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.