Welcome to my home-page !!!

 

 

Here you will find an extract of my collection of paper three-dimensional solids for you to explore: visit my collection page at http://www.polyhedra.it/collection.htm

After experimenting, in the past few years, I have chosen these paper made polyhedra for this web-site. Each solid is unique in terms of number of faces and color.

 

 

Here below you will find some basic information on polyhedra: visit my home-page at http://www.polyhedra.it/

 

 

A polyhedron (plural polyhedra or sometimes polyhedrons) is a three-dimensional solid which consists of a collection of polygons, usually joined at their edges.

Each face is a polygon: a flat shape with straight sides.

Cubes and pyramids are examples of polyhedra.

There are no curved surfaces; cones, spheres and cylinders are not polyhedrons.

 

The word  derives from the Greek poly - meaning "many" - plus the Indo-European hedron - meaning “face” or “seat”.

Polyhedra can be built up from different kinds of element, each associated with a different number of dimensions:

·                          3 dimensions: The interior is the volume bounded by the faces. It might or might not be realised as a solid body.

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

·                          1 dimension: An edge E 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 V (plural vertices) is a corner point.

 

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.

 

A convex polyhedron has flat polygonal faces, sharp corners or vertices and straight edges.

If we count the number of faces F, edges E, and vertices V of a convex polyhedron we verify Euler's Formula (or the "Polyhedra Formula"):

 

the number of faces plus the number of vertices minus the number of edges equals 2

or

F + V − E = 2

 

F= number of faces or flat surfaces

V= number of vertices or corner points

E= number of edges

In general, it is defined the Euler characteristic χ= F + V – E

For a convex polyhedron or more generally any simply connected polyhedron, χ = 2

For more complicated shapes, the Euler characteristic relates to the number of toroidal holes, handles and/or cross-caps in the surface and will be less than 2 (χ < 2).

Leonhard Euler's discovery of the characteristic, which bears his name, marked the beginning of the modern discipline of topology.

 

 

The first printed illustration of a rhombicuboctahedron, drawn by Leonardo da Vinci to illustrate a book by Luca Pacioli with title “De divina proportione”. Leonardo da Vinci drew the illustrations of the regular solids in “De divina proportione” (written in Milan in 1496–98, published in Venice in 1509) while he lived with and took mathematics lessons from Pacioli. Leonardo's drawings are probably the first illustrations of skeletonic solids, which allowed an easy distinction between front and back.

 

 

Polyhedra may be classified and are often named according to the number of faces F. The naming system is based on Classical Greek. The following table lists the name given to some polyhedra of F number of faces.

F

Polyhedron

F

Polyhedron

4

tetrahedron

12

dodecahedron

5

pentahedron

14

tetradecahedron

6

hexahedron

20

icosahedron

7

heptahedron

24

icositetrahedron

8

octahedron

30

triacontahedron

9

nonahedron

32

icosidodecahedron

10

decahedron

60

hexecontahedron

11

undecahedron

90

enneacontahedron

 

For every polyhedron there exists a dual polyhedron having:

·         faces in place of the original's vertices and vice versa,

·         the same number of edges

·         the same orientability and Euler characteristic χ= F + V − E

Dual polyhedra exist in pairs. The dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.

 

Many of the most studied polyhedra are highly symmetrical.

A symmetrical polyhedron can be rotated and superimposed on its original position such that its faces have changed position. All the elements which can be superimposed on each other in this way are said to lie in a given symmetry orbit.

Regular polyhedra are the most highly symmetrical. Altogether there are nine regular polyhedra. The five convex examples have been known since antiquity and are called the Platonic solids. These are the triangular pyramid or tetrahedron, cube (regular hexahedron), octahedron, dodecahedron andicosahedron.

There are also four regular star polyhedra, known as the Kepler-Poinsot polyhedra after their discoverers. The dual of a regular polyhedron is also regular.

 

There are several types of highly symmetric polyhedra, classified by which kind of element (V, E, F) belong to a single symmetry orbit:

·         Regular polyhedra are V-E-F transitive (i.e. 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 polyhedra are V-E transitive (i.e. vertex-transitive and edge-transitive) and hence has regular faces but not face-transitive. A quasi-regular dual polyhedron is face-transitive and edge-transitive and hence every vertex is regular but not vertex-transitive.

·         Semi-regular polyhedra are V transitive (i.e. vertex-transitive) but not edge-transitive, and every face is a regular polygon. These polyhedra include the semiregular prisms and antiprisms. A semi-regular dual polyhedron is face-transitive but not vertex-transitive, and every vertex is regular.

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

·         Noble polyhedra are V-F transitive (i.e. vertex-transitive and face-transitive) but not necessarily edge-transitive. The regular polyhedra are also noble; they are the only noble uniform polyhedra.

 

 

Please feel free to write me an e-mail: einardinima@gmail.com

 

 Web pages in English:

Visit my home-page in English: http://www.polyhedra.it/

Visit my collection-page in English: http://www.polyhedra.it/collection.htm

Some information about myself in English: http://www.polyhedra.it/author.htm

 

 Pagine web in Italiano:

Visita la mia pagina web in Italiano: http://www.polyhedra.it/indice.htm

Visita la pagina della mia collezione in Italiano http://www.polyhedra.it/collezione.htm

Alcune informazione su di me in Italiano http://www.polyhedra.it/autore.htm

 

 

 

 

I plan to update this web-site in the next future, possibly with the help of some friends.

This page was created on 04/10/2016.

This page was updated on 06/10/2016.