Table Of ContentFields Institute Communications 70
The Fields Institute for Research in Mathematical Sciences
Robert Connelly
Asia Ivic´ Weiss
Walter Whiteley
Editors
Rigidity and
Symmetry
Fields Institute Communications
VOLUME 70
The Fields Institute for Research in Mathematical Sciences
FieldsInstituteEditorialBoard:
CarlR.Riehm,ManagingEditor
EdwardBierstone,DirectoroftheInstitute
MatheusGrasselli,DeputyDirectoroftheInstitute
JamesG.Arthur,UniversityofToronto
KennethR.Davidson,UniversityofWaterloo
LisaJeffrey,UniversityofToronto
BarbaraLeeKeyfitz,OhioStateUniversity
ThomasS.Salisbury,YorkUniversity
NorikoYui,Queen’sUniversity
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Robert Connelly • Asia Ivic´ Weiss • Walter Whiteley
Editors
Rigidity and Symmetry
123
TheFieldsInstituteforResearch
intheMathematicalSciences
Editors
RobertConnelly AsiaIvic´Weiss
DepartmentofMathematics DepartmentofMathematicsandStatistics
CornellUniversity YorkUniversity
Ithaca,NY,USA Toronto,ON,Canada
WalterWhiteley
DepartmentofMathematicsandStatistics
YorkUniversity
Toronto,ON,Canada
ISSN1069-5265 ISSN2194-1564(electronic)
ISBN978-1-4939-0780-9 ISBN978-1-4939-0781-6(eBook)
DOI10.1007/978-1-4939-0781-6
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2014938202
MathematicsSubjectClassification(2010):05C,05E,20B,20D,20E,51E,51M,52B,52C
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Preface
The thematic program on Discrete Geometry and Applications took place at the
Fields Institute for Research in Mathematical Sciences in Toronto between July 1
and December 31, 2011. The papers included in this book are based on some
research conducted during the semester and on some of the lectures there, in
particularthoserelatedtothepartoftheprogramundertheheading“Rigidityand
Symmetry”.
This includes the study of the theory of rigidity as applied to discrete objects
suchasbarandjointframeworks,tensegrities,bodyandbarframeworksespecially
including such symmetric objects, periodic frameworks, and the combinatorics
when the objects are symmetric. When the configuration of points that define
the object is generic, the rigidity properties reduce to combinatorial properties
usuallyof some underlyinggraph.When the objectis symmetric,it automatically
becomes non-generic, but nevertheless it is possible to consider the case when
the configuration is generic modulo the symmetry group. This leads to a lot of
interesting and intricate theory. It is useful to keep in mind that there are two
approachestoasymmetricrigidobject.Incidentalrigidityiswhentheobjectisrigid
andsymmetric,butitisnotconstrainedtostayrigidunderaflex.Forcedrigidityis
when the object is rigid and symmetric, and the symmetry is part of constraints.
Bothsituationsoccurhere.
Another part concerns symmetry as applied to abstract as well as geometric
objects. Central to this theme are polytopes, the generalizations of polygons and
polyhedrato higherrank(theabstractanalogueofdimension).Severalarticlesare
devoted to regular maps on surfaces, which are just polyhedra in a general sense.
These usuallypermitoperations– replacingfaces bydifferentedge-circuits– that
change their combinatorial type, an important idea relating different maps. Such
operations can be applied in higher rank as well. Regular and chiral polytopes
(the latter roughly speaking half-regular) often correspond to interesting groups,
particularlysimple ones;such connexionsare exploredin severalpapers. Variants
of regularity, further weakening the condition, also lead to interesting questions.
Closely related to polytopes are graphs and complexes; these are the subject of
v
vi Preface
otherarticles.Moremetricalinscopearepapersonvolumeinnon-euclideanspaces,
symmetric configurationsin the plane, and a concept of rigidity of polytopesthat
providesabridgetothepreviouspart.
Ithaca,NY,USA RobertConnelly
Toronto,ON,Canada AsiaIvic´Weiss
WalterWhiteley
Contents
VolumesofPolytopesinSpacesofConstantCurvature ..................... 1
NikolayAbrosimovandAlexanderMednykh
CubicCayleyGraphsandSnarks.............................................. 27
AdemirHujdurovic´,KlavdijaKutnar,andDraganMarušicˇ
Local,DimensionalandUniversalRigidities:AUnifiedGram
MatrixApproach................................................................. 41
A.Y.Alfakih
GeometricConstructionsforSymmetric6-Configurations................. 61
LeahWrennBerman
OnExternalSymmetryGroupsofRegularMaps............................ 87
MarstonD.E.Conder,YoungSooKwon,andJozefŠiránˇ
VarianceGroupsandtheStructureofMixedPolytopes..................... 97
GabeCunningham
MobilityinSymmetry-RegularBar-and-JointFrameworks................ 117
P.W.Fowler,S.D.Guest,andB.Schulze
GenericGlobalRigidityinComplexandPseudo-EuclideanSpaces....... 131
StevenJ.GortlerandDylanP.Thurston
ChiralPolytopesandSuzukiSimpleGroups ................................. 155
IsabelHubardandDimitriLeemans
GloballyLinkedPairsofVerticesinRigidFrameworks .................... 177
BillJackson,TiborJordán,andZoltánSzabadka
BeauvilleSurfacesandGroups:ASurvey .................................... 205
GarethA.Jones
GenericRigiditywithForcedSymmetryandSparseColoredGraphs .... 227
JustinMalesteinandLouisTheran
vii
viii Contents
RigidityofRegularPolytopes................................................... 253
PeterMcMullen
HereditaryPolytopes ............................................................ 279
MarkMixer,EgonSchulte,andAsiaIvic´Weiss
OneBrickataTime:ASurveyofInductiveConstructionsin
RigidityTheory................................................................... 303
A.NixonandE.Ross
PolygonalComplexesandGraphsforCrystallographicGroups........... 325
DanielPellicerandEgonSchulte
TwoNotesonMapsandSurfaceSymmetry .................................. 345
ThomasW.Tucker
Buildingsands-TransitiveGraphs............................................. 357
RichardM.Weiss
Volumes of Polytopes in Spaces
of Constant Curvature
NikolayAbrosimovandAlexanderMednykh
Abstract We overviewvolumecalculationsforpolyhedrain Euclidean,spherical
andhyperbolicspaces.WeprovetheSforzaformulaforthevolumeofanarbitrary
tetrahedroninH3 andS3. We alsopresentsomeresults,whichprovideasolution
for the Seidel problem on the volume of non-Euclidean tetrahedron. Finally, we
consider a convex hyperbolic quadrilateral inscribed in a circle, horocycle or one
branch of equidistant curve. This is a natural hyperbolic analog of the cyclic
quadrilateralin the Euclideanplane.We find severalversionsof the Brahmagupta
formulafortheareaofsuchquadrilateral.Wealsopresentaformulafortheareaof
ahyperbolictrapezoid.
Keywords Volumes of polyhedra • Constant curvature spaces • Sforza formula
• Seidelproblem • Brahmaguptaformula
SubjectClassifications:Primary51M20;Secondary51M25,51M09,52B15
1 Volumes ofEuclideanPolyhedra
Calculatingvolumesofpolyhedraisaclassicalproblem,thathasbeenwellknown
sinceEuclidandremainsrelevantnowadays.Thisispartlydueto thefactthatthe
volumeofafundamentalpolyhedronisoneoftheinvariantsforathree-dimensional
manifold.
OneofthefirstresultsinthisdirectionwasobtainedbyTartaglia(1499–1557),
who had described an algorithm for calculating the height of a tetrahedron with
N.Abrosimov((cid:2))•A.Mednykh
SobolevInstituteofMathematics,NovosibirskStateUniversity,Novosibirsk,Russia
e-mail:[email protected];[email protected]
R.Connellyetal.(eds.),RigidityandSymmetry,FieldsInstitute 1
Communications70,DOI10.1007/978-1-4939-0781-6__1,
©SpringerScience+BusinessMediaNewYork2014
