Rhombus

In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.

Every rhombus is a parallelogram and a kite. A rhombus with right angles is a square.[1][2]

Rhombus
Rhombus
Two rhombi
Typequadrilateral, parallelogram, kite
Edges and vertices4
Schläfli symbol{ } + { }
Coxeter diagramCDel node 1.pngCDel sum.pngCDel node 1.png
Symmetry groupDihedral (D2), [2], (*22), order 4
Area (half the product of the diagonals)
Dual polygonrectangle
Propertiesconvex, isotoxal
Symmetries of square
The rhombus has a square as a special case, and is a special case of a kite and parallelogram.

Etymology

The word "rhombus" comes from Greek ῥόμβος (rhombos), meaning something that spins,[3] which derives from the verb ῥέμβω (rhembō), meaning "to turn round and round."[4] The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for two right circular cones sharing a common base.[5]

The surface we refer to as rhombus today is a cross section of this solid rhombus through the apex of each of the two cones.

Characterizations

A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:[6][7]

  • a parallelogram in which a diagonal bisects an interior angle
  • a parallelogram in which at least two consecutive sides are equal in length
  • a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram)
  • a quadrilateral with four sides of equal length (by definition)
  • a quadrilateral in which the diagonals are perpendicular and bisect each other
  • a quadrilateral in which each diagonal bisects two opposite interior angles
  • a quadrilateral ABCD possessing a point P in its plane such that the four triangles ABP, BCP, CDP, and DAP are all congruent[8]
  • a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point[9]

Basic properties

Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties:

The first property implies that every rhombus is a parallelogram. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel; adjacent angles are supplementary; the two diagonals bisect one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the sum of the squares of the diagonals (the parallelogram law). Thus denoting the common side as a and the diagonals as p and q, in every rhombus

Not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals (the second property) is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite. Every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus.

A rhombus is a tangential quadrilateral.[10] That is, it has an inscribed circle that is tangent to all four sides.

Area

Rhombus1
A rhombus. Each angle marked with a black dot is a right angle. The height h is the perpendicular distance between any two non-adjacent sides, which equals the diameter of the circle inscribed. The diagonals of lengths p and q are the red dotted line segments.

As for all parallelograms, the area K of a rhombus is the product of its base and its height (h). The base is simply any side length a:

The area can also be expressed as the base squared times the sine of any angle:

or in terms of the height and a vertex angle:

or as half the product of the diagonals p, q:

or as the semiperimeter times the radius of the circle inscribed in the rhombus (inradius):

Another way, in common with parallelograms, is to consider two adjacent sides as vectors, forming a bivector, so the area is the magnitude of the bivector (the magnitude of the vector product of the two vectors), which is the determinant of the two vectors' Cartesian coordinates: K = x1y2x2y1.[11]

Diagonals

The length of the diagonals p = AC and q = BD can be expressed in terms of the rhombus side a and one vertex angle α as

and

These formulas are a direct consequence of the law of cosines.

Inradius

The inradius (the radius of a circle inscribed in the rhombus), denoted by r, can be expressed in terms of the diagonals p and q as:[10]

Dual properties

The dual polygon of a rhombus is a rectangle:[12]

  • A rhombus has all sides equal, while a rectangle has all angles equal.
  • A rhombus has opposite angles equal, while a rectangle has opposite sides equal.
  • A rhombus has an inscribed circle, while a rectangle has a circumcircle.
  • A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides.
  • The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length.
  • The figure formed by joining the midpoints of the sides of a rhombus is a rectangle and vice versa.

Equation

The sides of a rhombus centered at the origin, with diagonals each falling on an axis, consist of all points (x, y) satisfying

The vertices are at and This is a special case of the superellipse, with exponent 1.

Other properties

As topological square tilings As 30-60 degree rhombille tiling
Isohedral tiling p4-55 Isohedral tiling p4-51c Rhombic star tiling
Some polyhedra with all rhombic faces
Identical rhombi Two types of rhombi
Rhombohedron Rhombicdodecahedron Rhombictriacontahedron Rhombic icosahedron Rhombic enneacontahedron
Rhombohedron Rhombic dodecahedron Rhombic triacontahedron Rhombic icosahedron Rhombic enneacontahedron

As the faces of a polyhedron

A rhombohedron is a three-dimensional figure like a cube, except that its six faces are rhombi instead of squares.

The rhombic dodecahedron is a convex polyhedron with 12 congruent rhombi as its faces.

The rhombic triacontahedron is a convex polyhedron with 30 golden rhombi (rhombi whose diagonals are in the golden ratio) as its faces.

The great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron with 30 intersecting rhombic faces.

The rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry.

The rhombic enneacontahedron is a polyhedron composed of 90 rhombic faces, with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim ones.

The trapezo-rhombic dodecahedron is a convex polyhedron with 6 rhombic and 6 trapezoidal faces.

The rhombic icosahedron is a polyhedron composed of 20 rhombic faces, of which three, four, or five meet at each vertex. It has 10 faces on the polar axis with 10 faces following the equator.

See also

References

  1. ^ Note: Euclid's original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition.
  2. ^ Weisstein, Eric W. "Square". MathWorld. inclusive usage
  3. ^ ῥόμβος, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  4. ^ ρέμβω, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  5. ^ The Origin of Rhombus
  6. ^ Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, pp. 55-56.
  7. ^ Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, p. 53.
  8. ^ Paris Pamfilos (2016), "A Characterization of the Rhombus", Forum Geometricorum 16, pp. 331–336, [1]
  9. ^ IMOmath, "26-th Brazilian Mathematical Olympiad 2004"
  10. ^ a b Weisstein, Eric W. "Rhombus". MathWorld.
  11. ^ WildLinAlg episode 4, Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube
  12. ^ de Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", Mathematical Gazette 95, March 2011, 102-107.

External links

4-5 kisrhombille

In geometry, the 4-5 kisrhombille or order-4 bisected pentagonal tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 8, and 10 triangles meeting at each vertex.

The name 4-5 kisrhombille is by Conway, seeing it as a 4-5 rhombic tiling, divided by a kis operator, adding a center point to each rhombus, and dividing into four triangles.

The image shows a Poincaré disk model projection of the hyperbolic plane.

It is labeled V4.8.10 because each right triangle face has three types of vertices: one with 4 triangles, one with 8 triangles, and one with 10 triangles.

Brill (fish)

The brill (Scophthalmus rhombus) is a species of flatfish in the turbot family (Scophthalmidae) of the order Pleuronectiformes. Brill can be found in the northeast Atlantic, Black Sea, Baltic Sea, and Mediterranean, primarily in deeper offshore waters.Brill have slender bodies, brown covered with lighter and darker coloured flecks, excluding the tailfin; the underside of the fish is usually cream coloured or pinkish white. Like other flatfish the brill has the ability to match its colour to the surroundings. Brill weigh up to 8 kg (18 lb) and can reach a length of 75 cm (2 ft 6 in), but are less than half that on average. Part of the dorsal fin of the fish is not connected to the fin membrane, giving the fish a frilly appearance. They are sometimes confused with the turbot (Scophthalmus maximus), which is more diamond-shaped. The two species are related and can produce hybrids.

On the west coast of Canada (outside the range of Scophthalmus rhombus) local fisherman refer to the petrale sole, Eopsetta jordani, as brill.

Golden rhombus

In geometry, a golden rhombus is a rhombus whose diagonals are in the ratio , where is the golden ratio.

Hotel Panorama

Hotel Panorama is located at 8A Hart Avenue, near Chatham Road South, in Tsim Sha Tsui, Hong Kong. It is managed by the Canadian Rhombus International Hotels Group, which also owns and manages LKF Hotel. It is next to the high-rise hotel Hyatt Regency Hong Kong, Tsim Sha Tsui.

John Frizzell

John B. Frizzell (born in Kingston, Ontario) is a Canadian screenwriter and film producer.

After several years writing, directing and co-producing the documentary series A Different Understanding for TVOntario, Frizzell joined partners Niv Fichman, Barbara Willis Sweete and Larry Weinstein to found the Canadian production company Rhombus Media. He left Rhombus in the mid-eighties to pursue a career in writing.

His credits include the television series Airwaves, The Rez, Twitch City, Angela Anaconda and Material World and the films A Winter Tan, Getting Married in Buffalo Jump, Life with Billy, Dance Me Outside, On My Own and Lapse of Memory. He was co-winner of a Writers Guild of Canada Award for Lucky Girl.

Kurt Rambis

Darrell Kurt Rambis (born February 25, 1958) is a Greek-American basketball coach and former player who is a senior basketball adviser for the Los Angeles Lakers of the National Basketball Association (NBA). As a player, he won four NBA championships while playing power forward for the Lakers. Rambis was a key member of the Showtime era Lakers and was extremely popular for his hard-nosed blue collar play. With his trademark black horned rim glasses, Rambis complemented the flashy Hollywood style of the Showtime era Lakers.Rambis played college basketball for the Santa Clara Broncos. As a senior in 1980, he was named the player of the year in the West Coast Conference (WCC). Rambis was selected by the New York Knicks in the third round of the 1980 NBA draft, but began his career in Greece with AEK Athens before joining the Lakers. He also played for the Charlotte Hornets, Phoenix Suns, and Sacramento Kings. Rambis became a coach, and has served as head coach for the Lakers, Minnesota Timberwolves and the Knicks. He also won two league championships as an assistant coach with the Lakers.

Lozenge

A lozenge (◊), often referred to as a diamond, is a form of rhombus. The definition of lozenge is not strictly fixed, and it is sometimes used simply as a synonym (from the French losange) for rhombus. Most often, though, lozenge refers to a thin rhombus—a rhombus with two acute and two obtuse angles, especially one with acute angles of 45°. The lozenge shape is often used in parquetry (with acute angles that are 360°/n with n being an integer higher than 4, because they can be used to form set of tiles of the same shape and size, reusable to cover the plane in various geometric patterns as the result of a tiling process called tessellation in mathematics) and as decoration on ceramics, silverware and textiles. It also features in heraldry and playing cards.

Merkel-Raute

The Merkel-Raute (German for "Merkel rhombus") is what has been termed Merkel diamond or Triangle of Power by English-speaking media: a hand gesture made by resting one's hands in front of the stomach so that the fingertips meet, with the thumbs and index fingers forming a rough quadrangular shape. This signature gesture of Angela Merkel, the current German Chancellor, has been described as "probably one of the most recognisable hand gestures in the world".Asked about how the Merkel-Raute was introduced as her trademark, Merkel stated that "there was always the question, what to do with your arms, and that's how it came about." She chose the gesture without having been assisted by a counsellor because "it contains a certain symmetry."

Penrose tiling

A Penrose tiling is an example of non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of prototiles implies that a shifted copy of a tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right.

Penrose tiling is non-periodic, which means that it lacks any translational symmetry. It is self-similar, so the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through "inflation" (or "deflation") and every finite patch from the tiling occurs infinitely many times. It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order.

Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions or subdivision rules, cut and project schemes and coverings.

Port Rhombus (EP)

Port Rhombus EP is a 1996 EP by Squarepusher. It was the first Squarepusher release on Warp Records. There is also a promo version, with identical tracks, but dated 1994. Also compiled in the US version of Big Loada on Nothing Records.

Psychonauts in the Rhombus of Ruin

Psychonauts in the Rhombus of Ruin is a first-person virtual reality adventure game developed by Double Fine Productions on the PlayStation 4 and Microsoft Windows for the HTC Vive and Oculus Rift. Released in 2017 on PS4 and on PC the following year, the game's story bridges the events between Psychonauts and Psychonauts 2.

Rectangle

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.

The word rectangle comes from the Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle).

A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a special case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.

Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons.

Rhombic Chess

Rhombic Chess is a chess variant for two players created by Tony Paletta in 1980. The gameboard has an overall hexagonal shape and comprises 72 rhombi in three alternating colors. Each player commands a full set of standard chess pieces.

The game was first published in Chess Spectrum Newsletter 2 by the inventor. It was included in World Game Review No. 10 edited by Michael Keller.

Rhomboid

Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled.

A parallelogram with sides of equal length (equilateral) is a rhombus but not a rhomboid.

A parallelogram with right angled corners is a rectangle but not a rhomboid.

The term rhomboid is now more often used for a rhombohedron or a more general parallelepiped, a solid figure with six faces in which each face is a parallelogram and pairs of opposite faces lie in parallel planes. Some crystals are formed in three-dimensional rhomboids. This solid is also sometimes called a rhombic prism. The term occurs frequently in science terminology referring to both its two- and three-dimensional meaning.

Sida rhombifolia

Sida rhombifolia (arrowleaf sida) ( Sanskrit : atibalā अतिबला ) is a perennial or sometimes annual plant in the Family Malvaceae, native to the New World tropics and subtropics. Other common names include rhombus-leaved sida, Paddy's lucerne, jelly leaf, and also somewhat confusingly as Cuban jute, Queensland-hemp, and Indian hemp (although S. rhombifolia is not related to either jute or hemp). Synonyms include Malva rhombifolia. It is used in Ayurvedic medicine, where it is known as kurumthotti.

The stems are erect to sprawling and branched, growing 50 to 120 centimeters in height, with the lower sections being woody. The dark green, diamond-shaped leaves are arranged alternately along the stem, 4 to 8 centimeters long, with petioles that are less than a third of the length of the leaves. The leaves are paler below, with short, grayish hairs. The apical half of the leaves have toothed or serrated margins while the remainder of the leaves are entire (untoothed). The petioles have small spiny stipules at their bases.

The moderately delicate flowers occur singly on flower stalks that arise from the area between the stems and leaf petioles. They consist of five petals that are 4 to 8 millimeters long, creamy to orange-yellow in color, and may be somewhat reddish in the center. Each of the five overlapping petals is asymmetric, having a long lobe on one side. The stamens unite in a short column. The fruit is a ribbed capsule, which breaks up into 8 to 10 segments. The plant blooms throughout the year.

This species is usually confined to waste ground, such as roadsides and rocky areas, stock camps or rabbit warrens, but can be competitive in pasture, because of its unpalatability to livestock.

Square

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.

Wajik

Wajik or wajid is a diamond-shaped kue or traditional snack made with steamed glutinous (sticky) rice and further cooked in palm sugar, coconut milk, and pandan leaves. The sweet sticky rice cake is commonly found in Indonesia, Malaysia and Brunei. It is called wajid in Brunei and Sabah.

In Indonesian language the term wajik is used to describe the shape of rhombus or diamond-shape, consequently in a card game, the carreaux (tiles or diamonds♦) is translated as a wajik.

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