What are polyhedra

Convex polyhedron

Definition: A geometric body is called convex, if with two points that belong to it, the line between these points also belongs completely to this body.

This excludes all bodies that contain "holes" or "dents".

Definition: A polyhedron (Polyhedron) is a geometric body, the surface of which consists of flat polygons.

This excludes all bodies that contain curved edges or surfaces, in particular spheres, cones and cylinders.

For convex polyhedra the applies Euler's polyhedron substitute ( Leonhard Euler):

Sentence: If f denotes the number of faces, k the number of edges and e the number of corners of a convex polyhedron, then the following applies

From this theorem useful conclusions can be drawn that can be applied to the description of convex polyhedra with certain properties.

Is called as Angular defect in a corner of a convex polyhedron is the difference between the full circle, that is, and the sum of all angles in the corners of those surfaces that meet in this polyhedron corner, then the following also applies Descartean formula (Rene Descartes):

Sentence: The sum of the angular defects of all corners of a convex polyhedron is

If one combines both results, then the sum S of the angle defects of any convex polyhedron always applies

Among the convex polyhedra are in particular:

the regular polyhedra,

the semi-regular polyhedra,

the quasi-regular polyhedra,

the deltahedron.

The deltahedra belong to the class of 92 Johnson solids. These are all non-regular and non-semi-regular convex polyhedra that can be formed from regular polygons.

Further important subclasses of convex polyhedra, which are formed from less regular polygons, are

the diamond bodies,

the zonahedra,

the parallelohedron.

There are other, non-convex regular polyhedra, the Kepler-Poinsot star bodies.

further reading

  • Peter R. Cromwell, Polyhedra, Cambridge University Press, Cambridge, 1997. ISBN 0-521-55432-2
  • Tiberiu Roman, regular and semi-regular polyhedra, VEB Deutscher Verlag der Wissenschaften, Berlin, 1987. ISBN 3-326-00192-4
  • Paul Adam, Arnold Wyss, Platonic and Archimedean bodies, their star shapes and polar structures, Verlag Freies Geistesleben, Stuttgart, 1994. ISBN 3-7725-0965-7
  • Magnus J. Wenninger, Polyhedron Models, Cambridge University Press, Cambridge, 1989. ISBN 0-521-09859-9
  • Magnus J. Wenninger, Dual Models, Cambridge University Press, Cambridge, 1983. ISBN 0-521-24524-9
  • Alan Holden, Shapes, Space and Symmetry, Dover Publ., New York, 1991. ISBN 0-486-26851-9