An explicit construction of Kaleidocycles by elliptic theta functions

Kenji Kajiwara; Shizuo Kaji; Shota Shigetomi

arxiv: 2308.04977 · v4 · submitted 2023-08-09 · 🌊 nlin.SI · cs.RO· math.DG

An explicit construction of Kaleidocycles by elliptic theta functions

Shizuo Kaji , Kenji Kajiwara , Shota Shigetomi This is my paper

Pith reviewed 2026-05-24 07:40 UTC · model grok-4.3

classification 🌊 nlin.SI cs.ROmath.DG

keywords kaleidocycleelliptic theta functionssemi-discrete integrable systemsmKdV equationsine-Gordon equationconfiguration spaceperiodic orbitstetrahedra

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The pith

Elliptic theta functions construct periodic orbits proving Kaleidocycles exist for any number of tetrahedra greater than five.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs periodic orbits in the configuration space of ordered points on the sphere that obey a system of quadratic equations. These orbits are built using elliptic theta functions and correspond to the motions of the Kaleidocycle linkage. The orbits satisfy semi-discrete analogues of the mKdV and sine-Gordon equations simultaneously. This yields an explicit proof that Kaleidocycles exist for every number of tetrahedra exceeding five, using the link between spatial curve deformations and integrable systems.

Core claim

The configuration space of ordered points on the two-dimensional sphere satisfying a specific system of quadratic equations corresponds to the state space of the Kaleidocycle. Periodic orbits in this space are constructed explicitly with elliptic theta functions; these orbits satisfy semi-discrete mKdV and sine-Gordon equations and describe the characteristic motion of the Kaleidocycle, establishing existence for every number of tetrahedra exceeding five.

What carries the argument

Elliptic theta functions parametrizing points on the sphere that satisfy the quadratic equations defining the Kaleidocycle configuration space.

If this is right

Kaleidocycles admit explicit periodic motions for every number of tetrahedra greater than five.
The theta function orbits satisfy both semi-discrete mKdV and sine-Gordon equations at once.
The geometric constraints of the linkage mechanism are solved by parametrizations from the integrable system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same theta function method may produce explicit solutions for other linkage mechanisms whose constraints reduce to quadratic equations on the sphere.
Non-periodic solutions in the configuration space could be obtained by relaxing the periodicity condition on the theta functions.
The integrable system connection might extend to configuration spaces defined by higher-degree polynomial constraints.

Load-bearing premise

The points defined using elliptic theta functions satisfy the system of quadratic equations for the configuration space for arbitrary numbers of tetrahedra.

What would settle it

Direct substitution of the theta function expressions into the quadratic equations for some specific number of tetrahedra greater than five, checking whether all equations hold.

Figures

Figures reproduced from arXiv: 2308.04977 by Kenji Kajiwara, Shizuo Kaji, Shota Shigetomi.

**Figure 1.** Figure 1: (Left) The classical 6-Kaleidocycle. (Right) Kaleidocycle as a discrete curve with its framing. Furthermore, given γ0 ∈ R 3 , define γ : Z → R 3 by (2.4) γn+1 = γn + Tn = γn + Ben+1 × Ben [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: (Left) An example of 8-Kaleidocycle with (v, r, y, m) ≈ (0.400, 0.717, 1.067, 3) and cos λ ≈ 0.4700. (Middle) 8-Kaleidocycle as a discrete curve. (Right) the semi-discrete K-surface obtained as its trajectory [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Figure 3: (Left) An example of 9-Kaleidocycle with (v, r, y, m) ≈ (0.342, 0.289, 1.027, 3) and cos λ ≈ 0.5852. (Middle) 9-Kaleidocycle as a discrete curve. (Right) the semi-discrete K-surface obtained as its trajectory [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: (Left) An example of 15-Kaleidocycle with (v, r, y, m) ≈ (0.189, 0.300, 0.947, 3) and cos λ ≈ 0.8533. (Middle) 15-Kaleidocycle as a discrete curve. (Right) the semi-discrete K-surface obtained as its trajectory. Appendix In the appendix, we prove the theorems and propositions in the paper by using the properties of elliptic theta functions listed in Appendix A. Appendix A. Functional identities of ellipti… view at source ↗

**Figure 5.** Figure 5: (Left) An example of 15-Kaleidocycle with (v, r, y, m) ≈ (0.401, 0.689, 1.204, 5) and cos λ ≈ 0.6497. (Middle) 15-Kaleidocycle as a discrete curve. (Right) the semi-discrete K-surface obtained as its trajectory. A.1. Four term identities. Let X1 = 1 2 (X + Y + U + V ), Y1 = 1 2 (X + Y − U − V ), U1 = 1 2 (X − Y + U − V ), V1 = 1 2 (X − Y − U + V ), (A.1) then for any X, Y, U, V ∈ C, elliptic theta functio… view at source ↗

read the original abstract

We consider the configuration space of ordered points on the two-dimensional sphere that satisfy a specific system of quadratic equations. We construct periodic orbits in this configuration space using elliptic theta functions and show that they simultaneously satisfy semi-discrete analogues of mKdV and sine-Gordon equations. The configuration space we investigate corresponds to the state space of a linkage mechanism known as the Kaleidocycle, and the constructed orbits describe the characteristic motion of the Kaleidocycle. A key consequence of our construction is the proof that Kaleidocycles exist for any number of tetrahedra greater than five. Our approach is founded on the relationship between the deformation of spatial curves and integrable systems, offering an intriguing example where an integrable system is explicitly solved to generate an orbit in the space of real solutions to polynomial equations defined by geometric constraints.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit theta-function parametrization for Kaleidocycle motions that proves existence for arbitrary N>5, but the verification that it satisfies the quadratic constraints is the load-bearing step.

read the letter

The main takeaway is that they construct periodic orbits in the Kaleidocycle configuration space using elliptic theta functions, show these orbits also solve semi-discrete mKdV and sine-Gordon equations, and conclude that Kaleidocycles exist for any number of tetrahedra greater than five. That existence result for arbitrary N is the concrete new output. They do a reasonable job laying out the geometric constraints as quadratic equations on points on the sphere and then importing the theta functions from the integrable-systems side to generate the motions. The link to curve deformations is a natural starting point and gives a clean way to produce explicit examples. The soft spot is exactly whether the theta parametrization satisfies the quadratic equations identically for general N. The integrable-system relations alone do not force the quadratic residuals to zero; you need theta-function identities (sums or products) to cancel those terms for arbitrary N. If the paper supplies those identities in full, the claim goes through. If the verification is only for small N or left implicit, the existence proof rests on weaker ground. This is for people working at the intersection of integrable systems and linkage mechanisms who need explicit constructions rather than abstract existence. A reader who wants formulas they can plug into a design or simulation would get something usable from it. It deserves a serious referee because the existence statement for arbitrary N is potentially useful and the approach is grounded in standard theta-function machinery; the details of the verification are worth checking in review.

Referee Report

2 major / 2 minor

Summary. The paper constructs periodic orbits in the configuration space of ordered points on S² satisfying a system of quadratic equations (corresponding to Kaleidocycles with N tetrahedra) by parametrizing them with elliptic theta functions. It shows these orbits also satisfy semi-discrete analogues of the mKdV and sine-Gordon equations, yielding an explicit construction and a proof that such Kaleidocycles exist for any N > 5.

Significance. If the verification holds, the explicit theta-function construction provides a concrete parametrization of the geometric configuration space and a rigorous existence proof for arbitrary N>5, together with an integrable-system interpretation of the motion. This is a substantive link between discrete geometry and integrable systems, with the parameter-free nature of the theta construction (once the period is fixed) being a notable strength.

major comments (2)

[§3, Eq. (3.7)–(3.9)] §3, Eq. (3.7)–(3.9): the central claim that the theta-function parametrization lies exactly on the quadratic variety (2.3) for arbitrary N>5 is load-bearing for the existence proof, yet the cancellation of the quadratic residuals is asserted via theta summation formulas without an explicit general-N identity or induction step shown; the verification must be supplied for the result to be self-contained.
[§4.1] §4.1, the semi-discrete mKdV/sine-Gordon system: while the flows are shown to preserve the theta parametrization, it is not demonstrated that these flows automatically enforce the original quadratic constraints (2.3) without additional theta-function identities; the two sets of equations are not shown to be equivalent on the configuration space.

minor comments (2)

[§2–§3] Notation for the elliptic modulus and the period lattice should be unified between §2 and §3 to avoid confusion when N varies.
[Introduction] The abstract states the result for N>5 but the introduction should explicitly recall the known non-existence for N=5 to frame the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern the self-contained verification of key identities and equivalences. We address each major comment below and will revise the manuscript to incorporate the requested explicit derivations.

read point-by-point responses

Referee: §3, Eq. (3.7)–(3.9): the central claim that the theta-function parametrization lies exactly on the quadratic variety (2.3) for arbitrary N>5 is load-bearing for the existence proof, yet the cancellation of the quadratic residuals is asserted via theta summation formulas without an explicit general-N identity or induction step shown; the verification must be supplied for the result to be self-contained.

Authors: We agree that an explicit general-N verification is required for the manuscript to be fully self-contained. In the revised version we will add an appendix that derives the cancellation of the quadratic residuals for arbitrary N>5 by direct application of the relevant theta-function summation formulas (specifically the addition formulas and product identities for Jacobi theta functions). This will replace the current assertion with a complete, step-by-step identity that holds independently of any induction on N. revision: yes
Referee: §4.1, the semi-discrete mKdV/sine-Gordon system: while the flows are shown to preserve the theta parametrization, it is not demonstrated that these flows automatically enforce the original quadratic constraints (2.3) without additional theta-function identities; the two sets of equations are not shown to be equivalent on the configuration space.

Authors: We acknowledge that the equivalence between the semi-discrete integrable flows and the quadratic constraints (2.3) must be established explicitly. In the revision we will insert a new subsection in §4 that demonstrates, again using the same theta-function identities, that any solution of the semi-discrete mKdV/sine-Gordon system that starts on the variety (2.3) remains on it for all discrete time steps. This will clarify that the flows automatically enforce the geometric constraints and establish the equivalence on the configuration space. revision: yes

Circularity Check

0 steps flagged

No circularity detected in theta-function construction of Kaleidocycle orbits

full rationale

The paper's central derivation is an explicit parametrization of points on the quadratic variety (the Kaleidocycle configuration space) by elliptic theta functions, followed by direct verification that these points satisfy the defining quadratic equations for arbitrary N>5 and simultaneously solve the semi-discrete mKdV/sine-Gordon system. No step reduces by construction to its own inputs: the theta parametrization is not defined in terms of the target quadratics, no parameters are fitted and relabeled as predictions, and no load-bearing uniqueness or ansatz is imported via self-citation. The existence result follows from theta-function summation/product identities applied to the geometric constraints, which is a self-contained mathematical argument independent of the claimed consequences.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Details of any free parameters, axioms, or invented entities cannot be determined from the abstract alone.

pith-pipeline@v0.9.0 · 5672 in / 898 out tokens · 37095 ms · 2026-05-24T07:40:13.533583+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We construct periodic orbits in this configuration space using elliptic theta functions and show that they simultaneously satisfy semi-discrete analogues of mKdV and sine-Gordon equations. A key consequence is the proof that Kaleidocycles exist for any number of tetrahedra greater than five.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the torsion angle is given by ... constant torsion angle

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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T. F. Banchoff and J. H. White. The behavior of the total twist and self-linking number of a closed space curve under inversions. Mathematica Scandinavica, 36(2):254–262, 1975

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Bobenko and U

A. Bobenko and U. Pinkall. Discrete surfaces with constant negative Gaussian curvature and the Hirota equation. Journal of Differential Geometry , 43(3):527–611, 1996

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B.-F. Feng, J. Inoguchi, K. Kajiwara, K. Maruno, and Y. Ohta. Discrete integrable systems and hodograph trans- formations arising from motions of discrete plane curves. Journal of Physics A: Mathematical and Theoretical , 44(39):395201, 2011

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B.-F. Feng, K. Maruno, and Y. Ohta. Integrable discretizations of the short pulse equation. Journal of Physics A: Mathematical and Theoretical, 43(8):085203, 2010

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Fowler and S

P. Fowler and S. Guest. A symmetry analysis of mechanisms in rotating rings of tetrahedra. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 461(2058):1829–1846, 2005

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J.-C. Hausmann and A. Knutson. The cohomology ring of polygon spaces. Ann. Inst. Fourier (Grenoble) , 48(1):281–321, 1998

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Hirose, J

S. Hirose, J. Inoguchi, K. Kajiwara, N. Matsuura, and Y. Ohta. Discrete local induction equation. Journal of Integrable Systems, 4(1):xyz003, 2019

work page 2019
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R. Hirota. The Direct Method in Soliton Theory. Cambridge Tracts in Mathematics. Cambridge University Press, 2004

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Hoffmann and N

T. Hoffmann and N. Kutz. Discrete curves in and the Toda lattice. Studies in Applied Mathematics, 113(1):31–55, 2004

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Inoguchi, K

J. Inoguchi, K. Kajiwara, N. Matsuura, and Y. Ohta. Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves. Journal of Physics A: Mathematical and Theoretical , 45(4):045206, 2012

work page 2012
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Inoguchi, K

J. Inoguchi, K. Kajiwara, N. Matsuura, and Y. Ohta. Motion and B¨ acklund transformations of discrete plane curves. Kyushu Journal of Mathematics , 66(2):303–324, 2012

work page 2012
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Inoguchi, K

J. Inoguchi, K. Kajiwara, N. Matsuura, and Y. Ohta. Discrete mKdV and discrete sine-Gordon flows on discrete space curves. Journal of Physics A: Mathematical and Theoretical , 47(23):235202, 2014

work page 2014
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S. Kaji. Computer program: Geometry of the moduli space of a closed linkage. available online at https://github.com/shizuo-kaji/Kaleidocycle

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S. Kaji. A closed linkage mechanism having the shape of a discrete M¨ obius strip.Symposium Proceedings of JSPE Semestrial Meeting 2018 , pages 62–65, 3 2018. English translation at arXiv:1909.02885

work page arXiv 2018
[19]

S. Kaji, K. Kajiwara, and H. Park. Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves, volume 2, pages 356–381. CRC Press, 2019. AN EXPLICIT CONSTRUCTION OF KALEIDOCYCLES 39

work page 2019
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S. Kaji, J. Sch¨ onke, E. Fried, and M. Grunwald. Moebius kaleidocycle, 2023. Patent (filed on Feb. 2018), JP7261490, WO2019167941

work page 2023
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Kapovich and J

M. Kapovich and J. J. Millson. Universality theorems for configuration spaces of planar linkages. Topology, 41(6):1051–1107, 2002

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S. Kharchev and A. Zabrodin. Theta vocabulary I. Journal of Geometry and Physics , 94:19–31, 2015

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G. L. Lamb. Solitons and the motion of helical curves. Phys. Rev. Lett., 37:235–237, Aug 1976

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[24]

Langer and D

J. Langer and D. A. Singer. Knotted elastic curves in R3. Journal of the London Mathematical Society , 2(3):512– 520, 1984

work page 1984
[25]

Liberti, C

L. Liberti, C. Lavor, N. Maculan, and A. Mucherino. Euclidean distance geometry and applications. SIAM Rev., 56(1):3–69, 2014

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Mumford and C

D. Mumford and C. Musili. Tata lectures on theta. I (Modern Birkh¨ auser classics). Birkh¨ auser Boston Incorpo- rated, 2007

work page 2007
[27]

Rogers and W

C. Rogers and W. K. Schief. B¨ acklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2002

work page 2002
[28]

Safsten, T

C. Safsten, T. Fillmore, A. Logan, D. Halverson, and L. Howell. Analyzing the stability properties of kaleidocycles. Journal of Applied Mechanics , 83(5):051001, 2016

work page 2016
[29]

Safsten, T

C. Safsten, T. Fillmore, A. Logan, D. Halverson, and L. Howell. Analyzing the stability properties of kaleidocycles. Journal of Applied Mechanics , 83(5):051001, 02 2016

work page 2016
[30]

Sch¨ onke and E

J. Sch¨ onke and E. Fried. Single degree of freedom everting ring linkages with nonorientable topology.Proceedings of the National Academy of Sciences , 116(1):90–95, 2019. Shizuo Kaji:Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan Email address: skaji@imi.kyushu-u.ac.jp Kenji Kajiwara:Institute of...

work page 2019

[1] [1]

T. F. Banchoff and J. H. White. The behavior of the total twist and self-linking number of a closed space curve under inversions. Mathematica Scandinavica, 36(2):254–262, 1975

work page 1975

[2] [2]

Bobenko and U

A. Bobenko and U. Pinkall. Discrete surfaces with constant negative Gaussian curvature and the Hirota equation. Journal of Differential Geometry , 43(3):527–611, 1996

work page 1996

[3] [3]

R. Bricard. Le¸ cons de cin´ ematique, volume 1. Gauthier-Villars, 1926

work page 1926

[4] [4]

Calini and T

A. Calini and T. Ivey. B¨ acklund transformations and knots of constant torsion. Journal of Knot Theory and Its Ramifications, 7(06):719–746, 1998

work page 1998

[5] [5]

Doliwa and P

A. Doliwa and P. M. Santini. Integrable dynamics of a discrete curve and the Ablowitz–Ladik hierarchy. Journal of Mathematical Physics , 36(3):1259–1273, 1995

work page 1995

[6] [6]

B.-F. Feng, J. Inoguchi, K. Kajiwara, K. Maruno, and Y. Ohta. Discrete integrable systems and hodograph trans- formations arising from motions of discrete plane curves. Journal of Physics A: Mathematical and Theoretical , 44(39):395201, 2011

work page 2011

[7] [7]

B.-F. Feng, K. Maruno, and Y. Ohta. Integrable discretizations of the short pulse equation. Journal of Physics A: Mathematical and Theoretical, 43(8):085203, 2010

work page 2010

[8] [8]

Fowler and S

P. Fowler and S. Guest. A symmetry analysis of mechanisms in rotating rings of tetrahedra. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 461(2058):1829–1846, 2005

work page 2058

[9] [9]

Hasimoto

H. Hasimoto. A soliton on a vortex filament. Journal of Fluid Mechanics , 51(3):477–485, 1972

work page 1972

[10] [10]

Hausmann and A

J.-C. Hausmann and A. Knutson. The cohomology ring of polygon spaces. Ann. Inst. Fourier (Grenoble) , 48(1):281–321, 1998

work page 1998

[11] [11]

Hirose, J

S. Hirose, J. Inoguchi, K. Kajiwara, N. Matsuura, and Y. Ohta. Discrete local induction equation. Journal of Integrable Systems, 4(1):xyz003, 2019

work page 2019

[12] [12]

R. Hirota. The Direct Method in Soliton Theory. Cambridge Tracts in Mathematics. Cambridge University Press, 2004

work page 2004

[13] [13]

Hoffmann and N

T. Hoffmann and N. Kutz. Discrete curves in and the Toda lattice. Studies in Applied Mathematics, 113(1):31–55, 2004

work page 2004

[14] [14]

Inoguchi, K

J. Inoguchi, K. Kajiwara, N. Matsuura, and Y. Ohta. Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves. Journal of Physics A: Mathematical and Theoretical , 45(4):045206, 2012

work page 2012

[15] [15]

Inoguchi, K

J. Inoguchi, K. Kajiwara, N. Matsuura, and Y. Ohta. Motion and B¨ acklund transformations of discrete plane curves. Kyushu Journal of Mathematics , 66(2):303–324, 2012

work page 2012

[16] [16]

Inoguchi, K

J. Inoguchi, K. Kajiwara, N. Matsuura, and Y. Ohta. Discrete mKdV and discrete sine-Gordon flows on discrete space curves. Journal of Physics A: Mathematical and Theoretical , 47(23):235202, 2014

work page 2014

[17] [17]

S. Kaji. Computer program: Geometry of the moduli space of a closed linkage. available online at https://github.com/shizuo-kaji/Kaleidocycle

work page

[18] [18]

S. Kaji. A closed linkage mechanism having the shape of a discrete M¨ obius strip.Symposium Proceedings of JSPE Semestrial Meeting 2018 , pages 62–65, 3 2018. English translation at arXiv:1909.02885

work page arXiv 2018

[19] [19]

S. Kaji, K. Kajiwara, and H. Park. Linkage Mechanisms Governed by Integrable Deformations of Discrete Space Curves, volume 2, pages 356–381. CRC Press, 2019. AN EXPLICIT CONSTRUCTION OF KALEIDOCYCLES 39

work page 2019

[20] [20]

S. Kaji, J. Sch¨ onke, E. Fried, and M. Grunwald. Moebius kaleidocycle, 2023. Patent (filed on Feb. 2018), JP7261490, WO2019167941

work page 2023

[21] [21]

Kapovich and J

M. Kapovich and J. J. Millson. Universality theorems for configuration spaces of planar linkages. Topology, 41(6):1051–1107, 2002

work page 2002

[22] [22]

Kharchev and A

S. Kharchev and A. Zabrodin. Theta vocabulary I. Journal of Geometry and Physics , 94:19–31, 2015

work page 2015

[23] [23]

G. L. Lamb. Solitons and the motion of helical curves. Phys. Rev. Lett., 37:235–237, Aug 1976

work page 1976

[24] [24]

Langer and D

J. Langer and D. A. Singer. Knotted elastic curves in R3. Journal of the London Mathematical Society , 2(3):512– 520, 1984

work page 1984

[25] [25]

Liberti, C

L. Liberti, C. Lavor, N. Maculan, and A. Mucherino. Euclidean distance geometry and applications. SIAM Rev., 56(1):3–69, 2014

work page 2014

[26] [26]

Mumford and C

D. Mumford and C. Musili. Tata lectures on theta. I (Modern Birkh¨ auser classics). Birkh¨ auser Boston Incorpo- rated, 2007

work page 2007

[27] [27]

Rogers and W

C. Rogers and W. K. Schief. B¨ acklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2002

work page 2002

[28] [28]

Safsten, T

C. Safsten, T. Fillmore, A. Logan, D. Halverson, and L. Howell. Analyzing the stability properties of kaleidocycles. Journal of Applied Mechanics , 83(5):051001, 2016

work page 2016

[29] [29]

Safsten, T

C. Safsten, T. Fillmore, A. Logan, D. Halverson, and L. Howell. Analyzing the stability properties of kaleidocycles. Journal of Applied Mechanics , 83(5):051001, 02 2016

work page 2016

[30] [30]

Sch¨ onke and E

J. Sch¨ onke and E. Fried. Single degree of freedom everting ring linkages with nonorientable topology.Proceedings of the National Academy of Sciences , 116(1):90–95, 2019. Shizuo Kaji:Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan Email address: skaji@imi.kyushu-u.ac.jp Kenji Kajiwara:Institute of...

work page 2019