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51-471: A cross is a compound geometrical figure consisting of two intersecting lines , usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of the Latin letter X , is termed a saltire in heraldic terminology. The cross has been widely recognized as a symbol of Christianity from an early period in that religion's history. Before then, it

102-450: A " p " have a different shape, at least when they are constrained to move within a two-dimensional space like the page on which they are written. Even though they have the same size, there's no way to perfectly superimpose them by translating and rotating them along the page. Similarly, within a three-dimensional space, a right hand and a left hand have a different shape, even if they are the mirror images of each other. Shapes may change if

153-410: A coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle. A described shape has external lines that you can see and make up the shape. If you were putting your coordinates on a coordinate graph you could draw lines to show where you can see a shape, however not every time you put coordinates in a graph as such you can make a shape. This shape has

204-405: A mirror is the same shape as the original, and not a distinct shape. Many two-dimensional geometric shapes can be defined by a set of points or vertices and lines connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called polygons and include triangles , squares , and pentagons . Other shapes may be bounded by curves such as the circle or

255-599: A outline and boundary so you can see it and is not just regular dots on a regular paper. The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of statistical shape analysis . In particular, Procrustes analysis is a technique used for comparing shapes of similar objects (e.g. bones of different animals), or measuring the deformation of a deformable object. Other methods are designed to work with non-rigid (bendable) objects, e.g. for posture independent shape retrieval (see for example Spectral shape analysis ). All similar triangles have

306-421: A reflection of each other, and hence they are congruent and similar, but in some contexts they are not regarded as having the same shape. Sometimes, only the outline or external boundary of the object is considered to determine its shape. For instance, a hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to determine whether or not two objects have

357-428: A set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is an equivalence relation , and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of subsets of a Euclidean space having the same shape. Mathematician and statistician David George Kendall writes: In this paper ‘shape’

408-435: A shape defined by n − 2 complex numbers S ( z j , z j + 1 , z j + 2 ) ,   j = 1 , . . . , n − 2. {\displaystyle S(z_{j},z_{j+1},z_{j+2}),\ j=1,...,n-2.} The polygon bounds a convex set when all these shape components have imaginary components of the same sign. Human vision relies on

459-488: A simple cross-shaped character, consisting of a horizontal and a vertical wedge ( 𒈦 ), read as maš "tax, yield, interest"; the superposition of two diagonal wedges results in a decussate cross ( 𒉽 ), read as pap "first, pre-eminent" (the superposition of these two types of crosses results in the eight-pointed star used as the sign for "sky" or "deity" ( 𒀭 ), DINGIR ). The cuneiform script has other, more complex, cruciform characters, consisting of an arrangement of boxes or

510-712: A simplification of a ligature for et "and" (introduced by Johannes Widmann in the late 15th century). The letter Aleph is cross-shaped in Aramaic and paleo-Hebrew . Egyptian hieroglyphs with cross-shapes include Gardiner Z9 – Z11 ("crossed sticks", "crossed planks"). Other, unrelated cross-shaped letters include Brahmi ka (predecessor of the Devanagari letter क) and Old Turkic (Orkhon) d² and Old Hungarian b , and Katakana ナ na and メ me . The multiplication sign (×), often attributed to William Oughtred (who first used it in an appendix to

561-570: A symbol in Buddhism , Jainism and Hinduism , and widely popular in the early 20th century as a symbol of good luck or prosperity before adopted as a symbol of Nazism in the 1920s and 30s. Cross shapes are made by a variety of physical gestures . Crossing the fingers of one hand is a common invocation of the symbol. The sign of the cross associated with Christian genuflection is made with one hand: in Eastern Orthodox tradition

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612-440: A triangle. The shape of a quadrilateral is associated with two complex numbers p , q . If the quadrilateral has vertices u , v , w , x , then p = S( u , v , w ) and q = S( v , w , x ) . Artzy proves these propositions about quadrilateral shapes: A polygon ( z 1 , z 2 , . . . z n ) {\displaystyle (z_{1},z_{2},...z_{n})} has

663-476: A wide range of shape representations. Some psychologists have theorized that humans mentally break down images into simple geometric shapes (e.g., cones and spheres) called geons . Meanwhile, others have suggested shapes are decomposed into features or dimensions that describe the way shapes tend to vary, like their segmentability , compactness and spikiness . When comparing shape similarity, however, at least 22 independent dimensions are needed to account for

714-403: Is by homeomorphisms . Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated mathematical joke is that topologists cannot tell their coffee cup from their donut, since a sufficiently pliable donut could be reshaped to the form of

765-611: Is divided into smaller categories; triangles can be equilateral , isosceles , obtuse , acute , scalene , etc. while quadrilaterals can be rectangles , rhombi , trapezoids , squares , etc. Other common shapes are points , lines , planes , and conic sections such as ellipses , circles , and parabolas . Among the most common 3-dimensional shapes are polyhedra , which are shapes with flat faces; ellipsoids , which are egg-shaped or sphere-shaped objects; cylinders ; and cones . If an object falls into one of these categories exactly or even approximately, we can use it to describe

816-480: Is preserved when one of the objects is uniformly scaled, while congruence is not. Thus, congruent objects are always geometrically similar, but similar objects may not be congruent, as they may have different size. A more flexible definition of shape takes into consideration the fact that realistic shapes are often deformable, e.g. a person in different postures, a tree bending in the wind or a hand with different finger positions. One way of modeling non-rigid movements

867-477: Is therefore congruent to its mirror image (even if it is not symmetric), but not to a scaled version. Two congruent objects always have either the same shape or mirror image shapes, and have the same size. Objects that have the same shape or mirror image shapes are called geometrically similar , whether or not they have the same size. Thus, objects that can be transformed into each other by rigid transformations, mirroring, and uniform scaling are similar. Similarity

918-406: Is used in the vulgar sense, and means what one would normally expect it to mean. [...] We here define ‘shape’ informally as ‘all the geometrical information that remains when location, scale and rotational effects are filtered out from an object.’ Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above. In particular, the shape does not depend on

969-698: The 2nd century AD to succeeding Ichthys in aftermaths of that new religion's separation from Judaism . Clement of Alexandria in the early 3rd century calls it τὸ κυριακὸν σημεῖον ("the Lord's sign") he repeats the idea, current as early as the Epistle of Barnabas , that the number 318 (in Greek numerals , ΤΙΗ) in Genesis 14:14 was a foreshadowing (a "type") of the cross (the letter Tau) and of Jesus (the letters Iota Eta ). Clement's contemporary Tertullian rejects

1020-729: The European Bronze Age the cross symbol appeared to carry a religious meaning , perhaps as a symbol of consecration, especially pertaining to burial. The cross sign occurs trivially in tally marks , and develops into a number symbol independently in the Roman numerals (X "ten"), the Chinese rod numerals ( 十 "ten") and the Brahmi numerals ("four", whence the numeral 4 ). In the Phoenician alphabet and derived scripts ,

1071-538: The Patriarchal cross (☦), Cross of Lorraine (☨) and Cross potent (☩, mistakenly labeled a " Cross of Jerusalem "). The following is a list of cross symbols, except for variants of the Christian cross and Heraldic crosses , for which see the dedicated lists at Christian cross variants and Crosses in heraldry , respectively. The swastika or crux gammata (in heraldry fylfot ), historically used as

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1122-446: The cardinal points , or the unity of a vertical axis mundi or celestial pole with the horizontal world (Koch, 1955). Speculation of this kind became especially popular in the mid- to late-19th century in the context of comparative mythology seeking to tie Christian mythology to ancient cosmological myths . Influential works in this vein included G. de Mortillet (1866), L. Müller (1865), W. W. Blake (1888), Ansault (1891), etc. In

1173-1596: The complex plane , z ↦ a z + b , a ≠ 0 , {\displaystyle z\mapsto az+b,\quad a\neq 0,}   a triangle is transformed but does not change its shape. Hence shape is an invariant of affine geometry . The shape p = S( u , v , w ) depends on the order of the arguments of function S, but permutations lead to related values. For instance, 1 − p = 1 − u − w u − v = w − v u − v = v − w v − u = S ( v , u , w ) . {\displaystyle 1-p=1-{\frac {u-w}{u-v}}={\frac {w-v}{u-v}}={\frac {v-w}{v-u}}=S(v,u,w).} Also p − 1 = S ( u , w , v ) . {\displaystyle p^{-1}=S(u,w,v).} Combining these permutations gives S ( v , w , u ) = ( 1 − p ) − 1 . {\displaystyle S(v,w,u)=(1-p)^{-1}.} Furthermore, p ( 1 − p ) − 1 = S ( u , v , w ) S ( v , w , u ) = u − w v − w = S ( w , v , u ) . {\displaystyle p(1-p)^{-1}=S(u,v,w)S(v,w,u)={\frac {u-w}{v-w}}=S(w,v,u).} These relations are "conversion rules" for shape of

1224-415: The ellipse . Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensional faces enclosed by those lines, as well as the resulting interior points. Such shapes are called polyhedrons and include cubes as well as pyramids such as tetrahedrons . Other three-dimensional shapes may be bounded by curved surfaces, such as the ellipsoid and

1275-620: The shape of triangle ( u , v , w ) . Then the shape of the equilateral triangle is 0 − 1 + i 3 2 0 − 1 = 1 + i 3 2 = cos ⁡ ( 60 ∘ ) + i sin ⁡ ( 60 ∘ ) = e i π / 3 . {\displaystyle {\frac {0-{\frac {1+i{\sqrt {3}}}{2}}}{0-1}}={\frac {1+i{\sqrt {3}}}{2}}=\cos(60^{\circ })+i\sin(60^{\circ })=e^{i\pi /3}.} For any affine transformation of

1326-404: The sphere . A shape is said to be convex if all of the points on a line segment between any two of its points are also part of the shape. There are multiple ways to compare the shapes of two objects: Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the letters " b " and " d " are

1377-596: The 10th century. A wide variation of cross symbols is introduced for the purposes of heraldry beginning in the age of the Crusades . The cross mark is used to mark a position, or as a check mark , but also to mark deletion . Derived from Greek Chi are the Latin letter X , Cyrillic Kha and possibly runic Gyfu . Egyptian hieroglyphs involving cross shapes include ankh "life", ndj "protect" and nfr "good; pleasant, beautiful". Sumerian cuneiform had

1428-509: The 1618 edition of John Napier's Descriptio ) apparently had been in occasional use since the mid 16th century. Other typographical symbols resembling crosses include the dagger or obelus (†), the Chinese ( 十 , Kangxi radical 24 ) and Roman (X ten). Unicode has a variety of cross symbols in the " Dingbat " block (U+2700–U+27BF): The Miscellaneous Symbols block (U+2626 to U+262F) adds three specific Christian cross variants , viz.

1479-602: The Latin crux (or its accusative crucem and its genitive crucis ), "stake, cross". The English verb to cross arises from the noun c.  1200 , first in the sense "to make the sign of the cross"; the generic meaning "to intersect" develops in the 15th century. The Latin word was influenced by popular etymology by a native Germanic word reconstructed as * krukjo (English crook , Old English crycce , Old Norse krokr , Old High German krucka ). This word, by conflation with Latin crux , gave rise to Old French crocier (modern French crosse ),

1530-570: The Roman world, furca replaced crux as the name of some cross-like instruments for lethal and temporary punishment, ranging from a forked cross to a gibbet or gallows . The field of etymology is of no help in any effort to trace a supposed original meaning of crux . A crux can be of various shapes: from a single beam used for impaling or suspending ( crux simplex ) to the various composite kinds of cross ( crux compacta ) made from more beams than one. The latter shapes include not only

1581-465: The accusation that Christians are crucis religiosi (i.e. "adorers of the gibbet"), and returns the accusation by likening the worship of pagan idols to the worship of poles or stakes. In his book De Corona , written in 204, Tertullian tells how it was already a tradition for Christians to trace repeatedly on their foreheads the sign of the cross . While early Christians used the T-shape to represent

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1632-620: The beginning of the Upper Paleolithic , and throughout prehistory to the Iron Age . Also of prehistoric age are numerous variants of the simple cross mark, including the crux gammata with curving or angular lines, and the Egyptian crux ansata with a loop. Speculation has associated the cross symbol – even in the prehistoric period – with astronomical or cosmological symbology involving " four elements " (Chevalier, 1997) or

1683-627: The cross in writing and gesture, the use of the Greek cross and Latin cross , i.e. crosses with intersecting beams, appears in Christian art towards the end of Late Antiquity . An early example of the cruciform halo , used to identify Christ in paintings, is found in the Miracles of the Loaves and Fishes mosaic of Sant'Apollinare Nuovo , Ravenna (6th century). The Patriarchal cross , a Latin cross with an additional horizontal bar, first appears in

1734-475: The cross symbol represented the phoneme /t/, i.e. the letter taw , which is the historical predecessor of Latin T . The letter name taw means "mark", presumably continuing the Egyptian hieroglyph "two crossed sticks" ( Gardiner Z9 ). The shape of the cross ( crux , stauros "stake, gibbet "), as represented by the Latin letter T , came to be used as a new symbol (seal) or emblem of Christianity since

1785-479: The fourfold arrangement of other characters, including the archaic cuneiform characters LAK -210, LAK-276, LAK-278, LAK-617 and the classical sign EZEN (𒂡). Phoenician tāw is still cross-shaped in Paleo-Hebrew alphabet and in some Old Italic scripts ( Raetic and Lepontic ), and its descendant T becomes again cross-shaped in the Latin minuscule t . The plus sign (+) is derived from Latin t via

1836-613: The gospels and always in the plural number to indicate a stake or pole. From the first century BC, it is used to indicate an instrument used in executions. The Greek word is used in descriptions in antiquity of the execution cross , which indicate that its normal shape was similar to the Greek letter tau ( Τ ). Due to the simplicity of the design (two intersecting lines), cross-shaped incisions make their appearance from deep prehistory; as petroglyphs in European cult caves, dating back to

1887-402: The naming convention of the Greek derived prefix with '-gon' suffix: Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon... See polygon In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations , rotations (together also called rigid transformations ), and uniform scalings . In other words, the shape of

1938-590: The number 10 ( 十 ) in Chinese number gestures . Unicode provides various cross symbols: Geometrical figure A plane shape or plane figure is constrained to lie on a plane , in contrast to solid 3D shapes. A two-dimensional shape or two-dimensional figure (also: 2D shape or 2D figure ) may lie on a more general curved surface (a two-dimensional space ). Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles , quadrilaterals , pentagons , etc. Each of these

1989-448: The object is scaled non-uniformly. For example, a sphere becomes an ellipsoid when scaled differently in the vertical and horizontal directions. In other words, preserving axes of symmetry (if they exist) is important for preserving shapes. Also, shape is determined by only the outer boundary of an object. Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are congruent . An object

2040-407: The physical world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by differential geometry , or as fractals . Some common shapes include: Circle , Square , Triangle , Rectangle , Oval , Star (polygon) , Rhombus , Semicircle . Regular polygons starting at pentagon follow

2091-461: The sacred marriage, as in Egyptian amulet Nefer with male cross and female orb, considered as an amulet of blessedness, a charm of sexual harmony. The word cross is recorded in 11th-century Old English as cros , exclusively for the instrument of Christ's crucifixion , replacing the native Old English word rood . The word's history is complicated; it appears to have entered English from Old Irish , possibly via Old Norse , ultimately from

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2142-422: The same shape, or to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same. Simple shapes can often be classified into basic geometric objects such as a line , a curve , a plane , a plane figure (e.g. square or circle ), or a solid figure (e.g. cube or sphere ). However, most shapes occurring in

2193-482: The same shape. These shapes can be classified using complex numbers u , v , w for the vertices, in a method advanced by J.A. Lester and Rafael Artzy . For example, an equilateral triangle can be expressed by the complex numbers 0, 1, (1 + i√3)/2 representing its vertices. Lester and Artzy call the ratio S ( u , v , w ) = u − w u − v {\displaystyle S(u,v,w)={\frac {u-w}{u-v}}}

2244-584: The sequence is head-heart-right shoulder-left shoulder, while in Oriental Orthodox, Catholic and Anglican tradition the sequence is head-heart-left-right. Crossing the index fingers of both hands represents and a charm against evil in European folklore. Other gestures involving more than one hand include the "cross my heart" movement associated with making a promise and the Tau shape of the referee 's "time out" hand signal. Crossed index fingers represent

2295-454: The shape of the object. Thus, we say that the shape of a manhole cover is a disk , because it is approximately the same geometric object as an actual geometric disk. A geometric shape consists of the geometric information which remains when location , scale , orientation and reflection are removed from the description of a geometric object . That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in

2346-403: The size and placement in space of the object. For instance, a " d " and a " p " have the same shape, as they can be perfectly superimposed if the " d " is translated to the right by a given distance, rotated upside down and magnified by a given factor (see Procrustes superimposition for details). However, a mirror image could be called a different shape. For instance, a " b " and

2397-471: The system of Greek numerals , it has a value of 300. The name in English is pronounced / t aʊ / or / t ɔː / , but in Greek it is [taf] . This is because the pronunciation of the combination of Greek letters αυ can have the pronunciation of either [ai] , [av] or [af] , depending on what follows and if a diaeresis is present on the second vowel (see Greek orthography ). Tau

2448-471: The term for a shepherd's crook , adopted in English as crosier . Latin crux referred to the gibbet where criminals were executed, a stake or pole, with or without transom , on which the condemned were impaled or hanged, but more particularly a cross or the pole of a carriage. The derived verb cruciāre means "to put to death on the cross" or, more frequently, "to put to the rack, to torture, torment", especially in reference to mental troubles. In

2499-528: The traditional †-shaped cross (the crux immissa ), but also the T-shaped cross (the crux commissa or tau cross ), which the descriptions in antiquity of the execution cross indicate as the normal form in use at that time, and the X-shaped cross (the crux decussata or saltire ). The Greek equivalent of Latin crux "stake, gibbet" is stauros , found in texts of four centuries or more before

2550-444: The way natural shapes vary. There is also clear evidence that shapes guide human attention . Tau Tau ( / ˈ t aʊ , ˈ t ɔː , ˈ t ɒ / ; uppercase Τ , lowercase τ or τ {\displaystyle {\boldsymbol {\tau }}} ; Greek : ταυ [taf] ) is the nineteenth letter of the Greek alphabet , representing the voiceless dental or alveolar plosive IPA: [t] . In

2601-532: Was used as a religious or cultural symbol throughout Europe, in western and south Asia (the latter, in the form of the original Swastika ); and in Egypt, where the Ankh was a hieroglyph that represented "life" and was used in the worship of the god Aten . The effigy of a man hanging on a cross was set up in the fields to protect the crops. It often appeared in conjunction with the female-genital circle or oval, to signify

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