The Classification of Platonic Solids I - Universal Hyperbolic Geometry
Exploring the classification of Platonic solids using modern notation and turn angles. Examines planar, spherical, and higher-dimensional cases, highlighting mathematical constructions and challenges in Euclid's proof.
Trigonometric Dual Laws and the Parallax Formula - Universal Hyperbolic Geometry
Explore trigonometric dual laws and the Right parallax formula in Universal Hyperbolic Geometry. Learn about Pythagoras' theorem, Cross dual law, and zero quadrance/spread theorems.
Menelaus, Ceva and the Laws of Proportion - Universal Hyperbolic Geometry
Explore classical theorems of Menelaus and Ceva in universal hyperbolic geometry, along with new results on triangle proportions and alternate spreads. Learn about ratios, distances, and quadrances in triangular geometry.
Thales' Theorem, Right Triangles + Napier's Rules - Universal Hyperbolic Geometry
Explores Thales' theorem and Napier's rules in universal hyperbolic geometry, focusing on right triangles. Covers key concepts like quadrea, spreads, and proofs, with practical exercises for deeper understanding.
The Cross Law in Universal Hyperbolic Geometry - Universal Hyperbolic Geometry - NJ Wildberger
Explores the Cross law in Universal Hyperbolic Geometry, proving it with a polynomial identity and demonstrating its application to equilateral triangles. Includes exercises and comparisons to Euclidean geometry.
The Spread Law in Universal Hyperbolic Geometry - Universal Hyperbolic Geometry
Explores the Spread law in universal hyperbolic geometry, including its definition, relation to Euclidean geometry, proof, and implications. Covers key concepts like quadrance, spread duality, and quadrea.
Visualizing Quadrance with Circles - Universal Hyperbolic Geometry - NJ Wildberger
Explore quadrance in universal hyperbolic geometry through circles, examining their appearance as ellipses, parabolas, or hyperbolas with different centers, enhancing understanding of this fundamental concept.
The Triple Quad Formula in Universal Hyperbolic Geometry
Explores the Triple quad formula in hyperbolic geometry, its proof, and related concepts. Compares with Euclidean geometry and discusses algebraic challenges, providing exercises for deeper understanding.
Pythagoras' Theorem in Universal Hyperbolic Geometry
Explore the hyperbolic version of Pythagoras' theorem, its importance in geometry, and the concept of quadrance. Learn about planar coordinates, formulas, and proofs in Universal Hyperbolic Geometry.
Quadrance and Spread - Universal Hyperbolic Geometry
Introduces algebraic definitions of quadrance and spread in Universal Hyperbolic Geometry, comparing with Rational Trigonometry and exploring main laws and exercises in this advanced mathematical framework.
Parallels and the Double Triangle - Universal Hyperbolic Geometry
Explores parallels, double triangles, and perspective centers in Universal Hyperbolic Geometry, offering new insights into Euclid's parallel postulate and its historical implications.
Medians, Midlines, Centroids and Circumcenters - Universal Hyperbolic Geometry
Explore triangle geometry in universal hyperbolic geometry, covering medians, midlines, centroids, and circumcenters. Discover unique properties and connections in this algebraic framework.
Apollonius and Polarity Revisited - Universal Hyperbolic Geometry
Explores Apollonius' polarity in hyperbolic geometry, focusing on null points, quadrangles, and the relationship between poles and polars. Includes explicit formulas and computational examples.
Theorems Using Perpendicularity - Universal Hyperbolic Geometry
Explore perpendicularity in hyperbolic geometry, including duality, triangles, and the Base center point. Learn a general theorem approach and investigate concepts using Standard Triangle 1.
Orthocenters Exist - Universal Hyperbolic Geometry
Explores orthocenters in universal hyperbolic geometry, introducing key concepts like triangles, altitudes, and Desargues' theorem. Culminates in defining orthic line, orthostar, and ortho-axis in hyperbolic triangle geometry.
Duality and Perpendicularity - Universal Hyperbolic Geometry
Explore perpendicularity in universal hyperbolic geometry, including duality, altitudes, and key theorems. Learn about unique aspects like perpendicular points and their applications in geometric analysis.
Apollonius and Harmonic Conjugates - Universal Hyperbolic Geometry 2 - NJ Wildberger
Explore harmonic conjugates in geometry, including their relation to circles, conics, and quadrangles. Learn about harmonic ranges, pencils, and vector combinations in this advanced mathematical discussion.
Null Points and Null Lines in Universal Hyperbolic Geometry - Lecture 12
Explore null points and null lines in universal hyperbolic geometry, including rational and projective parametrization, joins and meets theorems, and a triply nil triangle.
The Circle and Projective Homogeneous Coordinates in Universal Hyperbolic Geometry - Lecture 7b
Explores projective geometry as the foundation of universal hyperbolic geometry, focusing on the circle in projective homogeneous coordinates and its significance in mathematics and physics.
The Circle and Cartesian Coordinates in Universal Hyperbolic Geometry - Lecture 5
Explore Cartesian coordinates and the unit circle, covering key theorems on lines, points, and circles. Learn about determinants, parametrization, and the intersection of lines and circles in geometry.
Geometer's Sketchpad and Circles in Universal Hyperbolic Geometry - Lecture 25
Explore Geometer's Sketchpad for dynamic illustrations in universal hyperbolic geometry. Learn custom tools, construct point duals, and discover various circles, including equidistant curves, in hyperbolic space.