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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.

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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.

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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.

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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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