Symmetry: Culture and Science
Vol. 24, Nos. 1–4, 185-190, 2013
SYMMETRIC INTERLACE PATTERNS ON
POLYHEDRA USING GENERALIZED
TRUCHET TILES
David A. Reimann
Address :
David A. Reimann
Department of Mathematics and Computer Science
Albion College
Albion, Michigan, 49224, USA
[email protected]
Abstract: This work describes a technique for generating interlace patterns on
polyhedra using a generalization of Truchet tiles. A review of the notation for
describing a generalized Truchet tiles is given. Several examples of symmetric
patterns on paper models of polyhedra are then shown.
Keywords: symmetry, polyhedra, truchet tiles.
MSC: 00A66, 05B45, 52B70
1. INTRODUCTION
The aesthetics of polyhedra, especially those with a high degree of symmetry,
have fascinated people throughout history (Cromwell, 1999). Convex polyhedra
having regular faces, such as the Platonic, Archimedean, and Johnson Solids all
exhibit symmetry that arises because each individual face has dihedral symmetry. The possible symmetries of a polyhedron is given by one of 14 symmetry
group families (Conway, 2008). One can add to the visual appeal of a polyhedron by decorating its faces. Such decorations can lead to the decorated
polyhedron having symmetry which is a subgroup of the symmetry group of the
undecorated polyhedron.
Patterns created with modular units, such as tiles, have been used as the basis
of many patterns. Decorated tiles with simple motifs have been used to enhance
the visual appeal of tilings by creating additional patterns. Escher used decorated square tiles to create a variety of ribbon patterns and explored similar
patterns from other shapes (Schattschneider, 2004). Jablan (2001, 2002) has
also explored using decorated tiles for creating modular patterns. The commonly known Truchet tiles, square tiles decorated with quarter circle arcs, date
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Figure 1:
D.A. Reimann
Example tiles decorated with Bézier arcs. The seven geometrically unique crossing
pattern motifs, and notation, for decorating triangles using three Bézier arcs. For simplicity,
no arc crossings are depicted.
to paper by Smith (1987). Smith does not cite previous work for this idea, but
does refer to a Martin Garner column which describes similar arcs on Penrose
tiles (Gardner, 1977). Similar tiles are used for the puzzle KnoTiles (KnoTiles).
Browne (2008) presented an extension of Truchet tiles to triangles and hexagons.
Hexagonal interlace tiles have been used for the games Tantrix (Tantrix) and
Entanglement (Gopherwood).
The author has previously described a technique for creating interlace patterns
by decorating the polygons using a simple motif comprised of Bézier curves
that generalizes Truchet tiles (Reimann, 2011, 2012). Each n-gon is decorated
by subdividing its sides and placing d uniformly spaced endpoints along each
side, resulting in nd endpoints. Each endpoint can be assigned a unique integer
0, 1, . . . nd−1 starting with the rst side clockwise from a vertex. An arc pattern
in a single polygon can be described using the following notation:
(α, β|γ, δ|, ζ| · · · )[A/B, C/D · · · ]
where α, β, . . . represent the endpoint numbers of arcs in a the polygon and A/B
indicates arc A (from endpoints α to β ) overcrosses arc B (from endpoints γ to
δ ). A total of nd/2 arcs made from simple cubic Bézier curves connect pairs of
endpoints such that the tangent of each arc at the endpoints is perpendicular
to the polygon edge, resulting in a smooth continuous curves throughout the
structure.
2. THEORY
The set of the possible geometrically unique motifs (unique up to rotation) for
decorating a triangle with three arcs is shown in Figure 1. Tile faces must
be placed on a polyhedron such that the desired symmetry is obtained. Note
that tiles with dihedral symmetry are required if a polyhedron having mirror
symmetry across individual faces is desired. Similarly, mirror tile pattern pairs
are required if mirror symmetry across an edge is desired. This results in an
interesting interplay between the symmetries of the tiles and the polyhedron.
Symmetric interlace patterns on polyhedra
187
No simple formula is known for the number of geometrically unique patterns for
such a decorated n-gon. Since there are nd points around the perimeter of the
polygon, the number of patterns with m = nd/2 arcs is given by
f (m) =
(2m)!
.
2m m!
This formula does not distinguish rotated patterns. There is also no known
formula for determining the number of arc crossings for a given pattern.
3. RESULTS AND DISCUSSION
Paper models were manually constructed to illustrate several symmetric patterns on various polyhedra. Each polygon has side length of 3 cm. These
models are shown in Figures 27. This method can be used to decorate any
polyhedron having regular polygon faces, such as Platonic, Archimedean, and
Johnson solids.
Creating a symmetric pattern on a polyhedron from decorated regular polygonal
tiles depends on the symmetry groups of the tile decorations and the polyhedron.
The subgroup structure of the symmetry subgroups of the polyhedron gives all
possible symmetry groups. A variety of symmetric knotted and linked structures
is possible using a family of simple tile decorations.
A collection of decorated polyhedral models could make a useful tool for teaching symmetry or group structures. Future plans include producing all possible
patterns for a given polyhedron and tile family, then performing an analysis to
determine symmetry and a knot categorization.
Acknowledgments
This work was supported by a grant from the Hewlett-Mellon Fund for Faculty
Development at Albion College, Albion, MI.
REFERENCES
Cameron Browne. Truchet Curves and Surfaces. Computers & Graphics, 2008,
268281.
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D.A. Reimann
J.H. Conway, H. Burgiel, and C. Goodman-Strauss. The Symmetries of Things.
AK Peters Wellesley, MA, 2008.
Peter R. Cromwell. Polyhedra. Cambridge University Press, 1999.
Martin Gardner, Mathematical Games: Extraordinary nonperiodic tiling that
enriches the theory of tiles, Scientic American, January 1977, 110121.
Gopherwood Studios, Entanglement, http://entanglement.gopherwoodstudios.
com/.
Slavik V. Jablan. Modular Games. Bridges 2001 Art Exhibit, http://www.
bridgesmathart.org/art-exhibits/bridges2001/op/tiles/index.html.
Slavik V. Jablan. Symmetry, Ornament, and Modularity. World Scientic,
2002.
KnoTilesTM , Tesselations http://www.tessellations.com/.
David A. Reimann. Decorating regular tiles with arcs. In Reza Sarhangi and
Carlo Sequin, editors, Bridges Coimbra: Mathematics, Music, Art, Architecture,
Culture, pages 581584, Coimbra, Portugal, 2731 July 2011.
David A. Reimann. Modular construction of knots. Hyperseeing, Summer 2012,
63-69. The International Society of the Arts, Mathematics, and Architecture,
ISAMA '12, Chigago, Illinois, USA. 18-22 June 2012.
D. Schattschneider. M.C. Escher: Visions of Symmetry. Harry N. Abrams,
2004.
Cyril Stanley Smith, The Tiling Patterns of Sebastien Truchet and the Topology
of Structural Hierarchy, Leonardo, 1987, 373385.
Trax, Trax Game Ocial Website, http://www.traxgame.com/.
189
Symmetric interlace patterns on polyhedra
A
Figure 2:
B
C
Three views of a decorated cuboctahedron having symmetry 322. The views show
the 3-fold rotational point (A), the rst 2-fold rotational point (B), and the second 2-fold
rotational point (C). The interlace pattern is formed using six squares each having pattern
(0, 1|2, 3|4, 7|5, 6)[],
and eight triangles each having pattern
(0, 5|1, 2|3, 4)[].
Even though the
view in (B) appears to have mirror symmetry, the decorations of the tiles not seen result in
the overall pattern having 2-fold rotational symmetry about the center vertex.
A
Figure 3:
B
C
Three views of a decorated octahedron having symmetry 432. The 4-fold rotational
point is shown in (A) with the 3-fold and 2-fold and rotational points shown in (B) and
(C) respectively. The interlace pattern is formed using eight triangles each having pattern
(0, 3|1, 4|2, 5)[0/1, 1/2, 2/0].
The pattern suggests 8 interlinked rings centered at the vertices.
A
Figure 4:
B
C
Three views of a decorated dodecahedron having symmetry 532.
The views
show the 5-fold rotational point (A), the 3-fold rotational point (B), and the 2-fold rotational point (C). The interlace pattern is formed using twelve pentagons each having pattern
(0, 3|1, 8|2, 5|4, 7|6, 9)[0/1, 2/0, 3/2, 4/3, 1/4].
tered at the vertices.
The pattern suggests 20 interlinked rings cen-
190
D.A. Reimann
A
Figure 5:
B
C
Three views of a decorated tetrahedron having symmetry
2×.
The views show
the a typical view of the tetrahedron (A) and the two 2-fold rotational points (B) and (C).
The interlace pattern is formed using two triangles having pattern
two triangles having pattern
(0, 1|2, 4|3, 5)[1/2].
(0, 1|2, 4|3, 5)[2/1]
and
The two patterns are mirror images of one
another, resulting in an overall glide reection for the decorated tetrahedron.
A
Figure 6:
B
C
D
Three views of a decorated gyroelongated square pyramid (the Johnson solid
having symmetry
44.
J10 )
The views in (A) and (B) show the two 4-fold rotational points. Two
other views are shown in (C) and (D). The interlace pattern is formed using four triangles
having pattern
(0, 1|2, 4|3, 5)[2/1], four triangles having pattern (0, 2|1, 5|3, 4)[1/0], four tri(0, 2|1, 4|3, 5)[1/0, 2/1], and a square having pattern (0, 7|1, 2|3, 4|5, 6)[].
angles having pattern
A
Figure 7:
B
C
Three views of a decorated icosahedron having symmetry 225. The views show the
5-fold rotational point (A) and the two 2-fold rotational points (B) and (C). The interlace
pattern is formed using twenty triangles each having pattern
(0, 1|2, 4|3, 5)[2/1].
Note the
the interlace pattern is comprised of twelve disjoint links: two (5,2) torus knots, ve simple
circles, and ve twisted links.