Regular Polyhedra
The following two models are of regular polyhedra. They are
aranged so that they fit inside one anouther and share either
vertexes or the midpoints of an edge.
Terms
- Edge - A line segment. Edges connect vertexes and make up
the sides of a face.
- Vertex - A point. Vertexes are at the ends of edges and
at the corners of faces.
- Face - A plane. Faces have sides made of edges.
- Eulers Formula : vertexes + faces - edges = 2 . This is
true for all convex polyhedra .
- Regular polyhedra :
- All faces are the same shape
- All edges are the same length
- The number of edges at each vertex is the same
- It is possible to prove with simple alegebra that
there are only 5 regular polyhedra in 3
dimensions. It gets even more interesting in 4
dimensions.
- Regular polyhedra are also called regular
polytopes or platonic solids. Doing a search on
the web for any of these terms can find lots of
interesting stuff.
- Its amusing that the Greeks and the Pythagoreans
were real nuts about this stuff and integrated it
into their philosophy. Certain shapes represented
elements in nature such as fire, earth, water,
etc.
| Model Color |
Name |
Faces |
Vertexes |
Edges |
Edges At AVertex |
Edges On A Face |
| Blue |
Dodecahedron |
12 |
20 |
30 |
3 |
5 |
| Red |
Cube |
6 |
8 |
12 |
3 |
4 |
| Yellow |
Tetrahedron |
4 |
4 |
6 |
3 |
3 |
| Model Color |
Name |
Faces |
Vertexes |
Edges |
Edges At AVertex |
Edges On A Face |
| Blue |
Icosahedron |
20 |
12 |
30 |
5 |
3 |
| Red |
Octahedron |
8 |
6 |
12 |
4 |
3 |
| Yellow |
Tetrahedron |
4 |
4 |
6 |
3 |
3 |
- It is possible to fit 5 differnet Cubes in a Dodecahedron
in the way shown below. One edge is always on a face and
there are 5 ways of fitting a line inside of a petagon.
- It is possible to fit 5 differnet Octahedrons in a
Icosahedron in the way shown below.
- There are other ways to fit the shapes inside of one
anouther. For example some of the faces of an Octahedron
are parallel with the inside faces of a tetrahedron. Some
of the faces of an icosahedron are parallel with the the
inside of an octahedron. Be carefull it is possible to
strain your brain imagining how these things fit
toegether.
There is a nice shareware program called Poly that allows you
to view regular polyhedra and other shapes in differnet ways at http://www.peda.com/poly/Welcome.html
.
The following models were made with string, plastic straws,
and hot melt glue.
