Polyhedron Math Problem: Determining Regularity and Type

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In summary, P is a polyhedron with 12 inequalities representing its 12 faces. It is a dodecahedron, not an octahedron, as initially thought. To determine if a face is a regular polygon, we can change one inequality to an equation and work out its vertices, edge-lengths, and/or angles.
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wimma
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Homework Statement



P is a polyhedron defined by (+/-) x (+/-) z <= 1
(+/-) x (+/-) y <= 1
(+/-) y (+/-) z <= 1

These are 12 inequalities with every possible sign choice taken.

Is P a regular polyhedron? If so, which type?



Homework Equations



If we change one inequality to an equation, we get a face of the polyhedron. In order to see if the face is a regular polygon, work out its vertices and/or edge-lengths and/or angles


The Attempt at a Solution



I got an octahedron but I see a counterexample of (0.5,0.5,0.5), which seems to satisfy our equations, but not the formula for an octahedron.
 
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  • #2


Your 12 inequalities give 12 faces. This is a dodecahedron, not an octahedron.
 

1. What is a polyhedron?

A polyhedron is a three-dimensional shape with flat faces, straight edges, and sharp corners. Examples of polyhedrons include cubes, pyramids, and prisms.

2. What is the difference between a polyhedron and a polygon?

A polygon is a two-dimensional shape with straight sides and corners, while a polyhedron is a three-dimensional shape with flat faces and sharp corners. Polygons can be thought of as the 2D versions of polyhedrons.

3. How do you find the surface area of a polyhedron?

To find the surface area of a polyhedron, you need to find the area of each face and add them together. The formula for finding the area of a face depends on the shape of the face. For example, the area of a square face is length x width, while the area of a triangular face is 1/2 x base x height.

4. How many faces, edges, and vertices does a polyhedron have?

The number of faces, edges, and vertices of a polyhedron depends on its shape. For example, a cube has 6 faces, 12 edges, and 8 vertices. A triangular pyramid has 4 faces, 6 edges, and 4 vertices.

5. What is the Euler's formula for polyhedrons?

Euler's formula states that for any polyhedron, the number of faces (F), edges (E), and vertices (V) satisfy the equation F + V - E = 2. This formula is useful for solving problems involving polyhedrons, such as finding the number of faces or vertices when given the number of edges.

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