The vertex is geared

A degree-four vertex is a linkage on a sphere, and solving its closure gives the fold angle at every crease at every moment of its motion. That essay establishes the motion exists, that it has one degree of freedom, and that two theorems usually proved about the flat state turn up in the answer without being put there.

One degree of freedom is a statement about how many numbers describe the vertex: one. It says nothing about the shape of the relationship between the four fold angles, and the shape turns out to be much simpler than the motion looks.

What the fold angles do #

Watch a vertex close and the four fold angles do not look related by anything simple. On the vertex with sectors 60°, 90°, 120°, 90°, at one moment they are −58.8°, 17.2°, −58.8°, −17.2°; later −144.7°, 80.2°, −144.7°, −80.2°; later still −171.5°, 149.0°, −171.5°, −149.0°.

Two things are visible immediately. (The sign convention is the site’s: a negative fold angle is a mountain and a positive one a valley, so this vertex is three mountains and one valley throughout its motion, which is Maekawa’s split appearing on the way rather than only at the end.) Opposite creases carry equal angles, and one pair is much further along than the other at every moment — the first crease is at −58.8° when the second is at 17.2°, and at −171.5° when the second is at 149.0°. The ratio of the raw angles is 3.4, then 1.8, then 1.15: it moves a great deal.

Take the tangent of half of each and the ratio stops moving. The quantity tan(ρ₂/2) ÷ tan(ρ₁/2) is 0.267949192 at every one of the 199 points sampled across the motion, with a spread of 8 × 10⁻¹³.

The closed form, and why it is a check #

The number 0.267949192 is 2 − √3, which is tan 15°, and it is also cos((α + β)/2) ÷ cos((α − β)/2) with α = 60° and β = 90°: cos 75° ÷ cos 15° = 0.2588 ÷ 0.9659.

That form is a check rather than a derivation, because the solver never evaluates it. vertexState places the first three creases from the sector angles and a fold parameter, then solves for the fourth by intersecting two cones — the fourth crease must sit at a prescribed angle from each of its neighbours — and reads the fold angles off the resulting configuration by flattening each pair of neighbours into the plane square to their shared crease. Nowhere in that does a cosine of a half-sum appear.

So the agreement is two computations meeting: a spherical closure solved numerically, and a trigonometric expression, agreeing to 5 × 10⁻¹⁴ at every point of four different vertices.

Why the half-angle tangent #

The substitution deserves a paragraph, because a reader who has not met it will suspect it of being chosen to make the answer come out.

A spherical linkage’s closure condition is a relation among cosines and sines of the fold angles. Writing t = tan(ρ/2) turns every cosine into (1 − t²)/(1 + t²) and every sine into 2t/(1 + t²), so a trigonometric relation becomes a rational one — this is the standard substitution for exactly that reason, and it is older than the folding literature by a long way.

Having made the closure rational, the four-crease vertex’s particular relation turns out to be as simple as a rational relation can be: linear and homogeneous, which is to say a constant ratio. That is the content. The substitution is not chosen to produce the answer; it is the substitution that turns the problem into algebra, and the answer is what the algebra says.

A useful sanity check on that claim is what happens at the ends. At the flat state every fold angle is zero, so every tangent is zero and the ratio is 0/0 — undefined, and the figures start a little way in from it. At the fully folded state every angle is ±180°, every tangent is infinite, and the ratio is again indeterminate. The constant is the value throughout the interior and it is not a statement about either endpoint.

What “geared” means here #

A gear train has the property that turning one shaft turns another by a fixed factor, independent of how far it has already turned. The multiplier says a rigid vertex has exactly that property, in the coordinate tan(ρ/2) rather than in the angle itself.

In the raw fold angles the relationship is transcendental and moves; in the half-angle tangents it is a constant, and it is the second description that a machine is designed against.

So a crease pattern is a mechanism with its gear ratio built into its geometry, and the ratio is set by the sector angles alone — not by the crease lengths, not by how far it has folded, not by which branch it is on.

Why the opposite crease is exactly one #

The dashed line in the first figure sits at exactly 1 throughout, which is worth a sentence of its own.

The two creases opposite one another across the vertex fold by identical amounts at every moment — the ratio of their half-angle tangents is 1 to 10⁻¹³. That is not a separate fact; it is the same multiplier with α and β exchanged, and exchanging them leaves cos((α+β)/2) ÷ cos((α−β)/2) alone because cosine is even.

On a flat-foldable vertex the sectors come in the pattern α, β, π−α, π−β, so opposite sectors are supplementary and the vertex has a symmetry that the equality reflects. It is the reason a Miura fold’s creases can be described by one fold angle in each direction: the whole sheet has one freedom because every vertex in it has one and they all agree about which.

What this does for a designer #

The gear ratio is the quantity a mechanism designer actually wants, and it is the reason rigid origami has an engineering literature at all.

A deployable structure driven by one actuator needs to know what every other joint does when the actuator moves. “One degree of freedom” says the question has an answer; the multiplier is the answer, in closed form, computable from the crease pattern before anything is built. And because it does not vary along the motion, an actuator sized for one part of the travel is sized for all of it.

The contrast with a general linkage is the point. Most mechanisms have a transmission ratio that varies through their travel, which is why a linkage designer worries about where in the stroke the mechanism is fast and where it is strong. A rigid four-crease vertex has none of that structure in the half-angle coordinate, and its behaviour is the same at the first degree of the fold as at the last.

The Miura, and why one number drives a sheet #

The clearest consequence is the pattern the whole field is built on.

A Miura fold is one vertex repeated, so every vertex in it has the same sectors and therefore the same multiplier. Drive one crease and every crease in the sheet follows by a factor that is either the multiplier or one, depending on which family it belongs to — and the factors do not change as the sheet closes.

That is why the pattern’s packing behaviour is describable at all, and it is worth seeing as a chain: constant multipliers at every vertex, one degree of freedom for the sheet, a single parameter for the folded state, and therefore directional shrink factors that are functions of one variable. Each step depends on the one below it, and the bottom step is the one measured here.

Where the model stops #

This is one vertex. A pattern has many, and the multipliers compose along whatever path connects them. Whether the composition around a loop of vertices comes back to one is exactly the condition for the pattern to have a rigid folding at all, and it is a much harder question — a Miura fold moved by a thousandth of a cell has no isometric folded position whatever.

Panels, not paper. Everything above assumes flat rigid panels joined by ideal hinges. A real sheet bends between its creases, and the difference between panels and paper is where every practical rigid-folding difficulty lives.

The branch is fixed. A degree-four vertex has two branches — two ways to pop through — and the multiplier is computed along one of them. The other branch has its own multiplier, and which branch a vertex takes is not decided by the geometry.

Nothing here counts a freedom. The multiplier describes how the fold angles move together; it does not establish that there is exactly one degree of freedom, which the closure computation establishes, and it is not a mobility count. Counting mobility from a linkage’s structure is machinekinematics.xyz’s ground and no such count appears above.

The multiplier is a ratio and not a rate. It relates two fold angles to one another and it does not say how either of them depends on time, on an actuator’s travel, or on anything else outside the vertex. A mechanism designer wanting the second thing has to differentiate, and what comes out is not constant.

A thick panel is a different mechanism. Giving the panels thickness changes the axes the hinges sit on, which changes the vertex’s kinematics — sometimes destroying the motion entirely. Getting thickness round a corner is the standing account.

Who found this, and what it was for #

The relationship is old in the mechanisms literature and comparatively recent in the folding one, which is the usual pattern for this field.

A degree-four rigid vertex is a spherical four-bar linkage, and spherical four-bars were analysed thoroughly in the nineteenth and early twentieth centuries as machine elements. The half-angle relationship is in that literature under other names, applied to shafts and bevel gears rather than to paper. Rigid origami rediscovered it in the 1990s and 2000s — Tomohiro Tachi’s work on rigid-foldable patterns leans on it heavily, and it is what makes his software able to solve a pattern’s motion rather than sample it.

That is the same rediscovery pattern this site keeps documenting: an object studied in one subject for one purpose, found again in another because the geometry is the same. What origami added was the question. A machine designer asks what a four-bar does; a folder asks which crease patterns have one at every vertex simultaneously, which is a question about a pattern rather than about a linkage and is where the difficulty moved.

What the closed form says about which vertices are geared hardest #

The four measured multipliers can be read as a rule rather than as four numbers, and the rule is the opposite of the one a reader would reach for.

The expression is the cosine of half the sum of the two sector angles over the cosine of half their difference. So it falls as the sum grows and rises as the difference grows, and both movements are steep near the ends of their ranges.

Sort the four vertices by their sector sums. The vertex with sectors 40° and 75° has a sum of 115° and the largest multiplier, 0.5634. The two with sums of 150° — 60° and 90°, and 30° and 120° — come in at 0.2679 and 0.3660, and the second is larger because its sectors differ by ninety degrees rather than thirty. The vertex with 75° and 100° has a sum of 175° and the smallest multiplier of the four, 0.0447.

So a vertex of four nearly right angles is the most heavily geared object in the family, not the least. As the two sectors approach ninety degrees each, their sum approaches a straight angle, the cosine of half of it approaches nought, and the multiplier goes with it — meaning one crease folds almost the whole way while the other barely moves at all.

At exactly four right angles the multiplier is zero, and that is not a failure of the formula. A vertex whose sectors are all square folds along one line and leaves the other crease flat: it is a sheet folded in half, with a second crease drawn on it that never turns. The limit is telling the truth about a degenerate case.

And which are barely geared #

The other end is where the multiplier approaches one, and it is a shape nobody would call a typical vertex.

Unity needs the sum to be small and the difference smaller — two very acute sectors, and therefore two sectors close to a straight angle opposite them. A vertex with sectors of ten and twelve degrees has a multiplier of 0.985, so its two crease families fold at almost the same rate throughout the motion.

That gives the family a legible span. The gearing runs from nearly one at a vertex with two sharp spikes in it, down to nearly nothing at a vertex of four right angles, and the sector sum is the variable that carries it. Every practical crease pattern is somewhere in between, and the ones drawn on grids — where sectors are multiples of forty-five degrees and sums cluster near a straight angle — sit toward the heavily geared end.

That is worth a designer’s attention for a reason the essay’s engineering section states without the number. An actuator driving one crease of a near-square vertex is driving the other one hardly at all, so almost the whole of the motion is happening in one family and the mechanism’s travel is concentrated there. The multiplier is the ratio; where it is small, the mechanism is very unevenly loaded, and the sector sum says so from the drawing.

What the multiplier does not do #

Two things it would be easy to read into the constant are not there, and separating them keeps the claim the right size.

It does not say the vertex moves smoothly. The fold angles’ relationship is fixed and their relationship to the driving parameter is not: a vertex moves quickly through the middle of its range and slowly at the ends, which is why the four states in the motion figure are not evenly spaced in angle. The gearing is between the creases, not between a crease and time.

It does not say the vertex is easy to drive. How much torque is needed at each moment is a mechanics question about panels, hinges and whatever the sheet is made of, and none of it is in a ratio of tangents. What holds a fold shut and what a structure does under load are somebody else’s subject and are not implied by anything above.

The measurement, and how it could have gone wrong #

The figures assert three things before drawing, and each is a way the claim could fail.

The ratio must be constant. The spread over the sampled motion has to be below 10⁻⁹, and a generator whose vertex produced a moving ratio refuses rather than plotting a nearly-flat line. That is the check that guards against a coincidence at one fold angle being read as a law.

It must equal the closed form. The measured value has to agree with cos((α+β)/2) ÷ cos((α−β)/2) to the same tolerance, which is the check that guards against a solver bug producing a constant that is constant and wrong.

The opposite creases must be exactly equal. Their ratio has to be 1 to 10⁻⁹, which is a third statement about the same configuration and would break if the fold angles were being read off in the wrong order.

All three are conditions on a solved motion rather than on a formula, so all three would fail if the closure solver drifted, and none of them can be satisfied by a solver that returns a constant for the wrong reason.

Where the ladder goes next #

The immediate continuation is composition. A Miura’s vertices each have a multiplier, the multipliers compose along the sheet, and the pattern moves as one because the composition is consistent — which suggests the multiplier is the right object to state a rigid-foldability condition in. Whether the loop condition can be written as a product of multipliers returning to one is a computation this machinery could attempt.

The other direction is what happens to the constant when the vertex is not flat-foldable. Every vertex above has sectors α, β, π−α, π−β, which is the flat-foldable form; a rigid vertex need not be flat-foldable, and whether its multiplier is still constant is a question the same solver can answer for sector sets nobody has tried it on.