CPS

Part 2 – Platonic Solids

8. Introduction - Platonic Solids And CPS 9. The in CPS 10. The in CPS 11. The in CPS 12. The in CPS 13. The in CPS

Nick Trif

Ottawa, Ontario, Canada – 2018 www.platonicstructures.com CPS Geometry Part 2 – Platonic Solids – 13: The Dodecahedron from

YouTube: https://youtu.be/EGKENm2re4o As we have done, for the previous platonic solids, let's have a look first at the dodecahedron in CPS - as the proof of its existence as a perfect body. As we have done, for the previous platonic solids, let's have a look first at the dodecahedron in CPS - as the proof of its existence as a perfect body. As we have done, for the previous platonic solids, let's have a look first at the dodecahedron in CPS - as the proof of its existence as a perfect body. Let's have a look only to the outside layers of nodes that create a dodecahedron. Let's have a look only to the outside layers of nodes that create a dodecahedron. This helps us see though details that are critical to the understanding of this pattern. Let's have a look only to the outside layers of nodes that create a dodecahedron. This helps us see though details that are critical to the understanding of this pattern. Let's have a look only to the outside layers of nodes that create a dodecahedron. This helps us see though details that are critical to the understanding of this pattern. The icosahedron and dodecahedron are super-patterns. The icosahedron and dodecahedron are super-patterns. The icosahedron and dodecahedron are super-patterns. To come into existence these patterns need a relative large number of spheres in the CPS arrangement. The icosahedron and dodecahedron are super-patterns. To come into existence these patterns need a relative large number of spheres in the CPS arrangement. The icosahedron and dodecahedron are super-patterns. To come into existence these patterns need a relative large number of spheres in the CPS arrangement. The icosahedron and dodecahedron are super-patterns. To come into existence these patterns need a relative large number of spheres in the CPS arrangement. The icosahedron and dodecahedron are super-patterns. To come into existence these patterns need a relative large number of spheres in the CPS arrangement. The icosahedron and dodecahedron are super-patterns. To come into existence these patterns need a relative large number of spheres in the CPS arrangement. The icosahedron and dodecahedron are super-patterns. To come into existence these patterns need a relative large number of spheres in the CPS arrangement. For example, the dodecahedron pattern shown here, as a platonic structure, requires 10122 spheres and 56424 struts. For example, the dodecahedron pattern shown here, as a platonic structure, requires 10122 spheres and 56424 struts. As we can see, it gets difficult to display an object with such a big number of components. The easiest way to uncover the dodecahedron pattern in the CPS is to start from the icosahedron pattern, and use the fact that, the icosahedron and dodecahedron are dual. Let's have a look at the vertices of the icosahedron. All twelve vertices are identical. As the size of the icosahedron increases, the vertices are farther and farther away. All twelve vertices are identical. As the size of the icosahedron increases, the vertices are farther and farther away. It also helps to see, the vertices of four nested , in one view. It also helps to see, the vertices of four nested icosahedrons, in one view. Also, let's have a short look at the faces of the four nested icosahedrons. It also helps to see, the vertices of four nested icosahedrons, in one view. Also, let's have a short look at the faces of the four nested icosahedrons. Only the faces with the hexagonal lattice are shown. The edges of the icosahedron can be imagined as the connections between the centers of the dodecahedron's 12 faces. Inversing this, the faces of the dodecahedron develop around the vertices of the icosahedron. Inversing this, the faces of the dodecahedron develop around the vertices of the icosahedron. The intersections of these surfaces define the edges of the dodecahedron. Pentagonal faces are found as bases of the pentagonal pyramids, obtained by cutting the icosahedron with planes, under the corresponding angles. Pentagonal faces are found as bases of the pentagonal pyramids, obtained by cutting the icosahedron with planes, under the corresponding angles. To assemble the dodecahedron, these 12 faces are translated, in the right direction, and with the right amount, such that the center of each of them ends up in a of the icosahedron. To assemble the dodecahedron, these 12 faces are translated, in the right direction, and with the right amount, such that the center of each of them ends up in a vertex of the icosahedron. Please note that no rotation is required. Once this is done, the dodecahedron pattern reveals itself for one to contemplate and admire. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. The first few facts about the dodecahedron structure one notices are: 1. All the faces of the dodecahedron have the same underlining lattice. All twelve faces develop around the twelve identical vertices of the icosahedron. The first few facts about the dodecahedron structure one notices are: 1. All the faces of the dodecahedron have the same underlining lattice. All twelve faces develop around the twelve identical vertices of the icosahedron. 2. The underlining lattice of the pentagonal faces is not homogeneous; the pattern of the lattice is different along different directions. We will say more about this later in these videos. 2. The underlining lattice of the pentagonal faces is not homogeneous; the pattern of the lattice is different along different directions. We will say more about this later in these videos. 3. The dodecahedron has two types of vertices. This is the consequence of the way three pentagonal faces join to form a vertex. 3. The dodecahedron has two types of vertices. This is the consequence of the way three pentagonal faces join to form a vertex. The holium-magnesium-zinc quasi- is a quasi-crystal in the shape of a . This quasi-crystal has faces that are true regular . The holium-magnesium-zinc quasi-crystal is a quasi-crystal in the shape of a regular dodecahedron. This quasi-crystal has faces that are true regular pentagons. The process of constructing a dodecahedron from the icosahedron has also produced the great stelled dodecahedron structure. The process of constructing a dodecahedron from the icosahedron has also produced the great stelled dodecahedron structure. This again is an amazing fact; one needs a lot of imagination to see this pattern in the CPS. This object encapsulates very well the properties and characteristics of both: the icosahedron and dodecahedron. To see this great object in CPS, one has to re-use the same twelve surfaces created from the icosahedron. If you want to grasp the beauty of this object, I suggest you take the time and assemble it as a 3-D Platonic Structure. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. Our website provides detailed instructions of how to do this – see: www.platonicstructures.com Our website provides detailed instructions of how to do this – see: www.platonicstructures.com www.platonicstructures.com

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