The Icosahedron from Spheres

Part 2 – Platonic Solids

8. Introduction - Platonic Solids And CPS 9. The Tetrahedron in CPS 10. The Octahedron in CPS 11. The Cube in CPS 12. The Icosahedron in CPS 13. The Dodecahedron in CPS

Nick Trif

Ottawa, Ontario, Canada – 2018 www.platonicstructures.com CPS Geometry Part 2 – Platonic Solids – 12: The Icosahedron from Spheres

YouTube: https://youtu.be/7aBUxnYcQCg As we have seen in the previous videos, the tetrahedron, octahedron and cube can be easily assembled in CPS. How about the icosahedron? The larger the number the spheres one uses, the more precise platonic solid one gets. The larger the number the spheres one uses, the more precise platonic solid one gets. For these three solids, even a relatively small number of spheres, will give a very well defined platonic solid. The icosahedron does not follow the same rule.

To get a reasonable icosahedron shape, one needs a large number of spheres. Let's see two such icosahedrons. First presented here requires 892 spheres and the second one involves 5282 spheres. When more and more spheres are used, arranged in the same pattern, a more perfect icosahedron start to emerge. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. Only the shell of the icosahedron will be shown. We want to make sure the necessary details of the icosahedron's structure are clearly seen. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. Only the shell of the icosahedron will be shown. We want to make sure the necessary details of the icosahedron's structure are clearly seen. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. Only the shell of the icosahedron will be shown. We want to make sure the necessary details of the icosahedron's structure are clearly seen. The structures are presented as Platonic Structures first, and as combination of surfaces next. The structures are presented as platonic structures first, and as combination of surfaces next. The structures are presented as platonic structures first, and as combination of surfaces next. The structures are presented as platonic structures first, and as combination of surfaces next. The structures are presented as platonic structures first, and as combination of surfaces next. As expected, all faces of the icosahedron are equilateral triangles; this makes this solid an icosahedron. As expected, all faces of the icosahedron are equilateral triangles; this makes this solid an icosahedron. As expected, all faces of the icosahedron are equilateral triangles; this makes this solid an icosahedron. A preliminary investigation of the pattern reveals that these triangular faces have two underlining patterns. As expected, all faces of the icosahedron are equilateral triangles; this makes this solid an icosahedron. A preliminary investigation of the pattern reveals that these triangular faces have two underlining patterns. Let's concentrate first on the eight faces with the hexagonal pattern; we have already seen this pattern in tetrahedron. As expected, all faces of the icosahedron are equilateral triangles; this makes this solid an icosahedron. A preliminary investigation of the pattern reveals that these triangular faces have two underlining patterns. Let's concentrate first on the eight faces with the hexagonal pattern; we have already seen this pattern in tetrahedron. As expected, all faces of the icosahedron are equilateral triangles; this makes this solid an icosahedron. A preliminary investigation of the pattern reveals that these triangular faces have two underlining patterns. Let's concentrate first on the eight faces with the hexagonal pattern; we have already seen this pattern in tetrahedron. In CPS, the twelve directions, from a sphere to its neighbors, do not allow the existence of the five identical plane surfaces meeting in the vertex of the icosahedron. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. If one wants to proceed with the search for icosahedron in CPS, one must allow different types of underlining lattices for the five surfaces meeting in a vertex. Next, we can add the other 12 faces. All these faces also have the same underlining pattern, or lattice. This new pattern is specific to icosahedron and is different from both the hexagonal pattern or the square pattern seen before. The intersections of lines in this model are the nodes of the platonic structure. The lines are the rods. The icosahedron shell, presented next, can be built as a platonic structure using 6226 nodes and 20380 struts.

These videos and the 3-D models on our web site cannot show the full beauty of this structure. The only way to really appreciate this structure, in its entire splendor, is to build it as a real 3-D platonic structure.

I can promise you that you will not be disappointed. The only way to really appreciate this structure, in its entire splendor, is to build it as a real 3-D platonic structure.

I can promise you that you will not be disappointed. It is worth having a closer look to the vertices of the icosahedron.

All twelve vertices are identical. The nodes involve in defining these nodes are connected in exactly the same way. There are five surfaces meeting in each vertex - two based on a hexagonal lattice and three based on a new type of lattice.

We should get familiar with these connections and try to remember them. It is quite remarkable that the icosahedron can be assembled from identical spheres. This fact should strengthen our conviction that, the points in space, are distributed in a CPS arrangement. In the next slide, more and more similar facts will solidify this explanation. To start to understand this pattern, we will use four icosahedrons that can be assembled using larger and larger numbers of spheres. www.platonicstructures.com

Beauty makes beautiful things beautiful!