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Essays arrive in groups rather than one at a time, and a group usually opens up a subject the collection had not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.

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