Compound of dodecahedron and icosahedron - Wikipedia
What they are doing is having the edges of the dual figures bisect each other. As a result, the vertices of the icosahedron and the. obey the polar relationship as the ratio of the surface area of the solid and its dual, a property first noted by Apollonius for the dodecahedron and icosahedron. A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which Relation to the regular icosahedron; Relation to the nested cube; Relation to the . The regular dodecahedron has icosahedral symmetry Ih, Coxeter group [5,3], order , with an abstract group structure of A 5 × Z2.
Some face planes are divided into regions by edges. In such planes, different edges can be selected to form different faces within the plane, to yield different facetings. Those face sets which do not, on their own, close space to form a polyhedron must be combined with another face set.
Are all such faces allowable? In A, edges n can be selected to form a star pentagon. Alternatively, two adjacent m edges and a single n edge form an obtuse isosceles triangle. Five such triangles form a kind of composite face. Is this face allowable, and why? I make no attempt here to identify or systematize such faces, or to answer this question.
Mainly for this reason, the notation used here to describe the faces of facetings is incomplete. If we create a kind of complete faceted polyhedron including every possible edge and face, then its vertex figure will show all the possible inward structures at a vertex.
I will call this complete vertex figure the faceting diagram for the original polyhedron. The possible edges are those running from one vertex to another, so the faceting diagram will show edges running to all other vertices. The possible face planes are those bounded by three or more coplanar edges.
The faceting diagram of the dodecahedron is shown in Fig 3. For reasons of clarity I have not identified the faces. These are edges s and the hemi faces J and K. We will return to these later. Full stellation diagram of the icosahedron A stellation diagram shows a face plane of a polyhedron, giving the lines of intersection with the other face planes. These lines define the edges of the various stellations. The points where several lines intersect likewise define vertices.
Here is the stellation diagram for the icosahedron as traditionally drawn, with some additional information: I have identified sets of congruent vertices as A to H and sets of congruent edges as m to q avoiding the letter o. Edges p are divided into dextro right-handed and laevo left-handed forms, shown as pd and pl respectively. Each set of vertices lies on a circle, which I have also shown. As is often done, vertices H are chopped off the drawing to make more room for the inner detail, and edge segments which extend to infinity are entirely omitted.
I have used the same identifying letter for reciprocal features throughout the diagrams, to help make this clear. From these, we find that the stellation diagram of a polyhedron is reciprocal to the faceting diagram of its dual. The stellation diagram of the icosahedron is reciprocal to the faceting diagram of the dodecahedron.
Stellating the icosahedron and faceting the dodecahedron
Just as the face diagram of a given stellation is a unique subset of the stellation diagram, so the vertex figure of its reciprocal faceting is a unique subset of the faceting diagram. Examples are illustrated in Figs 8 and 9. In the faceting diagram above Fig 3note the point S corresponding to an edge s through the centre of the dodecahedron, also note lines corresponding to 'hemi' faces J, K also through the middle. The reciprocal features of these have been completely ignored in the traditional stellation diagram, because they would have to appear infinitely far from the centre.
Not all polyhedra have finite duals. If a face passes through the centre of the reciprocating sphere, then the corresponding vertex of the dual is located at infinity and the connecting edges describe an infinitely long prism. If we accept both the principle of duality and that hemi solids are polyhedra, we must also accept polyhedra with infinite faces.
Because of this, in the dual figure to the faceting diagram, the lines of intersection between the face planes are in truth infinitely long and end at vertices located at infinity.
The traditional stellation diagram is evidently not the full reciprocal of the faceting diagram; it is incomplete. We must find a way to complete it. Instead of projecting the icosahedron in ordinary flat, 3-dimensional Euclidean space, let us project it in 4-dimensional spherical space. This is rather like drawing a polygon on a ball instead of on a sheet of paper, but with an extra dimension added. Each face, instead of being a flat plane, now lies on the surface of a sphere, and all its lines of intersection are now great circles on the sphere.
I will call a line, drawn from the centre of the polyhedron and normal to a face plane, a ray. For a given face, the centre of the sphere must lie on the associated ray, and the sphere will intersect the polyhedron at the vertices and edges of the face. The stellation diagram now comprises closed curves instead of infinite lines, indeed any polyhedron projected in this manner is now finite in size.
The diagram is in fact duplicated on the other side of the sphere, so we may conveniently describe only a single hemisphere. If the above paragraph defeats you, just imagine seeing an ordinary flat stellation diagram through a wide-angle "fish-eye" lens or reflected in a hemispherical mirror. The end result is the same. So now, here is the new stellation diagram, showing the previously missing outer detail. I have in turn had to project the hemisphere onto the plane of this page, which has bent all the lines.
The traditional diagram is just the limiting case as the hemisphere's radius increases to infinity. I have numbered the outermost vertices within their sets for identification, as J1, J2, etc.
In 20th-century artdodecahedra appear in the work of M. Eschersuch as his lithographs Reptiles and Gravitation Gerard Caris based his entire artistic oeuvre on the regular dodecahedron and the pentagon, which is presented as a new art movement coined as Pentagonism. In modern role-playing gamesthe regular dodecahedron is often used as a twelve-sided die, one of the more common polyhedral dice. Some quasicrystals have dodecahedral shape see figure.
Some regular crystals such as garnet and diamond are also said to exhibit "dodecahedral" habitbut this statement actually refers to the rhombic dodecahedron shape. It is based on regular dodecahedron.
Compound of dodecahedron and icosahedron
The Megaminx twisty puzzle, alongside its larger and smaller order analogues, is in the shape of a regular dodecahedron. In the children's novel The Phantom Tollbooththe regular dodecahedron appears as a character in the land of Mathematics. Each of his faces wears a different expression — e.
- Dual Polyhedron
- Regular dodecahedron
- Platonic Relationships
Dodecahedron is the name of an avant-garde black metal band from Netherlands. This was proposed by Jean-Pierre Luminet and colleagues in  and an optimal orientation on the sky for the model was estimated in