Rectangle
From Wikipedia, the free encyclopedia
| Rectangle | |
|---|---|
 |
|
| Type | Quadrilateral |
| Edges and vertices | 4 |
| Schläfli symbol | {}x{} |
| Symmetry group | D2 (*2) |
| Coxeter-Dynkin diagram |   |
| Dual polygon | Rhombus |
| Properties | convex, isogonal, cyclic |
In Euclidean geometry, the term rectangle normally refers to a quadrilateral with four right angles. This is a simple rectangle.
A rectangle that is not simple is complex, but more clearly described as self-intersecting or crossed. It is defined as a self-intersecting quadrilateral with the same vertex arrangement as a simple rectangle.
In recreational mathematics a popular subject is the tiling of rectangles by polygons, ranging from simple puzzles to unsolved problems.
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[edit] Properties
- All angles are 90 degrees.
- Opposite sides are equal in length.
- Opposite sides are parallel.
- Diagonals are equal in length and bisect each other.
[edit] Related polygons
- A rectangle is a cyclic polygon.
- The dual polygon of a rectangle is a rhombus.
- When the length is equal to the width, the rectangle is a square.
- A rectangle is a special case of a parallelogram, which has two pairs of parallel opposite sides. A parallelogram, and hence also a rectangle, is a special case of a trapezium (known as a trapezoid in North America), which has at least one pair of parallel opposite sides.
[edit] Area, perimeter, and other facts
If a rectangle has length l and width w
.The term oblong is occasionally used to refer to a non-square rectangle. [1][2]
Two rectangles, neither of which will fit inside the other, are said to be incomparable.
[edit] Tessellations
The rectangle is used in many periodic tessellation patterns, in brickwork, for example, these isogonal tilings:
 Stacked bond |
 Running bond |
 Basket weave |
 Basket weave |
 Herringbone pattern |
[edit] Crossed rectangle
A crossed rectangle is a complex (self-intersecting) rectangle, also called a bow-tie rectangle or butterfly rectangle.
It has the same vertex arrangement as a simple rectangle with which it shares two edges. Its other two edges are the diagonals of the simple rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.
The interior of a crossed rectangle can have a polygon density of +/-1 in each half triangle, dependent upon the winding orientation as clockwise or counterclockwise.
[edit] Squared, perfect, and other tiled rectangles
A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangled" (or "triangulated") rectangle respectively. The tiled rectangle is perfect[3][4] if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is imperfect. In a perfect (or imperfect) triangled rectangle the triangles must be right triangles.
A rectangle has commensurable sides if and only if it is tilable by a finite number of unequal squares.[5][3] The same is true if the tiles are unequal isosceles right triangles.
The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent polyaboloes.
[edit] See also
[edit] References
- ^ http://www.mathsisfun.com/definitions/oblong.html
- ^ http://www.icoachmath.com/SiteMap/Oblong.html
- ^ a b R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte (1940). "The dissection of rectangles into squares". Duke Math. J. 7 (1): 312–340. doi:. http://projecteuclid.org/euclid.dmj/1077492259.
- ^ J.D. Skinner II, C.A.B. Smith and W.T. Tutte (November 2000). "On the Dissection of Rectangles into Right-Angled Isosceles Triangles". J. Combinatorial Theory Series B 80 (2): 277–319. doi:.
- ^ R. Sprague (1940). "Ũber die Zerlegung von Rechtecken in lauter verschiedene Quadrate". J. fũr die reine und angewandte Mathematik 182: 60–64.
[edit] External links
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Wikimedia Commons has media related to category: Rectangles |
- Weisstein, Eric W., "Rectangle" from MathWorld.
- Definition and properties of a rectangle With interactive animation
- Area of a rectangle with interactive animation
