Configuration space ofMoebius Kaleidocycle
HSE International Laboratory of
Algebraic Topology and Its Applications
Seminar, 8 Apr. 2022

Shizuo KAJI
(Kyushu University)
Articles, simulation codes, paper models,
3D printable models are available athttps://github.com/shizuo-kaji/KaleidocycleMy research on
Shape Generation/Analysis

Shape similarity
Shape generation

Shape deformation

Shape interpolation

4D visualisation

Image manipulation

My research on
Topological Data Analysis
Modelling ranking data withhyperplane arrangement(SIGKDD ‘21)

Configuration of geometricobjects as representation of data
in machine learning

Representing networks withconfiguration of based balls(ICML ‘20)
Pretraining CNN with topology
without natural images

Homological image features that arecomplementary to those obtained byconvolution
Image segmentation withpersistent homology(J Appl Physiol. 2021)

AdvertCubical Ripser(with T. Sudo and K. Ahara)• Fast and easy-to-use software for computing persistent homology ofimages and volume data.
https://github.com/shizuo-kaji/CubicalRipser_3dim
Tutorial for TDA libraries in Python• Concrete and simple example task demonstration for various TDA

packages that runs on Google Colab (no installation needed!)https://github.com/shizuo-kaji/TutorialTopologicalDataAnalysis
For Japanese

数学セミナー連載(2020/04 – 2021/03)

Introduction
Configuration of objects in the Euclidean spacenConfiguration of objects in R

Linkage (robot arm)
Fixed-angle chain (protein folding)
Moduli of polygon
Graph embedding
Geometric data representation
aa>b>c

From Harder et al. 2010
b>a>c

Circle-packing graph
a>c>b

b
c>a>b

b>c>a
hyperplane arrangement and ranking

c>b>a

c

Kaleidocycle

Video from Youtube

T. Tendo, 2019
The Variety of Kaleidocycles (excerpt)

Variation: Magic Cube

Video from Youtube

Variation: Mobius Kaleidocycle

S. Kaji, J. Schoenke, E. Fried, M. Grunwald, Moebius Kaleidocycle, Patent filed, JP2018-033395, Feb. 2018
The results of the joint patent were later published (without permission) in the following paper:
J. Schoenke , E. Fried, Single degree of freedom everting ring linkages with nonorientable topology, PNAS 116 (1), 2019.

Kaleidocycle plays a role in
Recreational maths
W. W. Rouse Ball (1939?)“Mathematical recreations and essays”
D. Schattschneider and W. M. Walker (1985)“M. C. Escher Kaleidocycles”
Theory of Polytopes
Rigidity theorems
Bellows theorem
Rigid Origami
Kinematics, Robotics
Bricard 6R linkage mechanism
violating Mobility formula

Bricard’s octahedronToday’s topic Mobius Kaleidocycle

Mathematics： Intersection of topology, geometry, algebraic geometry, integrable system
Engineering：Linkage mechanism with many desirable properties: 1-DoF, constant energy, falling cat
Outreach：fun and tangible object that convey various levels of mathematics

Outline
Kinematic Chain (KC)a maths model of transformable shape

Kaleidocycle:
a special family of KC<=>
constant torsional curve

Study of KC throughits Configuration Space

Motion analysis of Kaleidocycle
a flow on its configuration space

Kinematic Chains (KC)Kinematic Chain (linkage mechanism)
A kinematic chain is a collection of rigid bodies (links) connected by joints.
It is used to transfer/transform motion.

Oil drilling

A kinematic chain is closed if the underlying graph contains a cycle.
Definition: KC is a realisation of a filtered 1-dim
complex as (the 1-skeleton of) the VR complex.

Wiper

Watt’s parallel motion

Vehicle suspension

http://dynref.engr.illinois.edu/aml.html

How a pantograph works
The large green and small red triangles are always similar!

Pantogprah = Copier in Greek

components, the neural system can directly control mobility bymuscle activation. The concept of ‘freezing’ DoF is usedthroughout the motor coordination literature (e.g. Vereijken
et al., 1992; Newell and McDonald, 1994; Scholz et al., 2000;
Todorov and Jordan, 2002; Verrel et al., 2013), usually based on
the observation that motion is reduced or prohibited along aparticular joint axis. Freezing DoF is thought to be advantageouswhen a system has more DoF than needed to perform a particular
task (Todorov and Jordan, 2002; Domkin et al., 2005) or during
the early stages of learning a motor task when precise control of all
DoF is not yet possible (Berthouze and Lungarella, 2004). But
such observations do not test whether muscles are activated withthe direct objective of reducing system mobility, raising thequestion of whether mobility is itself a motor control parameter orsimply a consequence of other motor control strategies (Daley andBiewener, 2006; Biewener and Daley, 2007; Daley et al., 2007;
Nishikawa et al., 2007).

a loop between the upper and lower mandible (Fig. 5B) and forms a
Figure 4a) Ligaments within the CKC system. A stylized diagram of the human knee joint (lateral view) showing the cruciatetransient CKC (the reader can verify this with the aid of a soft foodligaments as part of a conversely-crossed 4-bar system that guides motion of the bones; and b) the ‘Jacob’s ladder’ toy
item, such as a grape). As long as the food contacts the teeth, thedemonstrating this principle.
joint between the food and teeth can be represented as a 3D sliding
Journal of Experimental Biology (2019) 222, jeb195735. doi:10.1242/jeb.195735
joint with five DoF, permitting the food item full rotational DoF and COMMENTARY
two translational DoF along the tooth surface (Fig. 5B, inset).
Applying Eqn 1, the resulting CKC has a total of seven DoF, five for
A Bass opercular mechanism and corresponding linkage model
B Ligament slack condition
D Ligament taut condition
food motion and two for jaw joint motion (Fig. 5B). As long as the1food is simply held and not punctured, the mandible can protrude or1
3
3
retrude and yaw but not depress or elevate substantially. In forming
Neurocranium
2
Suspensorium
2(and maintaining) this transient CKC, the mandible has lost a DoF.
For relatively tough foods, the mandible can continue to maintain
Operculum33
11−6(1)=5 3
3
this three-DoF system, exerting a force sufficiently strong to hold the11
food in place, but not so strong that the teeth begin to puncture the1
5food item. However, if the force exerted by the teeth on the food3
3
exceeds the puncture force, the system gains additional mobility as
the mandible can now elevate to drive the teeth through the food.
44Analogous to the previous examples of intrinsic compliant tissues
Jaw
can
be
depressed
independently
Operculum
drivenjaw
depression
CE(e.g. ligaments, muscles), the compliance of an extrinsic link (i.e.
Lower jaw
the food) creates a variable fi in the mobility equation (Eqn 1),
InteroperculoInteroperculum
giving this transient CKC conditional mobility; in this case,mandibularconditional on whether the bite force is sufficient to permit motion
ligamentof the teeth through the food. A CKC analysis brings an interesting
perspective to the mechanics of feeding. Firstly, the mobility of thisClosed Kinematic Chains in living creatures

Transient CKCs with conditional mobility

The CKC classification in Fig. 1 implies the existence of a fourthclass: a transient CKC with conditional mobility. Can such a class of
CKCs be found in nature? Feeding systems provide one example, as
they generally consist of an open kinematic chain that ‘closes’
around a compliant food item. Taking the human upper jaw and

Human mandible open chain
Mandible closed chain with food puncture
B Mandible closed chain without food puncture
igure 2 a)A The
coupling of multiple CKCs.
A fish skull (perch) showing two ofDthe
many coupled 4-bar mechanisms that
3 (large arrows) change the relative
Neurocranium3(Day, 1889); and b) & c) how
5 muscles
5
cilitate mouth opening
and sucking in of theprey
14−6(1)13−6(1)
F Human ankle agonist−antagonist muscles and linkage model
1 =8
osition of individual ‘bars’ (small arrows) and =7
alter the shape of multiple CKCs during
feeding.
Soleus m. and tendon
5

2

7

1

5
6
5

3

4

1

6

8

1

4

Journal of Experimental Biology

E Food puncture with closed chain

1

2

2

3

3

4

and tendon (mostly

C Food manipulation with closed chain3

and (mostlyTibialis
tendinous)
tertius as the bars of two distinct but connected 4-bar systems; c) the resulting ‘higher-order’
anteriorperoneusm.

Mandible

Mopen=3

3

and ligament system that guides motion of the more distal
joints (re-drawn
after Van den Bogert, 2003);1 b) the gastrocnemius
1
1

5
3

3

I Soleus isometric condition

Figure 5a - c)Tibia
Muscleswithin the CKC system. Schematic 3diagram of the hind limb of the horse showing: a) the muscle, tendon
and fibula

7

2

G Neither muscle isometric condition

2
3
2
3
3
obscuredtheby tibia)
CKC that couples
relatively small amounts of movementin the 3knee and ankle joints.
5d-f)
Schematic
diagram of the human15−6(2)=3

16−6(2)=4

lower limb showing: d) the rectus femoris and hamstring muscles as the bars of two distinct but connected 4-bar systems; e) the
H Plantarflexion and inversion

J Only inversion

higher-order 4-bar that couples flexion and extension of the hip and knee joints; and f) links with the gastrocnemius CKC.
Foot

Soleusshortens

Tibialis
anterior
lengthens

Tibialis
anterior
shortens

Medial viewFig. 5. The human mandible with a food item has a mobility that changes with food contact and manipulation. Without a food item, the mandible is asingle-link three-DoF open chain (A). If a food item is simply held between the teeth without puncture (B), the tooth–food joint acts as a 3D sliding joint with Fig. 4. Both the opercular mechanism in fishes and the human ankle joint have dynamic mobilities that depend on the mobilities of the constituent
five DoF (inset), and the system becomes a seven-DoF closed chain (orange arrows). This closed chain decreases mobility at the jaw joint to two DoF, permitting joints. The four-bar opercular linkage of largemouth bass (A) includes a link composed of ligamentous and bony elements ( purple). When the ligament isslack (B), the linkage has five DoF (orange arrows) and the lower jaw can be depressed independent of opercular rotations (C). If the ligament is taut and inprimarily yaw and protrusion (C). Food puncture adds at least one additional DoF, increasing the mobility to at least eight DoF (D) and restoring threetension, the mobility decreases to four DoF (D) and opercular elevation can drive lower jaw depression (E). An antagonistic muscle pair at the human ankle
rotational DoF to the jaw joint, including mandibular elevation (E).

• The significance of closed kinematic chains to biological movement and dynamic stability
Stephen M LevinSusan Lowell de SolórzanoSusan Lowell de SolórzanoGraham ScarrGraham Scarr, 20176
• A mobility-based classification of closed kinematic chains in biomechanics and implications for motor control
tension is transmitted through the ligament and not diverted into ankle rotation because the system is underactuated (i.e. a single
muscle controlling a two-DoF joint; Fig. 4H). However, if the soleus
lengthening of the ligament.
A. M. Olsen, 2019
Similar to a ligamentous link, an isometrically contracting muscle activates isometrically, it reduces the DoF of the system by one,l Biology(F) can be modeled as a multiloop linkage. When both muscles are non-isometric, the system has four DoF (G), including two rotational DoF at the ankle joint (H).
If one muscle contracts isometrically, the mobility decreases to three DoF (I), allowing the other muscle to control the remaining one DoF at the ankle (J).

Fundamental Questions of KC
1.
2.
3.
4.

How many independent ways to change the state?
Is the given state flexible or stiff?
Is the motion easy to control? (cf. Miura folding)Is there a continuous transition from the current state to the desired one?

These questions can be formulated and studied throughthe topology of the configuration spaceconsisting of all states of a given KC.

The configuration space of KC
is defined by equations(%" , &" )*(%! , &! )http://dynref.engr.illinois.edu/aml.html
(

)

fixed
(0,0)

fixed
(1,0)

States are identified with thesolution to a system of equations
4+, + 6+, − 8, = 0
4, − 1 , + 6,, − < , = 0
4+ − 4, , + 6, − 6+ , − = , = 0
4 variables satisfying 3 equations
We expect 1-dimensional family of solutions

Fundamental example of the configuration space
Consider the linkage below with 3 joints in R3
Let h>0 be a constant and the two ends

a = ( h, 0, 0),b = (h, 0, 0)x2<latexit sha1_base64="+QjSsg+KUy+hArNokSuRb0knQiA=">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</latexit>

are fixed to the wall. Then
x3

x1

a

b

Topology changes according to the parameter >2hTopology of the configuration space
Topological invariants of the configuration space tells a lot about the mechanics:
here’s a small dictionary
path => motion
fundamental group => non-trivial cyclic motion
topological complexity => minimum number of “if-statements” for motion planningsingular point => bifurcation, locking
number of connected components => number of mutually unreachable configurations
dimension => degree of freedom
The topic has been studied by many mathematicians including
Euler, Sylvester, Thurston, Niemann, Kapovich-Millson…
See Demaine & O’Rourke’s excellent book “Geometric Folding Algorithms”

Configuration space of a bar Linkage

CautionSelf-collision is ignored

A bar linkage L is a finite simple graph (V,E) whose edges are assigned lengths >The configuration space Mm(L) is the space of isometric embeddings of L in Rm
modulo the global symmetry.
A state is described by the real solution to a system of quadratic equations

Each connected component corresponds to “motion” of the linkage
The most notable result on bar linage isThm (Kempe's universality theorem, proposed in 19C with a wrong proof, proved in 21C)
Any bounded plane algebraic curve can be traced by some linkage(or “There is a linkage which can sign your name”)

Millennium Prize Problem
phrased in terms of kinematic chain
Even determining if the configuration space is non-empty is already difficult.
In fact, it is so even for 1-dimensional KC.

Can we place this closed KC on a line?

This problem is exactly the partition problem, which is NP-complete.
If you find an efficient solution, you will get $1 million!

Kinematic chains consisting of hinges

n: number of tetrahedra

Kaleidocycle

2n joints
5n bars

Kaleidocycle as KC

Bricard linkage
n hingesmore convenient
n barsto work with

Hinge joint
Hinge is a special type of joint which restricts the relative motion ofconnected rigid bodies to rotation around its axis.

Note: 2D bar linkages are 3D hinged
linkages with parallel hinges.

Sarrus linkage with 4 hinges

Remark: various joints (including high-dimensional ones) can be
specified by their invariant/fixed subspace.

Our definition of Kaleidocycle
An n-Kaleidocycle (Kn) is a KC consisting of n identical equifacial
tetrahedra connected by hinges.

Shape of bodies does not matter (we are ignoring self-collision)
Relation between hinges is essential.

Recap: theory of smooth curves

Adapted framed curve
A framed curve is a space curvewith an orthonomal frame assigned at each point.
A framed curve is said to be adapted if
the first vector is tangent to the curve.
Given a curve, there are canonical ways toassign framings such as
Frenet framing and Bishop framing.

Frenet frame of a discrete curve
Polyline

A- − A-.+?-.+ ≔|A- − A-.+|

Tangent
Hinge = Binormal

s.t. !! ⋅ #! = !! ⋅ #!"# = 0"−& "-/+!-/,$−$-/+

!-/+
ℓ-/+

!ℓ-

!-.+

Frenet frame(in mathematics)=
Danavit-Hartenberg
parameter(in mechanics)

A geometric model of the configuration space
Kaleidocycleó closed discrete curve"spine”of tetrahedra
hinge
Kaleidocycle

polygonal curve
binormal(up to sign)

constant torsion(congruence of tetrahedra)* Fixed-angle chain <=> constant curvatureAn algebraic model of the configuration space
Binormals (hinges)
reproduce the whole shape!! ×!!$##! ≔|!! ×!!$# |Think of a Kaleidocycle as
an arrangement of lines (hinges) in 3-space

an arrangement of planes in 4D projective space
"!"#

"

"!$#

ℓ!"#
ℓ!

ℓ!$#

a sub-variety of the product of the Grassmanians
G(4,2)

Note: we ignore the possibility of self-collision, as we can always avoid self-collision by considering thin tetrahedra(by setting the hinge length enough small compared to the length of the centre curve)

The configuration space of Kaleidocycles
Binormals determine a Kaleidocycle up to scaling. They satisfy the following equations!! ∈ *%&! ×&!"#<1 (EFGHI?HJ)
0 ≤ , ≤ - } #! ≔<0 = C−<1 (IEI EFGHI?HJ)|&! ×&!"# |(the centre polygon should be closed)

1.+

L ?-

1.+

= L <- ×<-/+ = 0-20(the angles between adjacent hinges are constant)n: numberof hinges-20(for all -)The moduli #$ (%) (resp. #" (n)) of (resp. non) oriented n-Kaleidocycles
is defined to be the space of the real solutions to the above quadratic equations
modulo the global symmetry (' 3 ↷ {+! })Configurations of a Kaleidocycle
Denote by #± %; / (⊂ #± % ) the subset of solutions
with a fixed +!"# ⋅ +! = / ∈ ℝ (cosine of the angle between adjacent hinges).
It corresponds to the space of all states of a particular Kaleidocycle.

an element of N. 6; 0
= a single state of 6-Kaleidocycle

elements of N/ 7; −0.29= motion of 7-Kaleidocycles

Panel-hinge Frameworks
Immobility of Panel-hinge bodyбெ(panel)ầἤὅἊ(hinge)੗ӳẰủẺನᡯཋ
Kaleidocycles are a special type of panel-hinge body studied in combinatorics and chemistry

Images taken from S. Tanigawa’s slides

Panel-hinge
framework:ሺ‫ܩ‬ǡ݄ሻ
Molecular conjecture (stated in Tay-Whiteley1984, solved in Kato-Tanigawas2011)provides a combinatorial characterisation for a generic panel-hinge body to be rigid ‫ ܩ‬ൌ ሺܸǡ (that‫ܧ‬ሻǣἂἻἧύ݄ǣǲ-coplanarityǳử฼ẺẴἤ
is, the configuration space is 0-dimensional)᪬ໜ ֞ ἣ἟Ἵ = Rdϋỉ࠯᩿Dimension for a generic Kaleidocycle
For a generic c, let us count the dimension of M(n;c) by

Dim of solutions = # of variables – # of equations
Fix two hinges to kill the global symmetry.
2

We have degree two freedom for each b2 , b3 , . . . , bn 1 2 S
and there the angle constraints bi · bi+1 = c which contributes –(n–1)<latexit sha1_base64="MLROl0nxt4kHDMXznI3hG55Gngg=">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</latexit>

and the closing constraints

X<latexit sha1_base64="JGjii3mdRg4Oy/kHlttqCupN2Vs=">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</latexit><latexit sha1_base64="tXRDgFp6zsoyUo3JR7qieR5UoFw=">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</latexit>bi ⇥ bi+1 = 0

which contributes –3

In total, the dimension should be2(n–2) – (n–1) – 3 = n–6

An n-Kaleidocycle has in generaln-6 degrees of freedom

Wait! 6-Kaleidocycle is mobile!
Because the equations are redundant due to its high symmetry
Such a system is said to be over-constrained
Fowler-Guest2005 gives a method to analyse over-constrained
systems using representation theory
Simple example of over-constrained system
Adding the blue bar (one quadratic equation)
has no impact on the configuration space(which is a point)Mobius Kaleidocycle= Kaleidocycle with a min/max twist angle

The space of the shape of tetrahedron(determined by the twisting angle acos(c))
1configuration space of all nonorientable n-Kaleidocycles
solutions to∑/! ×/!"# = 0/! ⋅ /!"# = 4 45678/! ⋅ /! = 1/$ = −/%

subspace corresponding to
the Mobius Kaleidocycle

cn: maximum
map c<- ⋅ <-/+ = V1

a point = a state of a Kaleidocycle
a path = motion of a KaleidocycleIm(c)inverse image-1

Open problem

Degree of Freedom of Mobius Kaleidocycles

Mobius Kaleidocycles have a single degree-of-freedom regardless of nConjecture (K-Schoenke, patented in 2018)
There is cn > 0 for each n≧7 such that/$ -; 1 is non-empty iff −1) ≤ 1 ≤ 1 (n:odd)/" -; 1 is non-empty iff −1 ≤ 1 ≤ 1) (n:odd)/" -; 1 is non-empty iff −1) ≤ 1 ≤ 1) (n:even)
Moreover,/± -; ±1) ≃ 6 # (when ±1) is the boundary value)
for each point, it is numerically checked thatthe ε-ball intersects exactly at two points

this S1 is the characteristic “everting” motion

It is very rare for a linkage to have a degenerate but non-trivial configuration spaceUnder-constrained system
They are under-constrained system: the dimension of solutions is less than
expected. This is peculiar to real solutions.
(as opposed to over-constrained systems, which exist for complex solutions as well)
Under-constrained linkages have not been studied well.
In fact, (to the best of my knowledge) Mobius Kaleidocycles are the only example
which are under-constrained and have a non-trivial configuration space.

A flow on the configuration space that
Governs the Motion of Kaleidocycles

The “everting” motion of Kaleidocycles
Goal: describe this motion as a 1-dim path in the configuration space

It looks like a soliton of some kind…

Idea: construct a flow on the moduli spacethat generates the everting motion
A solution of KdV

Physics Department of the University of Burgundy
Vidéo by Julien FATOME, Stéphane PITOIS et Guy MILLOT• Classically, the modified KdV equation
is known to generate an isoperimetricdeformation of smooth space curves.
• We consider a semi-discrete version of(discrete space & continuous time)
KdV: 7+ + 677, + 7,,, = 0
mKdV: 7+ + 67- 7, + 7,,, = 0

Motion = Curve deformation
CKC

closed discrete curve

Kaleidocycle

constant torsion and speed

motionpath in configuration space

infinitesimal motion

flow on the conf. sp.

We construct a flow by some (semi-discrete) integrable equations.

e condition (4.3) suggests the potential function ✓n in Proposition 1 asCurve deformation and Kaleidocycle’s motion

1. arc length and torsion arepreserved
✓n+1
✓n (ó
✓n ✓n+1
1 bars are rigid)n =, wn =,
2 osculating plane (è writhe
2 is preserved)
2. velocity at each vertex lies in thewrittenas (ó speed of motion is uniform at all✓ vertices) ◆3. equidistantd✓n+1 ✓n(✓n+1 + ✓n ) = 2↵ sin.
Theorem (K-Kajiwara-Park)dt
2
1. To meet the above assumptions, W- = cos ∠?- ?-.+ should satisfy the following
the authors’ knowledge, this is a novel form of the semi-discrete potentia
2✏2ct, the continuum limit ↵ = ✏ , X = ✏n + t, T = 12 t, ✏ ! 0 yields the potentia
2. In an appropriate limit, they yield the potential mKdV and the sine-Gordon equations

1✓T + (✓X )3 + ✓XXX = 0.
2
1-DoF
motion of a Mobius
ducing the potential functionin Proposition
such
that
:./ ✓−n sin: = 0. 2The
An explicit solution was constructed recently by Shigetomi

Kaleidocycle is generated by this flow.

!
An orbit in ! 9; 0.58"!

#!

Motion of a Mobius Kaleidocycle
The semi-discrete mKdV equation generates a 1-dim orbit in#$ %; / when n is odd and #" %; / when n is even for any c,where dim(#± %; / ) = % − 6 for a generic c.
Similarly, the semi-discrete sine-Gordon equation generates a 1-dim
orbit in #" %; / for any n and c.

Note the duality defined by <- ↦ −1 - <N. 2[ + 1, V ≃ N/ 2[ + 1, −V
N. 2[, V ≃ N. 2[, −V

If the 1-DoF conjecture is true,these orbits match the whole configuration space : #Conservation laws
Integrable systems have conserved quantities

Corollary to Theorem
For the deformation governed by the sine-Gordon equation, we havefor an oriented extreme Kaleidocycles,
the value itself seems to be zero

1

Introduction

Kaleidocycles represent a class of mechanisms which havepotential in applications of engineering design due to their unique
combination of characteristics: they are continuously rotatablemechanisms, they have the ability to possess multiple stable equilibrium states with a variety of characteristics, and they can be
designed as compliant mechanisms. An understanding of kaleidocycle behavior, especially the ability to characterize their stableequilibrium properties, has the potential of leading to new compact devices not previously possible. This can be particularly useful in applications, such as switches, relays, and autonomous
sensors. They also show promise as a way of improving durability
in applications where detents are currently used to achieve multiple stable states. Figures 1 and 2 show six-jointed kaleidocycles.
In this paper, we develop analytical solutions for finding thenumber of stable states of six-jointed kaleidocycles with respect
to a range of geometric parameters. Associated theorems and their
proofs are presented to support the analysis, and potential engineering applications are listed. The work is inspired by preliminary work that numerically approximated the solutions [1].
The popularization of the kaleidocycle is attributed to artist
Wallace Walker and mathematician Schattschneider who
published the book M. C. Escher Kaleidocycles in 1977 [2].
Kaleidocycles have been an object of fascination ever since, as
their continuous rotation without articulating hinges is accomplished in a visually complex pattern.
We consider six-jointed kaleidocycles consisting of a ring oftetrahedra for which the tetrahedra are all congruent. We also
require that each tetrahedron be joined along one edge to a congruent edge of another tetrahedron. Having a single degree-offreedom, the position of a kaleidocycle through its rotation can bemeasured by the angle between each consecutive pair of tetrahedra. The symmetry of the kaleidocycles causes each of the six
angles to have one of just two mutually dependent values. These
values alternate as we move around the kaleidocycle. This

Conservation laws: Bending energy
The bending (elastic) energy

can be discretised in many different ways but consider?&0)1 = @ log 1 + tan!

F!
2Fig. 1 A compliant-mechanism kaleidocycle that uses polypro
Safstenet“surrogate
al. 2016
the of the rotatingpylenefolds” analysed
to create the function
joints and the torsion springs
potential energy of K6 whose hinges
are attached torsional springs.

Theorem (K-Kajiwara-Park-Shigetomi)
Ebend is constant under the deformation by the sine-Gordon (mKdV) equation.
Contributed by the Applied Mechanics Division of ASME for publication in the
JOURNAL OF APPLIED MECHANICS. Manuscript received September 2, 2015; final
manuscript received January 21, 2016; published online February 10, 2016. Assoc.
Editor: Shaoxing Qu.

Journal of Applied Mechanics

C 2016 by ASME
Copyright V

Global quantities are mysteriously preserved!

Fig. 2 An n 5 6 kaleidocycle

MAY 2016, Vol. 83 / 051001-1

Downloaded From: http://asmedigitalcollection.asme.org/ on 02/10/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Open problem

Variational problem of bending energy
Conjecture
When non-oriented or odd n,
the minimisers of Ebend attain the extremum of the torsion.
Bending energy is determined by the curvature alone.
So the mysterious 1-DoF Kaleidocycles are characterised in two different ways;
one by the curvature and the other by the torsion.

Open problem

Coulomb & dipole energies
Imagine the centres γ of hinges are electrically charged. The potential of the system is

Imagine the hinges are dipoles. The potential of the system is

Note that thesedepend on the
global shape ofthe curve

Eclmb and Edipl are almost constant under deformation
Problem: What is the correct discritisation of the these energies
that are strictly conserved?

Topology makes it interesting!
Topological constraint has a large impact on the algebraic systemdefining the Kaleidocycles.

Topological invariants of a smooth curve
We assume the arc length of the strip is 2π.
There are two conformal invariants called twist and the writhe:

Calugareanu-White’s theorem#half twists := 2(Twist + Writhe)is an integer and an isotopy invariant

These invariants are also studied in“Topological Fluid Mechanics”.
See, for exameple,
Scheeler et al, “Complete measurement ofhelicity and its dynamics in vortex tubes"
Science 357, 2017

This type of results have beenstudied mainly in the context
of DNA supercoilingTwist and writhe for a discrete curve

Kaleidocycle with a large Tw

For a Kaleidocycle in motion
Tw remains unchanged by definition
so Wr should remain unchanged as well

Calugareanu-White’s theorem#half-twists := 2(Tw + Wr)
is an integer and an isotopy invariant
This puts a very strong topological constraint for a curve to be closed

帯に、
X において、ねじれの部分に図1 βの規則によって、+ 2 と − 2 を対応
帯. X+ 12 のねじれが α 個、− 12 のねじれが個あるとき、この合計、つまり、
11Computing Wr and Twn(X) =

1(α − β)2Twist is the average of ← over all projections
It is the difference of b(2π) and
帯 X のねじれ数という。
また帯 X において、交差の部分に図 2 の規則によって、+1 と図−11を対応させる。
b(0)parallel transported along the curveに、+ 12 のねじれが α 個、− 12 のねじれが β 個あるとき、この合計、つまり、Writhe is the average of ← over all projections.
It depends only on the centre curve

1n(X) =(α − β)図22The integration formula for Writhe and Tw are understood by the pictures below: Think of theのねじれ数という。
帯 X に、+1 の交差が x 個、−1 の交差が y 個あるとき、この合計、つまり、
angles with which one sees an intersection for a pair of infinitesimal curve portions.
310
Klenin and Langowski
X において、交差の部分に図2 の規則によって、+1 と −1 を対応させる。
can be found in R
w(X) = x − y

帯 X のライズ (writhe) という。

can be found in

total weight of the self-crossings not divided by two.
The additional factor 1/2 in Eq. (7) is due to the factthat in double integration over the same curve C every
pair of vectors dr1 and dr2 is taken into account twice.
For our further purposes, we also need an integral
representation of the twist. We assume that the spacecurve of the DNA axis is parameterized by the distance t traveled along its contour: r " r(t), 0 $ t$ L, r(0) " r(L), L being the total DNA length. Then
one of the strands can be represented in the forms " s(t) " r(t) & %a(t), where a(t) is a unit vector
perpendicular to the axis r(t), a(0) " a(L), and % has
the meaning of the DNA radius. As the twist does notdepend on %, we can formally pass to the limit % 3 0,

METHODS O
THE WRITHE

ねじれ数 n(X) は、リボンの場合は整数、メビウスの帯の場合は分数になるこに注意する。また、帯のねじれ数やライズは帯の向きの付け方によらないで決
図2る。
Klenin-Langowski2000

In computer sim
modeled by a p
segments. In thi
allowing the calcに、+1 の交差が x 個、−1 の交差が y 個あるとき、この合計、つまり、The Gauss doubof N segments c

The Writhe aOpen problem

Twist and Writhe•

Twist is the difference between b(2π) andb(0) parallel-transported along the curve.
It measures how the strip revolves around its centre curve.

•

Writhe is the number of self-intersections of the projected centre curve averaged
over all the projections.
It depends only on the centre curve and measures how the curve winds arounditself.

•None of them is an isotopy invariant but the sum is.
The sum is the number of half-twists of the strip.
If the number is odd, the strip is topologically a Mobius strip.

•

Mobius Kaleidocycles have the minimum Tw, so the maximum Wr
among Kaleidocycles with positive half-twists

Conjecture
the number of half-twists of a Kaleidocycle takes values between 3 and n-3.
(In particular, there is no Kaleidocycle with a single half-twist)

Open problem

Falling cat

Knot

The global orientation seems to evolve
without any external force(that is, keeping zero angular momentum)
Possible application to space antenna

Kaleidocycles can be made knotted.
Which knot types are realizable withwhat Tw and Wr?

c.f. Montgomery, Gauge theory of the falling catc.f. Naokawa, Extrinsically flat Möbius strips on givenknots in 3-dimensional spaceformSummary
Kaleidocycle provides links among
linkage, algebraic geometry (systems with geometric constraints),
discrete differential geometry, and integrable systems.
closed hinged linkage

discrete closed
framed curve

motionlength and torsion
preserving deformation

everting motion of
Kaleidocycle

for closed curves, topology
poses strong constraints

flow on the configuration space definedby a semi-discrete integrable systems

Applied Applied Maths

From “pure” vs “applied”
to “pure” + “applied”• Topology of a real algebraic
variety• Configuration space of a linkage• 1-dimensional singular fibre• A topological constraint• Discovery of a linkage with 1 DoF• 1 DoF is very important inEngineering

Maths.

(linking number) forces 1
DoF.

engineering.
• Everting motion• Physically interesting properties• Orbit governed by a semidiscrete integrable system• Conservation law

Pure maths and application on two sides of the same band, cycling together
make an ideal feedback loopØ

Many open problems!

Open problem

Prove that Mobius Kaleidocycles have a single-degree-of-freedomØØØThis is a problem in real algebraic geometry: the dimension of the real algebraic set
Techniques in symbolic algebra such as quantifier elimination and Groebner basis are not feasible due to the large number ofvariables (currently, only works for n≦5)
Homotopy continuation would be helpful but it does not give a solution for general nØFind conserved quantities during the eversion

ØFind an analytic expression for Kaleidocycles
Ø

Recently, a partial answer was given by S. Shigetomi (a PhD candidate at Kyushu University)ØCharacterise Mobius Kaleidocycles in terms of a variational problem of Kirchhoff rod (Langer-Singer’s theory)ØSymplectic structure on the moduli space? (c.f. Kapovich-Millson’s form on the moduli of polygons)ØWhat is the limit of nà∞ ?

ØGeneralise Mobius Kaleidocycles to find a wider family of under-constrained linkage systems

ØProve there exists no Kaleidocycle with a half twist (Mobius Kaleidocycles have 3-half twists)Ø(Stiefel) Realisation problem of persistent homology: find a Euclidean realisation of a complex which has the
specified persistent homology.

Ø

Find a (Mobius) Kaleidocycle in nature