UnivHypGeom4: First steps in hyperbolic geometry

This video outlines the basic framework of universal hyperbolic geometry---as the projective study of a circle, or later on the projective study of relativistic geometry. Perpendicularity is defined in terms of duality, the pole-polar correspondence introduced by Apollonius, and we explain that the three altitudes of a triangle meet in a point- the orthocenter H. The basic measurements of quadrance and spread in this geometry arise from the cross ratio of suitable points and lines. We state the main formulas: Pythagoras' theorem, the Triple quad formula, Pythagoras' dual theorem, the Triple spread formula, the Spread law and the Cross law and its dual. These are closely related to, but different from the corresponding laws in Rational Trigonometry.

CONTENT SUMMARY: notion of perpendicularity @04:48 Perpendicularity via duality @05:33 Do the altitudes of a triangle meet in a point? @10:54 Quadrance: m'ment beween points @15:14 exercise @18:41 remark on Beltrami-Klein model @19:11 Pythagoras' theorem and Triple quad formula @20:30 Spread: m'ment between lines and quadrance spread duality theorem @23:29 Remark on Beltrami-Klein model @26:45 Pythagoras' dual theorem @28:43 Main formulas for triangles that involve both quadrances and preads @31:13 (THANKS to EmptySpaceEnterprise)

Category:

Education

Tags:

• UnivHypGeom4
• hyperbolic
• geometry
• Wildberger
• duality
• perpendicular
• altitudes
• orthocenter
• Pythagoras'
• theorem
• relativistic
• Einstein
• Minkowski
• Triple
• quad
• formula
• spread
• dual
• pole
• polar
• quadrance
• cross
• law
• overview
• education

[EmptySpaceEnterprise]

Thanks for the show. Hope you will cut and paste these( I'm trying to do the same for all your vid's that I watch) into your vid summery text so the map doesn't drift into never/never_land. I had to slim them up to meet the 500 character restriction for a post otherwise the pointers would be in a column. I purchased some colored pencils and hope to complete your course and then return frequently to review. Best wishes.

[njwildberger]

@EmptySpaceEnterprise In fact I was going to ask you if it would be okay to paste your comments into the vid summary--so I will go ahead and do that, making a few small modifications here and there. That is much appreciated!

[EmptySpaceEnterprise]

notion of perpendicularity @04:48 Perpendicularity via duality @05:33 Do the altitudes of a triangle meet in a point? @10:54 Quadrance: m'ment beween points @15:14 exercise @18:41 remark on Beltrami-Klein model @19:11 Pythagoras' theorem and Triple quad formula @20:30 Spread: m'ment between lines and quadrance spread duality theorem @23:29 Remark on Beltrami-Klein model @26:45 Pythagoras' dual theorem @28:43 Main formulas for triangles that involve both quadrances and preads @31:13

[njwildberger]

@EmptySpaceEnterprise Thanks so much!

[xuanji07]

Hi Prof Norman, I was wondering what is the difference between classical hyperbolic geometry and your "universal hyperbolic geometry"? From earlier videos I thought that they are the same theory and your development of it is just a different exposition, but if orthocenters don't necessarily exist in classical hyperbolic geometry then it seems that the two theories are different.

[njwildberger]

@xuanji07 There are many differences. For one thing, universal hyperbolic geometry is a logically correct theory, while classical hyperbolic geometry is not. For another, UHG works over a general field, classical HG only over the `real numbers'. For another the formulas in UHG work also for points and lines outside the usual Beltrami Poincare disk. However the mathematical reality that both theories try to capture is essentially the same. UHG is just a lot more successful at capturing that.

[EmptySpaceEnterprise]

Thanks.

[brangelito]

I appreciated this lecture. Thank you!

[njwildberger]

Hi Junkbox09 The three points forming the triangle are general. ALmost any three points work---there are a few exceptions. But for any general three points the three altitudes meet in exactly one point, so each triangle has exactly one orthocenter.

[Junkbox09]

hi norman. just asking a question about the slide on the altitudes meeting in a point. is the choice of points of the triangle arbitrary? if yes, would that mean that there would be many orthocentres depending on the choice of points?