Abstract
Twisting a yarn tilts its fibres away from the load direction. A tilt reduces the resolved component, so twist ought to lower the strength——and yet a yarn with no twist has almost no strength at all. No new theorem or law is claimed.
First, with the tilt term $\cos^2\alpha$ alone the strength is maximal at $0$ degrees and falls monotonically (Section 2).
Second, adding the grip term $1-e^{-k\tan\alpha}$ makes the strength at $0$ degrees exactly $\mathbf{0.000000}$ (Section 2).
Third, and this is the core of the paper. The maximum stands at $\mathbf{34.55}$ degrees with the value $\mathbf{0.381464}$: a product of two monotone factors has an interior maximum (Section 3).
Fourth, even at $60$ degrees the strength is still $\mathbf{0.5734}$ of the maximum and does not fall below half (Section 3).
Fifth, remove the grip term and the maximum disappears, returning to $0$ degrees. What rises is not the tilt (Section 4).
Sixth, the separator is “are the fibres continuous or of finite length” (Section 5).
Scope (what is not claimed)
No new theorem or law is claimed——the twist-angle--strength relation, the gripping length and the $\cos^2$ rule are all known in fibre science. No yarn is designed——no optimal twist count or yarn number is given. Only the mechanism by which a maximum stands at an interior point is examined, in one model. No measured value is claimed——$34.55$ degrees is the value of this model, not the optimal twist angle of cotton yarn itself. Only the sameness of shape is asserted. No theory of friction is built——the inter-fibre friction coefficient and the distribution of lateral pressure are not entered. Failure is not treated——whether fibres slip out or break first is not the subject. Ropes are not treated——the construction and layering of twisted rope are not the subject. Relation to earlier papers: Paper 357 showed that maximum power and maximum efficiency are different points——here two monotone factors make one interior point. Paper 363 showed that the sweet spot is three points——here the maximum is a single point. Paper 300 showed that same-root and different-root can be decided——tilt and grip are different-root by that criterion. Paper 367 showed that pushing the sensitivity down pushes it up elsewhere——here twist lowers one thing and raises another. What is added is putting a number on the strength at ten twist angles, locating the maximum at $34.55$ degrees with the value $0.381464$, showing that $0$ degrees gives exactly $0.000000$, confirming that removing the grip term destroys the maximum, and placing the separator at “continuous or finite-length fibres”.
1. Notation and the Question
Setting: write the strength of a yarn of twist angle $\alpha$ in the following dimensionless model.
$$S(\alpha)=\underbrace{\cos^{2}\alpha}_{\textbf{tilt}}\times\underbrace{\left(1-e^{-k\tan\alpha}\right)}_{\textbf{grip}},\qquad k=1.2$$The first factor is the projection onto the load direction together with the strain correction; the second is the fraction of fibres that can take up force without slipping out.
Observation 1.1 the first factor decreases monotonically in $\alpha$. One at $0$ degrees, zero at $90$——on this factor alone, no twist is strongest.
Observation 1.2 the second factor increases monotonically in $\alpha$. Zero at $0$ degrees, approaching one at large angles——without twist the fibres do not grip one another.
Observation 1.3 both are monotone. Not one factor has a maximum——and yet the product has an interior maximum.
Question: if twist only tilts the fibres, why is an untwisted yarn weak?
2. The Strength at Zero Degrees Is Exactly Zero
Computation 2.1: strength $S(\alpha)$ against twist angle (computed in this paper).
| Twist angle [deg] | Tilt $\cos^2\alpha$ | Strength $S$ |
|---|---|---|
| $\mathbf{0}$ | $\mathbf{1.000000}$ | $\mathbf{0.000000}$ |
| $5$ | $0.992404$ | $0.098906$ |
| $10$ | $0.969846$ | $0.184955$ |
| $15$ | $0.933013$ | $0.256548$ |
| $20$ | $0.883022$ | $0.312481$ |
| $25$ | $0.821394$ | $0.352004$ |
| $30$ | $0.750000$ | $0.374877$ |
| $40$ | $0.586824$ | $0.372432$ |
| $50$ | $0.413176$ | $0.314310$ |
| $60$ | $0.250000$ | $0.218719$ |
Observation 2.2 the first row reads $1.000000$ and $0.000000$ side by side. The tilt term is at its maximum and the strength is zero——the fibres point exactly along the load and not one of them takes up any force.
Observation 2.3 the middle column falls monotonically. From $1.000000$ down to $0.250000$——on the tilt term alone, more twist is always worse.
Observation 2.4 the right column rises and then falls. It turns between $0.374877$ at $30$ degrees and $0.372432$ at $40$——the product of two monotone factors is not monotone.
Observation 2.5 even at $5$ degrees there is only $0.098906$. The tilt term is $0.992404$, almost untouched——the missing $90$ per cent is not the fault of the tilt.
3. The Maximum Stands at $34.55$ Degrees and Sixty Degrees Stays Above Half
Computation 3.1: position and value of the maximum (computed in this paper).
| Quantity | Value |
|---|---|
| Twist angle of the maximum | $\mathbf{34.55}$ deg |
| Maximum strength | $\mathbf{0.381464}$ |
| Strength at $60$ deg / maximum | $\mathbf{0.5734}$ |
| Strength at $0$ deg / maximum | $0.000000$ |
Observation 3.2 this is the core of the paper. The maximum stands not at an endpoint but at an interior point——though both factors are monotone, the product peaks at $34.55$ degrees with $0.381464$.
Observation 3.3 the maximum is flat. $0.374877$ at $30$ degrees, $0.372432$ at $40$——$98.3$ and $97.6$ per cent of the maximum, nearly the same over a $10$-degree span.
Observation 3.4 even at $60$ degrees there is still $0.5734$ of the maximum. Although the tilt term has fallen to $0.250000$——the grip term is nearly one, so the descent is gentle.
Observation 3.5 the two sides are not symmetric. A cliff on the $0$-degree side, a slope on the $60$-degree side——too little twist is far more dangerous than too much.
Observation 3.6 the optimal twist angle of real cotton yarn is also placed near $30$ degrees. An independently known range——the model’s $34.55$ degrees does not fall outside it. This paper does not claim agreement, only sameness of shape.
4. What Rises Is Not the Tilt
Computation 4.1: with the grip term removed (computed in this paper).
| Twist angle [deg] | With grip | Without grip ($\cos^2\alpha$ only) |
|---|---|---|
| $0$ | $0.000000$ | $\mathbf{1.000000}$ |
| $34.55$ | $\mathbf{0.381464}$ | $0.678$ |
| $60$ | $0.218719$ | $0.250000$ |
| Position of the maximum | $\mathbf{34.55}$ deg | $\mathbf{0}$ deg |
Observation 4.2 remove the grip and the maximum jumps from $34.55$ degrees to $0$. The interior maximum disappears——what makes the maximum is the grip term, not the tilt term.
Observation 4.3 the tilt term contributes nothing to the position of the maximum. It sets only the rate of decline——“twist tilts the fibres and weakens them” cannot explain the rising side.
Observation 4.4 the shape is that of Paper 357. There power and efficiency peaked at different points——here two monotone factors make one interior point. In both, the maximum comes from a property of the product rather than from a “compromise”.
Observation 4.5 the reason zero degrees gives zero is friction. Without twist there is no lateral pressure between fibres——without lateral pressure there is no friction, and without friction a finite-length fibre pulls out.
5. Separator: Are the Fibres Continuous or of Finite Length
Observation 5.1 separator——does a fibre run from one end of the yarn to the other, or does it stop partway.
| Fibre | Grip term | Optimal twist angle |
|---|---|---|
| Finite length (cotton, wool) | needed | interior ($34.55$ deg) |
| Continuous (silk, synthetic filament) | not needed | $0$ deg |
Observation 5.2 this separator can be tested. Continuous-filament yarns carry almost no twist——and that is in fact the case.
Observation 5.3 the same material moves its optimum when it is cut. Chop a synthetic filament short and spin it, and twist becomes necessary——what changed is not the material but the length.
Observation 5.4 they are different-root by the criterion of Paper 300. Tilt is set by geometry, grip by friction and fibre length——a single variable, the twist angle, moves two different-root quantities at once.
Observation 5.5 to be honest——$k=1.2$ is a constant of the model, not a measurement. The position of the maximum moves with $k$——what does not move is that $0$ degrees gives exactly zero, and that removing the grip sends the maximum to an endpoint.
Observation 5.6 one last thing——it is not obvious that a product of two monotone factors should fail to be monotone. It is precisely because one of them is exactly zero at $0$ degrees that an interior maximum stands——if the grip term retained even $0.1$ at $0$ degrees, the maximum would sit closer to the endpoint.
6. Roll Call and Closing
- Strength at $0$ degrees is $0.000000$ (Computation 2.1): though the tilt is maximal.
- The tilt term falls monotonically (Observation 2.3): $1.000000$ to $0.250000$.
- The strength turns between $30$ and $40$ degrees (Observation 2.4): $0.374877$ and $0.372432$.
- The maximum is $34.55$ degrees, $0.381464$ (Computation 3.1): interior.
- The maximum is flat (Observation 3.3): above $97.6$ per cent over $10$ degrees.
- $0.5734$ of the maximum at $60$ degrees (Computation 3.1): above half.
- The sides are asymmetric (Observation 3.5): too little twist is the danger.
- Remove the grip and the maximum jumps to $0$ degrees (Observation 4.2): the grip makes the maximum.
- The separator is fibre length (Observation 5.1): filament yarns are not twisted.
- $k$ is a model constant (Observation 5.5): the position moves; the zero and the jump do not.
Closing: twisting a yarn tilts its fibres away from the load direction, and a tilt reduces the resolved component——so twist ought to lower the strength. And indeed the tilt term $\cos^2\alpha$ falls monotonically——from $1.000000$ at $0$ degrees to $0.250000$ at $60$, without exception, so on this factor alone no twist is strongest and the maximum stands at the endpoint $0$ degrees. Yet the strength of an untwisted yarn is exactly zero——in the model $S(\alpha)=\cos^{2}\alpha (1-e^{-1.2\tan\alpha})$ it is $\mathbf{0.000000}$ at $0$ degrees. The tilt term is at its maximum of $1.000000$, and not one fibre takes up any force. Even at $5$ degrees there is only $0.098906$, while the tilt term is $0.992404$ and almost untouched, so the missing $90$ per cent is not the fault of the tilt. The maximum stands at an interior point——$\mathbf{0.381464}$ at $\mathbf{34.55}$ degrees. Both factors are monotone and the product is not. It is a flat peak, holding above $97.6$ per cent of the maximum across a $10$-degree span ($0.374877$ at $30$ degrees, $0.372432$ at $40$), and at $60$ degrees there is still $\mathbf{0.5734}$ of the maximum, above half. The sides are asymmetric——a cliff at $0$ degrees, a slope at $60$——so too little twist is far more dangerous than too much. What rises is not the tilt——remove the grip term and the maximum jumps from $34.55$ degrees to $0$, and the interior maximum disappears. The tilt term contributes nothing to the position of the maximum; it sets only the rate of decline. “Twist tilts the fibres and weakens them” cannot explain the rising side. One thing does the separating——does a fibre run from one end of the yarn to the other, or does it stop partway. For finite-length fibres (cotton, wool) the grip term is needed and the optimum stands at an interior point; for continuous filaments (silk, synthetic filament) it is not, and $0$ degrees will do. Filament yarns do in fact carry almost no twist, and the same synthetic fibre, chopped short and spun, comes to require twist——what changed is not the material but the length. They are different-root by the criterion of Paper 300: tilt is set by geometry, grip by friction and fibre length. To be honest——$k=1.2$ is a constant of the model rather than a measurement, and the position of the maximum moves with $k$. What does not move is that $0$ degrees gives exactly zero, and that removing the grip sends the maximum to an endpoint. One last thing——it is not obvious that a product of two monotone factors should fail to be monotone. It is precisely because one of them is exactly zero at $0$ degrees that an interior maximum stands. If the grip term retained even $0.1$ at $0$ degrees, the maximum would sit closer to the endpoint.
References
- Hearle, J. W. S., Grosberg, P., Backer, S. (1969). Structural Mechanics of Fibers, Yarns, and Fabrics. New York: Wiley-Interscience.
- Gegauff, C. (1907). Force et élasticité des filés en coton. Bulletin de la Société Industrielle de Mulhouse.
- Pan, N. (1992). Development of a constitutive theory for short-fiber yarns. Textile Research Journal.
- Morton, W. E., Hearle, J. W. S. (2008). Physical Properties of Textile Fibres (4th ed.). Cambridge: Woodhead.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). Maximum Power and Maximum Efficiency Are Not the Same Point [Paper 357]. Zenodo. 10.5281/zenodo.22697180
- Yamagishi, Y. (2026). A Bat Has Three Sweet Spots, and They Lie $\mathbf{23}$ cm Apart [Paper 363]. Zenodo. 10.5281/zenodo.22697505
- Yamagishi, Y. (2026). Push the Sensitivity Down Somewhere and It Must Rise Somewhere Else [Paper 367]. Zenodo. 10.5281/zenodo.22697709
Reference
Yamagishi, Y. (2026). Twisting Only Tilts the Fibres Away from the Load, Yet a Yarn with No Twist Has Zero Strength [Paper 385]. YOI: https://www.yamagishiyuuki.com/yoi/385
DOI: 10.5281/zenodo.22699091
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 385. This paper is a preprint and has not been peer reviewed.