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{\LARGE\textbf{GEOMETRIC SYMMETRY}} \\
\ \\
{\Large\textbf{SLIDES}} \\
\bigskip\bigskip\bigskip
ORDER FOR FRIEZE GROUPS
\medskip
\begin{tabular}{ll}
t & L\\
tg & L$\Gamma$\\
tm & V\\
mt & C\\
t2 & N\\
t2mg & V$\Lambda$\\
t2mm & H
\end{tabular}
\end{center}
\begin{center}
ORDER FOR WALLPAPER GROUPS \\
\medskip
\begin{tabular}{lllll}
p1 & p2 & pm & pg & cm \\
pmm & pmg & pgg & cmm \\
p4 & p4m & p4g \\
p3 & p3m1 & p31m \\
p6 & p6m
\end{tabular}
\end{center}
\newpage
\begin{center}
GEOMETRIC SYMMETRY---SLIDES
\end{center}
010
(Cypriot Vase with the Name of Thales).
\bigskip
020
SONNET ABOUT THE RAIN - Dobrivoje Jevti , 1978.
B. Pavlovi , ``On symmetry and asymmetry in literature'',
\textit{Comp.\ and Math.\ with Applics.}\ \textbf{12B} (1/2) (1986),
197--227, at p.208 = Istvan Hargittai, \textit{Symmetry Unifying Human
Understanding}, Pergamon 1986, 197--227, at p.208.
\bigskip
030
Hedgehog
Edward Rolland McLachlan, \textit{The Cartoons of McLachlan}, Private Eye 1973.
\bigskip
040
Palindrome.
THE END - Frederic Brown.
Martin Gardner, \textit{The Ambidextrous Universe}, Penguin 1970, end of
Chapter 5 (pp.50--51).
\bigskip
050
Palindrome
BLACK AND WHITE by J.A. Lindon.
\medskip
Lived as a dog - o no! God, as a devil
\quad Doom lives ever, it's astir, Eve's evil mood,
Live, O devil, revel ever, live, do evil!
\quad Do, O God, no evil deed, live on, do good!
\medskip
Howard W. Bergerson, \textit{Palindromes and Anagrams}, Dover 1973, p.117.
\bigskip
060
m
The human anatomy according to ideal principles. Venice, Academy.
[N.B. Leonardo's notebooks were written backwards, from right to left].
Bruno Santi, \textit{Leonardo}, Constable 1978, p.41.
\bigskip
070
m
Mixtec culture: a sacrificial obsidian knife with mosaic handle.
A two-handled mosaic serpent (below). These objects are believed to
have been among the treasure presented to Cort\'es by Moctezuma.
Hammond Innes, \textit{The Conquistadors}, Collins 1969,
repr.\ by Fontana 1972, p.110.
\bigskip
080
m
(Russian imperial eagle).
Alexander Pushkin, \textit{Boris Godunov} (Illustrated by Boris Zvorykin)
(Introduction by Peter Ustinov), Allen Lane 1982, Back endpaper.
\bigskip
090
m
Flags of Europe I: 1, Denmark; 2, Norway; 3, Iceland; 4, Sweden; 5, Finland;
6, Greece; 7, Czechoslovakia; 8, Poland.
Numbers 1, 2, 3, 4 (and 5 apart from the badge) have symmetry m but number
7 needs inversion of red and yellow to go with the mirror operation.
G.\ Campbell and T.O.\ Evans, \textit{The Book of Flags}, Oxford University
Press 1950, Plate XI.
\bigskip
100
2
Fig.\ 129. M83. This picture of a face-on spiral galaxy was made of
Kodak Vericolor II negative film, rather than by the three-colour
superposition technique and shows much less intense but similar colours
to the picture of NGC 2997 [Fig.\ 128 in the same book].
David Malin and Paul Murdin, \textit{Colours of the Stars}, Cambridge
University Press 1984, Fig.\ 129.
\bigskip
110
2m
Fig.\ 133. Centaurus A. Basically an elliptical galaxy with a population of
yellowish stars, Cen A is crossed by a band of dust which reddens the stars
still further. Formed from the dusty material, blue stars and pink H II
regions can be perceived along the rim of the dust band.
David Malin and Paul Murdin, \textit{Colours of the Stars}, Cambridge
University Press 1984, Fig.\ 133.
\bigskip
120
2 (colour !)
Back cover and p.13. The yin-yang symbol [tai-ji] surrounded by the
eight trigrams. This arrangement is known as the Sequence of Earlier
Heaven, or the Primal Arrangement (Roland Michaud).
Neil Powell, \textit{The Book of Changes: How to Understand and Use the
I Ching}, Orbis 1979 reprinted by Macdonald \& Co.\ under the Black Cat
imprint.
\bigskip
130
2 (colour !)
Figure 6.8 portrays two ancient symbols belonging to the same group, the
runic symbol for death and the well-known Chinese yin-yang symbol of
universal duality.
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, MIT Press 1974,
p.43, Figs.\ 6.8a and 6.8b.
\bigskip
140
2
m, 3m, 4m, (near bottom centre), 14m (top), 31m (or thereabouts)
An arrangement of diatoms from the South Pacific, photographed in polarized
$\text{light}\times150$.
Eric Linklater, \textit{The Voyage of the Challenger}, John Murray 1972
and Sphere 1974, p.228.
\bigskip
150
4
Fig.\ 20 \textit{Aurelia insulinda}. Example of an organism possessing
a four-fold symmetry axis (Haeckel).
A.V. Shubnikov and V.A. Koptsik, \textit{Symmetry in Science and Art},
Plenum 1974, Fig.\ 20, p.20.
\bigskip
160
4
8. A sprig of Codlins and Cream (Epilobium hirsutum)
The whole plant grows to five or six feet
Iolo A. Williams, \textit{Flowers of Marsh and Stream}, Penguin 1946
(King Penguin No. 27), Plate 8.
\bigskip
170
Fig.\ 1. The starfish Pentaceraster mamillatus, one of the most conspicuous
inhabitants of deeper sand bottoms, is 20 to 25cm across and bright red or
orange in colour. This individual was brought up from the bottom at 15m
and photographed on the beach . [Jana Island].
Philip W.\ Basson, John E.\ Burchard, Jr., John T.\ Hardy, and Andrew R.G.\
Price (illustrated and designed by Lisa Bobrowski), \textit{Biotopes of the
Western Arabian Gulf}, Aramco (Dharan, Saudi Arabia) 1977, Fig.\ 1.
\bigskip
180
6m
W.A.\ Bentley and W.J.\ Humphreys, \textit{Snow Crystals}, McGraw-Hill 1931
and Dover 1962.
\bigskip
190
2m, 3m, 6m
W.A.\ Bentley and W.J.\ Humphreys, \textit{Snow Crystals}, McGraw-Hill 1931
and Dover 1962.
\bigskip
200
7m and 5m
(top left) \textit{Antedon bifida} (top right) \textit{Luida ciliaris}
(always has 7 flattened rays)
(centre left) \textit{Astropecten aranciacus} (centre right)
\textit{Astropecton irregularis}
(bottom left) \textit{Ceramaster placenta} (bottom right)
\textit{Porania pulvillas}
A.C. Campbell (illustrated by James Nicholls), \textit{The Seashore and
Shallow Seas of Britain and Europe}, Collins 1976, p.241.
\bigskip
210
8m, 9m, 12m and 5m
(top left) \textit{Crossaster papposus} (8-13 blunt rays) (12 shown)
(top right) \textit{Solaster endeca} (7-13 rays) (9 shown)
(centre left) \textit{Marthasterias glacialis}
(centre right) \textit{Asteriasw rubens}
(bottom left) \textit{Stichastrella rosea}
(bottom right) \textit{Coscinasterias tenuispina} (6--10 rays often of very
different lengths) (8 shown)
A.C.\ Campbell (illustrated by James Nicholls), \textit{The Seashore and
Shallow Seas of Britain and Europe}, Collins 1946, p.\ 245.
\bigskip
220
12
9. \textit{Sea anemone} (\textit{Halcampa chrysanthellum}).
[It] is very much enlarged and inhabits sandy and muddy places, burying
its worm-like body and extending its tentacles at its surface.
T.A.\ Stephenson, \textit{Seashore life and pattern}, Penguin 1944
(King Penguin No. 15), Plate 9.
\bigskip
230
20m (approx.)
Floating sea slug attacking a jellyfish, Great Barrier Reef.
David Attenborough, \textit{Life on Earth}, Collins/British Broadcasting
Corporation 1979.
\bigskip
240
Fig.\ 14--6 Asante brass weights in geometric shapes, used for measuring
gold dust currency. Many of the designs have symbolic meaning; for example,
a zigzag line represents the fire of the sun. British Museum.
C. Zaslavsky, \textit{Africa Counts}, Prindle Weber and Schmidt 1973, p.176.
\bigskip
250
The Seven Friezes
\medskip
\begin{tabular}{lllllll}
L & L & L & L & L & L & L \\
L & $\Gamma$ & L & $\Gamma$ & L & $\Gamma$ & L \\
V & V & V & V & V & V & V \\
C & C & C & C & C & C & C \\
N & N & N & N & N & N & N \\
V & $\Lambda$ & V & $\Lambda$ & V & $\Lambda$ & V \\
H & H & H & H & H & H & H
\end{tabular}
\bigskip
260
Plate XV, Greek No. 1
Owen Jones, \textit{The Grammar of Ornament}, John Day 1856 and Van Nostrand
Reinhold 1982.
\bigskip
270
Plate VI. Egyptian No. 3
Owen Jones, \textit{The Grammar of Ornament}, John Day 1856 and Van Nostrand
Reinhold 1982.
\bigskip
280
Plate XCIII Leaves from Nature No. 3
Owen Jones, \textit{The Grammar of Ornament}, John Day 1856 and Van Nostrand
Reinhold 1982.
\bigskip
285
Granada. Sala de Reposo [del Ba¤o]
[Baths. The Rest Hall or Hall of Repose in the Comares Palace Bath]
(Postcard)
\bigskip
290
p1
Plate X, No. 11. Ornament on the walls, Hall of the Abencerrages.
Albert F. Calvert, \textit{The Alhambra}, John Lane, The Bodley Head 1906.
\bigskip
300
p2
24.9a Peruvian textile design
24.9b Peruvian textile design
24.9c Peruvian textile design
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, M.I.T.\ Press 1984.
\bigskip
310
pm (diamonds); cm (lozenges)
Plate 31
Plate 32
Victoria and Albert Colour Books. \textit{Decorative Endpapers}, Webb and
Bower 1985.
\bigskip
320
pg
M.C.\ Escher, Study for the lithograph ``Encounter'', pencil and ink 1944.
Note that in this case it is impossible to bring, e.g., the two white men
to coincidence by a rotation.
Caroline H.\ McGillavry, \textit{Symmetry Aspects of M.C.\ Escher's Periodic
Drawings}, International Union of Crystallography, Plate 3 = J.L.\ Locher,
\textit{The World of M.C.\ Escher}, Harry N. Abrams 1971, Fig.\ 123.
\bigskip
330
pg
Plate 9
Plate 10
Victoria and Albert Colour Books. \textit{Decorative Endpapers}, Webb and
Bower 1985.
\bigskip
340
cm
Plate V, No.\ 5. Ornament on the side of windows, Hall of the Two Sisters.
Albert F.\ Calvert, \textit{The Alhambra}, John Lane, The Bodley Head 1906.
\bigskip
350
cm
Plate IV, No. 4. Ornament at the entrance to the Ventana, Hall of the Two
Sisters.
Albert F.\ Calvert, \textit{The Alhambra}, John Lane, The Bodley Head 1906.
\bigskip
360
cm
Plate XI, No.\ 12. Ornament in panels on the walls, Court of the Mosque.
Albert F.\ Calvert, \textit{The Alhambra}, John Lane, The Bodley Head 1906.
\bigskip
370
p2mm
Figure 27.4
(a) (top right) American Indian, Nez Perc\'e
(b) (bottom left) Romanesque
(c) (bottom right) mosaic pavement, Florence baptistry
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, M.I.T.\ Press 1984.
\bigskip
380
p2mg
Chinese lattice designs
25.7 (a) (top) (Dye N2b, p.215)
\phantom{25.7} (b) (bottom left) (Dye I3b, p.157)
\phantom{25.7} (c) (top right) (Dye 7, p.307)
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, M.I.T.\ Press 1984.
\bigskip
390
p2gg
26.6 (a) Congo, Africa
26.6 (b) Italy, sixteenth century
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, M.I.T.\ Press 1984.
\bigskip
400
c2mm
28.4 (a) (top) Japanese
28.4 (b) (bottom left) Italian, sixteenth century
28.4 (c) (bottom right) medieval design
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, M.I.T.\ Press 1984.
\bigskip
410
p4
This pattern presents a tricky problem. It is formed by an array of
starfishes, clams and snail shells. At first sight, one would mark as
cell corners the points where four clams and four starfish meet. However,
on closer inspection it is seen that the edges of this square are not true
translations of the patterns; the snail shells halfway between these
pseudocell corners are not repeated in identical orientation by such
shifts\dots. On the other hand, the mid-point of the psueudo-cell, where
four starfishes and four snail shells meet, is indeed a true fourfold
rotation point\dots.
Caroline H.\ MacGillavry, \textit{Symmetry Aspects of M.C.\ Escher's Periodic
Drawings}, International Union of Crystallography 1965, Plate 13.
\bigskip
420
p4mm
Octagon allovers, Buddhist temples, Mount Omei, Szechuan 1800-1900 a.d.
Fig.\ H 10a (top right)
Fig.\ H 10b (top right)
Fig.\ H 10c (bottom right)
Fig.\ H 10d (bottom right)
Daniel Sheets Dye, \textit{Chinese Lattice Designs}, Harvard University
Press 1937 and Dover 1974, p.149.
\bigskip
430
p4mm
34.6 (a) (top) Byzantine design
34.6 (b) (bottom left) tartan motif
34.6 (c) (bottom right) Mayan motif, Uxmal, Yucatan
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, M.I.T.\ Press 1984,
p.308.
\bigskip
440
p4g
M.C.\ Escher, Study of Regular Division of the Plane with Angels and Devils,
Pencil, india ink, blue crayon and white gouache 1941.
Plate 6. The pattern has evidently fourfold rotation and
mirror-symmetry\dots. It is seen that near the bottom of the picture there
is a row of three fourfold rotation points. Of these, only the first and
third are equivalent by translation; the surroundings of the three points
are alternatively `left' and `right' since there are mirror lines running
between them. Choose the point in the middle of this row and find in the
pattern three other fourfold points in the same orientation. These outline
a square unit cell which is tilted at 45ø to the borders of the picture.
Caroline H.\ MacGillavry, \textit{Symmetry Aspects of M.C.\ Escher's
Periodic Drawings}, International Union of Crystallography 1965, Plate 6.
\bigskip
450
p4gm
33.6a (top) mother-of-pearl inlay motif, Turkey
33.6b (bottom left) Arabian
33.6c (bottom right) Arabian
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, M.I.T.\ Press 1984,
p.298.
\bigskip
460
p3
Plate XLVIII No.\ 58. Mosaic from the portico of the Generalife.
Albert F. Calvert, \textit{The Alhambra}, John Lane, The Bodley Head 1906.
\bigskip
465
p3
Postcard with caption ``La Alhambra (Granada) Estucado''.
\bigskip
470
p3m1
30.3 (b) (top left) Persian
30.3 (c) (bottom right) Chinese (Dye C12b, p.79)
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, M.I.T.\ Press 1984,
p.267.
\bigskip
480
p31m
31.4 (a) (top left) Chinese
31.4 (b) (top right) Chinese (Dye C5a, p.72)
31.4 (c) (bottom) Russian
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, M.I.T.\ Press 1984,
p.275.
\bigskip
490
12m or 4m (top); p6 (bottom)
Plate XLII , Moresque No. 4 (Do not confuse with Plate XLII*, Moresque No. 4*)
No.\ 6 (top) $\cong$ Albert F.\ Calvert, \textit{The Alhambra}, John Lane,
The Bodley Head 1906, Plate LXXVIII, No.\ 106, Part of the Portico of the
Court of the Fish Pond.
No.\ 5 (bottom) $\cong$ Calvert, \textit{op.\ cit.}, Plate LXXX, No. 108,
Ornaments on the walls, House of Sanchez.
Owen Jones, \textit{The Grammar of Ornament}, John Day 1856 and Van Nostrand
Reinhold 1982.
\bigskip
500
p6mm
36.6 (a) (left) An ancient design from the palace at Nimrud, Mesopotamia
36.6 (b) (right) A Japanese design
P.S.\ Stevens, \textit{Handbook of Regular Patterns}, M.I.T.\ Press 1984,
p.328.
510
\bigskip
p6mm
44. Some of the myriads of elegant basalt columns that make up the Giant's
Causeway in Antrim, Northern Ireland. Most, but not all, are hexagonal.
(Institute of Geological Sciences photograph).
Peter Francis, \textit{Volcanoes}, Penguin 1976, p.220.
\bigskip
520
p6mm
Plate 10 B. Polygonal jointing in Tholeiitic Basalt lava. Giant's Causeway.
H.E.\ Wilson, \textit{Regional Geology of Northern Ireland}, H.M.S.O.\ 1972.
\bigskip
530
p6mm
Plate 52. Workers of the dwarf honey bee gnawing wax off an abandoned comb
and putting it into their leg baskets (see p.95).
Karl von Frisch (with the collaboration of Otto von Frisch), \textit{Animal
Architecture} (translated by Lisbeth Gombrich), Hutchinson 1975, p.97.
\bigskip
540
p6mm
Plate 49 (top left) Comb built by bees whose antennal tips had been
amputated. The cells are irregular; their walls are excessively thick
in some places, excessively thin in others, and there are holes. (See p.92).
Plate 50 (top right) Bee with baskets filled with pollen. (See p.83).
Plate 51 (centre) Bee with baskets filled with propolis. (See p.94).
Plate 52 (bottom) Workers of the dwarf honey bee gnawing wax off an abandoned
comb and putting it into their leg baskets. (See p.95).
Karl von Frisch (with the collaboration of Otto von Frisch), \textit{Animal
Architecture} (translated by Lisbeth Gombrich), Hutchinson 1975, p.97.
\bigskip
550
pg$'$ (black and white)
Fig.\ 137 (above) M.C. Escher, Study of Regular Division of the Plane with
Horsemen, Indian Ink and Watercolour 1946 ( Magic Mirror, Fig.\ 225)
Fig.\ 138 (below), M.C. Escher, Horsemen, Woodcut in three colours
1946 ($\cong$ \textit{Magic Mirror}, Fig.\ 224 $\cong$ \textit{Graphic Work},
Fig.\ 6)
J.L.\ Locher (ed.), \textit{The World of M.C.\ Escher}, Harry N.\ Abrams 1971.
\bigskip
560
Colour plane symmetry
This is a more typical case of colour symmetry.
The pattern is composed of fishes in four different colours and orientations.
All the fishes of one colour have the same orientation and surroundings, so
there is one fish of each colour per cell\dots.
[Discussed also by Arthur L.\ Loeb, \textit{Color and Symmetry}, Wiley 1971,
Chap.\ 22, pp.162--165]
Caroline H.\ MacGillavry, \textit{Symmetry Aspects of M.C. Escher's Periodic
Drawings}, International Union of Crystallography 1965, Plate 37.
\bigskip
570
M.C.\ Escher, \textit{The Graphic Work of M.C.\ Escher}, Oldbourne 1961,
Fig.\ 20
Bruno Ernst, \textit{The Magic Mirror of M.C.\ Escher}, Ballantine 1976,
Fig.\ 244
J.L.\ Locher (ed.), \textit{The World of M.C. Escher}, Harry N.\ Abrams 1971,
Colour Plate VII and Fig.\ 239
\bigskip
580
Tilings
(Frontispiece) Johannes Kepler (1580-1630) is well known for his pioneering
work in astronomy. He also made fundamental contributions to the theory of
tilings, and some of his ideas have still not been fully investigated. We
reproduce here tilings from his book \textit{Harmonica Mundi} (volume 2)
published in 1619. The tiling marked Aa can be extended over the whole plane
as shown in Figure 2.0.1 and described in Section 2.5.
Branko Gr\"unbaum and G.C.\ Shephard, \textit{Tilings and Pattern}s, Freeman
1987.
\bigskip
590
Tilings
Figure 8.0.1. An attractive and ingenious pattern of squares from Islamic
art (Bourgoin [1879, Plate 13]) perfectly coloured in four colours. The
pattern is of type PP48A and it can be perfectly $k$-coloured if $k$ has
one of the values $3n$, $6n$, $n^2$, $2n^2$, $3n^2$ or $6n^2$ for a positive
integer $n$ (see Senechal [1979a]).
Branko Gr\"unbaum and G.C.\ Shephard, \textit{Tilings and Pattern}s, Freeman
1987.
\bigskip
600
Regular solids
M.C.\ Escher, Study for the wood engraving "Stars", Woodcut 1948.
J.L.\ Locher (ed.), \textit{The World of M.C.\ Escher}, Harry N. Abrams 1971,
Fig.\ 151.
\bigskip
610
Regular solids
M.C.\ Escher, Stars, Wood engraving 1948.
M.C.\ Escher, \textit{The Graphic Work of M.C.\ Escher}, Oldbourne 1961,
Fig.\ 51
Bruno Ernst, \textit{The Magic Mirror of M.C.\ Escher}, Ballantine 1976,
Fig.\ 209
J.L.\ Locher (ed.), \textit{The World of M.C.\ Escher}, Harry N.\ Abrams 1971,
Fig.\ 152
\bigskip
620
Regular solids
A.\ Holden, \textit{Shapes, Space and Symmetry}, Columbia University Press
1971, Plate opposite p.1.
\bigskip
630
Fig.\ 26. Skeletons of various radiolaria (after Haeckel).
BUT see H.S.M. Coxeter, Review of ``Symmetry'' by H. Weyl, \textit{American
Mathematical Monthly}, \textbf{60} (1953), 136--139.
A.F. Wells, \textit{The Third Dimension in Chemistry}, Oxford: Clarendon Press
1956.
\bigskip
640
Regular solids
Fig.\ 46. Here (Fig.\ 46) is his [Kepler's] construction, by which he believed
he had penetrated deeply into the secrets of the Creator. The six spheres
correspond to the six planets, Saturn, Jupiter, Mars, Earth, Venus, Mercurius,
separated in this order by cube, tetrahedron, octahedron, dodecahedron,
icosahedron.
H.\ Weyl, \textit{Symmetry}, Princeton University Press 1952.
\bigskip
650
Table 9. The thirteen semi-regular (Archimedean) polyhedra and their duals
(a) truncated octahedron (b) truncated icosahedron (c) great
rhombicuboctahedron or truncated cuboctahedron (d) snub cube (e) truncated
tetrahedron (f) cuboctahedron (g) icosidodecahedon (h) small
rhombidodecahedron (i) small rhombicosidodecahedron (j) truncated cube
(k) truncated dodecahedron (l) truncated dodecahedron (m) snub dodecahedron.
Open University Technology/Mathematics Course TM 361, \textit{Graphs, Networks
and Designs}, TM 361--13 Geometry, p.36.
\bigskip
660
Table 10. Some examples of space filling systems.
Open University Technology/Mathematics Course TM 361, \textit{Graphs, Networks
and Designs}, TM 361--13 Geometry, p.46.
\bigskip
670
Crystal
Fig.\ 191. The fourteen Bravais lattices. If we assume spherical symmetry
for the lattice points, the space groups of the fourteen Bravais lattices
will be (1) P1 (2) P2/m (3) C2/m (4) Pmmm (5) Cmmm (6) Immm (7) Fmmm (8)
P6/mmm (9) R3m (10) P4/mmm (11) I4/mmm (12) Pm3m (13) Im3m (14) Fm3m.
The capital letters denote the translation groups: P, R primitive lattices;
C lattices centred in the face cutting the edge c; F face centred lattices;
I body-centred lattices. (1) is triclinic, (2)--(3) monoclinic, (4)--(7)
orthorhombic (8) hexagonal (9) trigonal (10)--(11) tetragonal (12)--(14) cubic.
A.V.\ Shubnikov and V.A.\ Koptsik, \textit{Symmetry in Science and Art},
Plenum 1974, p.205.
\bigskip
680
Crystal
Fig.\ 73. The 47 simple forms which crystals may take: (1)--(7) Pyramids;
(8)--(14) bipyramids; (15)--(21) prisms (22), (23), (25) tetrahedra; (24),
(26), (28) trapezohedra; (27) rhombohedron; (34) scalene triangle; (33),
(35) scalenohedra; (31) dihedron; (32) pinacoid; (23), (29), (36)--(47)
simple forms of the cubic system; (23) tetrahedron; (29) cube; (30)
octahedron; (36) trigonal tristetrahedron; (38) pentagonal tristetrahedron;
(39) rhombic dodecahedron; (40) pentagonal dodecahedron; (41) tetrahexahedron;
(42) hexatetrahedron; (43) didodecahedron; (44) tetragonal trisoctahedron;
(45) trigonal trisoctahedron; (46) pentagonal trisoctahedron
(47) hexoctahedron.
A.V.\ Shubnikov and V.A.\ Koptsik, \textit{Symmetry in Science and Art},
Plenum 1974, p.205.
\bigskip
690
Figs.\ 69 \& 70. Here are two Laue diagrams (Figs.\ 69 and 70), both of
zinc-blende from Laue's origianl paper (1912); the pictures are taken in
such directions as to exhibit the symmetry around an axis of order 4 and 3
respectively.
H. Weyl, \textit{Symmetry}, Princeton University Press 1952.
\bigskip
700
Crystal
14. Fluorspar
15. Fluorspar
N.\ Wooster, \textit{Semi-precious Stones} (with sixteen coloured plates by
Arthur Smith), Penguin 1952 (King Penguin No.\ 65), Plates 14 \& 15.
\bigskip
710
Crystal
12. Iron pyrites
13. Fluorspar 'Bluejohn'
N.\ Wooster, \textit{Semi-precious Stones} (with sixteen coloured plates by
Arthur Smith), Penguin 1952 (King Penguin No.\ 65), Plates 14 \& 15.
\bigskip
720
Spiral
NAUTILOIDS, GONIATITES AND CERATITES [on left] 64 \textit{Orthoceras}
[top left]; 65 \textit{Nautilus} [top right of left]; 66 \textit{Glyphioceras}
[bottom left]; 67 \textit{Ceratites}.
LIASSIC AMMONITES [on right] 68 \textit{Promicroceras} [top right of left];
69 \textit{Phylloceras} [top right]; 70 \textit{Dactilioceras} [bottom right
of left]; 71 \textit{Harpoceras}.
J.F. Kirkaldy (Photographs by Michael Allman), \textit{Fossils in Colour},
Blandford Press 1967, revised 1975.
\bigskip
730
Spiral
Plate XXXV Shell, Logarithmic Spiral and Gnomonic Growth (Photographs:
Kodak Ltd)
Plate XXXV gives two examples of X-rayed shells, showing clearly the
directing spiral of \textit{Nautilus Pompilius} and the gnomonic growth of
\textit{Triton Tritonis} (p.97). Gnomonic growth means growth from inside
outwards as opposed to agglutination as is common in crystal [paraphrased
from p.90].
M.\ Ghyka, \textit{The Geometry of Art and Life}, Sheed and Ward 1946,
reprinted by Dover 1977, p.101.
\bigskip
740
Spiral
Fig.\ 44. For everybody looking at this picture (Fig.\ 44) of a giant
sunflower, \textit{Helianthus maximus}, the florets will naturally arrange
themselves into logarithmic spirals of opposite sense of coiling.
H.\ Weyl, \textit{Symmetry}, Princeton University Press 1952.
\bigskip
750
Helical
(334) No. 23 HIGH PETERGATE, c. 1779 (left) (helical)
(334) No. 62 LOW PETERGATE, c. 1725 (right)
Royal Commission on Historical Monuments, \textit{City of York. Volume 5: The
Central Area}, H.M.S.O.\ 1981, Plate 196.
\bigskip
760
Helical
J.D.\ Watson, \textit{The Double Helix: A New Critical Edition}, Weidenfeld
and Nicholson 1981, p.121.
\bigskip
770
Helical
G. Nass, \textit{The Molecules of Life}, Weidenfeld and Nicholson 1970, p.35.
\bigskip
780
Conical spiral
HOLOSTOMATOUS AND SIPHONOSTOMATOUS GASTROPODS
110. Left, \textit{Turitella}, Right \textit{Cerithium}
Bartonian (Upper Eocene), Gisois, Normandy, $\text{France}\times1\frac{1}{4}$.
J.F. Kirkaldy (Photographs by Michael Allman), \textit{Fossils in Colour},
Blandford Press 1967, revised 1975.
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