In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple hexagon is 720°. A regular hexagon has Schläfli symbol and can be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon, both equilateral and equiangular, it is bicentric, meaning that it is both tangential. The common length of the sides equals the radius of the circumscribed circle, which equals 2 3 times the apothem. All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, making up the dihedral group D6; the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, that the regular hexagon can be partitioned into six equilateral triangles. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, so are useful for constructing tessellations.
The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons, it is not considered a triambus, although it is equilateral. The maximal diameter, D, is twice the maximal radius or circumradius, R, which equals the side length, t; the minimal diameter or the diameter of the inscribed circle, d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor: 1 2 d = r = cos R = 3 2 R = 3 2 t and d = 3 2 D; the area of a regular hexagon A = 3 3 2 R 2 = 3 R r = 2 3 r 2 = 3 3 8 D 2 = 3 4 D d = 3 2 d 2 ≈ 2.598 R 2 ≈ 3.464 r 2 ≈ 0.6495 D 2 ≈ 0.866 d 2. For any regular polygon, the area can be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, p = 6 R = 4 r 3, so A = a p 2 = r ⋅ 4 r 3 2 = 2 r 2 3 ≈ 3.464 r 2. The regular hexagon fills the fraction 3 3 2 π ≈ 0.8270 of its circumscribed circle.
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumscribing circle between B and C PE + PF = PA + PB + PC + PD. The regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups: Dih3, Dih2, Dih1, 4 cyclic subgroups: Z6, Z3, Z2, Z1; these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a group order. R12 is full symmetry, a1 is no symmetry. P6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles; these two forms have half the symmetry order of the regular hexagon. The
Toshikazu Sunada is a Japanese mathematician and author of many books and essays on mathematics and mathematical sciences. He is professor of mathematics at Meiji University, is professor emeritus of Tohoku University, Japan. Before he joined Meiji University in 2003, he was professor of mathematics at Nagoya University, at the University of Tokyo, at Tohoku University. Sunada was involved in the creation of the School of Interdisciplinary Mathematical Sciences at Meiji University and is its first dean. Sunada's work covers complex analytic geometry, spectral geometry, dynamical systems, graph theory, discrete geometric analysis, mathematical crystallography. Among his numerous contributions, the most famous one is a general construction of isospectral manifolds, based on his geometric model of number theory, is considered to be a breakthrough in the problem proposed by Mark Kac in "Can one hear the shape of a drum?". Sunada's idea was taken up by Carolyn S. Gordon, David Webb, Scott A. Wolpert when they constructed a counterexample for Kac's problem.
For this work, Sunada was awarded the Iyanaga Prize of the Mathematical Society of Japan in 1987. He was awarded Publication Prize of MSJ in 2013, the Hiroshi Fujiwara Prize for Mathematical Sciences in 2017, the Prize for Science and Technology in 2018, the 1st Kodaira Kunihiko Prize in 2019. In a joint work with Atsushi Katsuda, Sunada established a geometric analogue of Dirichlet's theorem on arithmetic progressions in the context of dynamical systems. One can see, in this work as well as the one above, how the concepts and ideas in different fields are put together to formulate problems and to produce new results, his study of discrete geometric analysis includes a graph-theoretic interpretation of Ihara zeta functions, a discrete analogue of periodic magnetic Schrödinger operators as well as the large time asymptotic behaviors of random walk on crystal lattices. The study of random walk led him to the discovery of a "mathematical twin" of the diamond crystal out of an infinite universe of hypothetical crystals.
He named it the K4 crystal due to its mathematical relevance. What was noticed by him is that the K4 crystal has the "strong isotropy property", meaning that for any two vertices x and y of the crystal net, for any ordering of the edges adjacent to x and any ordering of the edges adjacent to y, there is a net-preserving congruence taking x to y and each x-edge to the ordered y-edge; this property is shared only by the diamond crystal. The K4 crystal and the diamond crystal as networks in space are examples of “standard realizations”, the notion introduced by Sunada and Motoko Kotani as a graph-theoretic version of Albanese maps in algebraic geometry. For his work, see Isospectral, Reinhardt domain, Ihara zeta function, Ramanujan graph, quantum ergodicity, quantum walk. T. Sunada, Holomorphic equivalence problem for bounded Reinhardt domains, Mathematische Annalen 235, 111–128 T. Sunada, Rigidity of certain harmonic mappings, Inventiones Mathematicae 51, 297–307 J. Noguchi and T. Sunada, Finiteness of the family of rational and meromorphic mappings into algebraic varieties, American Journal of Mathematics 104, 887–900 T.
Sunada, Riemannian coverings and isospectral manifolds, Annals of Mathematics 121, 169–186 T. Sunada, L-functions and some applications, Lecture Notes in Mathematics 1201, Springer-Verlag, 266–284 A. Katsuda and T. Sunada and closed geodesics in a compact Riemann surface, American Journal of Mathematics 110, 145–156 T. Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, Topology 28, 125–132 A. Katsuda and T. Sunada, Closed orbits in homology classes, Publications Mathématiques de l'IHÉS 71, 5–32 M. Nishio and T. Sunada, Trace formulae in Proc. ICM-90 Kyoto, Springer-Verlag, Tokyo, 577–585 T. Sunada, Quantum ergodicity, Trend in Mathematics, Birkhauser Verlag, Basel, 1997, 175–196 M. Kotani and T. Sunada, Albanese maps and an off diagonal long time asymptotic for the heat kernel, Communications in Mathematical Physics 209, 633–670 M. Kotani and T. Sunada, Spectral geometry of crystal lattices, Contemporary Mathematics 338, 271–305 T. Sunada, Crystals that nature might miss creating, Notices of the American Mathematical Society, 55, 208–215 T.
Sunada, Discrete geometric analysis, Proceedings of Symposia in Pure Mathematics, 77, 51–86 K. Shiga and T. Sunada, A Mathematical Gift, III, American Mathematical Society T. Sunada, Lecture on topological crystallography, Japan Journal of Mathematics 7, 1–39 T. Sunada, Topological Crystallography, With a View Towards Discrete Geometric Analysis, Springer, 2013, ISBN 978-4-431-54176-9 978-4-431-54177-6 T. Sunada, Generalized Riemann sums, in From Riemann to Differential Geometry and Relativity, Editors: Lizhen Ji, Athanase Papadopoulos, Sumio Yamada, Springer, 457-479 T. Sunada, Topics on mathematical crystallography, Proceedings of the symposium Groups and random walks, London Mathematical Society Lecture Note Series 436, Cambridge University Press, 2017, 473--513 Atsushi Katsuda
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
Cairo is the capital of Egypt. The city's metropolitan area is one of the largest in Africa, the largest in the Middle East, the 15th-largest in the world, is associated with ancient Egypt, as the famous Giza pyramid complex and the ancient city of Memphis are located in its geographical area. Located near the Nile Delta, modern Cairo was founded in 969 CE by the Fatimid dynasty, but the land composing the present-day city was the site of ancient national capitals whose remnants remain visible in parts of Old Cairo. Cairo has long been a centre of the region's political and cultural life, is titled "the city of a thousand minarets" for its preponderance of Islamic architecture. Cairo is considered a World City with a "Beta +" classification according to GaWC. Cairo has the oldest and largest film and music industries in the Middle East, as well as the world's second-oldest institution of higher learning, Al-Azhar University. Many international media and organizations have regional headquarters in the city.
With a population of over 9 million spread over 3,085 square kilometers, Cairo is by far the largest city in Egypt. An additional 9.5 million inhabitants live in close proximity to the city. Cairo, like many other megacities, suffers from high levels of traffic. Cairo's metro, one of two in Africa, ranks among the fifteen busiest in the world, with over 1 billion annual passenger rides; the economy of Cairo was ranked first in the Middle East in 2005, 43rd globally on Foreign Policy's 2010 Global Cities Index. Egyptians refer to Cairo as Maṣr, the Egyptian Arabic name for Egypt itself, emphasizing the city's importance for the country, its official name al-Qāhirah means "the Vanquisher" or "the Conqueror" due to the fact that the planet Mars, an-Najm al-Qāhir, was rising at the time when the city was founded also in reference to the much awaited arrival of the Fatimid Caliph Al-Mu'izz who reached Cairo in 973 from Mahdia, the old Fatimid capital. The location of the ancient city of Heliopolis is the suburb of Ain Shams.
The Coptic name of the city is Kashromi which means "man breaker", akin to Arabic al-Qāhirah . Sometimes the city is informally referred to as Kayro by people from Alexandria; the area around present-day Cairo Memphis, the old capital of Egypt, had long been a focal point of Ancient Egypt due to its strategic location just upstream from the Nile Delta. However, the origins of the modern city are traced back to a series of settlements in the first millennium. Around the turn of the 4th century, as Memphis was continuing to decline in importance, the Romans established a fortress town along the east bank of the Nile; this fortress, known as Babylon, was the nucleus of the Roman and the Byzantine city and is the oldest structure in the city today. It is situated at the nucleus of the Coptic Orthodox community, which separated from the Roman and Byzantine churches in the late 4th century. Many of Cairo's oldest Coptic churches, including the Hanging Church, are located along the fortress walls in a section of the city known as Coptic Cairo.
Following the Muslim conquest in 640 AD, the conqueror Amr ibn As settled to the north of the Babylon in an area that became known as al-Fustat. A tented camp Fustat became a permanent settlement and the first capital of Islamic Egypt. In 750, following the overthrow of the Umayyad caliphate by the Abbasids, the new rulers created their own settlement to the northeast of Fustat which became their capital; this was known as al-Askar. A rebellion in 869 by Ahmad ibn Tulun led to the abandonment of Al Askar and the building of another settlement, which became the seat of government; this was al-Qatta ` closer to the river. Al Qatta'i was centred around a ceremonial mosque, now known as the Mosque of ibn Tulun. In 905, the Abbasids re-asserted control of the country and their governor returned to Fustat, razing al-Qatta'i to the ground. Since 1860s, Cairo expanded west as far as what is called now In 968, the Fatimids were led by general Jawhar al-Siqilli to establish a new capital for the Fatimid dynasty.
Egypt was conquered from their base in Ifriqiya and a new fortified city northeast of Fustat was established. It took four years to build the city known as al-Manṣūriyyah, to serve as the new capital of the caliphate. During that time, Jawhar commissioned the construction of the al-Azhar Mosque by order of the Caliph, which developed into the third-oldest university in the world. Cairo would become a centre of learning, with the library of Cairo containing hundreds of thousands of books; when Caliph al-Mu'izz li Din Allah arrived from the old Fatimid capital of Mahdia in Tunisia in 973, he gave the city its present name, al-Qāhiratu. For nearly 200 years after Cairo was established, the administrative centre of Egypt remained in Fustat. However, in 1168 the Fatimids under the leadership of vizier Shawar set fire to Fustat to prevent Cairo's capture by the Crusaders. Egypt's capital was permanently moved to Cairo, expanded to include the ruins of Fustat and the previous capitals of
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
A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur in architecture and decorative art in textiles and tiles as well as wallpaper. A proof that there were only 17 distinct groups of possible patterns was first carried out by Evgraf Fedorov in 1891 and derived independently by George Pólya in 1924; the proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done. The seventeen possible wallpaper groups are listed below in § The seventeen groups. Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are different in style, scale or orientation may belong to the same group. Consider the following examples: Examples A and B have the same wallpaper group.
Example C has a different wallpaper group, called p4g or 4*2. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities. A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it looks the same after the transformation. For example, translational symmetry is present when the pattern can be translated some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe; the pattern is unchanged. Speaking, a true symmetry only exists in patterns that repeat and continue indefinitely. A set of only, five stripes does not have translational symmetry—when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, classification is applied to finite patterns, small imperfections may be ignored. Sometimes two categorizations are meaningful, one based on shapes alone and one including colors.
When colors are ignored there may be more symmetry. In black and white there are 17 wallpaper groups; the types of transformations that are relevant here are called Euclidean plane isometries. For example: If we shift example B one unit to the right, so that each square covers the square, adjacent to it the resulting pattern is the same as the pattern we started with; this type of symmetry is called a translation. Examples A and C are similar. If we turn example B clockwise by 90°, around the centre of one of the squares, again we obtain the same pattern; this is called a rotation. Examples A and C have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C. We can flip example B across a horizontal axis that runs across the middle of the image; this is called a reflection. Example B has reflections across a vertical axis, across two diagonal axes; the same can be said for A. However, example C is different, it only has reflections in vertical directions, not across diagonal axes.
If we flip across a diagonal line, we do not get the same pattern back. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C. Another transformation is "Glide", a combination of reflection and translation parallel to the line of reflection. Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations. Two such isometry groups are of the same type if they are the same up to an affine transformation of the plane, thus e.g. a translation of the plane does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry. Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those that preserve orientation, it follows from the Bieberbach theorem that all wallpaper groups are different as abstract groups.
2D patterns with double translational symmetry can be categorized according to their symmetry group type. Isometries of the Euclidean plane fall into four categories. Translations, denoted by Tv, where v is a vector in R2; this has the effect of shifting the plane applying displacement vector v. Rotations, denoted by Rc,θ, where c is a point in the plane, θ is the angle of rotation. Reflections, or mirror isometries, denoted by FL, where L is a line in R2.. This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror. Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance; this is a combination of a reflection in the line L and a translation along L by a distance d. The condition
In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" together with the Greek suffix "-agon" meaning angle. A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians, its Schläfli symbol is. The area of a regular heptagon of side length a is given by: A = 7 4 a 2 cot π 7 ≃ 3.634 a 2. This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, halving each triangle using the apothem as the common side; the apothem is half the cotangent of π / 7, the area of each of the 14 small triangles is one-fourth of the apothem. The exact algebraic expression, starting from the cubic polynomial x3 + x2 − 2x − 1 is given in complex numbers by: A = a 2 4 7 3, in which the imaginary parts offset each other leaving a real-valued expression; this expression cannot be algebraically rewritten without complex components, since the indicated cubic function is casus irreducibilis.
The area of a regular heptagon inscribed in a circle of radius R is 7 R 2 2 sin 2 π 7, while the area of the circle itself is π R 2. As 7 is a Pierpont prime but not a Fermat prime, the regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass; this type of construction is called a neusis construction. It is constructible with compass and angle trisector; the impossibility of straightedge and compass construction follows from the observation that 2 cos 2 π 7 ≈ 1.247 is a zero of the irreducible cubic x3 + x2 − 2x − 1. This polynomial is the minimal polynomial of 2cos, whereas the degree of the minimal polynomial for a constructible number must be a power of 2. An approximation for practical use with an error of about 0.2% is shown in the drawing. It is attributed to Albrecht Dürer. Let A lie on the circumference of the circumcircle. Draw arc BOC. B D = 1 2 B C gives an approximation for the edge of the heptagon; this approximation uses 3 2 ≈ 0.86603 for the side of the heptagon inscribed in the unit circle while the exact value is 2 sin π 7 ≈ 0.86777.
Example to illustrate the error: At a circumscribed circle radius r = 1 m, the absolute error of the 1st side would be -1.7 mm The regular heptagon belongs to the D7h point group, order 28. The symmetry elements are: a 7-fold proper rotation axis C7, a 7-fold improper rotation axis,S7, 7 vertical mirror planes, σv, 7 2-fold rotation axes, C2, in the plane of the heptagon and a horizontal mirror plane, σh in the heptagon's plane; the regular heptagon's side a, shorter diagonal b, longer diagonal c, with a<b<c, satisfy a 2 = c, b 2 = a, c 2 = b, 1 a = 1 b + 1 c and hence a b + a c