How Does the Hyperbolic Axiom Define a Maximal Triangle?

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Discussion Overview

The discussion revolves around the concept of a maximal triangle in the context of hyperbolic geometry, particularly focusing on the implications of the hyperbolic axiom regarding triangle formation and properties. Participants explore definitions and characteristics of triangles within this geometric framework.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that a triangle with parallel edges is considered a maximal triangle, though this raises questions about the definition of a triangle itself.
  • One participant questions the geometric context, suggesting that if edges are parallel, it may not conform to traditional definitions of a triangle, especially in Euclidean geometry.
  • Another participant identifies the geometry in question as Lobachevsky-Bolyai or hyperbolic geometry, which is relevant to the discussion of maximal triangles.
  • A participant explains the hyperbolic axiom, stating that given a line and a point not on that line, there are multiple lines through the point that do not intersect the original line, which can define a triangle with certain properties.
  • There is uncertainty regarding what "maximal" means in this context, with suggestions that it could refer to various measures such as circumference, area, or angles.

Areas of Agreement / Disagreement

Participants express differing views on the definition and properties of triangles in hyperbolic geometry, indicating that the discussion remains unresolved with multiple competing interpretations of what constitutes a maximal triangle.

Contextual Notes

Limitations include unclear definitions of "maximal" and the specific measures being referenced, as well as the potential confusion arising from the use of parallel edges in the context of triangle definitions.

Quantum Velocity
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I know that if a triangle have it edge // to each orther then it í the maximum triangle.

Pls explain i don't understand hơ thí even possible
 
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Quantum Velocity said:
I know that if a triangle have it edge // to each orther then it í the maximum triangle.

Pls explain i don't understand hơ thí even possible
With parallel sides, it isn't a triangle anymore. Except you're not using Euclidean geometry. So a) where are the triangles defined on, and b) what does "maximal" mean: circumference, area, one angle, a side?
 
it's some type of geometry that i don't remember the name
 
it's called Lobachevshy-Bolyai geometry
 
or sometime called Hyperbolic geometry
 
You simply apply the hyperbolic axiom:

For any given line R and point P not on R, in the plane containing both line R and point P
there are at least two distinct lines through P that do not intersect R.

This is a given fact. So with two points on R, and a point P, not on R, we have three different points which define a triangle. And the two lines through P with the line R are parallel and the sides of the triangle. (Not quite sure about the sides.)

What maximal means, still depends on your measure.
 

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