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Essays arrive in groups rather than one at a time. The most recent group is below in full, and every earlier one after it, newest first.

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.

5 November 2026

5 essays on designing a base, folding nobody designed, who found it, and when, tessellations and what it costs to know

Designing a base

The split that is worst on a rectangle

Two families of grafted strips share an allowance of shortfall, and the bill is a product of two sides whose sum the allowance fixes. So where the allowance goes is a question about a rectangle with a set perimeter: its area is largest when level and smallest when lopsided. On a sheet that may be any rectangle the split that levels the sides is the dearest there is, and it is barely dearer, because the gap enters squared. On a square sheet the same split is the cheapest, and the gap enters to first order, ninety-four times as heavily on five features. On a grid the first answer dissolves — the strip nearest a line decides, not the family — and the second does not.

7 figures Folding nobody designed

A crease held the other way

Hold a creased sheet on the far side of flat and its memory crosses zero after τ ln(1 + φ₀⁄φₕ), a time that contains no stiffness and no fold count. Asking what happens at that moment shows that two accounts of a corrugation's energy cannot both stand: a hinge whose memory can cross zero gives one parabola and no barrier at all. The barrier belongs to a degree-four vertex, whose two branches turn out to be at right angles in the space of its fold angles. Held on the other branch its wall dips and never falls, and its two resting states trade depth at nearly the same logarithm, with lengths in place of angles. Only the exact mirror takes every rest angle through zero together, which is the fastest way to erase a crease and the least forgiving.

7 figures Who found it, and when

Half the waterbomb's rules have no stack

The recipes for the waterbomb and the Yoshimura were written for the repeating rules that pass the vertex conditions, and nobody had asked those rules for a folded state. Asked, sixteen of the waterbomb's thirty-two have none, and four of the Yoshimura's twenty-six. Four panels are enough to refuse a waterbomb rule, in five lines, by forcing two creases on one folded line to interleave. And the waterbomb's surviving rules turn out to be exactly what a recipe of its kind was proved unable to describe: five same-and-differ clauses and no prohibition. Counting the layers shortens the waterbomb's recipe and lengthens the Yoshimura's, whose refusals are not local at all.

6 figures Tessellations

The Yoshimura folds flat at a few heights

The Yoshimura's row height has always been treated as a free dial below the equilateral: the sectors close, the counts hold, the lemma has nothing to forbid. Asked for an actual stack of layers, its usual lettering refuses nearly everywhere. Folded, its triangles turn round one centre by twice the base angle at each step, and they close on a ring only when that angle is π over a whole number: a triangle at sixty degrees, a square at forty-five, a pentagon at thirty-six. At any other height the rosette spirals, and seven panels are enough to prove it cannot pass itself. Below π over the number of rows plus two, it does not get far enough round to meet itself, so the small patch hides this too.

6 figures What it costs to know

The proportion everyone draws is the hard one

The Yoshimura is drawn equilateral, and every cost and count of it has been measured there. Moved a quarter of a step along its own height dial, a search for a stacking that took 268,650 steps takes 808, and a sheet with 9,790 folded states has 92. The equilateral pattern is the most expensive member of its family on every sheet measured, because exact coincidence removes the conditions that a panel lying across a crease would impose: from 1,024 conditions to none. A drawing chosen for being easy to construct is the one instance where almost nothing constrains the layers, and a population of one chosen drawing had been reporting that freedom as the difficulty of the pattern.

6 figures

Before that

Everything published earlier, newest first. Titles only — the cards are on the full listing.

3 November 2026

3 essays on who found it, and when, what it costs to know and rigid folding

A trapped run is holding four letters — who found it, and when
The torus keeps its tail — what it costs to know
A fragment counts creases, not length — rigid folding

30 October 2026

3 essays on tessellations, designing a base and axioms and construction

What refused the mirror was the rim — tessellations
Two corners compete across an edge — designing a base
Each new fold can cross every old one — axioms and construction

23 October 2026

3 essays on what it costs to know, curves and material and folding nobody designed

An odd strip can hook its own ends — what it costs to know
What a loop of creases carries — curves and material
Growth that only turns — folding nobody designed

21 October 2026

4 essays on who found it, and when, curves and material, rigid folding and tessellations

Two stacks shuffled in glued pairs — who found it, and when
An array is priced by its vertices — curves and material
No placement cleans every drawing — rigid folding
Clamped, a stack counts in pairs — tessellations

19 October 2026

5 essays on rigid folding, tessellations, designing a base, folding nobody designed and axioms and construction

A stack is a shared fate — rigid folding
Eight creases close the circle too — tessellations
A corner turned shows the sheet beneath — designing a base
A cut cannot tell which way a leaf grew — folding nobody designed
Two sides at sixty are enough — axioms and construction

14 October 2026

7 essays on rigid folding, axioms and construction, designing a base, folding nobody designed, who found it, and when, what it costs to know and curves and material

Splitting insures only the failures that come alone — rigid folding
The window has an exact edge — axioms and construction
Equal effort was counted in steps — designing a base
A circle sees only the average — folding nobody designed
A long run shows itself early — who found it, and when
The route built from the gluing — what it costs to know
A slit tip keeps one condition — curves and material

5 October 2026

4 essays on folding nobody designed, who found it, and when, what it costs to know and curves and material

A shared supply is the same bill — folding nobody designed
A line in the columns, a binomial in the rows — who found it, and when
The machine's misses have no short list — what it costs to know
A start is worth an inverse square — curves and material

2 October 2026

4 essays on tessellations, designing a base and axioms and construction

A twisted tube has two heights — tessellations
Four clicks make sixteen lengths — tessellations
Half the difference is a theorem — designing a base
The worst angle sets the price — axioms and construction

30 September 2026

6 essays on who found it, and when, curves and material, folding nobody designed, designing a base, rigid folding and what it costs to know

A resampled maximum can only fall — who found it, and when
Crossing the curve moves no start — curves and material
Reach taxes the extra lengths — folding nobody designed
Six creases belong to the square — designing a base
Parallel creases forgive every lean — rigid folding
A tube refuses round itself — what it costs to know

27 September 2026

6 essays on curves and material, tessellations, designing a base, axioms and construction, folding nobody designed and rigid folding

The paper reads strain — curves and material
A dial can turn the fold over — tessellations
The event is an incircle — designing a base
The rule has a width — axioms and construction
Supply decides whether lengths pay — folding nobody designed
A cut along the zigzag needs a tether — rigid folding

26 September 2026

8 essays on what it costs to know, axioms and construction, designing a base, who found it, and when, curves and material, rigid folding, tessellations and folding nobody designed

The long circles go round the middle — what it costs to know
A sheet rewards the multiples of its sides — axioms and construction
The grid belongs to the subject — designing a base
A proud hundredth buys almost nothing — who found it, and when
The floor at degree four — curves and material
The slant belongs to the line — rigid folding
A sheet that turns with the twist — tessellations
A lumen costs nothing until it does — folding nobody designed

25 September 2026

7 essays on curves and material, tessellations, rigid folding, designing a base, axioms and construction, who found it, and when and what it costs to know

Where two bands part — curves and material
Every panel holds a frame — tessellations
The slant that stacks in one depth — rigid folding
A grid chosen before the features — designing a base
An odd polygon fits like its double — axioms and construction
A perfect tree is the joins it leaves out — who found it, and when
A map refuses in small pieces — what it costs to know

24 September 2026

8 essays on designing a base, tessellations, curves and material, rigid folding, axioms and construction, who found it, and when, folding nobody designed and what it costs to know

A spacing costs one plus the other side — designing a base
The folded strip lies at its own slant — tessellations
A disc as deep as the degree — curves and material
The rows are free — rigid folding
A fold is worth what follows it — axioms and construction
No odd grid holds a perfect tree — who found it, and when
A corridor has two mouths — folding nobody designed
Nine stamps have piles of their own — what it costs to know

23 September 2026

8 essays on axioms and construction, designing a base, who found it, and when, rigid folding, folding nobody designed and what it costs to know

What the square saves — axioms and construction
One sheet down — designing a base
Turning is uphill all the way — axioms and construction
One choice with eleven answers — who found it, and when
Two faults, not four — rigid folding
The wedge belongs to one length — folding nobody designed
The deepest point pays for the paper — who found it, and when
The cheapest route crosses later — what it costs to know

19 September 2026

12 essays on tessellations, curves and material, folding nobody designed, what it costs to know and rigid folding

One number where the corners wanted four — tessellations
Every twist writes an equilibrium — tessellations
The sheet draws in crooked — tessellations
The plane the five points were in — tessellations
The direction that gets longer — tessellations
Only one side can run out — curves and material
The angle the eight does not know — folding nobody designed
In a tube the standing members lose — folding nobody designed
Each drawing has its own threshold — what it costs to know
The route, not the sheet — what it costs to know
A stub is never alone — rigid folding
An angle that turns faster than the crease — curves and material

16 September 2026

32 essays on folding nobody designed, what it costs to know, axioms and construction, designing a base, who found it, and when, curves and material, rigid folding and tessellations

A domain too short to be unique — folding nobody designed
A test that only knows one lattice — folding nobody designed
Deciding is not making — what it costs to know
The easiest strip needs the deepest reach — what it costs to know
Twos and threes run out — axioms and construction
Gauss's polygon is the expensive one — axioms and construction
The axiom that reaches furthest wastes most — axioms and construction
Six rectangles and one term — designing a base
A design that keeps its lines clear — designing a base
The width is charged in grid — designing a base
A sheet is as large as two arms — who found it, and when
The paper that will not hold a crease — who found it, and when
Eighty layers and the sheet decides the rest — who found it, and when
A length needs a scale — curves and material
The density a paper allows — curves and material
The crossing is as hard as the polygon — axioms and construction
The sheet a polygon fits exactly — axioms and construction
A paper limits spacing, not density — curves and material
A vertex creases the paper twice — curves and material
A shallow machine pays in states, not folds — what it costs to know
Fourteen states are one pile — what it costs to know
A price holds until the arrangement moves — designing a base
Rounding in the cheap direction — designing a base
A failure teaches a schedule nothing — who found it, and when
The field is empty where it would say nothing — who found it, and when
A gearing reflects stiffness squared — rigid folding
The deciding set does not move — rigid folding
Splitting a sheet buys area, not certainty — rigid folding
The pattern cheapest to trust — rigid folding
Holding a fold moves the force — folding nobody designed
A loop takes choices away — folding nobody designed
Closing the loops is not folding — tessellations

15 September 2026

20 essays on folding nobody designed, axioms and construction, designing a base, who found it, and when, curves and material and rigid folding

How far open is a question about the grip — folding nobody designed
Springs that disagree do not offer a choice — folding nobody designed
Standing up beats lying down by eight — folding nobody designed
The channel grows with what it feeds — folding nobody designed
Each fold needs its own two — axioms and construction
Counting operations is not counting power — axioms and construction
A crease that does not exist yet — axioms and construction
Every pair, not every circle — designing a base
What the condition does not decide — designing a base
The price of a limb is not its length — designing a base
The interval is wider than the number — who found it, and when
A question the record is too small to answer — who found it, and when
Some discoveries would make it worse — who found it, and when
Crowding outward costs almost nothing — curves and material
Three answers, one count — curves and material
Where a ring of divisions belongs — curves and material
Only four creases decide a Miura — rigid folding
Two drivers and one freedom — rigid folding
What a second deployment costs — rigid folding
The crease count is a reliability budget — rigid folding

13 September 2026

9 essays on who found it, and when, folding nobody designed, rigid folding, axioms and construction and curves and material

The border is where the cranes come apart — who found it, and when
The wall is the flat sheet — folding nobody designed
Two sentences for the Yoshimura — who found it, and when
Every cheap test misses a shape — folding nobody designed
A panel is not the unit of depth — rigid folding
Every even polygon beats every odd one — axioms and construction
Crowd the tucks toward the rim — curves and material
The prediction held at eight and broke at ten — who found it, and when
A chain of vertices switches all at once — folding nobody designed

12 September 2026

15 essays on folding nobody designed, who found it, and when, curves and material, rigid folding, axioms and construction and designing a base

A nest pays four a level — folding nobody designed
The fold count sets the spring — folding nobody designed
A corrugation has one resting state — folding nobody designed
The helix chooses the lattice — folding nobody designed
A row the route cannot leave — folding nobody designed
Three marks see nothing — folding nobody designed
A cut reads a slope — folding nobody designed
What the hindsight was worth — who found it, and when
Nearly every cutting fails at one crane — who found it, and when
A recipe needs degree four — who found it, and when
A tuck keeps what a gore cuts — curves and material
A straight tuck is a cone point — curves and material
Three kinds of pile — rigid folding
A stretch keeps crossings — axioms and construction
A graft needs a square line — designing a base

10 September 2026

20 essays on folding nobody designed, axioms and construction and who found it, and when

The census returns one — folding nobody designed
Four materials, four optima — folding nobody designed
The number is the angle — folding nobody designed
The surface has to be supplied — folding nobody designed
Two surfaces in one box — folding nobody designed
The container picks the member — folding nobody designed
The test measures the rim — folding nobody designed
Reachable is not cheap — axioms and construction
The field has no edge — axioms and construction
How many polygons a fold reaches — axioms and construction
What buys the reach costs the accuracy — axioms and construction
Twenty-two is a floor — axioms and construction
The square is in the answer — axioms and construction
A construction assumes its sheet — axioms and construction
A sheet has a size as well — who found it, and when
Which ceiling is binding — who found it, and when
How many wedges the paper allows — who found it, and when
Which cranes can stay joined — who found it, and when
When two of them are first attested together — who found it, and when
One lost source and the story changes — who found it, and when

5 September 2026

50 essays on flat-folding, tessellations, what it costs to know, curves and material, designing a base, rigid folding, axioms and construction, who found it, and when and folding nobody designed

The seam carries a sign — flat-folding
The band that needs an odd number — flat-folding
Closure is not the identity — flat-folding
Parity is not enough — flat-folding
The triangle a strip becomes — flat-folding
An alternating sum of angles — flat-folding
How rare a band that folds is — flat-folding
Half a rim — flat-folding
The rim adds up — flat-folding
Euler counts the gluing — flat-folding
A grid that will not close — flat-folding
The corrugation that closes on itself — tessellations
One node per panel, with the rim gone — tessellations
The turn a column costs — tessellations
The period nobody measured — tessellations
Half the slack — what it costs to know
Which pair is glued — what it costs to know
A cut is surgery — curves and material
Two holes are two conditions — curves and material
A proof in no nodes at all — what it costs to know
The cost of asking the wrong sheet — what it costs to know
One population, four sheets — what it costs to know
A map with no edges — what it costs to know
The tube a map makes — what it costs to know
A tessellation on a cylinder — tessellations
The seam that is not a symmetry — tessellations
A metamaterial with no edge — tessellations
A bottom layer on half a rim — flat-folding
The arc that arrived twice — flat-folding
The drawing does not say what is glued — flat-folding
A sheet with two edges — designing a base
The corner premium, with no corners — designing a base
A base needs an edge to point at — designing a base
A grid glued — designing a base
The symmetry a gluing adds — designing a base
A crease with no vertex to belong to — curves and material
The sixth thing that is not true — curves and material
The cut that changes nothing — curves and material
Two panels that are one panel — rigid folding
A mechanism that closes on itself — rigid folding
The tube that gets built — rigid folding
Thickness round a closed loop — rigid folding
Which of the seven survive — axioms and construction
A reference on a sheet with no corner — axioms and construction
Dividing a loop into n — axioms and construction
The proportion a band asks for — axioms and construction
Found by people not folding paper — who found it, and when
A theorem with an unstated hypothesis — who found it, and when
No format has a gluing — who found it, and when
Nothing grown has a seam — folding nobody designed

3 September 2026

20 essays on what it costs to know, flat-folding, tessellations, designing a base, curves and material, rigid folding, who found it, and when and folding nobody designed

What the rim was doing — what it costs to know
A loop that goes somewhere — flat-folding
The lettering that was proved impossible — flat-folding
A sheet with no edge — flat-folding
The rim is four letters a cell — flat-folding
The bottom layer is at the rim — flat-folding
One step per panel is a table size — flat-folding
Six creases and the same straight line — flat-folding
A knife edge nine decimals wide — flat-folding
Pruning on proofs alone — what it costs to know
The cost of proving something false — what it costs to know
Where you cut hardly matters — what it costs to know
The most decided vertex here — tessellations
Folding it flat is one similarity — tessellations
The edge was not what made it hard — designing a base
The designer's grid is the dearest thing here — designing a base
A count is not a length — curves and material
An order with no least element — rigid folding
A test imported without its hypothesis — who found it, and when
Nothing grown was cut out of anything — folding nobody designed

1 September 2026

20 essays on flat-folding, what it costs to know, tessellations, curves and material, rigid folding, designing a base, folding nobody designed and who found it, and when

The difficulty was in the coin — flat-folding
The order that is its own mirror — flat-folding
Which choice the cost lives in — what it costs to know
The order that proves nothing exists — what it costs to know
One witness or forty — flat-folding
The crease the drawing cannot show — flat-folding
A patch on a knife edge — flat-folding
A region with no lettering — flat-folding
A population nobody chose — what it costs to know
Restarting what cannot be restarted — what it costs to know
The dial and the tiling that is not alike — tessellations
A corrugation never backtracks — tessellations
The shortest crease is not a crease — curves and material
A crumple has no tail — curves and material
A search with nothing to reorder — rigid folding
The motion has no letters to choose — rigid folding
Ninety-nine in a hundred pass — designing a base
The edge is what makes it hard — designing a base
The plant's pattern is not a hard case — folding nobody designed
The cure was named first — who found it, and when

31 August 2026

20 essays on flat-folding, tessellations, what it costs to know, curves and material, axioms and construction, who found it, and when, rigid folding, designing a base and folding nobody designed

The lettering nobody could draw — flat-folding
Which condition does the refusing — flat-folding
One solution of a search nobody ran — flat-folding
Every move leaves the verdict — flat-folding
Twelve creases a micrometre long — flat-folding
Sixty-four rules, sixteen fold — tessellations
The loop is in the rule — tessellations
Four easy patches and one that is not — tessellations
Where a rule can close a loop — tessellations
Stopping is cheaper than finishing — what it costs to know
Four populations with nothing to separate — what it costs to know
Where a sector crosses sixty — flat-folding
The test that never fires on a map — what it costs to know
Rare is not hard — curves and material
The third fold cannot be listed — axioms and construction
Half the recipe is decoration — who found it, and when
The tail was named somewhere else — who found it, and when
Refused at one lettering — rigid folding
What a grid costs in circuits — designing a base
The leaf's rules are the Miura's — folding nobody designed

30 August 2026

20 essays on flat-folding, tessellations, what it costs to know, curves and material, rigid folding, designing a base, axioms and construction, who found it, and when and folding nobody designed

A proof in one pass — flat-folding
The loop a vertex cannot close — flat-folding
The loop is not the tangle — flat-folding
A contradiction is even — flat-folding
Consistent is not foldable — flat-folding
Letters that agree get rarer — tessellations
The ring is the loop — tessellations
A corrugation agrees with itself — tessellations
The rule that breaks the count — tessellations
The refusal that reads the list once — what it costs to know
A population that cannot fail — what it costs to know
The letters a crumple was given — curves and material
One cut removes one arc — curves and material
Two refusals that refuse differently — rigid folding
A tree cannot argue — designing a base
The axiom that names two folds — axioms and construction
How far from the nearest reference — axioms and construction
The file records no verdict — who found it, and when
The first thing about layers — who found it, and when
The taper decides nothing — folding nobody designed

29 August 2026

20 essays on flat-folding, tessellations, rigid folding, designing a base, what it costs to know, curves and material, axioms and construction, who found it, and when and folding nobody designed

The vertex the list does not have — flat-folding
Two creases that cross — flat-folding
Cutting a patch out of a plane — tessellations
Most of a patch is edge — flat-folding
The property a patch does not have — tessellations
The paper a pattern asks for — tessellations
One cut for a star — flat-folding
Two mechanisms at one point — rigid folding
How deep is a crossing — rigid folding
The corner that splits the shrink — designing a base
A notch is not a hole — designing a base
The order the refusals come in — what it costs to know
Drawn by the same hand — what it costs to know
Where the length sits — curves and material
A cut that reaches the edge — curves and material
An axiom may name no fold — axioms and construction
The edge was there first — axioms and construction
A file has no paper — who found it, and when
The reader decides the junction — who found it, and when
A leaf ends its pattern — folding nobody designed

27 August 2026

15 essays on flat-folding, tessellations, curves and material, designing a base, what it costs to know, folding nobody designed, rigid folding, axioms and construction and who found it, and when

No height to swap — flat-folding
The rim lies over less — flat-folding
The lettering that folds nowhere — flat-folding
The tiling the unit could not promise — tessellations
The crumple keeps its options — curves and material
Which side arrives — designing a base
The map counted from the layers — what it costs to know
Twice as thick where it is thickest — folding nobody designed
Spending the cheap paper — designing a base
How much line is on the paper — curves and material
The patterns a checker is tested on — what it costs to know
A collision is an order — rigid folding
Fourth of eight, and still not chosen for it — rigid folding
The grid a division makes — axioms and construction
Taught with a wrong reason — who found it, and when

26 August 2026

15 essays on flat-folding, rigid folding, designing a base, what it costs to know, curves and material, tessellations, axioms and construction, who found it, and when and folding nobody designed

The pieces without the list — flat-folding
Nothing slides past anything — flat-folding
A ring and a line — flat-folding
The allowance is spent at the end — rigid folding
The hardest instant — rigid folding
The symmetry the letters cannot keep — designing a base
A hole is cheap paper — designing a base
The count counts labels — what it costs to know
Where the machine catches up — what it costs to know
The decision a crumple has taken — curves and material
A cut is not local — curves and material
Thirty-two rules, thirty-two pieces — tessellations
A hole is an edge — axioms and construction
Two kinds of claim — who found it, and when
Two ceilings — folding nobody designed

24 August 2026

15 essays on flat-folding, rigid folding, designing a base, tessellations, curves and material, what it costs to know, axioms and construction, who found it, and when and folding nobody designed

The vertices nobody checks — flat-folding
Walking between two foldings — flat-folding
The creases that cannot move — flat-folding
Closing is not building — rigid folding
A tolerance is a direction — rigid folding
The pile, not the panel — rigid folding
The other grid — designing a base
Two packings, one radius — designing a base
Thirty-two rules, one object — tessellations
A cut is a licence — curves and material
Four ways to draw a pattern — what it costs to know
The cost is in the coincidences — what it costs to know
The sheet decides which points exist — axioms and construction
The half no notation records — who found it, and when
Four finders, one option — folding nobody designed

22 August 2026

15 essays on flat-folding, rigid folding, tessellations, designing a base, curves and material, axioms and construction, what it costs to know and who found it, and when

A tie is not a decision — flat-folding
The order decides the count — flat-folding
Which side is showing — flat-folding
Solving every face at once — rigid folding
Solved is not built — rigid folding
Which crease to push — rigid folding
The Miura folds two ways — tessellations
Nothing to average over — tessellations
Decided before the design — designing a base
The shapes the optimum has — designing a base
The fifth thing that is not true — curves and material
The gap between two curves — curves and material
Exact is not accurate — axioms and construction
Which vertices are the random ones — what it costs to know
The patterns nobody owns — who found it, and when

21 August 2026

15 essays on flat-folding, rigid folding, tessellations, designing a base, curves and material, axioms and construction and what it costs to know

Crimp it away and ask again — flat-folding
One marking, many objects — flat-folding
The order does not name it either — flat-folding
The condition that is not flat-foldability — rigid folding
One crease decides the sheet — rigid folding
Thickness has a sign — rigid folding
The propagation that never had to work — tessellations
The dial that decides nothing — tessellations
The whole alphabet of a grid — designing a base
The second term — designing a base
Where the rulings run out — curves and material
The sheet remembers — curves and material
Closer than a crease is wide — axioms and construction
Seven, and then twenty-two — axioms and construction
A short reason to say no — what it costs to know

20 August 2026

15 essays on flat-folding, curves and material, rigid folding, designing a base, tessellations, axioms and construction and what it costs to know

Even is not enough — flat-folding
Where the lemma says nothing — flat-folding
The shadow does not name the pattern — flat-folding
How little the conditions decide — flat-folding
A cut that removes no paper — curves and material
The crease that stops in the middle — curves and material
The family the Miura belongs to — rigid folding
Where an error goes — rigid folding
The flap nobody holds — designing a base
Spelling a tree on a grid — designing a base
The skeleton changes its mind — designing a base
The corrugation that curves — tessellations
What each axiom is worth — axioms and construction
The numbers a fold reaches — axioms and construction
The map that is not a rectangle — what it costs to know

19 August 2026

15 essays on tessellations, curves and material, designing a base, what it costs to know, flat-folding, axioms and construction and rigid folding

Any tiling makes a twist — tessellations
Where two twists share a pleat — tessellations
Fenced at both ends — tessellations
The creases a sheet gives itself — curves and material
Every facet is a layer — curves and material
One cut short of falling apart — curves and material
What the grid settles — designing a base
A no costs more than a yes — what it costs to know
A near miss is nearly as rare — flat-folding
The outline is mostly crease — flat-folding
One crossing, and then another — axioms and construction
One member of a family — axioms and construction
Cheap where it reaches — axioms and construction
Two directions that will not separate — what it costs to know
The vertex is geared — rigid folding

17 August 2026

15 essays on flat-folding, curves and material, rigid folding, axioms and construction, designing a base and tessellations

The lengths are free — flat-folding
More than one way to lie flat — flat-folding
Nothing meets at three — flat-folding
What one cut buys — curves and material
Bought with holes — curves and material
A curve has no panels — curves and material
Paper through paper — rigid folding
Nowhere to put the error — rigid folding
The only pattern that moves — rigid folding
A fold needs something to align — axioms and construction
The rectangle that keeps its shape — axioms and construction
Paying in paper — designing a base
The corner is worth four times the middle — designing a base
The cylinder the pattern chooses — tessellations
A shrink is two numbers — tessellations

16 August 2026

15 essays on flat-folding, designing a base, rigid folding, tessellations, axioms and construction and curves and material

The sheet has two sides — flat-folding
The paper is all still there — flat-folding
Almost every pattern fails — flat-folding
Where the paper stops — flat-folding
Bringing the other side to the front — designing a base
When symmetry costs — designing a base
The square is a choice — designing a base
Error is folded too — rigid folding
The base that tiles — tessellations
A unit that folds is not a tessellation — tessellations
What a corrugation costs — tessellations
Two creases at once — axioms and construction
The largest triangle in a square — axioms and construction
The biggest one that can also be folded — axioms and construction
Where curved creases meet — curves and material

14 August 2026

15 essays on folding nobody designed

A sheet that grows cannot lie flat — folding nobody designed
Which way the disc curves — folding nobody designed
The excess does not choose its waves — folding nobody designed
A crease carries no curvature — folding nobody designed
A leaf packs by corrugating — folding nobody designed
The bud chooses the pattern — folding nobody designed
Opening with nothing to pull — folding nobody designed
A wing that folds into nothing — folding nobody designed
No motor in the fold — folding nobody designed
Nothing in a body folds on a line — folding nobody designed
How much surface fits in a body — folding nobody designed
A sheet that routes itself — folding nobody designed
Two things called folding — folding nobody designed
The same corrugation in four places — folding nobody designed
The organism is not the model — folding nobody designed

13 August 2026

15 essays on who found it, and when

Nothing here is as old as it sounds — who found it, and when
The oldest book cuts the paper — who found it, and when
The name is not the date — who found it, and when
Fifty years in the wrong language — who found it, and when
A record is not a proof — who found it, and when
The kindergarten was a geometry class — who found it, and when
A schoolteacher's theorem — who found it, and when
What a dashed line can say — who found it, and when
Publishing the pattern instead of the sequence — who found it, and when
Found before it was designed — who found it, and when
The same vertex, found four times — who found it, and when
Two traditions and a merge — who found it, and when
The paper had to arrive first — who found it, and when
The star that was cut before it was proved — who found it, and when
From a shell to a solar array — who found it, and when

11 August 2026

15 essays on what it costs to know, axioms and construction, tessellations, designing a base and curves and material

The fold a machine can make — what it costs to know
The patient machine is the weak one — what it costs to know
A machine that can only crimp — what it costs to know
The machine that may choose — what it costs to know
Four questions about one sheet — what it costs to know
Hardness is about the worst one — what it costs to know
The answer is bigger than the question — what it costs to know
Where the exponent comes from — what it costs to know
What a checker cannot check — what it costs to know
What universality costs — what it costs to know
Getting close instead of getting it right — what it costs to know
The eleven-sided one nobody can fold — axioms and construction
Which polygons twist — tessellations
The molecule that does not exist — designing a base
How many times can it be halved — curves and material

10 August 2026

12 essays on flat-folding, what it costs to know, designing a base, axioms and construction, tessellations, rigid folding and curves and material

How many assignments fold — flat-folding
The gadgets that make it hard — flat-folding
The oldest open problem — what it costs to know
From a packing to a crease pattern — designing a base
The last free parameter — designing a base
Folding a strip into thirds — axioms and construction
Where the cubic comes from — axioms and construction
A property you can dial — tessellations
A state no motion reaches — rigid folding
Panels with somewhere to go — rigid folding
The crease has a radius — curves and material
One curve and one number — curves and material

7 August 2026

15 essays on tessellations, rigid folding and curves and material

A sheet with one freedom — tessellations
A square that turns — tessellations
A material made of creases — tessellations
Patterns nobody designed — tessellations
Panels instead of paper — rigid folding
What the vertex does on the way — rigid folding
The sheet has a thickness — rigid folding
Getting thickness round a corner — rigid folding
Paper that folds itself — rigid folding
Folding that gets built — rigid folding
A crease that curves — curves and material
The sculptors got there first — curves and material
What a flat sheet can become — curves and material
Four things that are not true — curves and material
Paper that stretches on purpose — curves and material

3 August 2026

19 essays on axioms and construction, flat-folding, designing a base and tessellations

One fold at a time, and there are exactly seven of them — axioms and construction
Folding beats the compass, by exactly one degree — axioms and construction
Why the list stops at seven — axioms and construction
Dividing without measuring — axioms and construction
The heptagon a compass cannot reach — axioms and construction
Two conditions at a point — flat-folding
Why the difference is two — flat-folding
The smallest sector decides — flat-folding
Local is not global — flat-folding
Which layer goes on top — flat-folding
A strip is decidable — flat-folding
One straight cut — flat-folding
A flap costs a circle — designing a base
Packing is the hard part — designing a base
What joins the flaps — designing a base
How much paper is wasted — designing a base
Designing on a grid — designing a base
Every flap on one axis — designing a base
One vertex, repeated — tessellations

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