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Rhomboid

From Wikipedia, the free encyclopedia
Rhomboid
A rhomboid is a parallelogram with two edge lengths and no right angles
Typequadrilateral, trapezium
Edges and vertices4
Symmetry groupC2, [2]+,
Areab × h (base × height);
ab sin θ (product of adjacent sides and sine of the vertex angle determined by them)
Propertiesconvex

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

The terms "rhomboid" and "parallelogram" are often erroneously conflated with each other (i.e, when most people refer to a "parallelogram" they almost always mean a rhomboid, a specific subtype of parallelogram); however, while all rhomboids are parallelograms, not all parallelograms are rhomboids.

A parallelogram with sides of equal length (equilateral) is called a rhombus but not a rhomboid. A parallelogram with right angled corners is a rectangle but not a rhomboid.

History

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Euclid introduced the term in his Elements in Book I, Definition 22,

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

— Translation from the page of D.E. Joyce, Dept. Math. & Comp. Sci., Clark University [1]

Euclid never used the definition of rhomboid again and introduced the word parallelogram in Proposition 34 of Book I; "In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas." Heath suggests that rhomboid was an older term already in use.

Symmetries

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The rhomboid has no line of symmetry, but it has rotational symmetry of order 2.

Occurrence

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In biology

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In biology, rhomboid may describe a geometric rhomboid (e.g. the rhomboid muscles) or a bilaterally-symmetrical kite-shaped or diamond-shaped outline, as in leaves or cephalopod fins.[1]

In medicine

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In a type of arthritis called pseudogout, crystals of calcium pyrophosphate dihydrate accumulate in the joint, causing inflammation. Aspiration of the joint fluid reveals rhomboid-shaped crystals under a microscope.

In anatomy, rhomboid-shaped muscles include the rhomboid major muscle and the rhomboid minor muscle.

References

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  1. ^ "Decapodiform Fin Shapes".
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