Three of these numerical identities can be seen if we examine a compound In this image is the intersection of the two components of the stella octangula.) 6 faces in a cube = 6 vertices in an octahedron, 8 vertices in a cube = 8 faces in an octahedron, 12 edges in a cube = 12 edges in an octahedron: Of the tetrahedron are the 6 vertices of the octahedron. Octahedron can be inscribed in a tetrahedron. This is a consequence of the fact that an Tetrahedra, or the stella octangula, Kepler's Latin term forĦ edges in a tetrahedron = 6 vertices in an octahedron: When two tetrahedraĪre combined in this manner, the result is called the compound of two No other Platonic solid has this property. Vertices of one tetrahedron are centered in the 4 faces of the other The two tetrahedra have a common center, so the 4 Tetrahedron can be superimposed with a copy of itself facing in the oppositeĭirection. Way of characterizing the same property is that a Have a face on top instead, a vertex is on top. Has the property that you can rest it face-down on a table and not In the other four Platonic solids, faces are oppositeįaces and vertices are opposite vertices, so the number of faces does not If there are 4 of one, there must beĤ of the other. This follows from the fact that in the tetrahedron,Įvery face is directly opposite a vertex, so there is a one-to-one relationīetween faces and vertices. Note that there are two different ways in which 4 of the 8 cube verticesĬould be chosen as the tetrahedron vertices.Ĥ faces in a tetrahedron = 4 vertices in a tetrahedron: Tetrahedron shows up as one of the diagonals of one of the 6 faces Relationship between faces of the cube and edges of the tetrahedron, so Stop and savorĮach of these connections, studying the accompanying figure until it isĬlear: 6 edges in a tetrahedron = 6 faces in a cube: Table there is a significant relationship to be understood. Relationships Between the Platonic SolidsĮvery time that a number shows up in two different places in the above These are not mere numerical coincidences. Observe that every number which appears somewhere in this table appearsĪt least twice. MakeĪ table with the fifteen answers and notice that only six different numbersįaces edges vertices tetrahedron 4 6 4 cube 6 12 8 octahedron 8 12 6 dodecahedron 12 30 20 icosahedron 20 30 12 ![]() Of faces, edges, and vertices found in each of these five models. Solids and the relationships between them.
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