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Solid geometry
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Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space).[1] A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its interior.
Solid geometry deals with the measurements of volumes of various solids, including pyramids, prisms, cubes (and other polyhedrons), cylinders, cones (including truncated) and other solids of revolution.[2]
History
[edit]The Pythagoreans dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height. He was probably also the discoverer of a proof that the volume enclosed by a sphere is proportional to the cube of its radius.[3]
Topics
[edit]Basic topics in solid geometry and stereometry include:
- incidence of planes and lines
- dihedral angle and solid angle
- the cube, cuboid, parallelepiped
- the tetrahedron and other pyramids
- prisms
- octahedron, dodecahedron, icosahedron
- cones and cylinders
- the sphere
- other quadrics: spheroid, ellipsoid, paraboloid and hyperboloids.
Advanced topics include:
- projective geometry of three dimensions (leading to a proof of Desargues' theorem by using an extra dimension)
- further polyhedra
- descriptive geometry.
List of solid figures
[edit]Whereas a sphere is the surface of a ball, for other solid figures it is sometimes ambiguous whether the term refers to the surface of the figure or the volume enclosed therein, notably for a cylinder.
| Figure | Definitions | Images | |
|---|---|---|---|
| Parallelepiped |
|
| |
| Rhombohedron |
|
| |
| Cuboid |
|
| |
| Polyhedron | Flat polygonal faces, straight edges and sharp corners or vertices |  Small stellated dodecahedron |
 Toroidal polyhedron |
| Uniform polyhedron | Regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other) |   (Regular) Tetrahedron and Cube |
 Uniform Snub dodecahedron |
| Pyramid | A polyhedron comprising an n-sided polygonal base and a vertex point |  square pyramid
| |
| Prism | A polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases |  hexagonal prism
| |
| Antiprism | A polyhedron comprising an n-sided polygonal base, a second base translated and rotated.sides]] of the two bases |  square antiprism
| |
| Bipyramid | A polyhedron comprising an n-sided polygonal center with two apexes. |  triangular bipyramid
| |
| Trapezohedron | A polyhedron with 2n kite faces around an axis, with half offsets |  tetragonal trapezohedron
| |
| Cone | Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex |  A right circular cone and an oblique circular cone | |
| Cylinder | Straight parallel sides and a circular or oval cross section |  A solid elliptic cylinder |
 A right and an oblique circular cylinder |
| Ellipsoid | A surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation |  Examples of ellipsoids |
sphere (top, a=b=c=4), spheroid (bottom left, a=b=5, c=3), |
| Lemon | A lens (or less than half of a circular arc) rotated about an axis passing through the endpoints of the lens (or arc)[6] | .png){kind=small}
| |
| Hyperboloid | A surface that is generated by rotating a hyperbola around one of its principal axes |
| |
Techniques
[edit]Various techniques and tools are used in solid geometry. Among them, analytic geometry and vector techniques have a major impact by allowing the systematic use of linear equations and matrix algebra, which are important for higher dimensions.
Applications
[edit]A major application of solid geometry and stereometry is in 3D computer graphics.
See also
[edit]Notes
[edit]- ^ The Britannica Guide to Geometry, Britannica Educational Publishing, 2010, pp. 67–68.
- ^ Kiselev 2008.
- ^ Paraphrased and taken in part from the 1911 Encyclopædia Britannica.
- ^ Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. p. 75. ISBN 9780521277396.
- ^ Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018.
- ^ Weisstein, Eric W. "Lemon". Wolfram MathWorld. Retrieved 2019-11-04.
References
[edit]- Robert Baldwin Hayward (1890) The Elements of Solid Geometry via Internet Archive
- Kiselev, A. P. (2008). Geometry. Vol. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat.
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Solid geometry
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Introduction
Definition and Scope
Solid geometry is a branch of Euclidean geometry that focuses on the study of three-dimensional objects characterized by length, width, and depth.[4][5] Unlike plane geometry, which examines two-dimensional figures, solid geometry addresses the properties, relationships, and measurements of shapes in three-dimensional space.[6] This field distinguishes between bounded solids, which are finite regions of space enclosed by surfaces and possessing limited volume, and unbounded spaces, such as the infinite expanse of three-dimensional Euclidean space.[7] Bounded solids form the primary objects of study, enabling analysis of their spatial configurations and interactions.[8] Core principles of solid geometry extend foundational concepts from plane geometry to three dimensions, including congruence—where solids are identical in shape and size—and similarity, where solids share proportional dimensions while maintaining the same form.[9][10] These principles rest on Euclidean axioms governing points, lines, and planes, such as the existence of a unique line through any two points and a unique plane through any three non-collinear points. In contrast, if three points are collinear (lie on the same straight line), they do not determine a unique plane; instead, they lie in infinitely many planes—specifically, every plane that contains the line passing through them.[11] Foundational shapes like the sphere, cube, and cylinder serve as primitive examples for developing these ideas.[7] Although originating in ancient Greece with early explorations of spatial forms, solid geometry's Euclidean framework has seen extensions to non-Euclidean contexts, such as the curved geometries in general relativity.[12][13]Historical Context Overview
The foundations of solid geometry were laid in ancient Greece, building upon earlier developments in plane geometry. Euclid of Alexandria, around 300 BCE, systematically organized the subject in his seminal work Elements, particularly in Books XI through XIII, which address solid angles, polyhedra, and methods for calculating volumes of three-dimensional figures.[14] A key advancement came from Archimedes of Syracuse (c. 287–212 BCE), who extended these ideas through innovative techniques in works such as On the Sphere and Cylinder. He employed the method of exhaustion to determine the volumes of spheres, cylinders, and cones, establishing foundational results like the volume of a sphere being two-thirds that of its circumscribing cylinder.[15] During the Renaissance, interest in solid geometry revived with astronomical applications. Johannes Kepler, in his 1596 treatise Mysterium Cosmographicum, proposed a model of the solar system where the five Platonic solids were inscribed and circumscribed between the spherical orbits of the planets, linking geometric solids to cosmic structure.[16] In the 19th century, the study of solid geometry expanded beyond Euclidean frameworks through explorations of curved spaces. Carl Friedrich Gauss initiated investigations into non-Euclidean geometry around 1792, questioning the parallel postulate and laying groundwork for geometries where space curvature affects solid properties. Bernhard Riemann further developed this in his 1854 habilitation lecture on the hypotheses of geometry, introducing Riemannian manifolds that generalized solid geometry to curved, higher-dimensional spaces.[17][18] The 20th century saw rigorous advancements in understanding polyhedral equivalence. At the 1900 International Congress of Mathematicians, David Hilbert posed his third problem, asking whether any two polyhedra of equal volume can be dissected into finitely many polyhedral pieces that can be reassembled via rigid motions to form the other. This was negatively resolved in 1900 by Max Dehn, who introduced Dehn invariants to show that such dissections are not always possible, profoundly influencing the algebraic topology of solids.[19]Fundamental Concepts
Basic Elements in Three Dimensions
In three-dimensional Euclidean space, the foundational elements are points, lines, and planes, which serve as the building blocks for all solid figures. A point is defined as an exact location in space with no size, dimension, or shape.[20] A line is an infinite set of points extending in both directions along a straight path, representing the shortest connection between any two points it contains.[20][21] A plane is a flat, two-dimensional surface consisting of all points that lie on the same level, extending infinitely in all directions and determined uniquely by any three non-collinear points.[20][21] Spatial relationships among these elements distinguish three-dimensional geometry from its two-dimensional counterpart. Two lines are parallel if they lie in the same plane and do not intersect, maintaining a constant separation.[22][21] Lines are perpendicular if they intersect at a right angle, forming 90-degree angles at their point of meeting.[22] Uniquely in three dimensions, skew lines are neither parallel nor intersecting and do not lie in the same plane, allowing them to exist without meeting in space.[22][21] For planes, parallelism means they never intersect and maintain constant distance apart, while perpendicular planes intersect such that their normal directions form right angles. Angles in three dimensions extend planar concepts to volumetric interactions. A dihedral angle is formed by the intersection of two planes along a common line (the edge), measured as the angle between two rays perpendicular to that edge, each lying in one of the planes.[23] At a vertex where multiple planes meet, a solid angle quantifies the three-dimensional extent of the region bounded by those planes, analogous to a plane angle but encompassing a portion of space, often measured in steradians as the area projected onto a unit sphere centered at the vertex.[24] Intersections of these elements define how they connect in space. A line and a plane either intersect at a single point, with the line contained entirely within the plane, or the line is parallel to the plane without intersecting it.[21] Two planes intersect along a line if not parallel, or coincide if identical.[21] Two lines intersect at a point if coplanar and non-parallel, coincide if the same line, or remain skew without intersection.[21] These elements combine to fill three-dimensional space without gaps or overlaps when arranged to bound regions, as seen in the formation of polyhedral cells where planes serve as faces, lines as edges, and points as vertices, creating closed solids that tessellate space in certain configurations.[3] Vectors can represent directions and positions of these elements for further analysis.[25]Coordinate Systems and Vectors
In solid geometry, the Cartesian coordinate system provides a foundational framework for representing points and objects in three-dimensional space. This system extends the two-dimensional plane by introducing a third mutually perpendicular axis, denoted as the z-axis, which is typically oriented vertically while the x-axis points horizontally to the right and the y-axis points horizontally forward, following the right-hand rule for orientation. A point in this space is uniquely identified by an ordered triple , where , , and represent the signed distances from the origin along the respective axes.[26][27] Vectors are essential tools in this coordinate system, serving as directed line segments that encode both position and direction in 3D space. A position vector from the origin to a point is simply , where , , and are the standard unit vectors along the positive x-, y-, and z-axes, respectively. Direction vectors, in contrast, describe the orientation of lines or displacements without specifying a starting point, such as . The magnitude of a vector quantifies its length, given by , while a unit vector normalizes it to unit length via , isolating pure direction.[28][29] Vector operations enable precise geometric computations, with the dot product yielding a scalar that measures the cosine of the angle between two vectors, useful for determining orthogonality when the result is zero. The cross product , defined component-wise as the determinant of the matrix formed by , , and the components of and , produces a vector perpendicular to both inputs with magnitude , representing the area of the parallelogram they span and following the right-hand rule for direction. These operations underpin angle and area calculations in 3D configurations.[30][31][32] Distances in 3D space are derived directly from vector norms, with the Euclidean distance between two points and given by . For the distance from a point to a line defined by a point and direction vector , it is , capturing the shortest perpendicular separation. The distance from a point to a plane is , where the denominator is the norm of the plane's normal vector.[33][34][35] Parametric equations offer a vector-based parameterization for lines and planes, facilitating intersection and traversal analyses. A line passing through point with direction vector is expressed as , or component-wise , , , where traces the line. For a plane containing point and spanned by two non-parallel vectors and , the parametric form is , with , providing a surface parameterization in 3D space.[36][37]Types of Solid Figures
Polyhedra and Platonic Solids
A polyhedron is a three-dimensional shape composed of flat polygonal faces, straight edges where the faces meet, and vertices where the edges intersect. Polyhedra can be classified as convex or concave; a convex polyhedron has all interior angles less than 180 degrees and contains the line segment between any two points within it, while a concave polyhedron has at least one interior angle greater than 180 degrees, creating indentations.[38] A fundamental relation for convex polyhedra is Euler's formula, which states that the number of vertices , edges , and faces satisfyThis holds for any convex polyhedron that is topologically equivalent to a sphere.[39] Platonic solids are the most symmetric convex polyhedra, defined by strict regularity criteria: all faces are congruent regular polygons, and the same number of faces meet at each vertex.[40] There are exactly five such solids, each with a distinct symmetry group that describes its rotational and reflectional invariances.[41] The table below summarizes their key features:
| Solid | Number of Faces | Face Type | Vertices | Edges | Symmetry Group (Rotational Order) |
|---|---|---|---|---|---|
| Tetrahedron | 4 | Triangle | 4 | 6 | Tetrahedral (, 12) |
| Cube | 6 | Square | 8 | 12 | Octahedral (, 24) |
| Octahedron | 8 | Triangle | 6 | 12 | Octahedral (, 24) |
| Dodecahedron | 12 | Pentagon | 20 | 30 | Icosahedral (, 60) |
| Icosahedron | 20 | Triangle | 12 | 30 | Icosahedral (, 60) |
Solids of Revolution and Curved Surfaces
Solids of revolution are three-dimensional figures generated by rotating a two-dimensional curve or region in the plane about a fixed axis, producing symmetric shapes with rotational invariance around that axis.[44] This process transforms planar geometry into volumetric forms, often resulting in surfaces that are smooth and continuous. Common examples include the sphere, formed by rotating a circle about its diameter; the cylinder, obtained by rotating a straight line segment parallel to the axis of rotation; the cone, created by rotating a slanted line about one end; and the torus, generated by rotating a circle in a plane offset from the axis.[44] Beyond basic solids of revolution, curved surfaces in solid geometry encompass a broader class of three-dimensional forms defined by quadratic equations, known as quadric surfaces, which extend conic sections into space.[45] The ellipsoid, for instance, is a closed, bounded surface resembling a stretched sphere, given by the equation
where , , and determine the semi-axes lengths along each direction; it reduces to a sphere when .[45] Paraboloids include the elliptic paraboloid, a bowl-shaped surface described by , which opens along the z-axis, and the hyperbolic paraboloid, a saddle-shaped form given by , extending infinitely in multiple directions.[45] Hyperboloids consist of the one-sheet variety, an hourglass-like surface with equation , and the two-sheet version, , comprising two separate unbounded sheets.[45]
A key classification among curved surfaces involves ruled surfaces, which are generated by the continuous motion of a straight line (called a ruling or generator) along a guiding curve (directrix) in space, as parametrized by .[46] Cylinders and cones exemplify ruled surfaces, where the rulings are parallel or converge to a point, respectively.[46] Within this category, developable surfaces are a subclass characterized by zero Gaussian curvature, meaning the tangent plane remains constant along each ruling, allowing the surface to be flattened onto a plane without distortion; principal curvatures satisfy and for mean curvature .[46] Non-developable ruled surfaces, such as certain hyperboloids, exhibit non-zero Gaussian curvature and cannot be isometrically mapped to a plane.[46]
These curved surfaces possess smooth, continuously varying boundaries that contrast sharply with the flat faces and edges of polyhedra, enabling their use in modeling organic and natural forms like celestial bodies or fluid dynamics.[45] For example, the hyperbolic paraboloid's infinite extent and saddle geometry make it suitable for approximating minimal surfaces in architecture, while ellipsoids represent planetary shapes with varying axial elongations.[45] In comparison to the highly regular, faceted Platonic solids, solids of revolution and curved surfaces emphasize fluid symmetry and continuity over discrete polygonal structure.[44]
Geometric Properties
Volume and Surface Area Calculations
Volume in solid geometry refers to the measure of the three-dimensional space enclosed by the boundary surfaces of a solid figure.[47] In mathematical terms, the volume of a bounded solid region in three-dimensional space is defined as the triple integral , which represents the limit of sums of volumes of infinitesimal elements filling the region.[48] Basic formulas exist for common solid figures, allowing direct computation based on their dimensions. For a cube with side length , the volume is .[49] The volume of a sphere with radius is .[49] A right circular cylinder with radius and height has volume .[49] For a right circular cone with base radius and height , the volume is .[49] Surface area quantifies the total area of the external faces or boundary of a solid, often distinguished as total surface area (including all faces) or lateral surface area (excluding bases for figures like cylinders and cones). For a cube, the total surface area is .[50] The surface area of a sphere is .[50] A cylinder's total surface area is , where is the lateral surface area.[50] Advanced methods provide ways to compute volumes without direct formulas, particularly for irregular or composite solids. Cavalieri's principle states that if two solids have the same height and every plane cross-section parallel to the bases has equal areas, then the solids have equal volumes.[51] For polyhedra, dissection involves decomposing the figure into simpler components, such as tetrahedra, whose volumes can be summed; the volume of a tetrahedron with vertices at coordinates for to is given by times the absolute value of the determinant of the matrix formed by these coordinates (adjusted for origin).[47] Volumes are measured in cubic units, such as cubic meters or cubic centimeters, reflecting the three-dimensional nature of the quantity. Under similarity transformations, where all linear dimensions are scaled by a factor , volumes scale by , preserving the shape but altering the size proportionally to the cube of the scaling factor.[52] Coordinate systems can be referenced briefly to set up the limits for such integral computations when needed.[48]Centers of Mass and Symmetry
In solid geometry, the center of mass of a solid object with uniform density, also known as the centroid, represents the average position of all points within the solid and serves as its balance point. For a solid with constant density , the mass where is the volume, and the centroid coordinates are given by
computed over the volume of the solid.[53][54] For polyhedra, these integrals can be evaluated by decomposing the solid into simpler elements such as tetrahedra, where the centroid of each tetrahedron is the average of its four vertices, and then taking a volume-weighted average of those centroids.[53]
Symmetry plays a crucial role in locating the centroid, as any symmetry operation of the solid must leave the centroid invariant, positioning it at the intersection of symmetry elements. The primary types of symmetry in three-dimensional solids include translational symmetry, which shifts the solid along a vector without altering its appearance; rotational symmetry about an axis through multiples of for integer ; reflectional symmetry across a plane that mirrors the solid; and inversion symmetry through a central point that maps each point to its antipode. These elements combine to form point groups, finite collections of symmetries that describe the overall symmetry class of the solid.[55]
Platonic solids exhibit particularly high degrees of symmetry, with their point groups encompassing multiple axes and planes. For instance, the icosahedron possesses the icosahedral rotation group , which includes 60 distinct rotational symmetries, including 6 five-fold axes, 10 three-fold axes, and 15 two-fold axes, reflecting its 20 triangular faces and 12 vertices.[56] This extensive symmetry ensures that the centroid coincides precisely with the geometric center, the common intersection of all symmetry axes and planes.
Such symmetry contributes to the stability of solid figures by ensuring even weight distribution around the centroid, allowing the solid to balance uniformly on any supporting plane passing through that point without preferential tilting. In symmetric designs, perturbations that displace the solid from equilibrium are counteracted by the restorative forces aligned with the symmetry axes, promoting inherent balance.[57]
For inhomogeneous solids where density varies, the center of mass shifts to a weighted average,
with similar expressions for and , where ; however, in purely geometric contexts, analysis often reverts to uniform density assumptions to focus on shape-determined averages.[53][54]
Analytical Techniques
Coordinate-Based Methods
Coordinate-based methods in solid geometry utilize a fixed Cartesian coordinate system to represent, analyze, and manipulate three-dimensional figures through algebraic equations and vector operations. This approach allows for precise definitions of points, lines, planes, and surfaces using coordinates (x, y, z), enabling computations of geometric properties without relying on transformations or projections. By assigning numerical coordinates to vertices and defining surfaces via equations, complex solids can be constructed and intersected systematically.[27] Planes, as fundamental elements in 3D space, are defined by the linear equation $ ax + by + cz = d $, where $ a, b, c $ are coefficients determining the normal vector $ \mathbf{n} = (a, b, c) $ perpendicular to the plane, and $ d $ relates to a point on the plane. This equation arises from the condition that the vector from a fixed point on the plane to any other point on it is orthogonal to the normal vector. More general surfaces, such as quadric surfaces, are represented by second-degree equations of the form $ Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0 $, encompassing ellipsoids, hyperboloids, paraboloids, cones, and cylinders. These quadrics provide a versatile framework for modeling curved solids, with specific forms obtained by simplifying coefficients.[58][45] Intersections between these surfaces are computed by substituting equations, yielding lower-dimensional loci. For instance, the intersection of a sphere $ x^2 + y^2 + z^2 = r^2 $ and a plane $ ax + by + cz = d $ forms a circle in the plane, with radius determined by the distance from the sphere's center to the plane. Similarly, the intersection of a line, parameterized as $ \mathbf{r}(t) = \mathbf{p} + t\mathbf{d} $, with a sphere leads to a quadratic equation in $ t $: $ at^2 + bt + c = 0 $, where solutions indicate entry and exit points (two real roots), tangency (one root), or no intersection (no real roots). These algebraic solutions facilitate ray-tracing and geometric querying in 3D models.[59][60] Polyhedra are modeled by specifying vertex coordinates and connecting them via edges and faces. For example, a cube centered at the origin with side length 2 has vertices at all combinations of $ (\pm 1, \pm 1, \pm 1) $, allowing edges to be represented as vectors between adjacent vertices, such as from $ (1,1,1) $ to $ (1,1,-1) $, which is $ (0,0,-2) $. This coordinate list defines the polyhedron's boundary, enabling further computations like face planes or convex hull verification. Edge vectors also support adjacency checks and traversal algorithms.[61] Distances between points or along edges are calculated using the Euclidean distance formula $ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} $, while angles, including dihedral angles between adjacent faces, employ vector operations. The dihedral angle between two planes is the angle between their normals $ \mathbf{n_1} $ and $ \mathbf{n_2} $, given by $ \cos \theta = \frac{|\mathbf{n_1} \cdot \mathbf{n_2}|}{||\mathbf{n_1}|| , ||\mathbf{n_2}||} $, where the dot product measures alignment. For oriented dihedrals, the cross product $ \mathbf{n_1} \times \mathbf{n_2} $ determines the angle's direction via its magnitude and sign. These computations are essential for assessing polyhedral regularity and stress analysis.[62] This coordinate framework underpins computer-aided design (CAD) systems, where solids are built from parametric equations and vertex lists for precise modeling and simulation.[63]Transformational Approaches
Transformational approaches in solid geometry examine how solids can be manipulated through various mappings that alter their position, orientation, or size while preserving specific geometric properties. These transformations, including rigid and similarity types, are fundamental for understanding the structure and symmetry of three-dimensional figures. Matrix representations are often employed in Cartesian coordinate systems to describe these operations precisely.[64] Rigid transformations maintain distances and angles between points, ensuring the solid's shape and size remain unchanged. Translations shift the entire solid by adding a fixed vector to every point , resulting in . Rotations reorient the solid around a specified axis by an angle ; for instance, a rotation about the z-axis uses the matrix
which preserves orientation and is orthogonal with determinant 1. Reflections reverse orientation by mirroring the solid across a plane, such as changing the sign of the coordinate perpendicular to the xy-plane for reflection over that plane. These operations collectively form the Euclidean group of isometries in three dimensions.[64][65][64]
Similarity transformations extend rigid motions by incorporating scaling, allowing proportional resizing while preserving shape and angles. Uniform scaling by a factor multiplies all coordinates by , transforming linear dimensions by , areas by , and volumes by . Compositions of similarities yield new similarities, maintaining the proportional relationship between figures.[66][67][68]
Complex transformations in three dimensions are efficiently composed using 4x4 matrices in homogeneous coordinates, where points are augmented as . A general transformation matrix combines rotation , scaling , and translation as
with the transformed point . Matrix multiplication enables sequential application, such as rotation followed by translation, facilitating the modeling of arbitrary rigid or similarity motions.[65]
Rigid transformations induce congruence, where transformed solids are identical in size and shape, while similarity transformations produce figures that are similar, with corresponding angles equal and sides proportional by the scaling factor. These preservation properties underpin classifications of solids under group actions. In group theory, the symmetries of a solid form a finite group under composition; for example, the rotation group of the cube, consisting of all orientation-preserving symmetries fixing its vertices, has order 24 and is isomorphic to the symmetric group .[64][66][40]