Trisecting angles in an alternate topology

In summary, the conversation discusses the possibility of trisecting an angle using only straight lines and circles on a two-dimensional manifold. The definition of lines, circles, and angles on this surface is also clarified. The question is then modified to ask if there is any topology where this is achievable.
  • #1
Loren Booda
3,125
4
Is there any topology where it is possible to trisect an angle using only straight lines and circles?
 
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  • #2
In other topologies, what do you mean by "straight lines" and "circles"? And how do measure angles?
 
  • #3
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?
 

Related to Trisecting angles in an alternate topology

1. How is trisecting angles in an alternate topology different from traditional geometry?

Trisecting angles in an alternate topology involves using different axioms and principles from traditional Euclidean geometry. This topology allows for curved lines and surfaces, rather than just straight lines and flat planes.

2. Is it possible to trisect any angle using alternate topology?

No, not all angles can be trisected in alternate topology. The angle must have a specific geometric structure that allows for trisection, such as being a multiple of π/3 radians or having certain symmetry properties.

3. Are there any practical applications for trisecting angles in alternate topology?

Yes, there are several applications in fields such as computer graphics, architecture, and physics. For example, trisecting angles in alternate topology can help with creating more realistic and accurate 3D models of curved surfaces.

4. What are some challenges in trisecting angles in alternate topology?

One challenge is finding the right geometric structure for the angle to be trisected. Another challenge is the lack of familiar geometric principles and tools in alternate topology, which may require new methods and techniques to be developed.

5. Are there any controversies surrounding trisecting angles in alternate topology?

Yes, there are ongoing debates about the validity and usefulness of alternate topology in mathematics and the practical applications of trisecting angles in this topology. Some argue that it is a valuable tool for solving complex geometric problems, while others believe it is unnecessary and overly complicated.

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