designcoding | dodecahedron
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. Exercise: Get to know the five Platonic solids and the relationships between them . Note that there are two different ways in which 4 of the 8 cube vertices could be chosen as the 12 faces of dodecahedron = 12 vertices of icosahedron. Interesting thing is it's close relationship with Dodecahedron, you should find the “tip” point of the Icosahedron by intersecting spheres from at.
Next, create a second pentagonal pyramid.Construct Transparent. Dodecahedron relationship to Hexahedron.
Finally, bring the two layers of extra sides together - the overlap tabs of one face will embrace the non-tabbed edge of another face, and you'll have another Platonic solid for your collection.
There is an alternate way of creating the icosahedron: In this case, the units are folded tightly, and the tab of one unit nestles between the two folded-together faces of another unit. You won't need tape for this construction. Since there are 3 faces per pyramid and 20 sides, you'll have to plan for 60 faces, so you'll need 30 business cards for this solid.
Both ways of constructing the icosahedron will result in a solid with the same dimensions except for minimal differences from the thickness of the cards. Geometrically, the dodecahedron and the icosahedron are related in all sorts of interesting ways. We'll actually create a cumulated dodecahedron, because the pentagonal faces will be represented by pentagonal pyramids. Hey, doesn't the cumulated icosahedron also have 60 triangular faces? You'll need tape for this shape, and you'll need to fold the tabs in the opposite direction of the main diagonal fold.
Here's a tip when assembling these pieces: If you look at the picture, you'll see that the diagonal fold of a unit is the edge between faces of the solid. The diagonal fold does not appear inside the pentagonal pyramid.
Just something to keep in mind to make sure you're building the shape correctly. To start, build a pentagonal pyramid with 5 units. Their purpose is not certain. 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. 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.
geometry - Cleverest construction of a dodecahedron / icosahedron? - Mathematics Stack Exchange
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.
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.
Stellating the icosahedron and faceting the dodecahedron
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.
- Platonic Relationships
- Dual Polyhedron
- Compound of dodecahedron and icosahedron
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.