Universal Hyperbolic Geometry 0: Introduction

Hyperbolic geometry, in this new series, is made simpler, more logical, more general and... more beautiful! The new approach will be called `Universal Hyperbolic Geometry', since it extends the subject in a number of directions. It works over general fields, it extends beyond the usual disk in the Beltrami Klein model, and it unifies hyperbolic and elliptic (and other) geometries.

This introduction outlines the differences between this course and that found in all the standard textbooks and courses at universities. We avoid ``real numbers'', avoid ``axioms'', avoid transcendental functions such as cos, sin, log, tanh, avoid absolute values. And treat square roots more algebraically. Our approach uses high school algebra, and is essentially projective geometry combined with a distinguished conic---the unit circle in the plane. We will also later see that this approach ties in very closely with the geometry of Einstein's special theory of relativity.

The series is developed and presented by N J Wildberger, also the originator of Rational Trigonometry.

Category:

Education

Tags:

• Hyperbolic
• Geometry
• Universal
• Wildberger
• mathematics
• non-Euclidean
• geometry
• rational
• trigonometry
• quadrance
• spread
• Apollonius
• conics
• Bolyai
• Lobachevsky
• Gauss
• Beltrami
• Klein
• Poincare
• Coxeter
• Escher
• UNSW
• projective
• unit
• circle
• geodesics
• Einstein
• special
• theory
• relativity
• Thurston
• parallel
• postulate
• course

[TheFlightguru]

I noticed how you said 'it's the way god wants to express the beauty of natural world'. I'm intrigued as if you are christian or any religion ( as I am christian). My maths teacher is an atheist and always tries to use maths to deny the existence of god when in fact I think such beauty and complexity could only be created by god. If you're an atheist, I have no problems with that, I am just intrigued.

[njwildberger]

@TheFlightguru I wouldn't say I am particularly religious in a standard way, but I certainly share your feeling that the beauty and complexity in the mathematical world suggests some kind of organizing principle/entity beyond what we can see directly---possible evidence of a god(s). See my paper Evolution versus Intelligent Design: A mathematician's view at my unsw webpages, under views.

[alexzarhin]

I have a question about the model that you build in this course. Is it you, who invented this "projective model" which you build in this course? I mean not the Klein model, but the "improved" model, which holds on the whole plane?

[njwildberger]

@alexzarhin Yes this is my model, building on the classical Beltrami Klein projective model, incorporating features of Rational Trigonometry to provide the metrical foundation.

[TupperWallace]

Professor Wildberger, congratulations on passing the 10^5 mark in the number of channel views! I'm one of the retired people interested in math you surmised might be in your audience. I still bitterly remember my disappointment in high school trig in October, 1963, when I asked Mr. Kennedy when we were going to get to the good stuff instead of memorizing formulas. He just shrugged. There is good fun stuff in trig but I had to wait all these years to see it in your videos. Best Regards.

[njwildberger]

@TupperWallace Hi, I think you are not alone in these sentiments. Up to now there hasn't been much of an alternative to the usual horrible trig. Now there is! Hope you enjoy the videos.

[EmptySpaceEnterprise]

Instead of distance and angle - quadrance and spread @09:35

formulas @10:23

Advantages of Universal Hyperbolic Geometry @11:44

A few words about the course @16:52

requirements: a straight edge and a compass (colored pencils) @17:36

computer geometry program @20:02

[Npowell01]

@njwildberger

Why must organization principles in nature necessarily point to a conscious originator of said principles? Is it not conceivable that the principles simply occur? Why must there be something comparable to a human mind behind it all? It is certainly comforting, but I see no evidence. Just human desire for cosmic companionship, which is completely understandable.

Ultimately there is little way of knowing either way, but why entertain the unfalsifiable/unprovable hypothesis?

[TheFlightguru]

@FluffyBunniesOnFire In my personal opinion, there is a great difference between theism and atheism. God is an intelligent entity, no amount of maths will bring a being other than what god has created. Now, without going off on a tangent (if you'll pardon the pun), I assume you understand conservation of mass. Therefore, how can our universe be created from nothing? I'm no finitist, but something infinitely small in nature (and not some imaginary domain) is nothing.

[FluffyBunniesOnFire]

@TheFlightguru When you get to a certain point of understanding the first difference between theism and atheism is merely the different terminology they use to describe the same thing. "God" is the unifying source concept, and it communicates to us through math.