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Glide reflection

In geometry, a glide reflection is an isometry of the Euclidean plane that combines a reflection across a line with a translation parallel to that line, resulting in a transformation that reverses orientation but preserves distances and angles.^[1]^[2] The order of the operations—whether the reflection precedes or follows the translation—does not affect the outcome, as the two commute when the translation vector is parallel to the reflection axis.^[3] Unlike pure reflections or translations, a glide reflection has no fixed points unless the translation component is zero, making it a distinct type of rigid motion.^[2] Glide reflections are fundamental in the classification of plane isometries, forming one of the four basic types alongside translations, rotations, and reflections; they are classified as opposite isometries due to their orientation-reversing nature.^[1]^[3] Mathematically, the square of a glide reflection yields a translation by twice the glide vector, highlighting its periodic behavior and utility in generating symmetries.^[2] This transformation preserves properties such as parallelism, collinearity, and midpoints, but alters the orientation of figures, such as reversing the clockwise order of vertices in a polygon.^[3] Beyond basic transformations, glide reflections play a key role in symmetry analysis, appearing in frieze groups for strip patterns and wallpaper groups for tilings, as well as in crystallography where they describe non-symmorphic space group operations like axial or diagonal glides.^[4] For instance, they model real-world phenomena such as alternating footprints in a trail, where each step combines a shift forward with a mirror-image placement.^[3] Their invariant line—the axis of reflection—remains the sole fixed set under the transformation, underscoring their structured geometric impact.^[2]

Definition and Properties

Formal Definition

A glide reflection is an isometry of the Euclidean plane defined as the composition of a reflection across a line, known as the axis, followed by a translation parallel to that axis by a nonzero distance. This operation combines the orientation-reversing property of the reflection with the displacement effect of the translation, resulting in an indirect isometry that cannot be decomposed into simpler direct isometries.^[5]^[6] Formally, a glide reflection \sigma is specified by its axis \ell and translation vector \mathbf{t} parallel to \ell, with \|\mathbf{t}\| > 0. For any point p in the plane, \sigma(p) = \tau_{\mathbf{t}} (\rho_{\ell}(p)), where \rho_{\ell} denotes the reflection over the line \ell and \tau_{\mathbf{t}} denotes the translation by \mathbf{t}. The order of composition is immaterial since the reflection and translation commute when the translation is parallel to the axis.^[6]^[7] This distinguishes a glide reflection from a pure reflection, which arises when \mathbf{t} = \mathbf{0} and fixes the axis pointwise, and from a pure translation, which involves no reflection and preserves orientation. The nonzero translation requirement ensures the glide reflection is irreducible to a reflection alone, maintaining its distinct classification among plane isometries.^[5]^[6]

Geometric Properties

A glide reflection is an orientation-reversing isometry of the Euclidean plane, meaning it reverses the handedness of figures, transforming clockwise orientations to counterclockwise and vice versa, in contrast to orientation-preserving isometries such as translations and rotations.^[8]^[9] This reversal arises from its composition, which includes an odd number of reflections, fundamentally altering chirality without preserving it.^[10] Unlike a pure reflection, which fixes an entire line (its axis) pointwise, a glide reflection has no fixed points in the plane; every point is displaced due to the combined effect of the reflection and the subsequent translation parallel to the axis.^[2]^[11] However, the axis itself remains invariant as a set, with points on it mapped to other points along the same line.^[11]^[2] This ensures that the transformation preserves the overall structure of the plane while enforcing motion everywhere. The order of a glide reflection is infinite unless the translation component is zero (reducing it to a reflection); applying it twice results in a pure translation by twice the glide vector along the axis direction.^[9]^[2] Furthermore, glide reflections are invariant under conjugation by translations parallel to the axis, meaning such a conjugation yields the same glide reflection with the unchanged axis and glide distance.^[10]^[12] While primarily defined in the two-dimensional Euclidean plane, the concept extends to higher dimensions, such as glide planes in three-dimensional space groups, where a reflection across a plane is combined with a translation parallel to that plane, maintaining similar orientation-reversing and fixed-point-free properties in the ambient space.^[13]^[7]

Algebraic Representation

A glide reflection in the Euclidean plane can be algebraically represented using coordinate transformations. For a horizontal axis at y = c and a translation distance d parallel to the axis, the glide reflection \sigma maps a point (x, y) to (x + d, 2c - y). This formula combines the reflection over the line y = c, which maps (x, y) to (x, 2c - y), with a subsequent translation by (d, 0).^[14] In matrix form using homogeneous coordinates, a glide reflection can be expressed as a 3×3 matrix. For the special case where the axis is the x-axis (c = 0) and the translation is by (d, 0), the matrix is

\begin{pmatrix} 1 & 0 & d \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.

This matrix is the product of the translation matrix \begin{pmatrix} 1 & 0 & d \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} and the reflection matrix over the x-axis \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, applied in sequence.^[15]^[14] Glide reflections belong to the isometry group E(2), which consists of all distance-preserving transformations of the plane and is generated by reflections and translations. In this group, a glide reflection is an orientation-reversing isometry of the form h(v) = A v + w, where A is an orthogonal matrix with \det A = -1 (representing a reflection) and w is a nonzero vector parallel to the reflection axis such that A w \neq -w (ensuring no fixed points).^[14] Regarding conjugation properties, within the group of orientation-reversing isometries, glide reflections with the same translation distance are conjugate to one another, but distinct from reflections due to the absence of fixed points in nontrivial glides.^[16]

Role in Symmetry Groups

Frieze Groups

Frieze groups represent the seven distinct infinite discrete symmetry groups that preserve patterns repeating periodically along an infinite strip in the Euclidean plane, capturing one-dimensional periodic symmetries extended to a bounded transverse direction.^[17] These groups are classified using international notation: p1 (translations only), p11g (glide reflections), p1m1 (horizontal reflections), p11m (vertical reflections), p2 (180° rotations), p2mg (rotations, vertical reflections, and glides), and p2mm (rotations and horizontal/vertical reflections).^[17] Glide reflections, as compositions of translations parallel to a reflection axis and the reflection itself, appear in two of the frieze groups, enabling symmetries that incorporate translational shifts along the strip's direction.^[18] Specifically, the p11g group features pure horizontal glide reflections without additional mirrors or rotations; and the p2mg group pairs glides with 180° rotations and vertical mirrors. These inclusions distinguish the groups from purely translational or rotational ones, allowing for more complex periodic motifs like those seen in architectural borders or decorative bands. The generators of these groups highlight the central role of glide reflections. In p11g, the group is generated solely by a single horizontal glide reflection, whose square yields the primitive translation.^[18] In p2mg, the glide reflection combines with a 180° rotation centered on a vertical mirror line, producing the full set of symmetries through their compositions.^[17] Algebraic representations from group theory confirm these generators, where the glide acts as an element of order 2 modulo translations. The classification of the seven frieze groups traces back to Evgraf Fedorov's foundational work in 1891 on crystallographic symmetries, where glide reflections proved essential for capturing non-orientable symmetries absent in pure reflection-based groups.^[19] This enumeration built on earlier analyses of periodic patterns, emphasizing glides as key to the two groups that exhibit them, bridging basic translations to richer strip symmetries.^[18]

Wallpaper Groups

Wallpaper groups comprise the 17 distinct classes of symmetry groups that govern periodic two-dimensional patterns, such as tilings, by combining translations with rotations, reflections, and glide reflections under the constraints of crystallographic restrictions. These groups classify all possible repeatable motifs in the plane, ensuring that the symmetries preserve the periodic lattice structure while allowing for decorative variety in designs like textiles or architectural tiles.^[20] Glide reflections are integral to eight of these wallpaper groups, where they contribute essential nonsymmorphic symmetries that cannot be generated solely by point group operations at lattice points. For instance, in the p2mg group, glide reflections occur along diagonal axes, integrating translation by half the lattice vector with reflection to achieve the group's characteristic 180-degree rotational symmetry and mirror lines. The pmg group incorporates horizontal and vertical glide reflections, which complement perpendicular mirror lines to form a rectangular lattice with no rotational centers at mirror intersections. Similarly, the pgg group relies on pure glide subgroups, featuring two sets of perpendicular glide reflections that, together with 180-degree rotations, generate the full symmetry without any pure reflections. Other groups, such as pg, cm, and p4mg, also feature glides adapted to their lattice types, like primitive or centered arrangements.^[21]^[22] In these groups, glide reflections typically pair with rotations or mirrors to generate the complete set of operations, ensuring closure under composition while maintaining translational periodicity; the minimal translation distance in a glide is often half a primitive lattice vector parallel to the reflection axis, linking the operation directly to the underlying lattice basis. This pairing distinguishes the groups' structures—for example, composing a glide with a rotation in pgg yields additional glides, expanding the symmetry network across the plane.^[20]^[23] Unlike pure translations, which solely shift the pattern without altering orientation, glide reflections incorporate a reflection component that inverts handedness, enabling richer symmetries in non-rectangular lattices where direct mirror symmetries might conflict with the lattice geometry, such as in rhombic or hexagonal tilings. This reflection aspect allows glide reflections to resolve apparent asymmetries in pattern motifs, facilitating periodic extensions that pure translations alone cannot achieve.^[22]^[21]

Space Groups

In three-dimensional crystallography, the 230 space groups classify the possible symmetry operations for periodic atomic arrangements in crystals.^[24] Glide reflections generalize to glide planes, which are nonsymmorphic operations combining a mirror reflection across a plane with a non-zero translation parallel to that plane.^[24] These operations reverse orientation and lack fixed points on the glide plane itself, analogous to their two-dimensional counterparts but extended to volume-filling lattices.^[25] The specific type of glide plane is denoted by a letter indicating the direction and fraction of the translation relative to the unit cell edges: an a-glide involves translation by a/2, a b-glide by b/2, a c-glide by c/2, an n-glide by (a + b)/2 or similar diagonal, and a d-glide by a quarter of a face or body diagonal.^[25] The translation always aligns with lattice directions to preserve periodicity, ensuring the operation maps the crystal lattice onto itself.^[24] In centered lattices, glide planes can produce equivalent translations in multiple directions, such as diamond (d)-glides in certain orthorhombic, tetragonal, and cubic groups.^[25] Glide planes occur in 149 of the 230 space groups, often combining with other elements like screw axes to form complex symmetries.^[26] For instance, in the monoclinic space group P2₁/c (No. 14), a c-glide plane perpendicular to the b-axis pairs with a 2₁ screw axis along b, enabling the structure to describe common mineral symmetries while enforcing specific systematic absences in diffraction patterns.^[27] Modern computational resources, such as the Bilbao Crystallographic Server developed since 2003, facilitate visualization and analysis of glide planes in 3D space groups by generating diagrams, symmetry operations, and subgroup relations. These tools extend beyond traditional tables, allowing interactive exploration of how glide planes contribute to overall crystallographic symmetry in higher dimensions.

Examples and Applications

Visual Examples

A classic visual example of a glide reflection in two dimensions is the pattern formed by footprints in sand. As a person walks along a straight path, each footprint is a reflection of the previous one across a horizontal axis midway between them, combined with a forward translation equal to the step length. This results in an alternating sequence of left and right footprints shifted progressively along the direction of motion, demonstrating the composite nature of the transformation without pure reflection or translation alone.^[3] In frieze patterns, which exhibit symmetries along a strip, certain meander patterns belong to the p11g group, characterized by translations and glide reflections. The pattern consists of interlocking rectangular motifs forming a continuous zigzag border, where the glide axis runs horizontally midway through the design; applying the glide reflection maps one segment of the meander to the next, producing the repeating effect with a half-unit shift and flip. This symmetry, lacking pure reflections or rotations, highlights the glide's role in generating the infinite extension.^[28] For wallpaper groups, which tile the full plane, parquet flooring in the p2mg group illustrates diagonal glide reflections amid 180-degree rotations and mirror symmetries. Typical herringbone parquet arrangements feature elongated rectangular tiles laid in a V-shaped pattern, where diagonal glide axes bisect the angles between tiles; a glide along this axis reflects and shifts a tile motif to an adjacent position, ensuring the overall tessellation without gaps or overlaps. This group, as classified in symmetry theory, combines these elements to create balanced, repeating floor designs.^[29] Extending to three dimensions, layered crystals like graphite exhibit glide planes between atomic sheets. Graphite's structure comprises stacked hexagonal layers of carbon atoms in the ABAB configuration, with glide planes parallel to the basal (001) faces; a glide operation across such a plane involves reflecting one layer relative to the adjacent one while translating it by half the lattice vector in the plane direction, resulting in the offset stacking that stabilizes the crystal lattice. Schematic views often depict these as parallel sheets with arrows indicating the glide vector perpendicular to the reflection plane.^[30] Standard diagrams of glide reflections frequently use arrow sequences to illustrate transformation paths. For instance, a figure might show an original shape, followed by arrows denoting the reflection line and parallel translation vector; the composite path traces the glide as a single curved or segmented arrow from pre-image to final image, emphasizing how the operation reverses orientation while shifting position. These visualizations, common in geometric texts, clarify the non-commutative order of the reflection and translation components.^[3]

Practical Applications

In crystallography, glide planes play a crucial role in describing the symmetry of mineral structures, such as those found in zeolites and diamond, facilitating the interpretation of X-ray diffraction patterns through systematic absences of certain reflections. For instance, the high-silica zeolite boggsite exhibits a body-centered lattice with coupled a- and m-glide planes normal to the c-axis, belonging to space group Imma, which helps in resolving its framework topology via diffraction analysis. Similarly, the germanosilicate zeolite HPM-14 is assigned to possible space groups such as C2/m, enabling the stacking of layers with extra-large pores that enhance its potential for catalytic applications. The diamond structure, with space group Fd-3m, features distinctive diamond glide planes (d-glides) involving translation by one-quarter of the unit cell along face diagonals, a symmetry element that contributes to its unique tetrahedral bonding and was instrumental in early 20th-century refinements of its atomic arrangement using X-ray methods. These glide symmetries aid in precise structure determination, as the translational component shifts reflection conditions, producing characteristic intensity distributions in diffraction data that distinguish nonsymmorphic groups from symmorphic ones. In biology, glide reflection symmetries appear in certain macromolecular assemblies and evolutionary structures, influencing packing efficiency and functional properties. Early biological forms, such as Ediacaran fossils from 635–541 million years ago, frequently displayed glide symmetries, which provided adaptive advantages in simple morphologies before evolutionary shifts favored mirror symmetries for organ specialization. In modern contexts, protein crystals often crystallize in space groups containing glide planes, such as those used in X-ray structure determination, where the symmetry operations generate equivalent positions that optimize packing density and reveal molecular interactions. For viral structures, while helical assemblies like the tobacco mosaic virus (TMV) primarily exhibit screw symmetry combining rotation and translation along the axis, their crystalline forms can incorporate glide planes; TMV, for example, crystallizes in space group I222, where body-centering translations interact with orthorhombic symmetries to produce glide-like effects in diffraction patterns, aiding resolution of its rod-like capsid. Glide reflections are integral to artistic and design traditions, particularly in creating infinite patterns with translational and reflective elements. In Islamic geometric art, many motifs from the Alhambra palace adhere to frieze groups like pmg2, which include glide-reflection symmetry along the translation axis combined with vertical mirrors and 180-degree rotations, enabling intricate border designs that repeat seamlessly. M.C. Escher's woodcut "Sky and Water I" (1938) employs glide reflections to transition birds into fish across the plane, where motifs slide and reflect to form a metamorphic tessellation based on a square lattice, exemplifying wallpaper group p4mg with embedded glide operations for visual ambiguity. Post-1980s advancements in materials science have leveraged glide-inspired symmetries in quasicrystals and metamaterials to achieve novel optical properties. Quasicrystals, discovered in 1982, often realize nonsymmorphic symmetries in higher dimensions, such as dodecagonal quasicrystals modeled with glide planes and screw axes in their 12-fold superspace groups, leading to forbidden diffraction symmetries that enhance thermal and electronic stability in alloys. In metamaterials, glide reflection symmetry enables ultrawideband suppression of coupling to radiation modes, as demonstrated in high-impedance surfaces where the symmetry enforces degenerate bands and flat dispersions, reducing losses for terahertz applications. Furthermore, glide-symmetric photonic crystals facilitate all-angle negative refraction without reflection losses, exploiting Weyl points in the band structure to guide light with high efficiency, a property exploited in cut-wire metasurfaces for advanced beam steering since the early 2000s. Computational modeling tools like CrystalMaker simulate glide reflection effects by constructing full crystal structures from space group symmetries, including nonsymmorphic operations, which is essential for visualizing glide planes in nanotechnology applications such as zeolite frameworks or quasicrystalline nanomaterials. The software generates atomic positions via symmetry operators, allowing real-time manipulation to predict diffraction patterns and defect behaviors in nanoscale devices, thereby supporting design of materials with tailored glide-induced properties like anisotropic conductivity.

References

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Glide Reflection - Interactive Mathematics Miscellany and Puzzles
Glide reflection is a composite transformation which is a translation followed by a reflection in line parallel to the direction of translation.
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Translations and Glide Reflections
A glide reflection is an isometry with no fixed points and one invariant line. A glide reflection is the composition of a line reflection Rm with a translation ...Missing: geometry | Show results with:geometry
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Glide Reflection - MathBitsNotebook(Geo)
A glide reflection is an opposite rigid transformation. A translation is a direct rigid transformation and a reflection is an opposite rigid transformation.
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Glide Reflection - an overview | ScienceDirect Topics
Glide reflection is defined as a symmetry operation that combines a reflection across a plane with a translation along that plane, specifically through a ...<|control11|><|separator|>
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[PDF] isometries of the plane - UChicago Math
Jul 21, 2009 · Definition 3.5. A glide-reflection is an isometry that is the product of a reflection and a translation in the direction of the axis of the ...
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A reflection about a line l followed by a transla- tion along l is called a glide reflection. In this definition, the order of reflection and translation does ...
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Glide reflection by reflecting across the line with equation px + qy + r = 0 followed by translation by the amount (tq,−tp), assuming that p2 + q2 = 1...
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[PDF] Geometry of linear transformations - Vipul Naik
The orientation-reversing self-isometries of R2 are precisely the reflections and glide reflections. Note that these do not form a group, but together with ...<|control11|><|separator|>
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If m is a glide reflection, it is a composition of reflection in some line ℓ, and translation by a vector b parallel to ℓ. Then m2 is a translation by 2b. In ...
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[PDF] isometries of the plane and complex numbers - Keith Conrad
Suppose h is a glide reflection, so it has no fixed points. Then ghg-1 also has no fixed points, which means ghg-1 is either a glide reflection or a translation ...
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Proposition 42 Let γ be a glide reflection with axis c. 1. γ has no fixed points. 2. The midpoint of point P and its image γ (P) lies on c.
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We can now determine the conjugates of a glide reflection. Theorem 127 The conjugate of a glide reflection is a glide reflection. If α is an isometry and γ ...
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### Summary of Conjugation Properties of Glide Reflections
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Here glide-reflection means symmetries that are composed of a trans- lation along the reflection axis followed-by a reflection about the axis. There are five ...
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May 10, 2019 · For glide reflections we have a combination of a translation and a horizontal reflection. The motion takes a figure and shifts it half of a ...
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E. S. Fedorov Promoting the Russian-German Scientific ... - MDPI
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main axis, where m means a reflection, g indicates a glide reflection axis, and 1 ... [5] https://en.wikipedia.org/wiki/Wallpaper group. [6] Martin von Gagern ...
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Oct 24, 2021 · Reflections are denoted by green lines, blue lines indicate glide reflections, red dots indicate order 2 rotation centers [10]. ... There are ...
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The others involve, in addition to translations, one or more of the other types of symmetries (rotations, reflections, glide-reflections). The Conway notation ...
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[PDF] Unique Visualizations: Exploring Symmetries with Complex-Valued ...
The frieze group F5 is generated by a glide reflection and a rotation. It is an infinite dihedral group that creates the pattern TVRG. The rotation points ...Missing: 1891 | Show results with:1891
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Glide Planes
In A-, B-, C-, and F-centred space groups, the combination of a centred lattice with a glide plane can lead to a situation where the glide direction exists as ...
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Monoclinic Space Groups
In this space group, the symmetry operator x,1/2-y,1/2+z corresponds to a c-glide plane at y = 1/4 (shown by the bent arrow in the bottom left-hand corner of ...
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Jan 27, 2009 · Has a glide reflection, with alternating vertical mirrors and 180° rotations. There are many naming systems for Frieze groups besides the IUC ...
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Oct 10, 2013 · This group contains glide reflections. The direction of the glide reflection is parallel to one axis of translation and perpendicular to the ...
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[PDF] Structural and Dynamic Investigations of Graphite Based Materials
... planes in the lattice. 63/m is a 63/ screw axis with a perpendicular mirror plane and the letters m and c describe other mirror and glide planes in the lattice.