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Color and Symmetry
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{{short description|1971 mathematics book by Arthur L. Loeb}}
{{short description|Bicycles on stamps}}
[[File:Mafeking 1d 1900.jpg|thumb|Stamp from the 1900 siege of Mafeking depicting Sgt. Major Goodyear on a bicycle]]
{{Infobox book
'''Bicycles on stamps'''
| italic title =
| name = Color and Symmetry
| image =
| image_size =
| border =
| alt =
| caption = Cover of the 1971 hardback edition
| author = [[Arthur Lee Loeb|Arthur L. Loeb]]
| country =
| language =
| subject = [[Dichromatic symmetry]], [[polychromatic symmetry]]
| publisher = [[Wiley (publisher)|Wiley Interscience]]
| translator =
| editor =
| pub_date = 1971
| media_type = Print
| pages = 179
| isbn = 978-0-471-54335-0
| oclc =
| dewey =
| congress =
| external_url =
| external_host =
}}
'''''Color and Symmetry''''' is a book by [[Arthur Lee Loeb|Arthur L. Loeb]] published by [[Wiley (publisher)|Wiley Interscience]] in 1971. The author adopts an unconventional algorithmic approach to generating the line and plane groups based on the concept of "rotocenter" (the invariant point of a rotation). He then introduces the concept of two or more colors to derive all of the plane [[dichromatic symmetry]] groups and some of the [[polychromatic symmetry]] groups.


==Structure and topics==
==Topic 1==
The book is divided into three parts. In the first part, chapters 1-7, the author introduces his "algorismic" (algorithmic) method based on "rotocenters" and "rotosimplexes" (a set of congruent rotocenters). He then derives the 7 [[frieze group]]s and the 17 [[wallpaper groups]].


In the second part, chapters 8-10, the dichromatic (black-and-white, two-colored) patterns are introduced and the 17 dichromatic line groups and the 46 black-and-white dichromatic plane groups are derived.


==Topic 2==
In the third part, chapters 11-22, polychromatic patterns (3 or more colors), polychromatic line groups, and polychromatic plane groups are derived and illustrated. Loeb's synthetic approach does not enable a comparison of colour symmetry concepts and definitions by other authors, and it is therefore not surprising that the number of polychromatic patterns he identifies are different from that published elsewhere.


==Audience==
Unusually, the author does not state the target audience for his book; his publisher, in their dust jacket blurb, say "''Color and Symmetry'' will be of primary interest on the one hand to crystallographers, chemists, material scientists, and mathematicians. On the other hand, this volume will serve the interests of those active in the fields of design, visual and environmental studies and architecture."


==Topic 3==
Only a school-level mathematical background is required to follow the author's logical development of his argument. Group theory is not used in the book, which is beneficial to readers without this specific mathematical background, but it makes some of the material more long-winded than it would be if it had been developed using standard group theory.<ref name=Klee>{{cite journal
| last1 = Klee
| first1 = W.E.
| author-link =
| date = 1972
| title = Color and Symmetry
| url = https://scripts.iucr.org/cgi-bin/paper?a08706
| journal = [[Acta Crystallographica]]
| volume = A28
| issue =
| pages = 364
| jstor =
| doi = 10.1107/S0567739472001020
| access-date = 4 April 2024
}}</ref>


Michael Holt in his review for [[Leonardo (journal)|Leonardo]] said: "In this erudite and handsomely presented monograph, then, designers should find a rich source of explicit rules for pattern-making and mathematicians and crystallographers a welcome and novel slant on symmetry operations with colours."<ref name=Holt>{{cite journal
| last1 = Holt
| first1 = Michael
| author-link =
| date = 1972
| title = Color and Symmetry
| url = https://www.muse.jhu.edu/article/597083
| journal = [[Leonardo (journal)|Leonardo]]
| volume = 5
| issue = 4
| pages = 361
| jstor = 1572601
| doi = 10.2307/1572601
| access-date = 4 April 2024
}}</ref>


==Reception==
==References==
{{Reflist}}
The book had a generally positive reception from contemporary reviewers. W.E. Klee in a review for ''[[Acta Crystallographica]]'' wrote: "''Color and Symmetry'' will surely stimulate new interest in colour symmetries and will be of special interest to crystallographers. People active in design may also profit from this book."<ref name=Klee/> [[David M. Brink|D.M. Brink]] in a review for ''Physics Bulletin'' published by the [[Institute of Physics]] said: "The book will be useful to workers with a technical interest in periodic structures and also to more general readers who are fascinated by symmetrical patterns. The illustrations encourage the reader to understand the mathematical structure underlying the patterns."<ref name=Brink>{{cite journal
| last1 = Brink
| first1 = D.M.
| author-link = David M. Brink
| date = 1972
| title = Color and Symmetry
| url =
| journal = Physics Bulletin
| volume = 23
| issue = 10
| pages = 607
| jstor =
| doi = 10.1088/0031-9112/23/10/015
| access-date =
}}</ref>


==External links==
J.D.H. Donney in a review for ''[[Physics Today]]'' said: "This book should prove useful to physicists, chemists, crystallographers
(of course), but also to decorators and designers, from textiles to ceramics. It will be enjoyed, not only by mathematicians, but by all lovers of orderliness, logic and beauty."<ref name=Donney>{{cite journal
| last1 = Donney
| first1 = J.D.H.
| author-link =
| date = 1972
| title = Color and Symmetry
| url =
| journal = [[Physics Today]]
| volume = 25
| issue = 12
| pages = 53,55
| jstor =
| doi = 10.1063/1.3071144
| access-date =
}}</ref> [[David Harker]] in a review for ''[[Science (journal)|Science]]'' said: "It may well be that this work will become a classic essay on planar color symmetry"<ref name=Harker>{{cite journal
| last1 = Harker
| first1 = David
| author-link = David Harker
| date = 1972
| title = Planes, Solids, and Nolids
| url =
| journal = [[Science (journal)|Science]]
| volume = 176
| issue = 4025
| pages = 653-654
| jstor = 1734491
| doi =
| access-date =
}}</ref>


* [https://www.bicyclestamps.de/ Bicycle Philately: The home of bicycle stamps collectors]
==Criticism==

The author's idiosyncratic approach was not adopted by researchers in the field, and later assessments of Loeb's contribution to color symmetry were more critical of his work than earlier reviewers had been. [[Marjorie Senechal]] said that Loeb's work on polychromatic patterns, whilst not wrong, imposed artificial restrictions which meant that some valid colored patterns with three or more colors were excluded from his lists.<ref name=Senechal1>{{cite journal
| last1 = Senechal
| first1 = Marjorie
| author-link = Marjorie Senechal
| date = 1975
| language =
| title = Point groups and color symmetry
| url =
| journal = [[Zeitschrift für Kristallographie – Crystalline Materials|Zeitschrift für Kristallographie]]
| volume = 142
| issue = 1-2
| pages = 1-23
| jstor =
| doi = 10.1524/zkri.1975.142.1-2.1
| access-date =
}}</ref><ref name=Senechal2>{{cite journal
| last1 = Senechal
| first1 = Marjorie
| author-link =
| date = 1979
| language =
| title = Color groups
| url = https://www.sciencedirect.com/science/article/pii/0166218X79900143/pdf?md5=08822e0e6dba4abfcf7b3f4631ffbb26&pid=1-s2.0-0166218X79900143-main.pdf
| journal = [[Discrete Applied Mathematics]]
| volume = 1
| issue = 1-2
| pages = 51-73
| jstor =
| doi = 10.1016/0166-218X(79)90014-3
| access-date = 4 April 2024
}}</ref><ref name=Senechal3>{{cite journal
| last1 = Senechal
| first1 = Marjorie
| author-link =
| date = 1983
| language =
| title = Color Symmetry and Colored Polyhedra
| url =
| journal = [[Acta Crystallographica]]
| volume = A39
| issue =
| pages = 505-511
| jstor =
| doi = 10.1107/S0108767383000987
| access-date =
}}</ref>

[[Rolph Ludwig Edward Schwarzenberger|R.L.E. Schwarzenberger]] in 1980 said: "The study of colour symmetry has been bedevilled by a lack of precise definitions when the number of colours is greater than two ... it is unfortunate that this paper<ref>{{cite journal
| last1 = van der Waerden
| first1 = B.L.
| last2 = Burkhardt
| first2 = J.J.
| author-link =
| date = 1961
| language =
| title = Farbgruppen
| url =
| journal = [[Zeitschrift für Kristallographie – Crystalline Materials|Zeitschrift für Kristallographie]]
| volume = 115
| issue = 3-4
| pages = 231-234
| jstor =
| doi = 10.1524/zkri.1961.115.3-4.231
| access-date =
}}</ref> was apparently ignored by [[Alexei Vasilievich Shubnikov|Shubnikov]] and Loeb whose books give incomplete and unsystematic listings."<ref name=Schwarzenberger1>{{cite book
| last1 = Schwarzenberger
| first1 = R.L.E.
| author-link = Rolph Ludwig Edward Schwarzenberger
| date = 1980
| title = N-dimensional crystallography
| url =
| url-access =
| location = New York
| publisher = [[Pearson plc#Publishing businesses 1921 to 1997|Pitman Publishing]]
| page = 134-135
| isbn = 978-0-8224-8468-4
}}</ref> In a 1984 review paper Schwarzenberger remarks: "... these authors [including Loeb] confine themselves to a restricted class of colour group ... for N&nbsp;>&nbsp;2 the effect is to dramatically limit the number of colour groups considered."<ref name=Schwarzenberger2>{{cite journal
| last1 = Schwarzenberger
| first1 = R.L.E.
| author-link =
| date = 1984
| language =
| title = Colour symmetry
| url =
| journal = [[London Mathematical Society#Publications|Bulletin of the London Mathematical Society]]
| volume = 16
| issue = 3
| pages = 209-240
| jstor =
| doi = 10.1112/blms/16.3.209
| access-date =
}}</ref>

[[Branko Grünbaum]] and [[Geoffrey Colin Shephard|G.C. Shephard]] in their book ''[[Tilings and patterns]]'' gave an assessment of previous work in the field. Commenting on ''Color and Symmetry'' they said:"Loeb gives an original, interesting and satisfactory account of the 2-color groups ... unfortunately when discussing multicolor patterns, Loeb restricts the admissible color changes so severely that he obtains a total of only 54 periodic ''k''-color configurations with k&nbsp;≥&nbsp;3."<ref name=Grünbaum>{{cite book
| last1 = Grünbaum
| first1 = Branko
| last2 = Shephard
| first2 = G.C.
| author-link =
| date = 1987
| title = Tilings and patterns
| url = https://archive.org/details/isbn_0716711931/page/462/mode/2up
| url-access = registration
| location = New York
| publisher = W.H. Freeman
| page = 463-470
| isbn = 978-0-716-71193-3
}}</ref> Later authors determined that the total number of ''k''-color configurations with 3&nbsp;≤&nbsp;k&nbsp;≤&nbsp;12 is 751.<ref name=Schwarzenberger3>{{cite journal
| last1 = Jarratt
| first1 = J.D.
| last2 = Schwarzenberger
| first2 = R.L.E.
| author-link =
| date = 1980
| language =
| title = Coloured plane groups
| url =
| journal = Acta Crystallographica
| volume = A36
| issue =
| pages = 884-888
| jstor =
| doi = 10.1107/S0567739480001866
| access-date =
}}</ref><ref name=Wieting>{{cite book
| last1 = Wieting
| first1 = T.W.
| author-link =
| date = 1982
| title = Mathematical theory of chromatic plane ornaments
| url =
| url-access =
| location = New York
| publisher = Marcel Dekker
| page =
| isbn = 978-0-824-71517-5
}}</ref>

==References==
{{Reflist}}


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[[Category:Symmetry]]
[[Category:Symmetry]]
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==Appendix: Subject coverage==

{{collapse top|''Color and Symmetry'' subject coverage}}
{| class="wikitable"
! scope="col" style="width: 15px;" | #
! scope="col" style="width: 175px;" | Chapter title
! scope="col" style="width: 680px;" | Relevant articles in Wikipedia
|-
| 1 || Introduction || [[Tessellation|Tiling]], [[Euclidean plane]], [[Packing problems|packing]], [[Cover (topology)|covering]], [[Ball (mathematics)|toplogical disk]], [[prototile]], [[Euclidean tilings by convex regular polygons#regular tiling|regular tiling]], [[Isohedral figure#k-isohedral figure|monohedral tiling]], [[Isohedral figure#k-isohedral figure|''k''-isohedral tiling]] (face-transitive), [[Symmetry in mathematics|symmetry]], [[isometry]], [[Rotation (mathematics)|rotation]], [[Translation (geometry)|translation]], [[Reflection (mathematics)|reflection]], [[glide reflection]], [[Group (mathematics)|group]], [[Transitive relation|transitivity]], [[Isogonal figure#k-isogonal and k-uniform figures|''k''-isogonal tiling]] (vertex-transitive), [[symmetry element]], [[isomorphism]], [[affine transformation]], [[frieze group]], [[wallpaper group]], [[fundamental domain]], [[space group]], [[rod group]]
|-
| 2 || Symmetry of one-sided rosettes || [[Uniform tiling]], [[Euclidean tilings by convex regular polygons#Archimedean, uniform or semiregular tilings|Archimedean tiling]], [[elongated triangular tiling]], [[snub square tiling]], [[truncated square tiling]], [[truncated hexagonal tiling]], [[trihexagonal tiling]], [[snub trihexagonal tiling]], [[rhombitrihexagonal tiling]], [[list of k-uniform tilings|list of ''k''-uniform tilings]], [[demiregular tiling]], [[3-4-3-12 tiling]], [[3-4-6-12 tiling]], [[33344-33434 tiling]], [[Isotoxal figure|''k''-isotoxal tiling]] (edge-transitive), [[Euclidean tilings by convex regular polygons#Tilings that are not edge-to-edge|tilings that are not edge-to-edge]], [[squaring the square]], [[star polygon]], [[Regular polygon#Regular star polygons|regular star polygon]], [[Polygram (geometry)|polygram]], [[Uniform tiling#Uniform tilings using star polygons|tilings using star polygons]], [[Penrose tiling#Development of the Penrose tilings|Kepler's star tiling]], [[pentagram]], [[pentacle]], [[Dissection puzzle|dissection tiling]], [[regular polygon]], [[List of Euclidean uniform tilings#Laves tilings|Laves tiling]], [[tetrakis square tiling]], [[rhombille tiling]], [[uniform coloring]], [[List of Euclidean uniform tilings#Uniform colorings|list of uniform colorings]], [[Graph coloring|Archimedean and uniform coloring]], [[Johannes Kepler|Johannes Kepler's]] [[Harmonices Mundi]]
|-
| 3 || Symmetry of figures with a singular point || [[Pathological (mathematics)|Well-behaved]], [[Singular point of a curve|singular point]], [[Locally finite space|locally finite]], [[Tessellation#Introduction to tessellations|normal tiling]], [[Euler characteristic#Plane graphs|Euler's theorem for tilings]], [[Tessellation|periodic tiling]], [[Heesch's problem]], [[Eberhard's theorem]], [[Karl Reinhardt (mathematician)|Karl Reinhardt]]
|-
| 4 || Symmetry of one-sided bands || [[Homeomorphism|Homeomorphism (topological equivalence)]], [[Combinatorial class|combinatorial equivalence]], [[Homotopy#Isotopy|isotopy]], ''[[Metamorphosis III]]'', [[Dual polygon|duality]], [[Pythagorean tiling]]
|-
| 5 || Symmetry of two-sided bands || [[Pattern]], [[Motif (visual arts)|motif]], [[group theory]], [[symmetry group]], [[subgroup]], [[Bravais lattice#In 2 dimensions|2-D lattice]], [[Peter Gustav Lejeune Dirichlet|Dirichlet tiling]], [[Topological group|continuous group]], [[Islamic geometric patterns]]
|-
| 6 || Symmetry of rods || [[Isohedral figure|Isohedral tiling]], [[Isogonal figure|isogonal tiling]], [[Isotoxal figure|isotoxal tiling]], [[list of isotoxal polyhedra and tilings#Isotoxal tilings of the Euclidean plane|list of isotoxal tilings]], [[Stripe (pattern)|striped pattern]], [[Evgraf Fedorov]], [[Alexei Vasilievich Shubnikov]], [[planigon]], [[Boris Delaunay|Boris Delone]]
|-
| 7 || Symmetry of network patterns || [[Conjugacy class|Conjugate element]], [[arrangement of lines]], [[Circle packing]], [[Dichromatic symmetry]], [[polychromatic symmetry]], [[Polychromatic symmetry#Group theory|perfect coloring]], [[Truchet tiles]], [[M.C. Escher]], [[Tessellation#Tessellations with polygons|Tilings by polygons]], [[triangular tiling]], [[square tiling#Topologically equivalent tilings|quadrilteral tiling]], [[pentagonal tiling]], [[hexagonal tiling]], [[parallelogon]], [[Concave polygon|non-convex polygon tilings]], [[anisohedral tiling]], [[polyomino]], [[heptomino]], [[polyiamond]], [[Polyhex (mathematics)|polyhex]], [[Voderberg tiling]], [[Marjorie Rice]], [[Similarity (geometry)|Similarity]], [[aperiodic tiling]], [[Raphael M. Robinson]], [[list of aperiodic sets of tiles]], [[Ammann A1 tilings]], [[Penrose tiling]], [[Golden ratio#Penrose tilings|golden ratio]], [[Ammann–Beenker tiling]], [[aperiodic set of prototiles]], [[Roger Penrose]], [[Robert Ammann]], [[John H. Conway]], [[Alan Lindsay Mackay]], [[Dan Shechtman]], [[Einstein problem]], [[Wang tile]], [[Hao Wang (academic)|Hao Wang]], [[Decidability (logic)|decidability]], [[Turing machine]], [[Cut point]], [[Anisohedral tiling#Isohedral numbers|disconnected tiles]], [[Uniform tiling#Expanded lists of uniform tilings|hollow tiling]], [[Vertex configuration#Vertex figures|vertex figure]], [[Riemann surface]], [[Harold Scott MacDonald Coxeter|H.S.M.&nbsp;Coxeter]]
|-
| 8 || Symmetry of layers ||
|-
| 9 || Symmetry of three dimensional spaces ||
|-
| 10 || Elements of group theory ||
|-
| 11 || Generalized symmetry, antisymmetry and colored symmetry ||
|-
| 12 || Symmetry in science and art, conservation laws ||
|-
|}
{{collapse bottom}}

<ref name=blank>{{cite web
| url = https://zbmath.org/0132.23302
| title = Color and Symmetry
| last = blank
| first =
| author-link =
| date = 1964
| language = German
| website =
| publisher = [[zbMATH Open]]
| zbl = 0132.23302
| access-date = 4 April 2024
}}</ref>

Revision as of 12:30, 28 April 2024

Stamp from the 1900 siege of Mafeking depicting Sgt. Major Goodyear on a bicycle

Bicycles on stamps

Topic 1

Topic 2

Topic 3

References

External links


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