Коцка

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Коцка

Коцка (правилни хексаедар, од грч. hexáedron - тело са шест површина) је један од пет правилних полиедара. Омеђена је са шест страница, квадратних површи спојених тако да образују тело са дванаест дужи (ивица) и осам темена. Коцка је специјалан случај квадра коме су све странице једнаке. Посебне врсте коцке за играње јесу коцкице и Рубикова коцка.

Уопштење

Коцка K у простору Rⁿ се може дефинисати помоћу једне тачке A = (a₁, ..., a_n) из Rⁿ, дужине ивице коцке a, као и са n вектора v₁, ..., v_n који чине једну позитивно оријентисану ортонормирану базу Rⁿ. Рецимо да је свака ивица коцке K паралелна са тачно једним различитим вектором те базе, као и да тачка A представља почетак координатног система кога граде ови вектори.

Свака тачка X = (x₁, ..., x_n) коцке K онда може бити представљена на следећи начин:

$X:A+\sum _{k=1}^{n}{\alpha _{k}\cdot v_{k}},\;\alpha _{k}\in [0,a]$

Уколико се за векторе v₁, ..., v_n узму вектори који чине канонску базу Rⁿ, добија се:

$X:\{x_{i}\in [a_{i},a_{i}+a],\;i=1,...,n\}$

Формуле

Следе неке од чешће коришћених формула које се везују за коцку.

Важнији елементи коцке

Површина	$P=6a^{2}\,$
Запремина	$V=a^{3}\,$
Мала дијагонала	$d_{1}=a{\sqrt {2}}$
Велика дијагонала	$D=a{\sqrt {3}}$
Полупречник уписане сфере	$r_{u}={\frac {a}{2}}$
Полупречник описане сфере	$r_{o}={\frac {d_{2}}{2}}=a{\frac {\sqrt {3}}{2}}$

Ортогоналне пројекције

The cube has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A₂ and B₂ Coxeter planes.

Orthogonal projections

Centered by	Face	Vertex
Coxeter planes	B2	A2
Projective symmetry	[4]	[6]
Tilted views

Сферно поплочавање

The cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Orthographic projection	Stereographic projection

Декартове координате

For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are

(±1, ±1, ±1)

while the interior consists of all points (x₀, x₁, x₂) with −1 < x_i < 1 for all i.

Уједначене боје и симетрија

Octahedral symmetry tree

The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123.

The cube has four classes of symmetry, which can be represented by vertex-transitive coloring the faces. The highest octahedral symmetry O_h has all the faces the same color. The dihedral symmetry D_4h comes from the cube being a solid, with all the six sides being different colors. The prismatic subsets D_2d has the same coloring as the previous one and D_2h has alternating colors for its sides for a total of three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol.

Name	Regular hexahedron	Square prism	Rectangular trapezoprism	Rectangular cuboid	Rhombic prism	Trigonal trapezohedron
Coxeter diagram
Schläfli symbol	{4,3}	{4}×{ } rr{4,2}	s2{2,4}	{ }3 tr{2,2}	{ }×2{ }
Wythoff symbol	3 | 4 2	4 2 | 2		2 2 2 |
Symmetry	Oh [4,3] (*432)	D4h [4,2] (*422)	D2d [4,2+] (2*2)	D2h [2,2] (*222)		D3d [6,2+] (2*3)
Symmetry order	24	16	8	8		12
Image (uniform coloring)	(111)	(112)	(112)	(123)	(112)	(111), (112)

Геометријски односи

The 11 nets of the cube.

These familiar six-sided dice are cube-shaped.

A cube has eleven nets (one shown above): that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the same color, one would need at least three colors.

The cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry).

The cube can be cut into six identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained (with pairs of coplanar triangles combined into rhombic faces).

Повезани полиедри

The dual of a cube is an octahedron, seen here with vertices at the center of the cube's square faces.

The hemicube is the 2-to-1 quotient of the cube.

The quotient of the cube by the antipodal map yields a projective polyhedron, the hemicube.

If the original cube has edge length 1, its dual polyhedron (an octahedron) has edge length $\scriptstyle {\sqrt {2}}/2$.

The cube is a special case in various classes of general polyhedra:

Name	Equal edge-lengths?	Equal angles?	Right angles?
Cube	Yes	Yes	Yes
Rhombohedron	Yes	Yes	No
Cuboid	No	Yes	Yes
Parallelepiped	No	Yes	No
quadrilaterally faced hexahedron	No	No	No

The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron; more generally this is referred to as a demicube. These two together form a regular compound, the stella octangula. The intersection of the two forms a regular octahedron. The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself; the other symmetries of the cube map the two to each other.

In uniform honeycombs and polychora

It is an element of 9 of 28 convex uniform honeycombs:

Cubic honeycomb	Truncated square prismatic honeycomb	Snub square prismatic honeycomb	Elongated triangular prismatic honeycomb	Gyroelongated triangular prismatic honeycomb
Cantellated cubic honeycomb	Cantitruncated cubic honeycomb	Runcitruncated cubic honeycomb	Runcinated alternated cubic honeycomb

It is also an element of five four-dimensional uniform polychora:

Tesseract	Cantellated 16-cell	Runcinated tesseract	Cantitruncated 16-cell	Runcitruncated 16-cell

Литература

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Спољашње везе

• Weisstein, Eric W. „Cube”. MathWorld. 
• Cube: Interactive Polyhedron Model*
• Volume of a cube, with interactive animation
• Cube (Robert Webb's site)