In mathematics, a Klein surface is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves over the complex numbers analytically, the former are used to study algebraic curves over the real numbers analytically. Klein surfaces were introduced by Felix Klein in 1882. A Klein surface is a surface (i.e., a differentiable manifold of real dimension 2) on which the notion of angle between two tangent vectors at a given point is well-defined, and so is the angle between two intersecting curves on the surface. These angles are in the range [0,π]; since the surface carries no notion of orientation, it is not possible to distinguish between the angles α and −α. (By contrast, on Riemann surfaces are oriented and angles in the range of (-π,π] can be meaningfully defined.) The length of curves, the area of submanifolds and the notion of geodesic are not defined on Klein surfaces. Two Klein surfaces X and Y are considered equivalent if there are conformal (i.e. angle-preserving but not necessarily orientation-preserving) differentiable maps f:X→Y and g:Y→X that map boundary to boundary and satisfy fg = idY and gf = idX. (Wikipedia).
This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/3kIo
From playlist 3D printing
https://github.com/timhutton/klein-quartic This is work in progress. The transition is linear at the moment, which causes a lot of self-intersection.
From playlist Geometry
Made from 24 heptagons. Source code and meshes here: https://github.com/timhutton/klein-quartic
From playlist Geometry
I made this video when I thought I had made a model of the Klein Quartic. But it is wrong, so please ignore it. You can find a corrected version here: https://www.youtube.com/watch?v=ADtwLnxLPTI
From playlist Geometry
This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/2p3Z
From playlist 3D printing
The three types of eight-fold way path on the Klein Quartic
Source code and mesh files here: https://github.com/timhutton/klein-quartic
From playlist Geometry
Hausdorff dimension of Kleinian group uniformization of Riemann surface... - Yong Hou
Topic: Hausdorff dimension of Kleinian group uniformization of Riemann surface and conformal rigidity Speaker: Yong Hou Date:Tuesday, November 24 For this talk I'll discuss uniformization of Riemann surfaces via Kleinian groups. In particular question of conformability by Hasudorff dimens
From playlist Mathematics
Introduction to Quadric Surfaces
http://mathispower4u.yolasite.com/
From playlist Quadric, Surfaces, Cylindrical Coordinates and Spherical Coordinates
Hunt for the Elusive 4th Klein Bottle - Numberphile
Carlo Séquin on his search for the elusive "fourth type of Klein bottle". More videos on Klein Bottles: http://bit.ly/KleinBottles More links & stuff in full description below ↓↓↓ Carlo's paper: http://bit.ly/Carlo_Klein Also featuring Carlo: Mobius House https://youtu.be/iwo7JReFTeg Tor
From playlist Carlo Séquin on Numberphile
Professor Carlo Séquin explains super bottles - and a super duper bottle. More Klein Bottle videos: http://bit.ly/KleinBottles More Carlo videos: http://bit.ly/carlo_videos More links & stuff in full description below ↓↓↓ Carlo Séquin is based at the University of California, Berkeley. S
From playlist Carlo Séquin on Numberphile
A mirror paradox, Klein bottles and Rubik's cubes
The Mathologer puts the latest $2000 addition to his Klein bottle collection to work. A couple of first-ever fun mathematical stunts in this video. This video finishes with a puzzle for you to think about. We posted a video with the solution on 1 August 2015: https://youtu.be/ZMC61C5tigA
From playlist Recent videos
Robert YOUNG - Quantifying nonorientability and filling multiples of embedded curves
Abstract: https://indico.math.cnrs.fr/event/2432/material/17/0.pdf
From playlist Riemannian Geometry Past, Present and Future: an homage to Marcel Berger
The 17-Klein Bottle - Numberphile
17 Klein Bottles become 1 - featuring Cliff Stoll and the glasswork of Lucas Clarke. More links & stuff in full description below ↓↓↓ More Cliff videos: http://bit.ly/Cliff_Videos More Klein Bottle videos: http://bit.ly/KleinBottles Buy glassware from Cliff (if you dare): https://www.kl
From playlist Klein Bottles on Numberphile
Quantifying nonorientability and filling multiples of embedded curves - Robert Young
Analysis Seminar Topic: Quantifying nonorientability and filling multiples of embedded curves Speaker: Robert Young Affiliation: New York University; von Neumann Fellow, School of Mathematics Date: October 5, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
Applied topology 4: An introduction to the torus and Klein bottle
Applied topology 4: An introduction to the torus and Klein bottle Abstract: We explain how the torus (the surface of the donut) is a 2-dimensional manifold that fits in 3D, whereas the Klein bottle is a 2-dimensional manifold that doesn't fit in 3D but does fit in 4D. This video accompan
From playlist Applied Topology - Henry Adams - 2021
Colloque d'histoire des sciences "Gaston Darboux (1842 - 1917)" - David Rowe - 17/11/17
En partenariat avec le séminaire d’histoire des mathématiques de l’IHP Wartime Memories of Gaston Darboux in Göttingen David Rowe, Université de Mayence, Allemagne À l’occasion du centenaire de la mort de Gaston Darboux, l’Institut Henri Poincaré souhaite retracer la figure du géomètre s
From playlist Colloque d'histoire des sciences "Gaston Darboux (1842 - 1917)" - 17/11/2017
Hypertwist: 2-sided Möbius strips and mirror universes
In this video the Mathologer sets out to track down the fabled 2-sided Möbius strips and Klein bottles inside some very exotic 3D universes. Also featuring 1-sided circles and cylinders and other strange mathematical creatures. Check out Jeffrey Weeks's amazing free "Torus Games" (play c
From playlist Recent videos
On Franco–German relations in mathematics, 1870–1920 – David Rowe – ICM2018
History of Mathematics Invited Lecture 19.1 On Franco–German relations in mathematics, 1870–1920 David Rowe Abstract: The first ICMs took place during a era when the longstanding rivalry between France and Germany strongly influenced European affairs. Relations between leading mathematic
From playlist History of Mathematics