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95-404: This small seated figure is the only known three dimensional depiction of Khufu which survives largely intact, though there are also several statue fragments. Most Egyptologists consider the statue contemporary with Khufu very likely from his reign. However, because of the unusual provenance, its dating has been repeatedly questioned. The Egyptologist Zahi Hawass doubts that the statuette dates to

190-414: A = 0 {\displaystyle a=0} . While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements i , j , k {\displaystyle i,j,k} , as well as the dot product and cross product , which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions. It

285-491: A solid figure . Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n -dimensional Euclidean space. The set of these n -tuples is commonly denoted R n , {\displaystyle \mathbb {R} ^{n},} and can be identified to the pair formed by a n -dimensional Euclidean space and a Cartesian coordinate system . When n = 3 , this space

380-462: A 6th Dynasty building complex. A number of objects from the 1st , 2nd , 6th and 30th Dynasties have been found in the Temple of Khenti-Amentiu, but nothing that can be surely dated to the 4th Dynasty. Furthermore, the temple does not seem to have been in use during this period. Petrie could not find any evidence of buildings from Khufu's time in his excavations, but he explained this with a reference to

475-487: A parallelogram , and hence are coplanar. A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P . The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball ). The volume of the ball is given by V = 4 3 π r 3 , {\displaystyle V={\frac {4}{3}}\pi r^{3},} and

570-444: A Persian subject, and it may be that the young Herodotus heard local eyewitness accounts of events within the empire and of Persian preparations for the invasion of Greece , including the movements of the local fleet under the command of Artemisia I of Caria . Inscriptions recently discovered at Halicarnassus indicate that Artemesia's grandson Lygdamis negotiated with a local assembly to settle disputes over seized property, which

665-473: A boy living on the island of Samos, to which he had fled with his family from the oppressions of Lygdamis, tyrant of Halicarnassus and grandson of Artemisia. Panyassis , the epic poet related to Herodotus, is reported to have taken part in a failed uprising. The Suda also states that Herodotus later returned home to lead the revolt that eventually overthrew the tyrant. Due to recent discoveries of inscriptions at Halicarnassus dated to about Herodotus's time, it

760-522: A choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space. Computationally, it is necessary to work with the more concrete description R 3 {\displaystyle \mathbb {R} ^{3}} in order to do concrete computations. A more abstract description still

855-525: A field , which is not commutative nor associative , but is a Lie algebra with the cross product being the Lie bracket. Specifically, the space together with the product, ( R 3 , × ) {\displaystyle (\mathbb {R} ^{3},\times )} is isomorphic to the Lie algebra of three-dimensional rotations, denoted s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} . In order to satisfy

950-412: A given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line. A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of

1045-646: A historical topic more in keeping with the Greek world-view: focused on the context of the polis or city-state. The interplay of civilizations was more relevant to Greeks living in Anatolia, such as Herodotus himself, for whom life within a foreign civilization was a recent memory. Before the Persian crisis, history had been represented among the Greeks only by local or family traditions. The "Wars of Liberation" had given to Herodotus

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1140-400: A hyperplane satisfy a single linear equation , so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in R 3 {\displaystyle \mathbb {R} ^{3}} form

1235-404: A literary critic of Augustan Rome , listed seven predecessors of Herodotus, describing their works as simple unadorned accounts of their own and other cities and people, Greek or foreign, including popular legends, sometimes melodramatic and naïve, often charming – all traits that can be found in the work of Herodotus himself. Modern historians regard the chronology as uncertain, but according to

1330-419: A plane curve about a fixed line in its plane as an axis is called a surface of revolution . The plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle. Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution

1425-504: A similar schema to the Khufu statuette, but its body is very slim and athletic and its execution is significantly more careful. The appearance of Khufu in the ivory statuette, however, is claimed not to be particularly well-worked. Khufu himself, in Hawass' view, would never have allowed such a comparatively crude item to be displayed in his palace or elsewhere. Further, Hawass alleges that the shape of

1520-442: A subtle way. By definition, there exists a basis B = { e 1 , e 2 , e 3 } {\displaystyle {\mathcal {B}}=\{e_{1},e_{2},e_{3}\}} for V {\displaystyle V} . This corresponds to an isomorphism between V {\displaystyle V} and R 3 {\displaystyle \mathbb {R} ^{3}} :

1615-432: A title conferred on him by the ancient Roman orator Cicero , and the " Father of Lies " by others. The Histories primarily cover the lives of prominent kings and famous battles such as Marathon , Thermopylae , Artemisium , Salamis , Plataea , and Mycale . His work deviates from the main topics to provide a cultural, ethnographical , geographical, and historiographical background that forms an essential part of

1710-445: A unique plane, so skew lines are lines that do not meet and do not lie in a common plane. Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can lie in

1805-472: A vector A is denoted by || A || . The dot product of a vector A = [ A 1 , A 2 , A 3 ] with itself is which gives the formula for the Euclidean length of the vector. Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by where θ is the angle between A and B . The cross product or vector product

1900-680: A version of the Histories written by "Herodotus of Thurium", and some passages in the Histories have been interpreted as proof that he wrote about Magna Graecia from personal experience there (IV, 15,99; VI, 127). According to Ptolemaeus Chennus , a late source summarized in the Library of Photius , Plesirrhous the Thessalian, the hymnographer, was the eromenos of Herodotus and his heir. This account has also led some historians to assume Herodotus died childless. Intimate knowledge of some events in

1995-510: A young Thucydides happened to be in the assembly with his father, and burst into tears during the recital. Herodotus observed prophetically to the boy's father: "Your son's soul yearns for knowledge." Eventually, Thucydides and Herodotus became close enough for both to be interred in Thucydides's tomb in Athens. Such at least was the opinion of Marcellinus in his Life of Thucydides . According to

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2090-627: Is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product A × B of the vectors A and B is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics , and engineering . In function language, the cross product is a function × : R 3 × R 3 → R 3 {\displaystyle \times :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}} . The components of

2185-439: Is a mathematical space in which three values ( coordinates ) are required to determine the position of a point . Most commonly, it is the three-dimensional Euclidean space , that is, the Euclidean space of dimension three, which models physical space . More general three-dimensional spaces are called 3-manifolds . The term may also refer colloquially to a subset of space, a three-dimensional region (or 3D domain ),

2280-758: Is a right circular cone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder . In analogy with the conic sections , the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0 , {\displaystyle Ax^{2}+By^{2}+Cz^{2}+Fxy+Gyz+Hxz+Jx+Ky+Lz+M=0,} where A , B , C , F , G , H , J , K , L and M are real numbers and not all of A , B , C , F , G and H are zero,

2375-572: Is angular and he does not wear the Pharaonic ceremonial beard. The king wears a short, pleated loincloth – his upper body is naked. On the right side, at Khufu's knee is the Horus name "Medjedu" and on the left side of the knee, the very faint traces of the end of his nomen "Khnum-Khufu" is visible in a cartouche . The artefact was found in 1903 by Flinders Petrie in the Kom el-Sultan necropolis at Abydos in one of

2470-466: Is called a quadric surface . There are six types of non-degenerate quadric surfaces: The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane π and all the lines of R through that conic that are normal to π ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well. Both

2565-416: Is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics , it serves as a model of the physical universe , in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of space-time . While this space remains the most compelling and useful way to model the world as it is experienced, it

2660-618: Is consistent with a tyrant under pressure. His name is not mentioned later in the tribute list of the Athenian Delian League , indicating that there might well have been a successful uprising against him some time before 454 BC. Herodotus wrote his Histories in the Ionian dialect , in spite of being born in a Dorian settlement. According to the Suda , Herodotus learned the Ionian dialect as

2755-406: Is found in linear algebra , where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors . A vector can be pictured as an arrow. The vector's magnitude

2850-486: Is generally assumed that he died not long afterwards, possibly before his sixtieth year. Herodotus would have made his researches known to the larger world through oral recitations to a public crowd. John Marincola writes in his introduction to the Penguin edition of the Histories that there are certain identifiable pieces in the early books of Herodotus's work which could be labeled as "performance pieces". These portions of

2945-425: Is its length, and its direction is the direction the arrow points. A vector in R 3 {\displaystyle \mathbb {R} ^{3}} can be represented by an ordered triple of real numbers. These numbers are called the components of the vector. The dot product of two vectors A = [ A 1 , A 2 , A 3 ] and B = [ B 1 , B 2 , B 3 ] is defined as: The magnitude of

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3040-794: Is now known that the Ionic dialect was used in Halicarnassus in some official documents, so there is no need to assume (like the Suda ) that he must have learned the dialect elsewhere. The Suda is the only source placing Herodotus as the heroic liberator of his birthplace, casting doubt upon the veracity of that romantic account. As Herodotus himself reveals, Halicarnassus, though a Dorian city, had ended its close relations with its Dorian neighbours after an unseemly quarrel (I, 144), and it had helped pioneer Greek trade with Egypt (II, 178). It was, therefore, an outward-looking, international-minded port within

3135-602: Is on account of the many strange stories and the folk-tales he reported that his critics have branded him "The Father of Lies". Even his own contemporaries found reason to scoff at his achievement. In fact, one modern scholar has wondered whether Herodotus left his home in Greek Anatolia , migrating westwards to Athens and beyond, because his own countrymen had ridiculed his work, a circumstance possibly hinted at in an epitaph said to have been dedicated to Herodotus at one of his three supposed resting places, Thuria : Herodotus

3230-753: Is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions ( coordinates ), any three directions can be chosen, provided that these directions do not lie in the same plane . Furthermore, if these directions are pairwise perpendicular , the three values are often labeled by the terms width /breadth , height /depth , and length . Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes

3325-664: Is the Kronecker delta . Written out in full, the standard basis is E 1 = ( 1 0 0 ) , E 2 = ( 0 1 0 ) , E 3 = ( 0 0 1 ) . {\displaystyle E_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},E_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},E_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}.} Therefore R 3 {\displaystyle \mathbb {R} ^{3}} can be viewed as

3420-528: Is the Levi-Civita symbol . It has the property that A × B = − B × A {\displaystyle \mathbf {A} \times \mathbf {B} =-\mathbf {B} \times \mathbf {A} } . Its magnitude is related to the angle θ {\displaystyle \theta } between A {\displaystyle \mathbf {A} } and B {\displaystyle \mathbf {B} } by

3515-418: Is to model physical space as a three-dimensional affine space E ( 3 ) {\displaystyle E(3)} over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of R 3 {\displaystyle \mathbb {R} ^{3}} ,

3610-478: Is unknown. The ivory figurine is about 7.5 cm high, 2.9 cm long and c. 2.6 cm wide and partially damaged. Its outer surface was originally smooth and polished to a sheen. The statuette depicts Khufu with the Red crown ( deshret ) of Lower Egypt . The King sits on a largely undecorated throne with a low back. In his right hand, which is placed over his breast, he holds a flail against his right shoulder with

3705-524: The Alcmaeonids , a clan whose history is featured frequently in his writing. According to Plutarch , Herodotus was granted a financial reward by the Athenian assembly in recognition of his work. Plutarch, using Diyllus as a source, says this was 10 talents . In 443 BC or shortly afterwards, he migrated to Thurii , in modern Calabria , as part of an Athenian-sponsored colony . Aristotle refers to

3800-562: The Byzantine Suda , an 11th-century encyclopedia which possibly took its information from traditional accounts. Still, the challenge is great: The data are so few – they rest upon such late and slight authority; they are so improbable or so contradictory, that to compile them into a biography is like building a house of cards, which the first breath of criticism will blow to the ground. Still, certain points may be approximately fixed ... Herodotus was, according to his own statement, at

3895-486: The Euphrates to Babylon . For some reason, possibly associated with local politics, he subsequently found himself unpopular in Halicarnassus, and sometime around 447 BC, migrated to Periclean Athens  – a city whose people and democratic institutions he openly admired (V, 78). Athens was also the place where he came to know the local topography (VI, 137; VIII, 52–55), as well as leading citizens such as

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3990-759: The Peloponnesian War on the abduction of some prostitutes – a mocking reference to Herodotus, who reported the Persians' account of their wars with Greece , beginning with the rapes of the mythical heroines Io , Europa , Medea , and Helen . Similarly, the Athenian historian Thucydides dismissed Herodotus as a story-teller. Thucydides, who had been trained in rhetoric , became the model for subsequent prose-writers as an author who seeks to appear firmly in control of his material, whereas with his frequent digressions Herodotus appeared to minimize (or possibly disguise) his authorial control. Moreover, Thucydides developed

4085-565: The Persian Empire , and the historian's family could well have had contacts in other countries under Persian rule, facilitating his travels and his researches. Herodotus's eyewitness accounts indicate that he traveled in Egypt in association with Athenians, probably sometime after 454 BC or possibly earlier, after an Athenian fleet had assisted the uprising against Persian rule in 460–454 BC. He probably traveled to Tyre next and then down

4180-472: The Predynastic period. Barry J. Kemp and William S. Smith further pointed out that the Khufu statuette's face most closely resembles those of statues from the time of Khasekhemwy , Djoser , and Sneferu in execution. The faces of Khasekhemwy and Snefru are also beardless and Khufu's facial expression seems to be modelled on that of Djoser's limestone statues. In particular, the broad nose, rounded face and

4275-569: The Suda , he was buried in Macedonian Pella and in the agora in Thurii. Herodotus announced the purpose and scope of his work at the beginning of his Histories: Here are presented the results of the inquiry carried out by Herodotus of Halicarnassus. The purpose is to prevent the traces of human events from being erased by time, and to preserve the fame of the important and remarkable achievements produced by both Greeks and non-Greeks; among

4370-561: The rose granite " Brooklyn Royal Head" (though it's also thought to depict Huni ) and the limestone "Munich Royal Head". Both heads show the king in the White crown of Upper Egypt . An unusual example is the front part of a polished basalt ram statue, with the Horus and Cartouche names of Khufu on it. The majority of Egyptologists put the statuette in the Old Kingdom at the time of Khufu. Petrie

4465-513: The 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton 's development of the quaternions . In fact, it was Hamilton who coined the terms scalar and vector , and they were first defined within his geometric framework for quaternions . Three dimensional space could then be described by quaternions q = a + u i + v j + w k {\displaystyle q=a+ui+vj+wk} which had vanishing scalar component, that is,

4560-410: The Greek historians Herodotus and Diodorus , who report that Khufu forbade the erection of temples and shrines to the gods during his reign. However, recently Richard Bussmann pointed to an unpublished limestone fragment at Abydos with Khufu's name, which shows at least some of the building activity at Abydos belongs to Khufu. Bussmann asks, therefore, whether Building K could have been a temple for

4655-434: The Old Kingdom at all. His argument that the statuette belongs to the 26th Dynasty has not received much credence, but has not yet been refuted. The ritual purpose of the statuette is also unclear. If it was contemporary with Khufu, it was either part of the traditional statue cult or mortuary cult . If the figurine is from a later period, it probably served (as claimed by Hawass) as a votive offering . The statuette's artist

4750-399: The above-mentioned systems. Two distinct points always determine a (straight) line . Three distinct points are either collinear or determine a unique plane . On the other hand, four distinct points can either be collinear, coplanar , or determine the entire space. Two distinct lines can either intersect, be parallel or be skew . Two parallel lines, or two intersecting lines , lie in

4845-490: The abstract vector space, together with the additional structure of a choice of basis. Conversely, V {\displaystyle V} can be obtained by starting with R 3 {\displaystyle \mathbb {R} ^{3}} and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis. As opposed to a general vector space V {\displaystyle V} ,

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4940-513: The affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes called Euclidean affine spaces for distinguishing them from Euclidean vector spaces. This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance. Herodotus Herodotus ( Ancient Greek : Ἡρόδοτος , romanized :  Hēródotos ; c.  484  – c.  425 BC)

5035-407: The ancient account, these predecessors included Dionysius of Miletus , Charon of Lampsacus, Hellanicus of Lesbos , Xanthus of Lydia and, the best attested of them all, Hecataeus of Miletus . Of these, only fragments of Hecataeus's works survived, and the authenticity of these is debatable, but they provide a glimpse into the kind of tradition within which Herodotus wrote his own Histories . It

5130-579: The audience. It was conventional in Herodotus's day for authors to "publish" their works by reciting them at popular festivals. According to Lucian , Herodotus took his finished work straight from Anatolia to the Olympic Games and read the entire Histories to the assembled spectators in one sitting, receiving rapturous applause at the end of it. According to a very different account by an ancient grammarian, Herodotus refused to begin reading his work at

5225-771: The axioms of a Lie algebra, instead of associativity the cross product satisfies the Jacobi identity . For any three vectors A , B {\displaystyle \mathbf {A} ,\mathbf {B} } and C {\displaystyle \mathbf {C} } A × ( B × C ) + B × ( C × A ) + C × ( A × B ) = 0 {\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )=0} One can in n dimensions take

5320-453: The beginning of his work, a native of Halicarnassus in Anatolia , and it is generally accepted that he was born there around 485 BC. The Suda says his family was influential, that he was the son of Lyxes and Dryo and the brother of Theodorus, and that he was also related to Panyassis – an epic poet of the time. Halicarnassus was then within the Persian Empire , making Herodotus

5415-578: The conclusion that the figure was probably sold to a pious citizen or pilgrim as an amulet or talisman in the 26th dynasty (or later). The figurine's presence in its find location would then be a result of use as a votive offering . Zahi Hawass is, finally, convinced that the Khufu statuette is most likely a replica of a life-size or over life-size statue. In his view the original was probably located in Memphis in Lower Egypt, which would explain why Khufu wears

5510-630: The construction for the isomorphism is found here . However, there is no 'preferred' or 'canonical basis' for V {\displaystyle V} . On the other hand, there is a preferred basis for R 3 {\displaystyle \mathbb {R} ^{3}} , which is due to its description as a Cartesian product of copies of R {\displaystyle \mathbb {R} } , that is, R 3 = R × R × R {\displaystyle \mathbb {R} ^{3}=\mathbb {R} \times \mathbb {R} \times \mathbb {R} } . This allows

5605-491: The construction of the five regular Platonic solids in a sphere. In the 17th century, three-dimensional space was described with Cartesian coordinates , with the advent of analytic geometry developed by René Descartes in his work La Géométrie and Pierre de Fermat in the manuscript Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space. In

5700-880: The cross product are A × B = [ A 2 B 3 − B 2 A 3 , A 3 B 1 − B 3 A 1 , A 1 B 2 − B 1 A 2 ] {\displaystyle \mathbf {A} \times \mathbf {B} =[A_{2}B_{3}-B_{2}A_{3},A_{3}B_{1}-B_{3}A_{1},A_{1}B_{2}-B_{1}A_{2}]} , and can also be written in components, using Einstein summation convention as ( A × B ) i = ε i j k A j B k {\displaystyle (\mathbf {A} \times \mathbf {B} )_{i}=\varepsilon _{ijk}A_{j}B_{k}} where ε i j k {\displaystyle \varepsilon _{ijk}}

5795-429: The cult of Khufu. The statuette is the only complete three dimensional object which depicts Khufu. It is often claimed that the little ivory figurine is the only surviving statue of Khufu. However, there are also several alabaster fragments of seated statues, which were found by George Reisner during his excavations at Giza . Altogether, Rainer Stadelmann estimates that around fifty statues of Khufu must have stood in

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5890-516: The definition of canonical projections, π i : R 3 → R {\displaystyle \pi _{i}:\mathbb {R} ^{3}\rightarrow \mathbb {R} } , where 1 ≤ i ≤ 3 {\displaystyle 1\leq i\leq 3} . For example, π 1 ( x 1 , x 2 , x 3 ) = x {\displaystyle \pi _{1}(x_{1},x_{2},x_{3})=x} . This then allows

5985-493: The definition of the standard basis B Standard = { E 1 , E 2 , E 3 } {\displaystyle {\mathcal {B}}_{\text{Standard}}=\{E_{1},E_{2},E_{3}\}} defined by π i ( E j ) = δ i j {\displaystyle \pi _{i}(E_{j})=\delta _{ij}} where δ i j {\displaystyle \delta _{ij}}

6080-534: The face of the Khufu statuette most closely resembles the black granite heads of King Taharqa. Citing the work of William S. Smith, Hawass claims that statues of the Old Kingdom Kings were mass-produced in later time, that this probably also applies to the Khufu statuette and that the rather sloppy form of the statuette corroborates this. Three dimension In geometry , a three-dimensional space ( 3D space , 3-space or, rarely, tri-dimensional space )

6175-439: The festival of Olympia until some clouds offered him a bit of shade – by which time the assembly had dispersed. (Hence the proverbial expression "Herodotus and his shade" to describe someone who misses an opportunity through delay.) Herodotus's recitation at Olympia was a favourite theme among ancient writers, and there is another interesting variation on the story to be found in the Suda : that of Photius and Tzetzes , in which

6270-407: The find circumstances and points out that Khufu's face is unusually round and chubby and shows no emotion whatsoever. In contrast to Petrie and Margaret Alice Murray , who described the figurine's face as "powerful" and "intimidating" (in accordance with Greek traditions about Khufu), Hawass saw the face of a very young, possibly underage man. Hawass compares the facial appearance of the statuette with

6365-448: The find situation as a strong argument for his doubts about the dating of the figure. He argues that no buildings which certainly date from the fourth dynasty have ever been excavated at Abydos or Kom el-Sultan and that Petrie was strictly speaking only convinced that Room C must have been a Fourth Dynasty temple or shrine because of the discovery of the Khufu statuette. But building K (next to the magazines) has since turned out to be part of

6460-422: The first genuinely historical inspiration felt by a Greek. These wars showed him that there was a corporate life, higher than that of the city, of which the story might be told; and they offered to him as a subject the drama of the collision between East and West. With him, the spirit of history was born into Greece; and his work, called after the nine Muses, was indeed the first utterance of Clio . Though Herodotus

6555-500: The first years of the Peloponnesian War (VI, 91; VII, 133, 233; IX, 73) suggests that he returned to Athens, in which case it is possible that he died there during an outbreak of the plague. It is also possible he died in Macedonia instead, after obtaining the patronage of the court there; or else he died back in Thurii. There is nothing in the Histories that can be dated to later than 430 BC with any certainty, and it

6650-426: The flail lying over his upper arm. His left arm is bent with his lower arm resting on his left thigh. The left hand is open, with the palm resting on his left knee. His feet have broken away, along with the pedestal. The red crown is damaged – both the ridge at the back and the decorative spiral at the front have broken off. His head is slightly over proportioned relative to his body, with large, projecting ears. His chin

6745-502: The fragments from a small seated statue the king's feet survive up to the ankles. To the right of his feet is the syllable "fu" in a cartouche , which can easily be reconstructed as the name of king "Khufu". The Palermo stone fragment C2 reports the creation of two colossal standing statues of the king - one of copper and the other of pure gold . Several statue heads also survive, which are sometimes attributed to Khufu on account of their stylistic features. The best known of these are

6840-421: The hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces , meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a regulus . Another way of viewing three-dimensional space

6935-460: The identity ‖ A × B ‖ = ‖ A ‖ ⋅ ‖ B ‖ ⋅ | sin ⁡ θ | . {\displaystyle \left\|\mathbf {A} \times \mathbf {B} \right\|=\left\|\mathbf {A} \right\|\cdot \left\|\mathbf {B} \right\|\cdot \left|\sin \theta \right|.} The space and product form an algebra over

7030-434: The king's mortuary temple originally. He estimated that twenty-one to twenty-five statues were taken over by Khufu's successor Djedefre . On the bases of the statues of Khufu, however, the complete royal titulary of the king was inscribed; today the names only survive in fragments, but they are enough to enable a certain identification. These used the full name ( Khnum-Khufu ) as often as the shortened form ( Khufu ). On one of

7125-476: The kings include the ceremonial beard. The artistic elaboration of the ivory figure has been universally acclaimed by researchers as "masterful" and "professional". It is to this day the earliest known Egyptian sculpture showing a king wearing the Red crown. This becomes more common under Khafre. Zahi Hawass on the other hand doubts the statuette is contemporary with Khufu. He considers Petrie's dating suspect on account of

7220-424: The matters covered is, in particular, the cause of the hostilities between Greeks and non-Greeks. His record of the achievements of others was an achievement in itself, though the extent of it has been debated. Herodotus's place in history and his significance may be understood according to the traditions within which he worked. His work is the earliest Greek prose to have survived intact. Dionysius of Halicarnassus ,

7315-612: The narrative and provides readers with a wellspring of additional information. Herodotus was criticized in ancient times for his inclusion of "legends and fanciful accounts" in his work. The contemporaneous historian Thucydides accused him of making up stories for entertainment. He retorted that he reported what he could see and was told. A sizable portion of the Histories has since been confirmed by modern historians and archaeologists . Modern scholars generally turn to Herodotus's own writing for reliable information about his life, supplemented with ancient yet much later sources, such as

7410-555: The position of any point in three-dimensional space is given by an ordered triple of real numbers , each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates , though there are an infinite number of possible methods. For more, see Euclidean space . Below are images of

7505-472: The product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions . It can be useful to describe three-dimensional space as a three-dimensional vector space V {\displaystyle V} over the real numbers. This differs from R 3 {\displaystyle \mathbb {R} ^{3}} in

7600-435: The rather schematic eyebrows are clearly inspired by the artistic style of the 3rd dynasty . The slightly protruding ears recall those on the statues of Khasekhemwy. With this facial composition, the portrait of Khufu is stylistically in transition from the archaic form to the classical Old Kingdom style. This artistic style can no longer be perceived in the artworks of any king after Djedefre ; from King Khafre , depictions of

7695-542: The red crown. This assumption also underpinned his dating to the 26th dynasty: at that time, homages to the Old Kingdom were very popular; old, long-forgotten deities were portrayed in reliefs and statues and miniatures of royal statues made and sold as talismans or votive offerings and old, long-forgotten titles of the Old Kingdom were reprised and awarded to officials. For example, the temple of King Taharqa contains reliefs which are modelled after Old Kingdom murals from entirely different contexts. Finally, Hawass maintains that

7790-468: The research seem independent and "almost detachable", so that they might have been set aside by the author for the purposes of an oral performance. The intellectual matrix of the 5th century, Marincola suggests, comprised many oral performances in which philosophers would dramatically recite such detachable pieces of their work. The idea was to criticize previous arguments on a topic and emphatically and enthusiastically insert their own in order to win over

7885-518: The rooms of "Magazine C" of the large, heavily ruined temple complex of Osiris - Khenti-Amentiu (labelled on the excavation plan as "Building K") in the southern sector. The temple of Kom el-Sultan was dedicated to the jackal god Khenti-Amentiu from the Early Dynastic Period until about the middle of the 3rd Dynasty . In the Middle Kingdom a sanctuary in honour of the mummiform god Osiris

7980-611: The significance of the discovery, he had all work stopped and announced a reward for the recovery of the head. Three weeks later the head was found among the debris from the room after an intensive sieving. Today, the restored statuette is in the Egyptian Museum in Cairo , in Room 32 with the inventory number JE 36143 . The circumstances of the Khufu statuette's discovery have been called "unusual" and "contradicting". Zahi Hawass in particular sees

8075-470: The son of Sphynx lies; in Ionic history without peer; a Dorian born, who fled from slander's brand and made in Thuria his new native land. Yet it was in Athens where his most formidable contemporary critics could be found. In 425 BC, which is about the time that Herodotus is thought by many scholars to have died, the Athenian comic dramatist Aristophanes created The Acharnians , in which he blames

8170-530: The space R 3 {\displaystyle \mathbb {R} ^{3}} is sometimes referred to as a coordinate space. Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} assumes

8265-482: The statues of other contemporary kings (such as Snefru, Khafre, and Menkaure ). These three kings' faces are of more normal proportions, thin and friendly - they conform to the ideal form which consciously diverges from reality. In particular, an ivory statuette of King Menkaure, now on display in the Boston Museum under the number Boston 11.280a-b , excites Hawass' interest. Although now headless, this figure displays

8360-418: The surface area of the sphere is A = 4 π r 2 . {\displaystyle A=4\pi r^{2}.} Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere : points equidistant to the origin of the euclidean space R . If a point has coordinates, P ( x , y , z , w ) , then x + y + z + w = 1 characterizes those points on

8455-440: The throne has no counterpart in Old Kingdom art: In the Old Kingdom, the back of the royal throne rose to the neck of the ruler. For Hawass, a conclusive proof that the statue must be a reproduction from a later time is the so-called Nehenekh flail in Khufu's left hand. Sculptural depictions of a king with such a flail as a ceremonial insignia do not appear chronologically before the Middle Kingdom . Zahi Hawass, therefore, comes to

8550-446: The unit 3-sphere centered at the origin. This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra . A surface generated by revolving

8645-491: The work of Hermann Grassmann and Giuseppe Peano , the latter of whom first gave the modern definition of vector spaces as an algebraic structure. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin , the point at which they cross. They are usually labeled x , y , and z . Relative to these axes,

8740-558: Was a Greek historian and geographer from the Greek city of Halicarnassus , part of the Persian Empire (now Bodrum , Turkey) and a later citizen of Thurii in modern Calabria , Italy. He wrote the Histories , a detailed account of the Greco-Persian Wars , and was the first writer to apply a scientific method to historical events. He has been described as " The Father of History ",

8835-468: Was built on the site. Khenti-Amentiu and Osiris merged with each other very early on and the temple complex was seen as the Sanctuary of Osiris-Khenti-Amentiu. Plaster remains of wooden statues from the same period were also found in the aforementioned room of Magazine C. The Khufu statuette was initially headless; Petrie ascribed this damage to some form of accident during the excavation. When Petrie realised

8930-400: Was especially sure that the figure had to derive from the 4th dynasty. The main argument for dating it to the 4th dynasty is the name of Khufu on the statuette. The style of the statuette in comparison with the artworks of the same period and earlier dynasties was cited as further evidence. Rainer Stadelmann pointed out that the figurine's throne is modelled on the short-backed, cubic throne of

9025-466: Was not until Josiah Willard Gibbs that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbook Vector Analysis written by Edwin Bidwell Wilson based on Gibbs' lectures. Also during the 19th century came developments in the abstract formalism of vector spaces, with

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