Hubbry Logo
Rotational symmetryRotational symmetryMain
Open search
Rotational symmetry
Community hub
Rotational symmetry
logo
8 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Rotational symmetry
Rotational symmetry
Rotational symmetry
View on Wikipedia
from Wikipedia

The triskelion appearing on the Isle of Man flag has rotational symmetry because it appears the same when rotated by one third of a full turn about its center. Because its appearance is identical in three distinct orientations, its rotational symmetry is three-fold.

Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.

Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids.[1][2]

Formal treatment

[edit]
See also: Rotational invariance

Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of E +(m) (see Euclidean group).

Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is homogeneous, and the symmetry group is the whole E(m). With the modified notion of symmetry for vector fields the symmetry group can also be E +(m).

For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO(m), the group of m × m orthogonal matrices with determinant 1. For m = 3 this is the rotation group SO(3).

In another definition of the word, the rotation group of an object is the symmetry group within E +(n), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.

Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem, the rotational symmetry of a physical system is equivalent to the angular momentum conservation law.

Discrete rotational symmetry

[edit]

Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the nth order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of ⁠⁠ (180°, 120°, 90°, 72°, 60°, 51 3⁄7°, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all objects look alike after a rotation of 360°).

The notation for n-fold symmetry is Cn or simply n. The actual symmetry group is specified by the point or axis of symmetry, together with the n. For each point or axis of symmetry, the abstract group type is cyclic group of order n, Zn. Although for the latter also the notation Cn is used, the geometric and abstract Cn should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see cyclic symmetry groups in 3D.

The fundamental domain is a sector of ⁠⁠

Examples without additional reflection symmetry:

  • n = 2, 180°: the dyad; letters Z, N, S; the outlines, albeit not the colors, of the yin and yang symbol; the Union Jack (as divided along the flag's diagonal and rotated about the flag's center point)
  • n = 3, 120°: triad, triskelion, Borromean rings; sometimes the term trilateral symmetry is used;
  • n = 4, 90°: tetrad, swastika
  • n = 5, 72°: pentad, pentagram, regular pentagon; 5-fold symmetry is not possible in periodic crystals.
  • n = 6, 60°: hexad, Star of David (this one has additional reflection symmetry)
  • n = 8, 45°: octad, Octagonal muqarnas, computer-generated (CG), ceiling

Cn is the rotation group of a regular n-sided polygon in 2D and of a regular n-sided pyramid in 3D.

If there is e.g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°.

A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is a propeller.

Examples

[edit]
C2 (more) C3 (more) C4 (more) C5 (more) C6 (more)

Double Pendulum fractal
Roundabout traffic sign
Taiwan's recycling symbol
US Bicentennial Star
Emblem of Fujieda, Shizuoka

The starting position in shogi
Snoldelev Stone's interlocked drinking horns design

Multiple symmetry axes through the same point

[edit]

For discrete symmetry with multiple symmetry axes through the same point, there are the following possibilities:

  • In addition to an n-fold axis, n perpendicular 2-fold axes: the dihedral groups Dn of order 2n (n ≥ 2). This is the rotation group of a regular prism, or regular bipyramid. Although the same notation is used, the geometric and abstract Dn should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see dihedral symmetry groups in 3D.
  • 4×3-fold and 3×2-fold axes: the rotation group T of order 12 of a regular tetrahedron. The group is isomorphic to alternating group A4.
  • 3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group O of order 24 of a cube and a regular octahedron. The group is isomorphic to symmetric group S4.
  • 6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group I of order 60 of a dodecahedron and an icosahedron. The group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5 (rotational symmetries like prisms and antiprisms).

In the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, and the number of them is half the number of edges. The other axes are through opposite vertices and through centers of opposite faces, except in the case of the tetrahedron, where the 3-fold axes are each through one vertex and the center of one face.

Rotational symmetry with respect to any angle

[edit]

Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry. The fundamental domain is a half-line.

In three dimensions we can distinguish cylindrical symmetry and spherical symmetry (no change when rotating about one axis, or for any rotation). That is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, respectively. Axisymmetric and axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry (i.e. rotational symmetry with respect to a central axis) like a doughnut (torus). An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties).

In 4D, continuous or discrete rotational symmetry about a plane corresponds to corresponding 2D rotational symmetry in every perpendicular plane, about the point of intersection. An object can also have rotational symmetry about two perpendicular planes, e.g. if it is the Cartesian product of two rotationally symmetry 2D figures, as in the case of e.g. the duocylinder and various regular duoprisms.

Rotational symmetry with translational symmetry

[edit]

Arrangement within a primitive cell of 2- and 4-fold rotocenters. A fundamental domain is indicated in yellow.
Arrangement within a primitive cell of 2-, 3-, and 6-fold rotocenters, alone or in combination (consider the 6-fold symbol as a combination of a 2- and a 3-fold symbol); in the case of 2-fold symmetry only, the shape of the parallelogram can be different. For the case p6, a fundamental domain is indicated in yellow.

2-fold rotational symmetry together with single translational symmetry is one of the Frieze groups. A rotocenter is the fixed, or invariant, point of a rotation.[3] There are two rotocenters per primitive cell.

Together with double translational symmetry the rotation groups are the following wallpaper groups, with axes per primitive cell:

  • p2 (2222): 4×2-fold; rotation group of a parallelogrammic, rectangular, and rhombic lattice.
  • p3 (333): 3×3-fold; not the rotation group of any lattice (every lattice is upside-down the same, but that does not apply for this symmetry); it is e.g. the rotation group of the regular triangular tiling with the equilateral triangles alternatingly colored.
  • p4 (442): 2×4-fold, 2×2-fold; rotation group of a square lattice.
  • p6 (632): 1×6-fold, 2×3-fold, 3×2-fold; rotation group of a hexagonal lattice.
  • 2-fold rotocenters (including possible 4-fold and 6-fold), if present at all, form the translate of a lattice equal to the translational lattice, scaled by a factor 1/2. In the case translational symmetry in one dimension, a similar property applies, though the term "lattice" does not apply.
  • 3-fold rotocenters (including possible 6-fold), if present at all, form a regular hexagonal lattice equal to the translational lattice, rotated by 30° (or equivalently 90°), and scaled by a factor
  • 4-fold rotocenters, if present at all, form a regular square lattice equal to the translational lattice, rotated by 45°, and scaled by a factor
  • 6-fold rotocenters, if present at all, form a regular hexagonal lattice which is the translate of the translational lattice.

Scaling of a lattice divides the number of points per unit area by the square of the scale factor. Therefore, the number of 2-, 3-, 4-, and 6-fold rotocenters per primitive cell is 4, 3, 2, and 1, respectively, again including 4-fold as a special case of 2-fold, etc.

3-fold rotational symmetry at one point and 2-fold at another one (or ditto in 3D with respect to parallel axes) implies rotation group p6, i.e. double translational symmetry and 6-fold rotational symmetry at some point (or, in 3D, parallel axis). The translation distance for the symmetry generated by one such pair of rotocenters is times their distance.

Euclidean plane Hyperbolic plane

Hexakis triangular tiling, an example of p6, [6,3]+, (632) (with colors) and p6m, [6,3], (*632) (without colors); the lines are reflection axes if colors are ignored, and a special kind of symmetry axis if colors are not ignored: reflection reverts the colors. Rectangular line grids in three orientations can be distinguished.
Order 3-7 kisrhombille, an example of [7,3]+ (732) symmetry and [7,3], (*732) (without colors)

See also

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Rotational symmetry

View on Grokipedia
from Grokipedia
Rotational symmetry is a fundamental geometric property where a figure or object appears identical to itself after around a fixed point, known as the center of . In the discrete case, this occurs for rotations by multiples of 360°/n, where n is the finite order of symmetry; in the continuous case, the figure is invariant under any (infinite order). This invariance under distinguishes it from other symmetries like reflection, as rotations preserve the orientation of the figure. For discrete rotational symmetry, the order n quantifies the symmetry, indicating the number of distinct rotations—ranging from 0° (identity) to multiples of 360°/n—that map the figure onto itself. For example, an has order 3, remaining unchanged under rotations of 120° and 240°; a , by contrast, exhibits continuous rotational symmetry of infinite order. In group theory, the set of discrete rotations forming rotational symmetry constitutes a Cn, closed under composition, with the and inverses ensuring the group's structure. Properties include associativity of rotations and a single fixed point at . Beyond two dimensions, rotational symmetry generalizes to m-dimensional via the special SO(m), where SO(3) in three dimensions underlies the conservation of in physical laws. These symmetries are crucial in , , and physics for analyzing invariant structures.

Fundamentals

Definition

Rotational symmetry is a geometric property exhibited by a figure or object that remains invariant—appearing unchanged—when rotated by specific angles around a center of rotation. In two dimensions, this center is a fixed point; in three dimensions, it is a fixed axis. This transformation preserves the distances and angles between points, mapping the object precisely onto itself without altering its overall appearance. In the Euclidean plane, such rotations are isometries that fix the center point while moving all other points along circular arcs. In three-dimensional space, rotations are isometries that fix all points along the axis while moving other points along circular paths perpendicular to the axis. For instance, a square demonstrates this symmetry, as it looks identical to its original orientation after a 90-degree rotation about its center. This form of symmetry differs fundamentally from other types, such as , which involves mirroring the object across a line or plane, potentially reversing its orientation and eliminating — the property of existing in non-superimposable mirror-image forms. In contrast, rotational symmetry maintains the object's and does not require a mirror line; objects with pure rotational symmetry can thus be chiral. Similarly, it is distinct from translation symmetry, which shifts the entire object along a straight line without fixing any point, resulting in no central invariance. At its core, rotational symmetry relies on the concept of invariance under a transformation, where the —a —leaves the object's essential features unaltered, even if individual points are repositioned. This property is fundamental in and extends to both two-dimensional figures in the plane and three-dimensional objects in space, provided the rotation axis passes through the designated center.

Order of Symmetry

The order of rotational symmetry, denoted as nn, is a quantitative measure that indicates the number of times a figure can be rotated by equal angles around its center while appearing unchanged, completing a full 360° rotation after nn such steps. This finite value applies to discrete rotational symmetries, where the object maps onto itself only for specific rotation angles. For instance, an has an order of 3, as it coincides with itself after rotations of 120°, 240°, and 360°. The smallest rotation angle θ\theta that preserves the figure is given by θ=360∘n=2πn radians.\theta = \frac{360^\circ}{n} = \frac{2\pi}{n} \text{ radians}. This angle divides the full circle evenly, and higher orders correspond to finer divisions, such as n=6n = 6 for a regular hexagon with θ=60∘\theta = 60^\circ. The order thus characterizes the periodicity of the symmetry, with n=1n = 1 indicating no non-trivial rotational symmetry beyond the full turn. In contrast, objects exhibiting continuous rotational symmetry, such as a , possess an infinite order because they remain invariant under rotations by any arbitrary , not limited to discrete positions. This infinite order reflects the absence of preferred orientations, allowing the figure to overlay itself perfectly for every possible rotation within 360°.

Mathematical Framework

Discrete Rotational Symmetry

Discrete rotational symmetry describes the property of an object or figure that remains invariant under rotations by multiples of a fixed 2πn\frac{2\pi}{n} radians around a central point or axis, where nn is a positive greater than or equal to 1. This symmetry is formalized mathematically as the CnC_n, which is the generated by a single element of order nn. The elements of CnC_n are the nn distinct rotations rkr^k for k=0,1,…,n−1k = 0, 1, \dots, n-1, where rr denotes the generator by 2πn\frac{2\pi}{n}, and rnr^n is the identity. As an abstract group, CnC_n satisfies the group axioms: it is closed under composition, meaning the composition of any two rotations in the group yields another rotation in the group; it contains the identity element corresponding to a 0° (or 2π2\pi) rotation; and every element has an inverse, given by the rotation in the opposite direction by the same angle. These properties ensure that the set of discrete rotations forms a well-defined algebraic structure, with the order of the group equal to nn, referencing the order of rotational symmetry discussed earlier. For instance, a square exhibits C4C_4 symmetry, invariant under rotations by 90°, 180°, and 270° around its center. In two dimensions, discrete rotational symmetry around a point is represented by the standard applied to coordinate vectors. For a rotation by θ=2πkn\theta = \frac{2\pi k}{n}, the transformation is given by (cos⁡θ−sin⁡θsin⁡θcos⁡θ),\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}, which preserves distances and orientations in the plane. This matrix form arises from the linear transformation that rotates the basis vectors, ensuring the figure maps onto itself for each kk. Such symmetries contribute to the point groups in two and three dimensions. In 2D, pure discrete rotational symmetries generate the cyclic subgroups of rosette groups, which describe symmetries fixing a point in the plane, such as those of regular polygons. In contrast, 3D point groups incorporating discrete rotations, like those of polyhedra (e.g., the icosahedral group with 5-fold axes), extend to finite rotations around one or more axes, but the fundamental single-axis case remains cyclic CnC_n.

Continuous Rotational Symmetry

Continuous rotational symmetry refers to the invariance of an object or system under rotations by any arbitrary angle θ within the interval [0, 2π), distinguishing it from discrete cases by allowing full rotational freedom without a minimal nonzero angle. In two dimensions, this symmetry is embodied by the special SO(2), which consists of all 2×2 orthogonal matrices with 1, representing rotations around the origin in the . Similarly, in three dimensions, continuous rotational symmetry is captured by the group SO(3), comprising all 3×3 orthogonal matrices with 1, which describe rotations in . The mathematical structure underlying continuous rotational symmetry is that of a , where the group operations are smooth and the parameter space—typically the angle θ—is continuous and infinite-dimensional in the sense of admitting a smooth manifold . For SO(2), the so(2) is one-dimensional, generated by rotations, reflecting the single degree of freedom in planar rotations. In contrast, so(3) for SO(3) is three-dimensional, corresponding to rotations about three independent axes, and exhibits a non-abelian structure due to the non-commutativity of successive rotations in 3D. A fundamental representation of continuous rotations in 2D utilizes the complex plane, where multiplication by the unit complex number eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta effects a counterclockwise rotation by angle θ around the origin. This exponential form arises from Euler's formula and directly parametrizes elements of SO(2). In 3D, rotations in SO(3) can be represented using unit quaternions, which provide a compact, singularity-free parametrization via the map from the unit sphere in four dimensions to the rotation group, leveraging the double-covering homomorphism from SU(2) to SO(3). These representations highlight the smooth, continuous nature of the symmetry, enabling interpolation between rotations and facilitating computations in fields requiring arbitrary angular transformations.

Multiple Axes and Invariance

In , objects can possess rotational about multiple axes that intersect at a single point, such as of the object, leading to a richer set of invariance properties compared to single-axis cases. For instance, a exhibits threefold rotational about axes passing through opposite vertices (four such axes, with rotations of 120° and 240°), fourfold about axes through the centers of opposite faces (three axes, with 90°, 180°, and 270° rotations), and twofold about axes through the midpoints of opposite edges (six axes, with 180° rotations). These multiple axes collectively generate the full rotational group of the , known as the octahedral group OO, which has order 24 and is isomorphic to the S4S_4. The invariance under combined rotations around these axes arises from the group structure formed by composing individual rotations, where the overall is generated by cyclic subgroups corresponding to each axis type, often involving semidirect products rather than direct products due to non-commutativity. For polyhedral symmetries, this results in finite groups such as the tetrahedral group TT (order 12, isomorphic to A4A_4) for the , featuring threefold axes through vertices and twofold axes through edges, and the icosahedral group II (order 60, isomorphic to A5A_5) for the or , with fivefold axes through vertices, threefold through faces, and twofold through edges. The composition of successive s R1(θ1)R_1(\theta_1) around axis u1\mathbf{u}_1 and R2(θ2)R_2(\theta_2) around axis u2\mathbf{u}_2 is represented by the matrix product R=R2R1R = R_2 R_1, which yields another rotation in the special orthogonal group SO(3)SO(3), but generally R1R2≠R2R1R_1 R_2 \neq R_2 R_1 unless the axes coincide. In two dimensions, rotational symmetries are confined to a single axis perpendicular to the plane, limiting the structure to cyclic or dihedral groups that are abelian, whereas three-dimensional space permits multiple non-collinear axes to intersect, enabling non-abelian polyhedral rotation groups like TT, OO, and II. This dimensionality difference restricts 2D objects to invariances under rotations about one effective axis, while 3D allows for the complex interplay of axes that defines higher-order symmetries in platonic solids.

Examples and Occurrences

Geometric Figures

Rotational symmetry in geometric figures is prominently exhibited by regular polygons in two dimensions and Platonic solids in three dimensions, where rotations about a central axis map the figure onto itself while preserving distances and angles. These shapes serve as canonical examples due to their uniform construction from congruent regular polygonal faces meeting identically at each vertex. In two dimensions, an possesses rotational symmetry of order 3, meaning it appears unchanged after rotations of 120°, 240°, and 360° about its ; visually, each such cycles the three vertices to the position of the next, forming a closed path that traces the perimeter. exhibits order 4 rotational symmetry, invariant under 90°, 180°, 270°, and 360° rotations about its , where vertices map sequentially around the boundary, and opposite sides align perfectly after 180° turns. Similarly, a regular pentagon has order 5 symmetry, remaining superimposed after rotations of 72°, 144°, 216°, 288°, and 360° about its , with each advancing the five vertices to adjacent positions in a star-like or circumferential path. The circle represents the limiting case of infinite rotational symmetry, appearing identical under any angle of about its , as every point on the maps continuously to another without discrete steps. Extending to three dimensions, Platonic solids demonstrate rotational symmetries along multiple axes passing through their centers, with orders determined by the figure's regularity. The tetrahedron features four axes of order 3, each passing through a vertex and the centroid of the opposite face; a 120° rotation about such an axis permutes the three adjacent vertices cyclically while fixing the opposite face's orientation. The cube and its dual, the octahedron, share rotational symmetries including three axes of order 4 (through opposite face centers, allowing 90°, 180°, and 270° rotations that cycle four edges or faces), four axes of order 3 (through opposite vertices, cycling three faces), and six axes of order 2 (through midpoints of opposite edges, swapping pairs of faces). The dodecahedron and icosahedron possess even richer symmetries, with six axes of order 5 (through opposite vertices, rotating by 72° increments to map five adjacent faces or vertices), ten axes of order 3 (through face centers), and fifteen axes of order 2 (through edge midpoints). In each case, rotations map vertices to vertices and faces to faces, preserving the overall structure. Archimedean solids, which incorporate regular polygons of multiple types in a vertex-transitive arrangement, inherit similar high rotational symmetries from the Platonic solids but exhibit semi-regularity, allowing rotations that permute faces of different shapes while maintaining uniformity at vertices.

Natural and Artistic Patterns

In nature, rotational symmetry manifests prominently in biological structures, particularly through radial arrangements that enhance functionality. , or sea stars, exemplify discrete rotational symmetry of order 5, with their five arms radiating from a central disk, a pentaradial pattern that evolved in adult echinoderms for efficient environmental interaction. Similarly, many flowers display rotational symmetry in their petals, typically of orders 3 to 8; for instance, lilies often exhibit order 3 rotational symmetry with six identical tepals (three petals and three sepals), while sunflowers approximate continuous rotational symmetry through densely packed florets arranged in spirals. The shell provides an example of approximate continuous rotational symmetry via its growth, where each chamber expands outward in a self-similar curve that maintains angular consistency over rotations, approximating invariance under arbitrary angles. Artistic and cultural creations frequently incorporate rotational symmetry to evoke harmony and introspection. Mandalas, originating in Hindu and Buddhist traditions, often feature discrete rotational symmetry of orders 4 to 12, with intricate patterns radiating from a center to symbolize the universe's cyclical nature and aid meditation. In Islamic art, rosette patterns in geometric tiles demonstrate rotational symmetry, such as 8-fold or 10-fold arrangements in mosque decorations, where star-like motifs repeat under specific rotations to create infinite, non-figural designs that reflect divine order. Leonardo da Vinci's Vitruvian Man (c. 1490) approximates order 4 rotational symmetry through the superposition of a circle and square, with the human figure's limbs positioned to align under 90-degree rotations, illustrating Renaissance ideals of proportional balance in the human form. Radial rotational symmetry in often serves evolutionary roles, particularly in sessile or slow-moving organisms, by enabling omnidirectional sensing that aids in predator avoidance; for example, the uniform arm distribution in allows threat detection from any angle without directional bias. Aesthetically, rotational symmetry holds cultural significance across societies, symbolizing balance and cosmic equilibrium in art, as seen in mandalas where symmetrical repetition fosters a of stability and spiritual unity. Real-world instances of rotational symmetry are typically approximate due to environmental imperfections, deviating from ideal discrete orders. Snowflakes, for instance, ideally possess order 6 rotational symmetry from the of ice crystals, but vapor fluctuations during formation cause slight irregularities, resulting in unique, non-perfect patterns while retaining overall sixfold invariance.

Physical and Crystallographic Applications

In , the rotational symmetries compatible with periodic lattice structures in three dimensions are restricted to rotation axes of orders 1, 2, 3, 4, and 6, resulting in 32 distinct s that describe the possible symmetry operations of . This limitation arises from the , which demonstrates that rotations of order 5 or higher cannot tile space periodically without gaps or overlaps, ensuring compatibility with translational lattice symmetries. For instance, (SiO₂) belongs to point group 32 and features a principal 3-fold axis aligned with its c-crystallographic axis, allowing the crystal to appear identical after a 120° , which contributes to its piezoelectric properties. In physics, continuous rotational invariance of the Lagrangian or action principle implies the conservation of through , a foundational result linking symmetries to conserved quantities. This symmetry underpins the rotational dynamics of isolated systems, such as planetary orbits or motion, where total remains constant absent external torques. In , rotational invariance extends to the intrinsic property of spin, representing an internal for elementary particles like electrons, which transforms under rotations according to the particle's spin representation and contributes to the total of composite systems. The quantum mechanical manifestation of rotational invariance is captured by the commutation relation between the Hamiltonian H^\hat{H} and the L^\hat{\mathbf{L}}: [H^,L^]=0,[\hat{H}, \hat{\mathbf{L}}] = 0, indicating that energy eigenstates can be simultaneously eigenstates of components, facilitating the in central potential problems like the . This ensures the conservation of in time evolution, as the SO(3) acts unitarily on the . In modern applications, rotational symmetry governs the structure of molecular orbitals, where the quantum number ℓ\ell determines the orbital's rotational character (e.g., s-orbitals with ℓ=0\ell=0 are spherically symmetric, while p-orbitals with ℓ=1\ell=1 exhibit directional lobes). In , fullerenes such as C₆₀ exemplify high-order rotational symmetry, possessing icosahedral (Iₕ) symmetry with 5-fold, 3-fold, and 2-fold rotation axes that dictate their closed-shell electronic structure and stability, enabling applications in carbon-based nanomaterials for electronics and . This symmetry influences the degenerate molecular orbitals of fullerenes, leading to unique optoelectronic properties exploited in photovoltaic devices.

Extensions and Relations

Combined with Other Symmetries

Rotational symmetry often combines with reflection symmetry to form dihedral groups, which describe the full set of symmetries for regular polygons in the plane. The dihedral group DnD_n consists of nn rotations by multiples of 360∘/n360^\circ/n around the center, paired with nn reflections across axes passing through the center and vertices or midpoints of sides, yielding a total order of 2n2n. This structure arises as a semidirect product Cn⋊C2C_n \rtimes C_2, where CnC_n is the cyclic group of rotations and C2C_2 generates the reflections, with the reflection conjugating rotations to their inverses. For instance, the square's symmetries form D4D_4, encompassing four rotations (0°, 90°, 180°, 270°) and four reflections, for a group of order 8. In two dimensions, rotational symmetry combines with to produce frieze groups, which govern infinite strip patterns repeating along one direction, such as decorative borders. The frieze group denoted p2 features translations along the strip axis combined with 180° rotations about points midway between motif centers, without reflections or glides, enabling rotational motifs like alternating S-shapes in friezes. These seven frieze groups collectively incorporate rotations up to order 2 with translations, alongside possible reflections and glide reflections (translations composed with reflections parallel to the strip). Extending to three dimensions, rotational symmetry pairs with along the same axis to define screw axes, fundamental to crystallographic structures. A screw operation applies a by θ\theta about an axis followed by a by distance tt parallel to that axis, denoted as nmn_m where nn is the order (θ=360∘/n\theta = 360^\circ/n) and m/nm/n is the fractional relative to the lattice repeat. Mathematically, for a point x\mathbf{x} relative to the axis, the transformed position is given by x′=Rθ(x−p)+p+tu^,\mathbf{x}' = R_{\theta} (\mathbf{x} - \mathbf{p}) + \mathbf{p} + t \hat{\mathbf{u}}, where RθR_{\theta} is the rotation matrix by θ\theta, p\mathbf{p} a point on the axis, and u^\hat{\mathbf{u}} the unit vector along the axis; after nn applications, the net effect is a full lattice translation. Glide operations, meanwhile, combine reflection with translation parallel to the reflection plane, further enriching these symmetries. In , these combinations—rotations with translations, reflections, screws, and glides—generate the full set of 230 groups, which classify all possible periodic crystal symmetries. These groups build upon the 32 point groups (pure rotations and reflections) by incorporating lattice translations, with screw axes and glide planes accounting for the majority of the 230 distinct types, as tabulated in the International Tables for Crystallography.

Group-Theoretic Representation

In group theory, rotational symmetry is formalized through the action of rotation groups, which capture the structure of transformations preserving orientation and distances in Euclidean space. These groups provide a unified algebraic framework for both discrete and continuous rotations, enabling the study of symmetries via abstract algebraic tools rather than geometric descriptions alone. The representation theory of these groups further allows symmetries to be realized as linear transformations on vector spaces, facilitating applications in diverse fields such as quantum mechanics. Rotational groups are realized as Lie subgroups of the O(n)O(n), which consists of all n×nn \times n orthogonal matrices preserving the Euclidean inner product, with the special SO(n)SO(n) specifically comprising the orientation-preserving rotations (those with 1). For instance, in three dimensions, SO(3)SO(3) parameterizes all possible rotations around the origin. Irreducible representations of these groups decompose the space of functions or states into fundamental building blocks invariant under rotations; in quantum applications, the irreducible representations of SO(3)SO(3) correspond to multiplets, where the dimension of the representation is 2ℓ+12\ell + 1 for integer or ℓ\ell, underpinning the classification of particle states and selection rules in . Finite rotational symmetries correspond to discrete subgroups, such as the cyclic group CnC_n generated by a rotation by 2π/n2\pi/n radians around a fixed axis, while continuous symmetries are modeled by infinite Lie groups like SO(3)SO(3), which is compact and non-abelian. Point groups, which include rotations about multiple axes, admit finite discrete subgroups classified by their character tables—tabular summaries of traces of representation matrices under group elements. For the point group C3vC_{3v}, which describes symmetries of an equilateral triangle with vertical mirror planes (e.g., rotations by 0, 120∘120^\circ, 240∘240^\circ and three reflections), the character table is as follows:
C3vC_{3v}EE2C32C_33σv3\sigma_vFunctions
A1A_1111zz, z2z^2
A2A_211-1RzR_z
EE2-10(x,y)(x,y), (xz,yz)(xz, yz)
This table reveals the one-dimensional totally symmetric representation A1A_1 and the two-dimensional irreducible EE, essential for decomposing vibrational modes in molecular symmetry analysis. For finite rotational groups like CnC_n, there exists an to a of the SnS_n (the group of permutations of nn elements), via , which embeds any finite group into a acting on itself by left multiplication; this realizes rotations as permutations of equivalent positions in a symmetric object. In the continuous case, on compact groups such as SO(3)SO(3) generalizes classical through the Peter-Weyl theorem, decomposing square-integrable functions into matrix-valued coefficients over irreducible representations, enabling harmonic expansions for signals on the rotation manifold. The formalization of symmetry groups, including rotational ones, traces to the late , with Arthur Cayley's development of abstract in the 1850s providing the algebraic foundation, and Felix Klein's (1872) classifying geometries by their transformation groups, thereby integrating rotations into a broader framework that influenced modern .

References

  1. https://www2.math.upenn.edu/~mlazar/math170/notes07.pdf
  2. https://www.energy.gov/science/doe-explainssymmetry-physics
  3. https://www.ms.uky.edu/~lee/ma111fa09/slides11.pdf
  4. https://mathresearch.utsa.edu/wiki/index.php?title=Symmetry
  5. https://www.classe.cornell.edu/~dms79/xrd/xtallography/Symmetry%20around%20a%20Point%20in%20the%20Plane.htm
  6. https://www.math.brown.edu/tbanchof/Yale/project05/rotation.htm
  7. https://ximera.osu.edu/m4t/elementaryTeachersTwo/elementaryReading/MakeArt/Symmetry
  8. https://math.hws.edu/eck/math110_s08/symmetries/index.html
  9. https://www2.tulane.edu/~sanelson/eens211/introsymmetry.htm
  10. https://www.cuemath.com/geometry/rotational-symmetry/
  11. https://byjus.com/maths/rotational-symmetry/
  12. https://mathworld.wolfram.com/RotationalSymmetry.html
  13. https://www.basic-mathematics.com/order-of-rotational-symmetry.html
  14. https://www.cuemath.com/geometry/symmetry-of-any-circle/
  15. https://www.onlinemath4all.com/order-of-rotational-symmetry-of-a-circle.html
  16. https://www.jmilne.org/math/CourseNotes/GT.pdf
  17. https://scipp.ucsc.edu/~haber/ph116A/RotationMatrices.pdf
  18. https://web.mit.edu/5.03/www/readings/point_groups/point_groups.pdf
  19. https://cmth.ph.ic.ac.uk/people/d.vvedensky/groups/Chapter7.pdf
  20. https://xiechen.caltech.edu/documents/27821/lecture1109.pdf
  21. https://www.cis.upenn.edu/~cis5150/gma-v2-chap9.pdf
  22. https://www.math.toronto.edu/mein/teaching/LectureNotes/lie.pdf
  23. https://math.umd.edu/~immortal/MATH431/book/ch_complex.pdf
  24. https://www.cis.upenn.edu/~cis5150/ws-book-Ib.pdf
  25. https://maths.nuigalway.ie/~rquinlan/groups/exhibition/ferns_murphy_muth.pdf
  26. https://sivulka.github.io/symmetry-groups-platonic.pdf
  27. https://motion.cs.illinois.edu/RoboticSystems/CoordinateTransformations.html
  28. https://math.ou.edu/~nbrady/teaching/s10-4513/SO3.PDF
  29. https://mathworld.wolfram.com/RegularPolygon.html
  30. https://mathworld.wolfram.com/PlatonicSolid.html
  31. https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_%28Denton%29/01%253A_Symmetry/1.02%253A_Counting_Symmetries
  32. https://www.math.brown.edu/~res/MFS/handout3.pdf
  33. https://mathworld.wolfram.com/ArchimedeanSolid.html
  34. https://www.science.org/content/article/genetically-speaking-starfish-have-no-arms-only-head
  35. https://eschermath.org/wiki/Rotational_Symmetry.html
  36. https://earthsky.org/human-world/nautilus-shell-fibonacci-logarithmic-spiral-golden-spiral/
  37. https://flexbooks.ck12.org/cbook/ck-12-cbse-maths-class-6-as-per-nep-2020/section/9.3/primary/lesson/rotation-symmetry/
  38. https://www.nature.com/articles/s40494-022-00852-w
  39. https://link.springer.com/article/10.1007/s00004-015-0247-7
  40. https://biologydirect.biomedcentral.com/articles/10.1186/1745-6150-7-22
  41. https://smarthistory.org/balance-symmetry-and-emphasis/
  42. https://www.snowcrystals.com/science/science.html
  43. https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Chemical_Group_Theory_%28Miller%29/02%253A_Rotational_Symmetry/2.04%253A_Crystallographic_Point_Groups
  44. https://www.utoledo.edu/nsm/chemistry/people/Webpages/pdf/3-plane-and-space-groups.pdf
  45. https://www.xtal.iqf.csic.es/Cristalografia/parte_03_1_1-en.html
  46. http://www.quartzpage.de/crs_intro.html
  47. https://opengeology.org/Mineralogy/10-crystal-morphology-and-symmetry/
  48. https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_%28Cline%29/07%253A_Symmetries_Invariance_and_the_Hamiltonian/7.04%253A_Rotational_invariance_and_conservation_of_angular_momentum
  49. https://www.feynmanlectures.caltech.edu/III_17.html
  50. https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/e765f050b2fd50a9d87d2ae801e9c52f_MIT8_05F13_Chap_09.pdf
  51. https://www.mdpi.com/2079-4991/14/19/1576
  52. https://www.nature.com/articles/s41598-024-59481-x
  53. https://www.ccsenet.org/journal/index.php/jmsr/article/download/76257/42214
  54. https://kconrad.math.uconn.edu/blurbs/grouptheory/dihedral.pdf
  55. https://www.whitman.edu/Documents/Academics/Mathematics/2019/Landau-Balof.pdf
  56. https://crystalsymmetry.wordpress.com/wp-content/uploads/2016/06/script_unit_4-4.pdf
  57. https://www.iucr.org/resources/commissions/international-tables/volume-a
  58. https://fardila.com/Clase/RepTh/smith.pdf
  59. https://cahill.people.unm.edu/467-21/ch11.pdf
  60. https://www.sas.rochester.edu/pas/assets/pdf/undergraduate/representations_of_the_rotation_groups_so-n.pdf
  61. http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=403&option=4
  62. https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_%28Doerr_and_Levasseur%29/15%253A_Group_Theory_and_Applications/15.03%253A_Permutation_Groups
  63. https://math.mit.edu/~dav/compactrev.pdf
  64. https://www.math.lsu.edu/~adkins/m7200/GroupHistory.pdf
Add your contribution
Related Hubs
User Avatar
No comments yet.