Mathematics and OrigamiIrregular polyhedra do not openly conform with to equality conditions of regular polyhedra.
Neither do the pseudorregular, but not so openly: they may have even a regular appearance. Later on
we shall study two of these: the rhombic-dodecahedron and the trapezohedron.
As for stellate polyhedra, we shall devote special chapters only to the regular ones.
A polyhedron is named convex when the plane containing any of its faces leaves the whole
polyhedron in one of its two hemispaces. Conversely it is a concave one if its volume is scattered
within the two hemispaces. The kneading-trough we ́ll see here after is one example of a concave ir-
regular polyhedron.
The conventional –and simplest- classification of polyhedra is: regular and irregular, the prism
and pyramid being part of the latter.
A prismatoid is an irregular polyhedron bounded by two polygons (bases) situated in parallel
planes, and several lateral faces shaped as triangles or trapeziums (in case their four sides were copla-
nar). If both bases have the same number of sides, the prismatoid becomes a prismoid.
If one of the bases of a prismatoid is reduced to a point, the prismatoid becomes a pyramid.
Therefore a pyramid is a polyhedron whose base may be any polygon; its lateral faces (as many as the
sides of the base) are triangles that meet at the vertex forming there a polyhedral angle.
The pyramid may be named triangular, quadrangular, etc. when its base is a triangle, a quad-
rilateral, etc. Moreover, if the base is a regular polygon and the lateral faces are congruent, we have a
regular pyramid.
A prism is like a prismoid with equal bases; the other faces (lateral) should be parallelograms;
their sides will belong to the bases or to the lateral faces.
A prism is named right if it has its lateral sides at right angles with the base; if not, it is an
oblique prism.
A prism will be triangular, quadrangular, etc. according to the polygon of its base. If such a
base consists of a regular polygon, we ́ll have a regular prism. The bases of an irregular prism are
irregular polygons.
A prism is named parallelepiped if its bases are parallelograms: in other words, it will have 6
faces such that each pair of the opposite ones are equal and parallel. A right parallelepiped derives
from a right prism.
A rectangular parallelepiped is a right one whose base is a rectangle.
When a plane oblique or parallel to the base cuts a pyramid, two solids are obtained: a small,
new pyramid and a frustum.
If a prism is cut off by a plane oblique to its base we get two truncated prisms.
Two regular polyhedra are conjugate if the number of faces (or vertices) in one is equal to the
number of vertices (or faces) in the other: cube and octahedron; pentagon-dodecahedron and icosahe-
dron.
We shall show the mathematical background of the solids to be studied here on. Paperfolding
will also play its roll through folding schemes that convey to ultimate figures, also shown in perspec-
tive. Folding does not produce interlocked assemblies; on the contrary, it requires the help of glue or
sticking paper (transparent) to fix the union laps (not always shown).
18.1 A KNEADING-TROUGH
It is an example of an irregular concave polyhedron that fulfils Euler ́s theorem: C = 18; V =
20; A = 36: 18 + 20 = 36 + 2
Fig. 1 is a perspective view with its two orthogonal transversal sections.
The design may be taken as a model to construct a kneading-trough made out of five equally
thick boards: the base and four lateral faces. Those are represented in the terminated figure by five
empty virtual boards of paper.