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THURBER / LALO Designs
A Symmetry Sampler
Arthur L. Loeb
In 1963 in Rome, Art and Science met at a congress of the
International Union of Crystallographers. Here Carolina
MacGillavry of the University of Amsterdam was commissioned
by the Teaching Commision to write a book on the Graphic Art
of M.C.Escher. This volume was intended to be used in the
teaching of symmetry, at that time principally of interest to
crystallographers. MacGillavry and I had been old friends, and
since the crystallography congress in Cambridge, UK in 1960 I
had been friendly with Escher, who visited us in Cambridge
Massachusetts, and lectured at MIT.
At that conference in Rome, Philippe LeCorbeiller and I
presented a paper relating all rotational-symmetry values in
the plane to a simple diophantine equation
(1/k) + (1/l) + (1/m) = 1,
using as illustrations a set of slides created by photographing
colored plastic forms on a black velvet background.
This equation has the following five solutions:
k:
1
2
2
2
3
l:
oo
2
3
4
3
m:
oo
oo
6
4
3
∞
Here k, l and m represent the symmetry values of centers of
rotational symmetry coexisting in a plane, and oo (infinity)
stands for translational symmetry.
The largest number of distinct rotocenters which may coexist
in a plane is four , and their symmetry values equal 2.
The accompanying slides were liked sufficiently that an
exhibit was spontaneously arranged. That same year, I used
collages to illustrate our approach to plane symmetry in a joint
exhibition with Duncan Stuart Symmetry and Transformations
at Harvard’s new Carpenter Center for the Visual Arts. I used
these collages in my courses on Design Science and Visual
Mathematics until they wore out. A set of slk screen prints in
collaboration with Holly Alderman was exhibited in Budapest
on the occasion of the first Symmetry Congress in 1989, but
unfortunately was never completed.
Whereas crystallographers were using a notation based on the
unit cell, and placed translational symmetry primary, we felt
that rotational symmetry was most fundamental, and used a
notation which has been more useful for designers and visual
artists. Rotocenters may or may not lie on a mirror line. If they
do, then we underline their symmetry value: k. Two distinct
rotocenters having the same symmetry value are distinguished
by a prime: k and k’, 4 and 4’, 3, 3’ and 3". If a pattern has
reflection symmetry, then every rotocenter not on a mirror line
will have a twin to which it is related by reflection symmetry;
such a pair is designated by a circumflex: k and k ^ are called
enantiomorphs.
In
the
present
sampler
both
the
crystallographic notations and my own are listed underneath
each pattern.
Symmetrical patterns in the plane may be periodic (i.e. repeat)
in one direction only, or they may repeat in two non-parallel
directions. The former are called ribbons or friezes. Such
patterns may be, but are not necessarily finite in width. The
frieze groups are limited to the symmetry values 1, 2 and
infinity.
Computer Technology has, for my purposes, overtaken
silkscreening, and by a slight rearrangement of my name I
have established THURBER / lalo DESIGNS* with its archive of
diverse patterns. It is perhaps not too surprising that these
archives contain examples of each of the twenty-eight
combinations of symmetry elements in the plane. You are
invited to enjoy these designs, and those interested in a more
profound theoretical background are referred to my Color and
Symmetry, Wiley 1971 and Concepts and Images, Birkäuser
1993. It is my hope and intention that the present visual fiction
may stimulate a broader interest in symmetry, and expand the
visual repertoire of artists, designers and planners, and to
allow the scientist to use her visual sense and intuition in
problem solving. The THURBER / lalo archive is intended to
illustrate the diversity of applicability of symmetry theory,
be whimsical and, like any work of fiction, be somewhat
*
COPYRIGHT 1997
autobiographical. The two designs for the University Lutheran
Church are real, but otherwise any resemblance to real
persons and places is, of course, totally coincidental, and the
reponsibility of THURBER / lalo Designs*.
*
COPYRIGHT 1997
PROCESSION
From a banner triptych
for the University Lutheran Church, Cambridge, MA
1
No symmetry
O MAGNUM MYSTERIUM
From a banner triptych
for the University Lutheran Church, Cambridge, MA
1m
A single mirror
WITCHES’ COVE
Print by Beol Ruhtra
1m
A single mirror. This print portrays a waterscape having mirror symmetry.
However, the print itself is not symmetrical: multiple reflections give
Ruhtra’s print a surrealistic quality
BELT
Advertisement by the Belletrie Department Store
oo
11
Translational symmetry in a single direction.
THE FALL OF THE LEAF
oo mm’
m1
Horizontal mirrors. Two alternating sets of distinct mirrors
TALL SHIPS
DESIGN FOR A SCREEN IN THE BAR OF THE VLIEBURG
YACHTCLUB
oo m
1m
Translation symmetry in one direction. A single mirror line parallel to the
direction of translation symmetry
SCORPIONS
From The Plagues of Egypt (Hatikvah Synagogue)
oo g
1g
Translation symmetry along a single glide line.
BANNER FOR THE WITCHES’ COVE INN
22’oo
12
Ribbon group. Alternating twofold rotocenters along the direction of
translation symmetry
WALL STENCIL
Jethro Winsloe House, Witches’ Cove
1 oo oo’mg
cm
Vertical alternating mirror and glide lines. Translation symmetry parallel and
perpendicular to the reflection lines
SEALS: EAST, WEST, HOME’S BEST
Wall Stencil, Jethro Winsloe House, Witches’ Cove
1 oo oo’gg’
pg
Vertical glide lines through and between columns of seals. Translation
symmetry parallel and perpendicular to the glide lines„
RINGS
Show window backdrop for jewellers VAN EIJSDEN & HASPELS
1
'gg'
pg
Vertical glide lines through and between columns of rings. Translation
symmetry parallel and perpendicular to the glide lines
TEXTILE TAFFETA DESIGN
1 oo oo'
p1
Translation symmetry in two non-parallel directions
DANCE OF THE IRISES:
Curtain design for the ballet
LE PRINTEMPS DE LA REINE CLAUDE
1 oo oo’mm’
pm
Alternating vertical mirrors,. Translation symmetry parallel and
perpendicular to the mirrors
Logo for CORN LIE DERMAETE, Cosmetics
2 C2
Single twofold rotocenter
Door for CORN LIE DERMAETE Cosmetics
2 D2
Single twofold rotocenter at the intersection of two mutually perpendicular
mirror lines
GEODES
Floor for CORN LIE DERMAETE Cosmetics showroom:
geodes set in basalt.
244’
p4
Between the geodes there are three types of interstices. Two have a square
outline, and the third a parallelogram. The former contain the centers of four
-fold rotational symmetry, 4 and 4’, the parallelogram twofold rotocenters.
As is the case in so many of these patterns, the limits of handicraft make
the symmetry somewhat imperfect; these imperfections are one of their
attractions, because perfect symmetry is static. In the present instance, the
large ’squares’ could be appropriately said to contain a four-fold rotocenter.
If not, they would still be twofold symmetrical, but the parallelgrams would
then lose their symmetry altogether. In this case, there would still be the
truly four-fold symmetrical ’square’, but they would now be of two distinct
types, so that the overall symmetry would still be 244’.
CASHMERE PONCHO
22’oo
12
Ribbon group. Alternating twofold rotocenters along the direction of
translation symmetry
DRAGONFLIES
22’oo
mm
Ribbon group. Alternating twofold rotocenters on a mirror line pallel to the
direction of translation symmetry. Mirrors intersecting that unique mirror
line perpendicularly at the rotocenters
CONTRADANCE
22^oo
mg
The pink and black figures are related to each other by a vertical glide line.
Inside each region enclosed by black and pink curves there are twofold
mutually enantiomorphic rotocenters.
VLIEDRECHT TILES in Enzovoort Castle
236
p6
Although it is generally assumed that ’Enzovoort’ stands for ’Etcetera’,
archeologist Eric Quellinck, realizing that the suffix ’-voort’ frequently
designates a geographic loccation, uncovered the ruins of the castle of
Enzovoort, near the village of Vliedrecht, and in them well-preserved tilings.
Although in texture and coloring quite traditional, the Enzovoort tiles are
unusual in their regular trangular shape: six meet at a sixfold rotocenter.
Each has a twofold rotocenter in the middle of one of its sides. One of the
vertices of each triangle is located at a sixfold rotocenter, the other two
contain mutually equivalent threefold rotocenters.
TILES AT ENZOVOORT CASTLE
3 3 ' 3’^
p31m
The large triangular white spaces contain a set of threefold rotocenters on
mirrors. The blue pinwheels all contain threefold rotocenters, which belong
to two enantiomorphic sets. Note again the remarkable triangular shape of
these Enzovoot tiles.
A MYSTERY AT ENZOVOORT CASTLE
oo mm’
m1
Eric Quellinck noted that this pattern differed substantially from the
previous one. The large white spaces were no longer identical, hence there
is no symmetry in the vertical direction, but vertical mirror lines, therefore
horizontal translation symmetry . What caused the tile setters to change the
orientation of the tiles in such a haphazard fashion? Rebellion? Faulty
instructions? Or is there some secret message in this pattern?
PATIO QUELLINCK HOME
236
p6
This example is made up of pentagonal tiles having four angles equal to
1200, the fifth one equal to 600... Sixfold rotocenters are located at the
meeting points of six 600 angles. Where three 1200 angles meet, we find
threefold rotocenters. Twofold rotocenters are found halfway between pairs
of nearest sixfold rotocenters. This tiling has no reflection symmetry, even
though each individual tile does.
DRAPES
in the Quellinck Home
QUASI-SYMMETRY
Although the symmetry of this pattern is not evident, there is a definite
structure here. A quasi-symmetrical string in two variables x and y has the
sequence x y x x y x y x x y x x y ...In this pattern two motifs were used
which are each others’ mirror image. There is quasi-symmetry in both
horizontal and vertical directions.
FLORENCE QUELLINCK’S GARDEN PARTY
Textile Design
22^2’2'^g/g’
pgg
This design for a summer gown has the same symmetry: glide lines but no
mirrors. The reader is invited to locate the rotocenters.
BELLETRIE DEPARTMENT STORE
Skylight: metal and stained glass
236
p6m
The Art Nouveau department store designed by architect Henri Delage has a
spectacular central courtyard lit by this skylight . All two-, three- and sixfold
rotocenters lie on mirrors.
(continued)
BELLETRIE DEPARTMENT STORE
FLOOR UNDERNEATH SKYLIGHT: MARBLE AND METAL INLAY
SYMMETRY AS FOR SKYLIGHT
BELLETRIE DEPARTMENT STORE
Autumn coaster display
33’3’’
p3
The black triangles pointing to the left all contain threefold rotocenters,
whereas the ones pointing to the right do not. Any vertical line passing
through a threefold rotocenter will also pass through rotocomplexes
distinct from the other two.
AUTUMN WHIRLS
Belletrie Department Store
33’3’’
p3m1
In its autumn program the store used the theme of turning, whirling leaves.
This textile design has three distinct sets of threefold rotocenters all lying
on mirrors. There are glide lines halfway between the mirrors.
DRAPERY DESIGN
Belletrie Department Store
22’2’’2’’’
pmm
In this whimsical drapery design, the motif is deliberately made
asymmetrical, containing only a vertical mirror line. There are horizontal
mirror lines between the rows of motifs, and vertical mirror lines through
as well as between the motifs. There is a twofold rotocenter at every
intersection between horizontal and vertical mirror lines.
THE TURNING SEASON
Belletrie Department Store
22^2’2'^g/g
pgg
The store included in its autumn collection a textile design featuring the
leaves’ color change. This is the only symmetry group periodic in two
directions which has glide lines, but no mirror lines. Thre are two sets of
enantiomorphically paired twofold rotocenters. The vertical glide lines pass
through the columns of leaves, the horizontal ones between the rows of
leaves . The twofold rotocenters lie between the leaves along the horizontal
rows of leaves. The glide lines form a rectangular grid with the two-fold
rotocenters at the centers of the rectangular meshes of the grid.
TILED FLOOR, CASA DORADA
Plaza de los Estados
244’
p4m
The Casa Dorada is not actually guilded: it is said that the original owner
called his villa after his daughters Dora and Ada. In this classic design the
tiles are set diagonally to the sides of the tableau; each tile contains a fourfold center on a mirror line at its center. The other fourfold rotocenters
occur where four tiles meet. The twofold rotocenters lie at the midpoints of
the edges, between the ends of the orange petals
AGELAOS’S PALACE: Floor tiles
244^
p4g
This symmetry group is very attractive, having balanced enantiomorphic
pairs of fourfold rotocnters in addition to twofold rotocenters on mirrors
SUBDIVISION OF VLIEBURG COMMON
Display in the Vlieburg courthouse
244'
p4
When the Vlieburg grazing common was subdivided among the hereditary
shareholders, the stipulation was that all plots should be identical, and that
the coastal boundaries should be reproduced as inland boundaries. Guido
Qellinck, a mathematical prodigy, designed this plan, a tiling of irregular
pentagons. The crosses mark Guido’s reference points, fourfold
rotocenters.
TARTAN QUILT AT CRIMSON HEATHER
SCOTTISH IMPORTS
22’2’’2’’’
p2
This quilt is stitched from ireregularly shaped identical quadrilateral
patches. The plane may be tiled by any straight-edged quadrilateral. Twofold
rotocenters are located at the centers of the stitched edges, belonging to
four distinct sets.
Screen for the Vlieburg Yachtclub
22^2’2’’
cmm
Alternating vertical as well as horizontal mirror and glide lines. Two sets of
twofold rotocenters at the intersections between mirror lines. Two
enantiomorphic sets of twofold rotocenters relate the motivs diagonally
above and below each other.
TransFLEVO AIRLINE:
Stewardesses’ scarves and stewards’ neckties
22^2’2’’
cmm
Two sets of rotocenters on mirrors, giving stability, and two sets
enantiomorphically paired, giving dynamism, This group is very attractive,
occurring several times in the archive.
LANTERNS ON POLE
Kukifoto Hotel
22^2’2'^g/g
pgg
Horizontal mirror lines through the lanterns, vertical glide lines through the
serpentine poles and halfway in between.
CHAPEL, TRINITY SCHOOL
Stained glass window
33’3’’
p3m1
(continued)
CHAPEL, TRINITY SCHOOL
Inlaid floor
Like the chapel window, the floor with its three distinct three-fold sets of
rotocenters signifies the theme of the Trinity. Note that, although there are
locations where six motifs meet, adjacent pairs of motifs are not related by
rotational symmetry because a line runs through them which causes such a
pair to be related by reflection symmetry. The meeting point of the three
pairs of motifs is therfore a threefold rotocenter.
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