Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art of M. C. Escher. Tessellations are seen throughout art history, from ancient architecture to modern art.

Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons.We can't show the entire plane, but imagine that these are pieces taken from planes that have been tiled. Here are examples of

a tessellation of triangles

a tessellation of squares

a tessellation of hexagons

When you look at these three samples you can easily notice that the squares are lined up with each other while the triangles and hexagons are not. Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other - a slide (or a glide!) is involved.

You can work out the interior measure of the angles for each of these polygons:

Shapetrianglesquarepentagonhexagonmore than six sides

   

Angle measure in degrees6090

108120

more than 120 degrees

Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane.

Semi-regular Tessellations

You can also use a variety of regular polygons to make semi-regular tessellations. A semiregular tessellation has two properties which are:

1. It is formed by regular polygons.2. The arrangement of polygons at every vertex point is identical.

Here are the eight semi-regular tessellations:

   

   

 

Part one 2D tessellation

1. Shape using pentagon

2. Fill with gray colour and a flower is adding into the shape.

3. The picture is reflected downward.

4. The picture is reflected again to the right hand side.

5. Then, the picture is rotated in 90°, 180°, 240° and finally 360°

6. The picture is repeated downward and reflected to right hand side as shown in below.

Explaination:

1. The operations that I used to create this tessellation are rotation, reflection and translation.

2. The basic of this tessellation is a square shape with drawing inside.

3. To create this tessellation, this basic is reflected vertical and horizontal and rotated 180°

4. Then this big square which builds by four small squares translates to the right and bottom.

5. Finally this tessellation become to the other big square.

Polyhedral

A polyhedron (plural polyhedra or polyhedrons) is often defined as a geometric solid with flat faces and straight edges (the word polyhedron comes from the Classical Greek πολύεδρον, from poly-, stem of πολύς, "many," + -edron, form of εδρον, "base", "seat", or "face").

Any polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions:

3 dimensions: The body is bounded by the faces, and is usually the volume enclosed by them.

2 dimensions: A face is a polygon bounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedral surface.

1 dimension: An edge joins one vertex to another and one face to another, and is usually a line segment. The edges together make up the polyhedral skeleton.

0 dimensions: A vertex (plural vertices) is a corner point.

-1 dimension: The nullity is a kind of non-entity required by abstract theories.

More generally in mathematics and other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely algebraic or abstract.

A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.

A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions.

Naming polyhedra

Polyhedra are often named according to the number of faces. The naming system is again based on Classical Greek, for example tetrahedron (4), pentahedron (5), hexahedron (6), heptahedron (7), triacontahedron (30), and so on.

Often this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. the Pentagonal dodecahedron.

Other common names indicate that some operation has been performed on a simpler polyhedron, for example the truncated cube looks like a cube with its corners cut off, and has 14 faces (so it is also an example of a tetrakaidecahedron).

Some special polyhedra have grown their own names over the years, such as Miller's monster or the Szilassi polyhedron.

Edges

Edges have two important characteristics (unless the polyhedron is complex):

An edge joins just two vertices. An edge joins just two faces.

These two characteristics are dual to each other.

Euler characteristic

The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:

χ = V - E + F.

For a simply connected polyhedron, χ = 2. For a detailed discussion, see Proofs and Refutations by Imre Lakatos.

Duality

For every polyhedron there is a dual polyhedron having faces in place of the original's vertices and vice versa. For a convex polyhedron the dual can be obtained by the process of polar reciprocation.

Vertex figure

For every vertex one can define a vertex figure consisting of the vertices joined to it. The vertex is said to be regular if this is a regular polygon and symmetrical with respect to the whole polyhedron.

[edit] Traditional polyhedra

A dodecahedron

In geometry, a polyhedron is traditionally a three-dimensional shape that is made up of a finite number of polygonal faces which are parts of planes; the faces meet in pairs along edges which are straight-line segments, and the edges meet in points called vertices. Cubes, prisms and pyramids are examples of polyhedra. The polyhedron surrounds a bounded volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges.

A polyhedron is said to be convex if its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface.

[edit] Symmetrical polyhedra

Many of the most studied polyhedra are highly symmetrical.

Of course it is easy to distort such polyhedra so they are no longer symmetrical. But where a polyhedral name is given, such as icosidodecahedron, the most symmetrical geometry is almost always implied, unless otherwise stated.

Some of the most common names in particular are often used with "regular" in front or implied because for each there are different types which have little in common except for having the same number of faces. These are the triangular pyramid or tetrahedron, cube or hexahedron, octahedron, dodecahedron and icosahedron:

Polyhedra of the highest symmetries have all of some kind of element - faces, edges and/or vertices, within a single symmetry orbit. There are various classes of such polyhedra:

Isogonal or Vertex-transitive if all vertices are the same, in the sense that for any two vertices there exists a symmetry of the polyhedron mapping the first isometrically onto the second.

Isotoxal or Edge-transitive if all edges are the same, in the sense that for any two edges there exists a symmetry of the polyhedron mapping the first isometrically onto the second.

Isohedral or Face-transitive if all faces are the same, in the sense that for any two faces there exists a symmetry of the polyhedron mapping the first isometrically onto the second.

Regular if it is vertex-transitive, edge-transitive and face-transitive (this implies that every face is the same regular polygon; it also implies that every vertex is regular).

Quasi-regular if it is vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive. A quasi-regular dual is face-transitive and edge-transitive (and hence every vertex is regular) but not vertex-transitive.

Semi-regular if it is vertex-transitive but not edge-transitive, and every face is a regular polygon. (This is one of several definitions of the term, depending on author. Some definitions overlap with the quasi-regular class). A semi-regular dual is face-transitive but not vertex-transitive, and every vertex is regular.

Uniform if it is vertex-transitive and every face is a regular polygon, i.e. it is regular, quasi-regular or semi-regular. A uniform dual is face-transitive and has regular vertices, but is not necessarily vertex-transitive).

Noble if it is face-transitive and vertex-transitive (but not necessarily edge-transitive). The regular polyhedra are also noble; they are the only noble uniform polyhedra.

A polyhedron can belong to the same overall symmetry group as one of higher symmetry, but will have several groups of elements (for example faces) in different symmetry orbits.

Some Polyhedra Dodecahedron

(Regular polyhedron) Small stellated dodecahedron

(Regular star) Icosidodecahedron

(Uniform) Great cubicuboctahedron

(Uniform star) Rhombic triacontahedron

(Uniform dual) Elongated pentagonal cupola

(Convex regular-faced) Octagonal prism

(Uniform prism) Square antiprism(Uniform antiprism)

Dodecahedron

A general dodecahedron is a polyhedron having 12 faces. Examples include the decagonal prism, elongated square dipyramid (Johnson solid ), hexagonal dipyramid, metabidiminished icosahedron ( ), pentagonal antiprism, pentagonal cupola ( ), (regular) dodecahedron, rhombic dodecahedron, snub disphenoid ( ), triakis tetrahedron, and undecagonal pyramid. Crystals of pyrite ( ) resemble slightly distorted dodecahedra (Steinhaus 1999, pp. 207-208), and sphalerite (ZnS) crystals are irregular dodecahedra bounded by congruent deltoids (Steinhaus 1999, pp. 207 and 209). The hexagonal scalenohedron is another irregular dodecahedron.

The regular dodecahedron is the Platonic solid composed of 20 polyhedron vertices, 30 polyhedron edges, and 12 pentagonal faces, . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and the Wythoff symbol .

There are 43380 distinct nets for the dodecahedron, the same number as for the icosahedron (Bouzette and Vandamme, Hippenmeyer 1979, Buekenhout and Parker 1998). Questions of polyhedron coloring of the dodecahedron can be addressed using the Pólya enumeration theorem.

The image above shows an origami dodecahedron constructed using six dodecahedron units, each consisting of a single sheet of paper (Kasahara and Takahama 1987, pp. 86-87).

A dodecahedron appears as part of the staircase being ascending by alligator-like lizards in Escher's 1943 lithograph "Reptiles" (Bool et al. 1982, p. 284; Forty 2003, Plate 32). Two dodecahedra also appear as polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43). The IPV pod that transports Ellie Arroway (Jodi Foster) through a network of wormholes in the 1997 film Contact was enclosed in a dodecahedral framework.

A 40-foot high sculpture (Nath 1999) known as Eclipse is displayed in the Hyatt Regency Hotel in San Francisco. It was constructed by Charles Perry, and is composed of pieces of anodized aluminum tubes and assembled over a period of four months (Kraeuter 1999). The layered sculpture begins with a regular dodecahedron, but each face then rotates outward. At the midpoint of the rotation, it forms an icosidodecahedron. Then, as the 12 pentagons continue to rotate outward, it forms a small rhombicosidodecahedron.

Dodecahedra were known to the Greeks, and 90 models of dodecahedra with knobbed vertices have been found in a number of archaeological excavations in Europe dating from the Gallo-Roman period in locations ranging from military camps to public bath houses to treasure chests (Schuur).

The dodecahedron has the icosahedral group of symmetries. The connectivity of the vertices is given by the dodecahedral graph. There are three dodecahedron stellations.

The dual polyhedron of a dodecahedron with unit edge lengths is an icosahedron with edge lengths , where is the golden ratio. As a result, the centers of the faces of an icosahedron form a dodecahedron, and vice versa (Steinhaus 1999, pp. 199-201).

A plane perpendicular to a axis of a dodecahedron cuts the solid in a regular hexagonal cross section (Holden 1991, p. 27). A plane perpendicular to a axis of a dodecahedron cuts the solid in a regular decagonal cross section (Holden 1991, p. 24).

A cube can be constructed from the dodecahedron's vertices taken eight at a time (above left figure; Steinhaus 1999, pp. 198-199; Wells 1991). Five such cubes can be constructed, forming the cube 5-compound. In addition, joining the centers of the faces gives three mutually perpendicular golden rectangles (right figure; Wells 1991).

The short diagonals of the faces of the rhombic triacontahedron give the edges of a dodecahedron (Steinhaus 1999, pp. 209-210).

The following table gives polyhedra which can be constructed by cumulation of a dodecahedron by pyramids of given heights .

result

60-faced dimpled deltahedron

pentakis dodecahedron

60-faced star deltahedron

small stellated dodecahedron

When the dodecahedron with edge length is oriented with two opposite faces parallel to the -plane, the vertices of the top and bottom faces lie at and the other polyhedron vertices lie at , where is the golden ratio. The explicit coordinates are

(1)

(2)

with , 1, ..., 4, where is the golden ratio.

Eight dodecahedra can be place in a closed ring, as illustrated above (Kabai 2002, pp. 177-178).

The polyhedron vertices of a dodecahedron can be given in a simple form for a dodecahedron of

side length by (0, , ), ( , 0, ), ( , , 0), and ( , , ).

For a dodecahedron of unit edge length , the circumradius and inradius of a pentagonal face are

(3)

(4)

The sagitta is then given by

(5)

Now consider the following figure.

Using the Pythagorean theorem on the figure then gives

(6)

(7)

(8)

Equation (8) can be written

(9)

Solving (6), (7), and (9) simultaneously gives

(10) (11) (12)

The inradius of the dodecahedron is then given by

(13)

so

(14)

and solving for gives

(15)

Now,

(16)

so the circumradius is

(17)

The midradius is given by

(18)

so

(19)

The dihedral angle is

(20)

The area of a single face is the area of a pentagon of unit edge length

(21)

so the surface area is 12 times this value, namely

(22)

The volume of the dodecahedron can be computed by summing the volume of the 12 constituent pentagonal pyramids,

(23)

Apollonius showed that for an icosahedron and a dodecahedron with the same inradius,

(24)

where is the volume and the surface area, with the actual ratio being

Icosahedrons

Icosahedron

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In geometry, an icosahedron (Greek: εικοσάεδρον, from eikosi twenty + hedron seat; pronounced /ˌa ɪ k ɵ səˈhi ː drən/ or /a ɪ ˌk ɒ səˈhi ː drən/ ; plural: -drons, -dra /-drə/) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces.

The regular icosahedron is one of the five Platonic solids. It is a convex regular polyhedron composed of twenty triangular faces, with five meeting at each of the twelve vertices. It has 30 edges and 12 vertices. Its dual polyhedron is the dodecahedron.

It has 43,380 nets. If one were to colour the icosahedron such that no two adjacent faces had the same colour, one would need to use 3 colours.

If the original icosahedron has edge length 1, its dual

dodecahedron has edge length , one divided by the golden ratio.

Related polyhedra and polytopes

The icosahedron can be transformed by a truncation sequence into its dual, the dodecahedron:

Picture

Icosahedron

Truncated

icosahed

Icosidodecahedron

Truncated

dodecah

Dodecahedron

Regular Icosahedron

(Click here for rotating model)

Type Platonic solid

ElementsF = 20, E = 30V = 12 (χ = 2)

Faces by sides 20{3}

Schläfli symbol {3,5} and s{3,3}

Wythoff symbol5 | 2 3| 3 3 2

Coxeter-Dynkin

Symmetry Ih (*532)

References U22, C25, W4

PropertiesRegular convex deltahedron

Dihedral angle 138.189685°

3.3.3.3.3 Dodecahedron

ron edronCoxeter-Dynkin

The icosahedron shares its vertex arrangement with three Kepler-Poinsot solids. The great dodecahedron also has the same edge arrangement.

Picture

Great dodecahedron Small stellated dodecahedron Great icosahedron

Coxeter-Dynkin

The icosahedron can tessellate hyperbolic space in the order-3 icosahedral honeycomb, with 3 icosahedra around each edge, 12 icosahedra around each vertex, with Schläfli symbol {3,5,3}. It is one of four regular tessellations in the hyperbolic 3-space.

The Five Platonic Solids Tetrahedron Cube

Tetrahedron Cube(or Hexahedron)

Octahedron Dodecahedron Icosahedron

http://mathworld.wolfram.com/Dodecahedron.html