Icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries; the set of orientation-preserving symmetries forms a group referred to as A5, the full symmetry group is the product A5 × Z2. The latter group is known as the Coxeter group H3, is represented by Coxeter notation, Coxeter diagram. Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries with the largest symmetry groups. Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups. Presentations corresponding to the above are: I: ⟨ s, t ∣ s 2, t 3, 5 ⟩ I h: ⟨ s, t ∣ s 3 − 2, t 5 − 2 ⟩; these correspond to the icosahedral groups being the triangle groups.
The first presentation was given by William Rowan Hamilton in 1856, in his paper on icosian calculus. Note that other presentations are possible, for instance as an alternating group; the icosahedral rotation group I is of order 60. The group I is isomorphic to A5, the alternating group of permutations of five objects; this isomorphism can be realized by I acting on various compounds, notably the compound of five cubes, the compound of five octahedra, or either of the two compounds of five tetrahedra. The group contains 5 versions of Th with 20 versions of D3, 6 versions of D5; the full icosahedral group Ih has order 120. It has I as normal subgroup of index 2; the group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the center corresponding to element, where Z2 is written multiplicatively. Ih acts on the compound of five cubes and the compound of five octahedra, but −1 acts as the identity, it acts on the compound of ten tetrahedra: I acts on the two chiral halves, −1 interchanges the two halves.
Notably, it does not act as S5, these groups are not isomorphic. The group contains 6 versions of D5d. I is isomorphic to PSL2, but Ih is not isomorphic to SL2; the following groups all have order 120, but are not isomorphic: S5, the symmetric group on 5 elements Ih, the full icosahedral group 2I, the binary icosahedral groupThey correspond to the following short exact sequences and product 1 → A 5 → S 5 → Z 2 → 1 I h = A 5 × Z 2 1 → Z 2 → 2 I → A 5 → 1 In words, A 5 is a normal subgroup of S 5 A 5 is a factor of I h, a direct product A 5 is a quotient group of 2 I Note that A 5 has an exceptional irreducible 3-dimensional representation, but S 5 does not have an irreducible 3-dimensional representation, corresponding to the full icosahedral group not being the symmetric group. These can be related to linear groups over the finite field with five elements, which exhibit the subgroups and covering groups directly. In computational contexts, the rotation icosahedral group I above can be explicitly represented by the following 60 rotatio
Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
Point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O itself is a subgroup of the Euclidean group E of all isometries. Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them; the symmetry group of an object is sometimes called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral; the point groups in three dimensions are used in chemistry to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, in this context they are called molecular point groups.
Finite Coxeter groups are a special set of point groups generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group is represented by a Coxeter -- Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. SO is a subgroup of E +, which consists of i.e. isometries preserving orientation. O is the direct product of SO and the group generated by inversion: O = SO × Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. There is a 1-to-1 correspondence between all groups of direct isometries H in O and all groups K of isometries in O that contain inversion: K = H × H = K ∩ SOFor instance, if H is C2 K is C2h, or if H is C3 K is S6. If a group of direct isometries H has a subgroup L of index 2 apart from the corresponding group containing inversion there is a corresponding group that contains indirect isometries but no inversion: M = L ∪ where isometry is identified with A.
An example would be C4 for H and S4 for M. Thus M is obtained from H by inverting the isometries in H ∖ L; this group M is as abstract group isomorphic with H. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries; this is clarifying when see below. In 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O and SO. Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations is normal both in the group obtained by adding reflections in planes through the axis and in the group obtained by adding a reflection plane perpendicular to the axis; the isometries of R3 that leave the origin fixed, forming the group O, can be categorized as follows: SO: identity rotation about an axis through the origin by an angle not equal to 180° rotation about an axis through the origin by an angle of 180° the same with inversion, i.e. respectively: inversion rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis reflection in a plane through the originThe 4th and 5th in particular, in a wider sense the 6th are called improper rotations.
See the similar overview including translations. When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O. For example, two 3D objects have the same symmetry type: if both have mirror symmetry, but with respect to a different mirror plane if both have 3-fold rotational symmetry, but with respect to a different axis. In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type if and only if there is a rotation mapping the whole structure of the first symmetry group to that of the second; the conjugacy definition would allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. There are many infinite isometry groups.
We may create non-cyclical abelian groups by adding more rotations around the same axis. There are non-abelian groups generated by rotations around different axes; these are free groups. They will be infinite. All the infinite groups mentioned so far are not closed as topological subgroups of O. We now discuss
Dihedral symmetry in three dimensions
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn. There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations: Schönflies notation, Coxeter notation, orbifold notation. Chiral Dn, +, of order 2n – dihedral symmetry or para-n-gonal group Achiral Dnh, of order 4n – prismatic symmetry or full ortho-n-gonal group Dnd, of order 4n – antiprismatic symmetry or full gyro-n-gonal group For a given n, all three have n-fold rotational symmetry about one axis, 2-fold about a perpendicular axis, hence about n of those. For n = ∞ they correspond to three frieze groups. Schönflies notation is used, with Coxeter notation in brackets, orbifold notation in parentheses; the term horizontal is used with respect to a vertical axis of rotation. In 2D the symmetry group Dn includes reflections in lines; when the 2D plane is embedded horizontally in a 3D space, such a reflection can either be viewed as the restriction to that plane of a reflection in a vertical plane, or as the restriction to the plane of a rotation about the reflection line, by 180°.
In 3D the two operations are distinguished: the group Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order. With reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis we have Dnh. Dnd, has vertical mirror planes between the horizontal rotation axes, not through them; as a result the vertical axis is a 2n-fold rotoreflection axis. Dnh is the symmetry group for a regular n-sided prisms and for a regular n-sided bipyramid. Dnd is the symmetry group for a regular n-sided antiprism, for a regular n-sided trapezohedron. Dn is the symmetry group of a rotated prism. N = 1 is not included because the three symmetries are equal to other ones: D1 and C2: group of order 2 with a single 180° rotation D1h and C2v: group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane D1d and C2h: group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that planeFor n = 2 there is not one main axes and two additional axes, but there are three equivalent ones.
D2 +, of order 4 is one of the three symmetry group types with the Klein four-group as abstract group. It has three perpendicular 2-fold rotation axes, it is the symmetry group of a cuboid with an S written on two opposite faces, in the same orientation. D2h, of order 8 is the symmetry group of a cuboid D2d, of order 8 is the symmetry group of e.g.: a square cuboid with a diagonal drawn on one square face, a perpendicular diagonal on the other one a regular tetrahedron scaled in the direction of a line connecting the midpoints of two opposite edges. For Dnh, order 4n Cnh, order 2n Cnv, order 2n Dn, +, order 2nFor Dnd, order 4n S2n, order 2n Cnv, order 2n Dn, +, order 2nDnd is subgroup of D2nh. Dnh,: D5h,: D4d,: D5d,: D17d,: List of spherical symmetry groups Point groups in three dimensions Cyclic symmetry in three dimensions Coxeter, H. S. M. and Moser, W. O. J.. Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9. CS1 maint: Multiple names: authors list N.
W. Johnson: Geometries and Transformations, ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups Conway, John Horton. "The Orbifold Notation for Two-Dimensional Groups", Structural Chemistry, Springer Netherlands, 13: 247–257, doi:10.1023/A:1015851621002 Graphic overview of the 32 crystallographic point groups – form the first parts of the 7 infinite series and 5 of the 7 separate 3D point groups
Tetrahedral symmetry
A regular tetrahedron has 12 rotational symmetries, a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is one such symmetry for each permutation of the vertices of the tetrahedron; the set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4. Chiral and full are discrete point symmetries, they are among the crystallographic point groups of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles in the plane; each of these 6 circles represent a mirror line in tetrahedral symmetry. The intersection of these circles meet at 3 gyration points. T, 332, +, or 23, of order 12 – chiral or rotational tetrahedral symmetry. There are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D2 or 222, with in addition four 3-fold axes, centered between the three orthogonal directions.
This group is isomorphic to the alternating group on 4 elements. The conjugacy classes of T are: identity 4 × rotation by 120° clockwise:, 4 × rotation by 120° counterclockwise 3 × rotation by 180°The rotations by 180°, together with the identity, form a normal subgroup of type Dih2, with quotient group of type Z3; the three elements of the latter are the identity, "clockwise rotation", "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. A4 is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G|, there does not exist a subgroup of G with order d: the group G = A4 has no subgroup of order 6. Although it is a property for the abstract group in general, it is clear from the isometry group of chiral tetrahedral symmetry: because of the chirality the subgroup would have to be C6 or D3, but neither applies. Td, *332, or 43m, of order 24 – achiral or full tetrahedral symmetry known as the triangle group.
This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S4 axes. Td and O are isomorphic as abstract groups: they both correspond to S4, the symmetric group on 4 objects. Td is the set obtained by combining each element of O \ T with inversion. See the isometries of the regular tetrahedron; the conjugacy classes of Td are: identity 8 × rotation by 120° 3 × rotation by 180° 6 × reflection in a plane through two rotation axes 6 × rotoreflection by 90° Th, 3*2, or m3, of order 24 – pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions; the 3-fold axes are now S6 axes, there is a central inversion symmetry. Th is isomorphic to T × Z2: every element of Th is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is a normal subgroup D2h, of type Dih2 × Z2 = Z2 × Z2 × Z2, it is the direct product of the normal subgroup of T with Ci.
The quotient group is the same as above: of type Z3. The three elements of the latter are the identity, "clockwise rotation", "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation, it is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the permutations of the body diagonals and the same combined with inversion, it is the symmetry of a pyritohedron, similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side. It is a subgroup of the full icosahedral symmetry group, with 4 of the 10 3-fold axes; the conjugacy classes of Th include those of T, with the two classes of 4 combined, each with inversion: identity 8 × rotation by 120° 3 × rotation by 180° inversion 8 × rotoreflection by 60° 3 × reflection in a plane The Icosahedron colored as a snub tetrahedron has chiral symmetry.
Octahedral symmetry Icosahedral symmetry Binary tetrahedral group Symmetric group S4 Peter R. Cromwell, Polyhedra, p. 295 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 N. W. Johnson: Geometries and Transformations, ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups Weisstein, Eric W. "Tetrahedral group". MathWorld
Cushioning
Package cushioning is used to protect items during shipment. Vibration and impact shock during shipment and loading/unloading are controlled by cushioning to reduce the chance of product damage. Cushioning is inside a shipping container such as a corrugated box, it is designed to absorb shock by crushing and deforming, to dampen vibration, rather than transmitting the shock and vibration to the protected item. Depending on the specific situation, package cushioning is between 50 and 75 millimeters thick. Internal packaging materials are used for functions other than cushioning, such as to immobilize the products in the box and lock them in place, or to fill a void; when designing packaging the choice of cushioning depends on many factors, including but not limited to: effective protection of product from shock and vibration resilience resistance to creep – cushion deformation under static load material costs labor costs and productivity effects of temperature and air pressure on cushioning cleanliness of cushioning effect on size of external shipping container environmental and recycling issues sensitivity of product to static electricity Loose fill – Some cushion products are flowable and are packed loosely around the items in the box.
The box is closed to tighten the pack. This includes expanded polystyrene foam pieces, similar pieces made of starch-based foams, common popcorn; the amount of loose fill material required and the transmitted shock levels vary with the specific type of material. Paper – Paper can be manually or mechanically wadded up and used as a cushioning material. Heavier grades of paper provide more weight-bearing ability than old newspapers. Creped cellulose wadding is available. Corrugated fiberboard pads – Multi-layer or cut-and-folded shapes of corrugated board can be used as cushions; these structures are designed to crush and deform under shock stress and provide some degree of cushioning. Paperboard composite honeycomb structures are used for cushioning. Foam structures – Several types of polymeric foams are used for cushioning; the most common are: Expanded Polystyrene, polypropylene and polyurethane. These can be molded engineered sheets which are cut and glued into cushion structures. Convoluted foams sometimes used.
Some degradable foams are available. Foam-in-place is another method of using polyurethane foams; these fill the box encapsulating the product to immobilize it. It is used to form engineered structures. Molded pulp – Pulp can be molded into shapes suitable for cushioning and for immobilizing products in a package. Molded pulp is recyclable. Inflated products – Bubble wrap consists of sheets of plastic film with enclosed “bubbles” of air; these sheets can be wrapped around items to be shipped. A variety of engineered inflatable air cushions are available. Note that inflated air pillows used for void-fill are not suited for cushioning. Other – Several other types of cushioning are available including suspension cushions, thermoformed end caps, various types of shock mounts. Proper performance of cushioning is dependent on its proper use, it is best to use a trained packaging engineer, reputable vendor, consultant, or independent laboratory. An engineer needs to know the severity of shock to protect against.
This can be based on an existing specification, published industry standards and publications, field studies, etc. Knowledge of the product to be packaged is critical. Field experience may indicate the types of damage experienced. Laboratory analysis can help quantify the fragility of the item reported in g's. Engineering judgment can be an excellent starting point. Sometimes a product can be made more rugged or can be supported to make it less susceptible to breakage; the amount of shock transmitted by a particular cushioning material is dependent on the thickness of the cushion, the drop height, the load-bearing area of the cushion. A cushion must deform under shock for it to function. If a product is on a large load-bearing area, the cushion may not deform and will not cushion the shock. If the load-bearing area is too small, the product may “bottom out” during a shock. Engineers use “cushion curves” to choose the best thickness and load-bearing area for a cushioning material. Two to three inches of cushioning are needed to protect fragile items.
Cushion design requires care to prevent shock amplification caused by the cushioned shock pulse duration being close to the natural frequency of the cushioned item. The process for vibration protection involves similar considerations as that for shock. Cushions can be thought of as performing like springs. Depending on cushion thickness and load-bearing area and on the forcing vibration frequency, the cushion may 1) not have any influence on input vibration, 2) amplify the input vibration at resonance, or 3) isolate the product from the vibration. Proper design is critical for cushion performance. Verification and validation of prototype designs are required; the design of a package and its cushioning is an iterative process involving several designs, redesigns, etc. Several published package testing protocols are available to evaluate the performance of a proposed package. Field performance should be monitored for feedback into the design process. D1596 Standard Test Method for D
Pentagonal pyramid
In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point. Like any pyramid, it is self-dual; the regular pentagonal pyramid has a base, a regular pentagon and lateral faces that are equilateral triangles. It is one of the Johnson solids, its height H, from the midpoint of the pentagonal face to the apex, can be computed as: H = 5 − 5 10 a ≈ 0.5257 a. Its surface area, A, can be computed as the area of pentagonal base plus five times the area of one triangle: A = a 2 ≈ 3.8855 a 2. Its volume when an edge length is known can be figured out with this formula: V = 5 + 5 24 a 3 ≈ 0.3015 a 3. It can be seen as the "lid" of an icosahedron. More an order-2 vertex-uniform pentagonal pyramid can be defined with a regular pentagonal base and 5 isosceles triangle sides of any height; the pentagrammic star pyramid has the same vertex arrangement, but connected onto a pentagram base: The pentagonal pyramid is topologically a self-dual polyhedron.
The dual edge lengths are different due to the polar reciprocation. Eric W. Weisstein, Pentagonal pyramid at MathWorld. Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra