Symmetry, Chemistry,
and Escher’s Tiles
In general, highly symmetrical molecules
are hydrophobic ("water-fearing"), while hydrophilic molecules ("water-loving")
have a unique direction, or polarity. For example, water (H–O–H or dihydrogen
oxide) itself has a bent shape, so the electronic trends of its two O–H
bonds can add together (figure 1a). In contrast, several non-polar, hydrophobic
molecules have high symmetry. For example, O=C=O (carbon dioxide) is completely
linear (Figure 1b). The net electronic trend is also zero in C
So, symmetry affects the polarity and
thus the solubility of substances. For related reasons, other properties
also are affected by symmetry, such as vapor pressure, melting point, boiling
point, crystal diffraction of
Many of the drawings of M. C. Escher demonstrate
certain symmetry relationships [2]. One technique he
used follows the method of tiling or tessellation, which has been used
for centuries in different cultures. This is the regular filling of a surface
with identical ( While molecules are three-dimensional objects, and tiling patterns are generally on a flat surface, it is possible to illustrate some symmetry relationships with tessellations. Artistic patterns can inspire and interest those who examine chemistry or mathematics, while less complex tilings with the same symmetries can clarify some relationships of molecules. For at least half a century, people have realized that tessellations demonstrate two-dimensional analogs of the three-dimensional symmetries of solid molecules, following the principles of crystallography [3,4,5]. The present work is not focused on the
repetition of molecules in a crystal, but on the inherent symmetry of individual
molecules, called The elements of point-group symmetry are
rotation, reflection, and inversion of all atoms in the molecule, in three
dimensions about a fixed central point, with no net change to the molecule.
In contrast, the elements of tessellation symmetry are rotation, reflection,
glide reflection, and translation of a side of a polygon to one other side,
to create a tile that can exactly cover the surface [8].
While these tessellation symmetries thus do not correspond to molecular
symmetries, they can suggest them, without requiring three dimensions.
M. C. Escher wrote, in regard to regular tessellation, that "it is the richest source of inspiration that I have ever tapped, and it has by no means dried up yet." [12] Escher twice visited the Alhambra in Spain, and made several sketches of the tile designs found there. About this he wrote that "The Moors were masters in the filling of a surface with congruent figures and left no gaps over." There can be no doubt about the great significance of this art form to Escher, when he wrote that "Periodic drawings are . . . not subjective; they are objective. . . . For once one has crossed over the threshold of the early stages, this activity takes on more worth than any other form of decorative art" [12]. For two-dimensional tilings of a surface, there are 17 different plane symmetry groups, cataloged by the Hungarian mathematician G. Pólya in 1924 [9]. These can be categorized by asking questions as to the smallest symmetric rotation, and the presence of translation, reflections, and glide-reflections, and their relationships. These 17 symmetry groups yield 81 possible isohedral tiles [10]. Here, a tiling is isohedral if, "for every choice of two tiles in the tiling, there is an element in the symmetry group of the tiling that sends one of the tiles onto the other" [11]. The German mathematicians Heinrich Heesch
and O. Kienzele later analyzed the types of asymmetric tiles that can form
a tessellation without reflections (Table 1) [13]. This
leaves seven of the 17 plane symmetry groups (called
Each tile has 3 to 6 edges. Each edge
has a symmetry relationship within itself or with another edge. Either
one half of an edge coincides with its other half by a half-turn rotation
(called G).
For each tile, the shape can be thought of as a basic figure (square, triangle,
quadrilateral, pentagon, or hexagon), combined with modifications or distortions
of the edges, following the rules of its symmetry group.
_{n}The translation ( n.
A glide-reflection,
G, is also a translation to the
opposite side of the figure, but one in which the shape of the edge is
reflected across the line of translation. So, a bump out on the upper left
becomes a bump in on the lower right.
_{n}The Heesch symbol that categorizes a tile
"is obtained by traveling a circuit around its boundary and associating
to each edge the appropriate letter. Edges with a center of half-turn symmetry
( Now that these symmetry categories have
been established, visual examples are needed to illustrate them. These
examples are chosen based on the symmetry of selected molecules (Table
2). Each of the representative molecules in Table 2 is planar, and represents
either a different point-group symmetry or chemical behavior, or both.
The table lists the Schoenflies three-dimensional point group symbol used by chemists, and a simplified three-dimensional international symbol used by crystallographers. The table also lists the two-dimensional international symbol of the tessellation group, and the specific tile symmetry under Heesch’s system, chosen by its analogy to three-dimensional symmetry, while avoiding reflection operations. These examples can now be illustrated,
both with geometrical symbols, and with tessellation artwork. In this artwork,
an individual shape has one or more sides modified according to Heesch’s
applicable symmetry rules. The shape is given patterns or colors, then
is replicated by again applying these rules of symmetry. Escher also developed
and applied patterns of shading or The planar-triangular HgCl
Benzene, C
Here, benzene is tessellated as a flower with a
Ethylene, H
Tetrachloroplatinum, PtCl
These results serve simultaneously to
illustrate tessellation symmetries and three-dimensional molecular symmetries.
Some analogies are more obvious than others, and further examples are possible.
A future paper will discuss these in relation to the symmetries of frieze
border patterns.
[2a] [2b] See also figures 8 and 9 in our
other paper in these proceedings, [3] M. C. Escher’s brother, Beer, a teacher
of crystallography, proposed in 1948 to write a book on this subject, with
his brother illustrating [2]. Their project never materialized,
though Escher later helped to produce such a work: [4] A key crystallography reference is
[5] G. H. Stout and L. H. Jensen, [6] F. A. Cotton, [7] F. A. Cotton and G. Wilkinson, [8] Schattschneider, [9] G. Pólya, [10] B. Grünbaum, G. C. Shephard,
[11] Schattschneider, [12] M. C. Escher, [13] H. Heesch and O. Kienzele, [14] All tessellations herein were created
by the authors or by Jill Etheridge with the program ^{©}MECC,
Minnesota, 1994. |