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August 2nd, 2019, 01:01 PM   #1
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Math Focus: Combinatorics
Non-Euclidean Geometry (NEG): New Outlook(?)

Non-Euclidean Geometry (NEG) is a special branch of geometry involving postulates about parallel lines or intersecting lines that are 'alternatives' to standard Euclidean Geometry (EG) which is why we see 'irregular' or 'multi-definitional' shapes in NEG analysis/exploration.

NEG inquiries may involve hyperbolic geometry or spherical geometry that evade the standard concepts of planar lines intersecting at perfect right angles. NEG shapes may be studies with more complex forms of mathematical inquiry, perhaps involving topology or nonlinear analysis.

While EG math gives us congruent right-triangles, parallelograms, and ideal cubes, NEG may give us asymmetrical 13-sided polygons, pierced spheres, and opposing joined cones forming a 'strange dumbbell' object.

NEG analysis and observations can help us dissect complex shapes in nature such as whirlpools and cirrus clouds as well as complex shapes in constructive conceptual geometry such as disintegrating vortexes and 'imperfect cubes.'

Many shapes in NEG are purely 'imaginary' but can help mathematicians and logicians study complex phenomena such as event horizons, black hole entropy, and sphere-tangent divergence.

Looking at the convergence/divergence of tangential lines in contact on the surface of a sphere, for example, may help NEG mathematicians/logicians 'measure' the smoothness of an 'irregular sphere' (if such an imaginary shape exists!) and therefore make theories about the existence of irregular curvilinear spaces that do not actually give rise to perfect spheres but rather to elliptical objects or 'deformed spheres.'

Applications of NEG are found in studies of nonlinear decomposition rates as well as deformation/resilience analyses (which are useful to engineers measuring the tensile strength and impact resistance of designed materials!).

However, NEG remains a largely 'conceptual' field of study, which is why many number theorists and mathematicians studying complex math (e.g., combinatorics, chaos math) can be drawn to it(!).

What's really interesting to me is how NEG seems to offer insights into the nature of shape construction and integrity maintenance in dynamical systems. This is something very useful to people studying the entropic behaviors concerning black holes, something explored incidentally in the recent movie The Theory of Everything!

I wonder therefore how much academic funding NEG mathematicians/logicians will receive in this new age of much 'complex analysis' intrigue/buzz(!).

I'd love to hear other people's thoughts on what intrigues you about NEG in the 21st Century...


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Last edited by Abisha100; August 2nd, 2019 at 01:10 PM.
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August 2nd, 2019, 02:38 PM   #2
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I've always found this one interesting (useless, but interesting).

The Case of the Scorpling Flugs.

Undefined terms: Scorple, Flug

Axioms

1) If A and B are distinct flugs, then A scorples B or B scorples A ( the possibility of both happening is not excluded)

2) No flug scorples itself

3) If A, B and C are flugs such that A scorples B and B scorples C, then A scorples C.

4) There are exactly four flugs.
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Last edited by skipjack; August 2nd, 2019 at 03:25 PM.
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