Trisecting angles in an alternate topology

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SUMMARY

The discussion centers on the feasibility of angle trisection using straight lines and circles within various topologies, specifically on arbitrary two-dimensional manifolds. Participants define "straight lines" as curves with minimal geodesics between points and "circles" as sets of equidistant points from a center. The conversation emphasizes the importance of curvature in measuring angles, suggesting that traditional methods of angle trisection may not apply in non-Euclidean geometries.

PREREQUISITES
  • Understanding of two-dimensional manifolds
  • Familiarity with concepts of geodesics
  • Knowledge of angle measurement in non-Euclidean geometry
  • Basic principles of topology
NEXT STEPS
  • Research the properties of geodesics in different topological spaces
  • Explore the implications of curvature on angle measurement
  • Study the limitations of classical constructions in non-Euclidean geometry
  • Investigate alternative methods for angle trisection in various topologies
USEFUL FOR

Mathematicians, geometry enthusiasts, and students of topology seeking to understand the complexities of angle trisection in non-Euclidean contexts.

Loren Booda
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Is there any topology where it is possible to trisect an angle using only straight lines and circles?
 
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In other topologies, what do you mean by "straight lines" and "circles"? And how do measure angles?
 
Lines are curves such that between every pair of points on the line, the segment there is a minimal geodesic; circles are a set of points all equidistant on lines from a center point; angles are measured regarding the curvature from local triangles.

Changing the problem slightly, all of these constructs are assumed to be on an arbitrary two-dimensional manifold. Is there any such topology where it is possible to trisect an angle using only lines and circles?
 

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