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The following polyhedra are neither regular (Platonic) nor semi-regular (Archimedean), but are very common all around us.
Let's start with pyramids. A pyramid is a structure in which all of the outer surfaces (excluding the base) are triangular and converge at a single point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape. The following are two examples:
The left one is completely irregular, while the right one has a rectangular base and two different couples of isosceles triangles.
Let's now limit out attention to those pyramids that have regular polygon faces only (that is, regular polygons bases and only equilateral triangle sides):
The left one has an equilateral triangle base: as the side faces are equilateral triangles too, this pyramid is also a Tetrahedron (one of the five Platonic, regular polyhedra). The right one has a squared base, so that it's very similar to Egyptian pyramids.
Now observe that it's only possible to add one single edge to the base polygon if we want the resulting polyhedra have only equilateral triangle lateral faces:
The left one is a regular pentagonal pyramid. But trying with an hexagonal base we get... a flat pyramid (the one shown above on the right): so the only regular pyramids can be built are those with triangular, square and pentagonal bases.
The polyhedra can be built with only regular polygons are called "Johnson Solids", after Norman Johnson, who in 1966 published a list which included 92 solids. They are strictly convex polyhedron and each face of which is a regular polygon; prisms and antiprisms (see below), Platonic and Archimedean solids are excluded. So only two kind of pyramids are also Johnson Solids: those which have square and pentagonal bases (as already stated, the one with the triangular base is a Tetrahedron, which is a Platonic solid).
Pyramids can be built with more than five sided bases (on the left the base is an hexagon, on the right a decagon); but the side faces can't be any longer equilateral triangles: they are isosceles instead.
A bipyramid is is a polyhedron formed by joining a simple pyramid and its mirror image base-to-base. Below the only two possible bipyramids that are also Johnson Solids (a stated before, with all the faces that are regular polygons):
The following are not Johnson Solids:
In fact the left one is made up by eight equilateral triangle faces, so is an Octahedron (another of the Platonic solids), while the right one has isosceles triangular faces.
Prisms. In geometry, a prism is a polyhedron with an polygonal base, a translated copy (not in the same plane as the first), and some edges joining corresponding vertices of the two bases. All cross-sections parallel to the base faces are the same.
The following are the simplest prism series:
They have regular polygonal bases and square sides: from left to right, the bases are triangular, squared, pentagonal, and heptagonal. This series of solids can grow up to infinite sides, and this is the reason these polyhedra have been excluded from the Jonson solids list. Note that the squared base prism simply is... a cube: in fact it has two bases and four faces that all are squares.
The following are two other kind of prisms: the left one is a rectangular cuboid, while the right one is a triangular prism with rectangular (not squared) faces.
The base of a prism can also be a not regular polygon: the model below on the left is an example that has squared faces (a prism can have also different length base edges, with or without one or more squared faces).
The model above on the right is a prism too, even it has two faces are not rectangular. So the prisms family has a lot of different shape solids... but be careful: not all seems to be a prism actually is!
The model above on the left has to squared bases perfectly aligned the one with respect the other, but the edges don't connect the corresponding vertices: the lateral faces are skew polygons, that is they don't lie in a flat plane, but zigzags in three dimensions. The model above on the right doesn't show this problem (the face edges are flat), but the cross-sections parallel to the base faces are not all the same (at mid height the section is a square).
A final definition about prisms: the Parallelepipeds are those with six parallel faces (as the cube or, in general, the rectangular cuboids we saw above). Let's see the following model:
This model of Prism (or more precisely of Parallelepiped) with six rhombic faces is a Rhombohedron: I show it in two versions, where the right one has yellow panels installed to better show the shape of the faces.
Antiprisms are polyhedra composed of two parallel copies of some regular polygon, connected by an alternating band of triangles. Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals. Here are two examples, with squared and pentagonal bases:
A particular case is the triangular antiprism, which has a total of eight triangles. Being all these triangles equilateral, the polyhedron actually is an Octahedron (a Platonic solid). I show it with horizontal bases (on the left) and raised up to look similar to what we normally imagine to be an octahedron:
As for the prisms, also antiprisms can grow up to infinite sides (actually both prisms and antiprisms are excluded from the Jonson solids list).
Two models more:
The left one is an antiprism where the triangular faces are not equilateral. The right model has the bases aren't regular polygons, so it can be easily seen the two bases are not identical each other: in fact, this IS NOT an antiprism!
As a conclusion of this article I'd like to collect some information about Platonic Solids.
- Tetrahedron: also an triangular pyramid
- Hexaedron (Cube): also a square prism
- Octahedron: also a square bypyramid, or a triangular antiprism
- Icosahedron: also the sum of a pentagonal bipyramid and a pentagonal antiprism
- Dodecahedron: unfortunately there is not a direct connection with this solid and the polyhedra classes described here.
So here is the list of all the Pyramd and Prism related polyhedra:
1 - Pyramid
2 - Bipyramid
3 - Prism
4 - Antiprism
Article written by Guest Blogger Aldo Cavini (aka aldoaldoz)
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