number
- A diagram turned on its side — number
10 August 2026
12 essays on geometry, analysis, algebra, discrete, topology and probability
- An angle that does not care where it stands — geometry
- The plane, divided by whoever is nearest — geometry
- Euclid proves it without moving anything — geometry
- The same terms, in a different order, adding to whatever is asked — analysis
- Area is the undoing of slope — analysis
- Completing the square, by completing a square — algebra
- Numbers that wrap — discrete
- Six people at a party — discrete
- Three moves, and what they cannot undo — topology
- Nothing on a sphere can be combed flat — topology
- A walk that always comes home, until it does not — probability
- The door that was not opened — probability
6 August 2026
15 essays on geometry, analysis, algebra, discrete, topology and probability
- The rectangle that eats itself — geometry
- A circle unrolled into a triangle — geometry
- Round is not the only way to be the same width — geometry
- A sum whose terms vanish and whose total does not — analysis
- One point's worth of information — analysis
- The slope of a single point — analysis
- The dot product is a shadow — algebra
- The directions a map leaves alone — algebra
- More things than boxes — discrete
- Four colours, and a proof nobody can read — discrete
- One sequence, counting everything — discrete
- Every corner pays for itself — topology
- Something always stays put — topology
- Twenty-three people — probability
- Bayes' theorem is a picture of a square — probability
2 August 2026
18 essays on geometry, analysis, algebra, discrete, topology and probability
- Two squares, four triangles, and no algebra — geometry
- Every square is a stack of odd numbers — geometry
- The oldest algorithm, drawn as a tiling — geometry
- Why the list of perfect solids stops at five — geometry
- One cone, four curves — geometry
- A sine wave is a circle seen from the side — analysis
- Adding up rectangles until they stop being rectangles — analysis
- A square wave built entirely out of round ones — analysis
- The curve that is its own slope — analysis
- A matrix is a picture of what happens to the grid — algebra
- Multiplying is turning — algebra
- Seven bridges, and the invention of throwing things away — discrete
- Pascal's triangle, in two colours — discrete
- The primes on a spiral, and a pattern nobody ordered — discrete
- The surface with one side, and what happens when it is cut — topology
- A sphere is a plane plus one point — topology
- A bell curve assembled out of coin flips — probability
- Getting pi by dropping needles on the floor — probability