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Truncated Cube from Spheres CPS Geometry Part 4 – Archimedean Solids 18. The Cuboctahedron in CPS 19. Truncated Tetrahedron in CPS 20. Truncated Octahedron in CPS 21. Truncated Cube in CPS 22. Truncated Icosahedron in CPS Nick Trif Ottawa, Ontario, Canada – 2018 www.platonicstructures.com CPS Geometry Part 4 – Archimedean Solids – 21: Truncated Cube from Spheres YouTube: https://youtu.be/1kJ6yEC-QcQ In the classical geometry, a truncated cube is generated from a cube using the same approach used for generating the truncated tetrahedron or the truncated octahedron. In the classical geometry, a truncated cube is generated from a cube using the same approach used for generating the truncated tetrahedron or the truncated octahedron. 1. Divide each edge of the cube in three parts; In the classical geometry, a truncated cube is generated from a cube using the same approach used for generating the truncated tetrahedron or the truncated octahedron. 1. Divide each edge of the cube in three parts; 2. Cut the vertexes of the cube with planes determined by the divisions done in the first step. In the case of the cube, the division will not have to generate three equal parts. Applying simple geometry, one can determine the relationship between the size of the initial cube AD and the size of the truncated cube BC. The presence of square 2 shows that these two dimensions are incommensurable with each other. For this reason, we will use another approach to identify the Truncated Cube pattern in CPS. We will do that by looking first to the Catalan Octahedron, the dual of the Truncated Cube. For this reason, we will use another approach to identify the Truncated Cube pattern in CPS. We will do that by looking first to the Catalan Octahedron, the dual of the Truncated Cube. We have already seen a Catalan solid, namely the Catalan Tetrahedron while presenting minimum surfaces related solids in CPS – see video 16 in this series. We have already seen a Catalan solid, namely the Catalan Tetrahedron while presenting minimum surfaces related solids in CPS – see video 16 in this series. Now, let’s look closer to the Catalan Octahedron in CPS. Starting from an octahedron, by adding triangular pyramids to each face, one can generate a Catalan Octahedron. This pattern can be spotted very easily in the CPS arrangement. Let’s look closer to the shape and structure of the triangular pyramids. One can see that the pyramids grow in height very slowly. When all the spheres forming this pattern are shown, the beauty of this structure comes fully into the light. When all the spheres forming this pattern are shown, the beauty of this structure comes fully into the light. Let us contemplate it for a moment. We have already seen these triangular pyramids as the components of the soap film, produced by a tetrahedral frame. This is the best known manifestation of the minimum surfaces and is also found in the close packing of spheres. To better see this structure, the spheres defining the vertexes have been colored in red. Also, we have decided to show only half of the Catalan octahedron. This view de-clutters the structure and lets us contemplates the essential components. When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 2 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 3 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 4 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 5 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 6 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 7 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 8 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 9 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 10 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 12 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 14 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 16 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 17 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 18 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 19 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 20 Layers When all the components are close together, the Catalan Octahedron pattern emerges. The view presented here shows the internal configuration of the structure along the planes with an underlining square lattice. 21 Layers Two instantiations of this pattern are shown next. First, let us look at the Catalan Octahedron corresponding to the octahedron formed from triangle pattern with a size of 22 spheres. Two instantiations of this pattern are shown next. First, let us look at the Catalan Octahedron corresponding to the octahedron formed from triangle pattern with a size of 22 spheres. Next, let us see a more developed version of this pattern. This Catalan octahedron pattern corresponds to the Octahedron of size 29. This Catalan octahedron pattern corresponds to the Octahedron of size 29. The next few views show this structure in its entire splendor. As seen before, complex patterns in CPS are formed by a relative large numbers of spheres. As seen before, complex patterns in CPS are formed by a relative large numbers of spheres. Assembled as a Platonic Structure, this pattern requires 3842 nodes and 11592 struts. As seen before, complex patterns in CPS are formed by a relative large numbers of spheres. Assembled as a Platonic Structure, this pattern requires 3842 nodes and 11592 struts. As seen before, complex patterns in CPS are formed by a relative large numbers of spheres. Assembled as a Platonic Structure, this pattern requires 3842 nodes and 11592 struts. As seen before, complex patterns in CPS are formed by a relative large numbers of spheres. Assembled as a Platonic Structure, this pattern requires 3842 nodes and 11592 struts. Let’s go back to the Catalan Octahedron of size 22 and use it to reveal its corresponding dual solid, namely the Truncated Cube. The vertexes of the Truncated Cube are located in the center of the faces of the Catalan Octahedron. As we can see, the center lays between the three red spheres. This tells us that a vertex of a perfect Truncated cube is not defined by one sphere. This was something to be expected – remember, the size of the cube is incommensurable with the size of the truncated cube. By removing the unwanted spheres from the Catalan Octahedron, one is left with the corresponding Truncated Cube. By removing the unwanted spheres from the Catalan Octahedron, one is left with the corresponding Truncated Cube. We have chosen to show the duality of Catalan Octahedron and Truncated Cube using a handful of views. Try to imagine the missing halves of both these solids, the Truncated cube and the Catalan Octahedron, and their dual relationship. Try to imagine the missing halves of both these solids, the Truncated cube and the Catalan Octahedron, and their dual relationship. Try to imagine the missing halves of both these solids, the Truncated cube and the Catalan Octahedron, and their dual relationship. Try to imagine the missing halves of both these solids, the Truncated cube and the Catalan Octahedron, and their dual relationship. Finally, we have got to the point where the complete Truncated Cube pattern can be exposed. Finally, we have got to the point where the complete Truncated Cube pattern can be exposed. Note that both the vertexes and the edges are not determined by a single sphere.
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