The Cuboctahedron from Spheres

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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 – 18: The Cuboctahedron from Spheres YouTube: https://youtu.be/SIl7zn1_ncE A cuboctahedron is a solid body with 8 triangular faces and 6 square faces. A cuboctahedron is a solid body with 8 triangular faces and 6 square faces. It has 12 identical vertices and 24 identical edges. As we have seen already, a sphere can be completely surrounded by exactly twelve identical spheres. These 13 spheres define the simplest cuboctohedron in the close packing of spheres arrangement Let's have a look at the evolution of this pattern, by adding more and more layers of spheres. Let's have a look at the evolution of this pattern, by adding more and more layers of spheres. As the number of spheres increases, the cuboctahedron shape becomes clearer. Buckminster Fuler has given a special place to this pattern in his investigation of close packing of spheres. Buckminster Fuler has given a special place to this pattern in his investigation of close packing of spheres. He introduced the term "vector equilibrium" to refer to this arrangement. All 12 distances from the center sphere to the spheres defining each vertex, are equal. The faces of this pattern can be used to see the seven planes identified before in the CPS arrangement. One can easily imagine variation of this pattern - for example a wire frame. We have mentioned before that all the patterns in CPS, when viewed as Platonic Structures, are expandable. The lengths of the struts do not contribute in any way to the shape of a Platonic Structure. Two Platonic Structures assembled from struts of different lengths, are Similar Structures. The Euclidian geometry is based on the similarity theorem, also known as the Proportionality Theorem, or Thales Theorem. The Euclidian geometry is based on the similarity theorem, also known as the Proportionality Theorem, or Thales Theorem. The preservation of patterns in CPS provides a nice explanation of this fundamental property of space. The Platonic Structure View, where the spheres are connected using identical struts, provides a better visualization of the nature of these patterns and the inside connections between nodes. The Platonic Structure View, where the spheres are connected using identical struts, provides a better visualization of the nature of these patterns and the inside connections between nodes. www.platonicstructures.com Beauty makes beautiful things beautiful!.

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Archimedean Solids
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat).
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