Spherical Geometry Equilateral Pentagon

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Discussion Overview

The discussion revolves around the properties of spherical geometry, specifically focusing on the possibility of constructing an equilateral pentagon with four right angles. Participants explore the implications of angle measures in spherical geometry.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions the assertion that an equilateral pentagon cannot have four right angles in spherical geometry, seeking clarification on the reasoning behind this claim.
  • Another participant suggests that the angles in a spherical pentagon must be greater than 108 degrees, implying a restriction on the possibility of having four right angles.
  • A subsequent reply confirms the previous point, reiterating that if all angles must exceed 108 degrees, then having four angles of 90 degrees is not feasible.

Areas of Agreement / Disagreement

Participants appear to agree on the notion that having four right angles in a spherical equilateral pentagon is not possible, but the reasoning behind this assertion is still being clarified and debated.

Contextual Notes

The discussion does not fully resolve the underlying assumptions about angle measures in spherical geometry or the specific characteristics of Lambert quadrilaterals.

GeometryIsHARD
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Hey PF! I'm going through a textbook right now and it just said "obviously, you can't have an equilateral pentagon with 4 right angles in spherical geometry (Lambert quadrilaterals).

However, I am not able to make the connection. can somebody help me understand why this is?
 
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So every angle must be greater than 108 degrees and therefore it's quite obvious you can't have four of them being 90 degree's. Is that what you are saying?
 
GeometryIsHARD said:
So every angle must be greater than 108 degrees and therefore it's quite obvious you can't have four of them being 90 degree's. Is that what you are saying?

Yes.
 

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