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Duality is a property in mathematics where the role of two classes of objects can be swapped. Provided this is done consistently, many of the properties such as incidence or intersection in the original structure are true in the swapped structure. In the context of polyhedra, duality is about vertices and faces.
One way to find a dual of a polyhedron is to replace the centre of every face by the vertex of the shape that will become the dual. Whenever two faces of the cube share an edge, join up the centres of these faces to form an edge of the dual. Although it may be somewhat difficult to visualise, the dual of the cube is a regular octahedron.
A cube has 8 vertices, 6 faces [and 12 edges]. There are 3 faces meeting at each vertex, each face has 4 vertices.
An octahedron has 8 faces, 6 vertices [and 12 edges]. There are 3 vertices on each face, there are 4 faces meeting at each vertex.
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How many apex does an octahedron?
A regular octahedron has 6 apexes as it can be rotated evenly as each face is congruent.
What solids have 12 edges?
A cube has 12 edges as does an octahedron and those are the two platonic solids (convex polyhedra with congruent regular polygons as faces where the same number of faces meet at each vertice) with 12 edges.
How many edges has an octahedron?
It has 12. This solid has 8 sides, each of which is an equilateral triangle (in the case of the regular octahedron). Use the link and surf on over to Wikipedia to see a picture.
What shape is each side of a cube?
Each side of a cube is square.
Each side of a cube is 42.6 mm long What is the volume of the cube in cm3?
If each side of the cube is 42.6 mm long, then the cube's volume is 77.31 cm3
This one's a tricky one. whoever solves this is my hero... if each edge of a cube is three-forths as long as each edge of a regular octahedron which shape has the larger volume?
if you would be kind enough to give it to me in real numbers i could solve it for you I think it is the octahedron
What are the dual pairs for each of the platonic solids?
A tetrahedron is its own dual. An octahedron is the dual of a cube and conversely. An icosahedron is the dual of a dodecahedron and conversely.
What is a dual?
A dual is almost like the opposite of a given polytope. For example, a regular octahedron is the dual of a cube.Look at the similarities between duals with the example of a cube and regular octahedron:Cube:Vertices: 8Edges: 12Faces: 6Edges per vertex: 3Type of face: square (4-sided)Regular Octahedron:Vertices: 6Edges: 12Faces: 8Edges per vertex: 4Type of face: triangle (3-sided)Both of these shapes can be put together to form a compound and can be rectified to form the same new shape: a cuboctahedron.Also, duals can fit perfectly inside another where each edge touches the face of the other.In this same way a regular dodecahedron is the dual of a regular icosahedron. Some polyhedra like the regular tetrahedron is the dual of itself. All polyhedra have duals. Polygons, polychora, and other polytopes can also have duals in a similar fashion.
Each face is what polygon on a octahedron?
A regular octahedron is one of the platonic solids. Each of its faces is an equilateral triangle.
What is a cube parallel edges?
Opposite edges are parallel to each other in a cube
How many apex does an octahedron?
A regular octahedron has 6 apexes as it can be rotated evenly as each face is congruent.
What is the Name of each face of an octahedron?
Each face is an equilateral triangle.
Which 2 solids have the same number of edges?
A rectangular prism (cuboid) and a hexagon-based pyramid, for example, both have 12 edges. Of the five Platonic solids, an octahedron and a cube each have 12 edges.
What shape is the face on a cube?
All faces on a cube are equal to each other the shape is of the face on a cube is a square
What does an octahedron look like?
An octahedron is a polyhedron with eight faces. A regular octahedron can be composed of eight equilateral triangles, four forming each half, and meeting in the middle at a square. Imagine a pyramid, with a square base. Rising up from each edge is an equilateral triangle that meets the other four triangles at the apex. Now, turn the pyramid over and add four more triangles to the other side the same way. You now have an octahedron. For more information, and a rotating picture, please see the related link below.
Why does a Platonic solid look the same no matter which vertex you position at the top?
The quick answer: because of the high degree of symmetry inherent in the Platonic solids. They are vertex-uniform, edge-uniform and face-uniform. If you hold several models of the same shape up by any vertex, all the models will appear the same. The same goes for holding the models up by any edge, or by any face. Read the following for a little more detail. Many solids that are not Platonic have symmetry as well, but the Platonic solids have some special symmetrical properties. You can create what are called 'dual polyhedrons' for solids, but the duals for Platonic solids are unique. You can form a Platonic solid's dual polyhedron by making the midpoint of every face of the original Platonic solid a vertex of the dual solid within the original. If you start with a cube, a hexahedron really, and make a new solid within it having vertexes at the centers of the square faces of the cube, the solid within will be an octahedron. Tetrahedrons are self-dual, squares and octahedrons are dual with one another, and dodecahedrons and icosahedrons are dual with one another. The dual polyhedron of a Platonic solid is always another Platonic solid. This is difficult to visualize without aid. See link for some clarification. On the dual relationship of a cube [6 faces, 8 vertexes] and octahedron [8 faces, 6 vertexes] breaking down the numbers of faces and vertexes might help. Each of the 6 faces of a cube contains one of the vertexes of the octahedron, and each of the vertexes of a cube will be at the center of one of the faces of the octahedron.
What octahedron triangle for all their faces?
A regular octahedron is a Platonic solid with equilateral triangles for each of the faces. A heptagonal pyramid is an octahedron with one heptagon (a seven sided figure) as its base and 7 triangles, one attached to each side, meeting at the either vertex.
