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biological macromolecules
quantum crystallography
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electron crystallography
high pressure
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congress

2020 iucr xxv
2017 iucr xxiv
2014 iucr xxiii
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2008 iucr xxi
2005 iucr xx
2002 iucr xix
1999 iucr xviii
1996 iucr xvii
1993 iucr xvi
1990 iucr xv
1987 iucr xiv
1984 iucr xiii
1981 iucr xii
1978 iucr xi
1975 iucr x
1972 iucr ix
1969 iucr viii
1966 iucr vii
1963 iucr vi
1960 iucr v
1957 iucr iv
1954 iucr iii
1951 iucr ii
1948 iucr i

people

nobel prize

all
agre
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charpak
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curl
davisson
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geim
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karle
karplus
kendrew
klug
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levitt
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michel
novoselov
pauling
perutz
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roentgen
shechtman
shull
skou
smalley
steitz
sumner
thomson
walker
warshel
watson
wilkins
yonath

resources

commissions

aperiodic crystals
biological macromolecules
quantum crystallography
crystal growth and characterization of materials
crystallographic computing
crystallographic nomenclature
crystallographic teaching
crystallography in art and cultural heritage
crystallography of materials
electron crystallography
high pressure
inorganic and mineral structures
international tables
journals
magnetic structures
mathematical and theoretical crystallography
neutron scattering
NMR crystallography
powder diffraction
small-angle scattering
structural chemistry
synchrotron radiation
xafs

outreach

openlabs

calendar
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Alicia Boole Stott, the third daughter of mathematician George Boole, is probably best known for establishing the term "polytope" for a convex solid in four dimensions. Alicia was also a long time collaborator of HSM Coxeter, one of the greatest geometers of the 20th Century.

Platonic solids are regular bodies in three dimensions, such as the cube and icosahedron, and have been known for millennia. They feature prominently in the natural world wherever geometry and symmetry are important, for instance in lattices and quasi-crystals, as well as fullerenes and viruses (see the recent paper by Pierre-Philippe Dechant from Durham University and Reidun Twarock's group in York [Dechant *et al.* (2014). *Acta Cryst*. A**70**, 162-167; doi: 10.1107/S2053273313034220].

Platonic solids have counterparts in four dimensions, and the Swiss mathematician Ludwig Schlaefli and Alicia Boole Stott showed that there are six of them, five of which have very strange symmetries. Stott had a unique intuition into the geometry of four dimensions, which she visualised via three-dimensional cross-sections.

A paper by Dechant published in *Acta Crystallographica Section A: Foundations and Advances* [Dechant (2013). *Acta Cryst.* A**69**, 592-602; doi: 10.1107/S0108767313021442] shows how regular convex 4-polytopes, the analogues of the Platonic solids in four dimensions, can be constructed from three-dimensional considerations concerning the Platonic solids alone. The rotations of the Platonic solids can be interpreted naturally as four-dimensional objects, known as spinors.

These in turn generate symmetry (Coxeter) groups in four dimensions, and yield the analogues of the Platonic solids in four dimensions. In particular, this spinorial construction works for any three-dimensional symmetry (Coxeter) group, such that these cases explain all "exceptional objects" in four dimensions, i.e. those four-dimensional phenomena that do not have counterparts in arbitrary higher dimensions. This also has connections to other "exceptional phenomena" *via* Arnold's trinities and the McKay correspondence. This spinorial understanding of four-dimensional geometry explains for the first time the strange symmetries that these four dimensional objects have.

This connection between the geometry of four dimension and that of three dimensions *via* rotations/spinors has not been noticed for centuries despite many of the great mathematicians working on Platonic solids and their symmetries, and is very different from Alicia Boole Stott's way of visualizing three-dimensional sections. It sheds new light on both sides, from real three-dimensional systems with polyhedral symmetries such as (quasi) crystals, viruses and fullerenes to four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.

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