In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in passing through the origin. The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. (This cannot be done in three-dimensional space without the surface intersecting itself.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1. Since the Möbius strip, in turn, can be constructed from a square by gluing two of its sides together with a half-twist, the real projective plane can thus be represented as a unit square (that is, [0, 1] × [0,1]) with its sides identified by the following equivalence relations: (0, y) ~ (1, 1 − y) for 0 ≤ y ≤ 1 and (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1, as in the leftmost diagram shown here. (Wikipedia).
Introduction to Projective Geometry (Part 1)
The first video in a series on projective geometry. We discuss the motivation for studying projective planes, and list the axioms of affine planes.
From playlist Introduction to Projective Geometry
Introduction to Projective Geometry (Part 2)
The second video in a series about projective geometry. We list the axioms for projective planes, give an examle of a projective plane with finitely many points, and define the real projective plane.
From playlist Introduction to Projective Geometry
The circle and projective homogeneous coordinates | Universal Hyperbolic Geometry 7a | NJ Wildberger
Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine
From playlist Universal Hyperbolic Geometry
The circle and projective homogeneous coordinates (cont.) | Universal Hyperbolic Geometry 7b
Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine
From playlist Universal Hyperbolic Geometry
Arithmetic Fake Compact Hermitian Symmetric Spaces - Gopal Prasad
Gopal Prasad University of Michigan February 16, 2012 A fake projective plane is a smooth complex projective algebraic surface whose Betti numbers are same as those of the complex projective plane but which is not the complex projective plane. The first fake projective plane was constructe
From playlist Mathematics
algebraic geometry 15 Projective space
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It introduces projective space and describes the synthetic and analytic approaches to projective geometry
From playlist Algebraic geometry I: Varieties
Imaginary Numbers Are Real [Part 6: The Complex Plane]
For full problem statement, check out: http://www.welchlabs.com/blog/2015/10/2/imaginary-numbers-are-real-part-6-the-complex-plane More information and resources: http://www.welchlabs.com Imaginary numbers are not some wild invention, they are the deep and natural result of extending our
From playlist Imaginary Numbers are Real
Imaginary Numbers Are Real [Part 12: Riemann's Solution]
Want to experiment with Riemann's idea yourself? You can download your very own copy of of the final w-planes to experiment with here: http://www.welchlabs.com/blog/2016/6/30/imaginary-numbers-are-real-part-12-riemanns-solution Supporting Code: https://github.com/stephencwelch/Imaginary-N
From playlist Imaginary Numbers are Real
Visualizing quaternions (4d numbers) with stereographic projection
How to think about this 4d number system in our 3d space. Part 2: https://youtu.be/zjMuIxRvygQ Interactive version of these visuals: https://eater.net/quaternions Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of t
From playlist Explainers
Perspectives in Math and Art by Supurna Sinha
KAAPI WITH KURIOSITY PERSPECTIVES IN MATH AND ART SPEAKER: Supurna Sinha (Raman Research Institute, Bengaluru) WHEN: 4:00 pm to 5:30 pm Sunday, 24 April 2022 WHERE: Jawaharlal Nehru Planetarium, Bengaluru Abstract: The European renaissance saw the merging of mathematics and art in th
From playlist Kaapi With Kuriosity (A Monthly Public Lecture Series)
On Ultra-Parallel Complex Hyperbolic Triangle Groups by Anna Pratoussevitch
SURFACE GROUP REPRESENTATIONS AND GEOMETRIC STRUCTURES DATE: 27 November 2017 to 30 November 2017 VENUE:Ramanujan Lecture Hall, ICTS Bangalore The focus of this discussion meeting will be geometric aspects of the representation spaces of surface groups into semi-simple Lie groups. Classi
From playlist Surface Group Representations and Geometric Structures
Siggraph2019 Geometric Algebra
**Programmer focused part** starts at 18:00 Try the examples here https://enkimute.github.io/ganja.js/examples/coffeeshop.html The Geometric Algebra course at Siggraph 2019. Intro : Charles Gunn (00:00 - 18:00) Course : Steven De Keninck (18:00 - end) Course notes, slides, software, disc
From playlist Bivector.net
Elliptic curves: point at infinity in the projective plane
This video depicts point addition and doubling on elliptic curve in simple Weierstrass form in the projective plane depicted using stereographic projection where the point at infinity can actually be seen. Explanation is in the accompanying article https://trustica.cz/2018/04/05/elliptic-
From playlist Elliptic Curves - Number Theory and Applications
Illusion: The City That Never Was
World War 1: Paris is the target of nightly German bombings that become more and more deadly. In a time where radars don’t yet exist and pilots can be fooled by false illuminations, the French General Staff secretly builds a fake illuminated Paris to save people’s lives. After being put in
From playlist Complete List of Members-Only Videos
Seminar on Applied Geometry and Algebra (SIAM SAGA): Rekha Thomas
Date: Tuesday, November 10 at 11:00am EST (5:00pm CET) Speaker: Rekha Thomas, University of Washington Title: When Two Cameras Meet a Cubic Surface Abstract: The set of images captured by an arrangement of pinhole cameras is usually modeled by the multiview variety. The true set is in f
From playlist Seminar on Applied Geometry and Algebra (SIAM SAGA)
Ciro Ciliberto, Enumeration in geometry - 15 Novembre 2017
https://www.sns.it/eventi/enumeration-geometry Colloqui della Classe di Scienze Ciro Ciliberto, Università di Roma “Tor Vergata” Enumeration in geometry Abstract: Enumeration of geometric objects verifying some specific properties is an old and venerable subject. In this talk I will
From playlist Colloqui della Classe di Scienze
Projective view of conics and quadrics | Differential Geometry 9 | NJ Wildberger
In this video we introduce projective geometry into the study of conics and quadrics. Our point of view follows Mobius and Plucker: the projective plane is considered as the space of one-dimensional subspaces of a three dimensional vector space, or in other words lines through the origin.
From playlist Differential Geometry
Michael Eastwood: Twistor theory for LQG
Twistor Theory was proposed in the late 1960s by Roger Penrose as a potential geometric unification of general relativity and quantum mechanics. During the past 50 years, there have been many mathematical advances and achievements in twistor theory. In physics, however, there are aspirati
From playlist Mathematical Physics