Polyhedron Math Problem: Determining Regularity and Type

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SUMMARY

The polyhedron P defined by the inequalities (+/-) x (+/-) z <= 1, (+/-) x (+/-) y <= 1, and (+/-) y (+/-) z <= 1 is determined to be a dodecahedron rather than an octahedron. The discussion highlights that while the inequalities produce 12 faces, the presence of the point (0.5, 0.5, 0.5) serves as a counterexample, confirming that P does not conform to the properties of an octahedron. The analysis emphasizes the importance of evaluating vertices, edge lengths, and angles to ascertain the regularity and type of polyhedra.

PREREQUISITES
  • Understanding of polyhedron properties and classifications
  • Familiarity with inequalities and their geometric interpretations
  • Knowledge of regular polygons and their characteristics
  • Basic skills in solving systems of equations
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  • Study the properties of dodecahedra and their geometric characteristics
  • Learn about the relationship between inequalities and polyhedral faces
  • Explore methods for determining the regularity of polyhedra
  • Investigate the implications of counterexamples in geometric proofs
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Mathematics students, geometry enthusiasts, and educators focusing on polyhedral theory and geometric problem-solving.

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

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