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Odd-4 Graph, Fano Planes, and the Coxeter Graph

http://demonstrations.wolfram.com/Odd4GraphFanoPlanesAndTheCoxeterGraph/ The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. The odd-4 graph is constructed by taking triplets from {1, 2, 3, 4, 5, 6, 7}, then connecting the triplets that share no values. Ignoring the colored edges, the entire 35-vertex graph is the odd-4 graph. The odd-3 graph is the Petersen graph, and the od... Contributed by: Ed Pegg Jr

published: 11 Jul 2009
views: 710

Dimensiones by Alexandre Rodichevski

(en) Dimensions by Alexandre Rodichevski The dimension is the number of degrees of freedom available for movements in a space. The world we live in has three spatial dimensions. In mathematics spaces with any number of dimensions, even infinite, are used. (it) Dimensioni di Alexandre Rodichevski La dimensione è il numero di gradi di libertà disponibili per i movimenti in uno spazio. Nel mondo in cui viviamo disponiamo di tre dimensioni spaziali. In matematica vengono utilizzati gli spazi con qualsiasi numero di dimensioni, anche infinito. (ru) Измерения Александра Родишевского Измерение -- это число степеней свободы перемещения в пространстве. В мире, в котором мы живем, имеются три пространственных измерения. Математические теории используют пространства с любым число измерений, даже бесконечным. (lt) Dimensiones (gr) Διαστάσεις (en) The following images are used to create this work. License type is indicated in parentheses. "Suprematist Composition: White on White" by Kazimir Malevich (public domain) "Suprematist Painting: Eight Red Rectangles" by Kazimir Malevich (public domain) "Spatial Construction n. 12" by Aleksandr Rodchenko (CC BY-SA 3.0) "Monologue" by Elisha Ben Yitzhak (CC BY-SA 3.0) "Number line" by Andeggs (public domain) "Five point stencil illustration" by Oleg Alexandrov (public domain) "The unknot" by Jim.belk (public domain) "A trefoil knot" by Jim.belk (public domain) "A figure-eight knot" by Jim.belk (public domain) "A cinquefoil knot" by Jim.belk (public domain) "A three-twist (5_2) knot" by Jim.belk (public domain) "A stevedore knot" by Jim.belk (public domain) "A 6 2 knot" by Jim.belk (public domain) "A 6 3 knot" by Jim.belk (public domain) "Cartesian coordinate system" by K. Bolino (public domain) "A finite set of points in two-dimensional Euclidean space" by Jean-Luc W (CC BY-SA 3.0) "Sphere closed path" by Luca Antonelli (CC BY-SA 3.0) "Lorentzian wormhole" by AllenMcC (CC BY-SA 3.0) "3D coordinate system" by Sakurambo (CC BY-SA 3.0) "Illustration of a piecewise linear function" by Oleg Alexandrov (public domain) "Golden rectangles in an icosahedron" by Fropuff, Mysid (public domain) "Illustration of Bicubic interpolation on a dataset" by Berland (public domain) "4-cube Petrie projecton in 3D" by Jgmoxness (CC BY-SA 3.0) "Hydrogen Density Plots for n up to 4" by PoorLeno (public domain) "6-simplex family graph in A6 Coxeter plane" by Tomruen (public domain) "6-orthoplexl family graph in B6 Coxeter plane" by Tomruen (public domain) "6-orthoplexl family graph in B6 Coxeter plane" by Tomruen (public domain) "6-demicube family polytopes in Coxeter planes symmetry" by Tomruen (public domain) "2_21 polytope family polytopes in Coxeter plane symmetry" by Tomruen (public domain) "1_22 polytope family polytopes in Coxeter plane symmetry" by Tomruen (public domain) "Calabi-Yau quintic" by Andrew J. Hanson (CC BY-SA 3.0) "CMS Higgs-event" by Lucas Taylor (CC BY-SA 3.0) "Schwarzschild wormhole which bridges two different universes" by Alain r (CC BY-SA 3.0) "Messier 82" by NASA (public domain) The images of this video are subject of the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0). See http://creativecommons.org/licenses/by-sa/4.0/ Music © Alexandre Rodichevski http://www.morfoedro.it

published: 05 Apr 2014
views: 156

The Vieta Mapping for the Coxeter Group A_2

http://demonstrations.wolfram.com/TheVietaMappingForTheCoxeterGroupA_2 The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. This Demonstration shows the real part of the Vieta mapping, which is important in singularity theory. The domain is the plane x+y+z=0 containing the green hexagon; it has been rotated around the origin to make it horizontal for the purpose of the Demon ... Contributed by: Andrzej Kozlowski Audio created with WolframTones: http://tones.wolfram.com

published: 12 Nov 2012
views: 99

chilli plane spotting

via YouTube Capture

published: 08 Aug 2013
views: 38

Coxeter discusses the math behind Escher's circle limit

HSM Coxeter talks about the Poincare disk model of hyperbolic geometry. He also talks about complex inversions. Clip from the "fantastic wrld of Escher"

published: 27 Dec 2010
views: 14343

UHG52: Petrie polygons of a polyhedron

John Flinders Petrie was the son of the famed Egyptologist Flinders Petrie, and a good friend of Donald Coxeter. He discovered lovely polygonal paths on polytopes or polyhedra, that in the case of the Platonic solids have remarkable properties when we project these solids orthogonally onto particular planes. We also expand our view of the Platonic solids by considering more general tesselations of a surface, including the three famous regular tesselations of the plane. The Schlafli symbol of a tesselation is also introduced. These will have important consequences when we extend this thinking to the hyperbolic situation.

published: 29 Nov 2015
views: 1391

Lecture 1 . Coxeter Groups (Federico Ardila)

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogota, Colombia) Spring 2008 Partially supported by: US National Science Foundation CAREER Award DMS-0956178 US National Science Foundation Grant DMS-0801075 SFSU-Colombia Combinatorics Initiative.

published: 07 Sep 2012
views: 2604

MACS JO717 THE CRYSTALLINE GALAXY CLUSTER

THE SECRET OF MACS JO717 , HOW IT ALLOWS MASSIVE GALAXIES TO TRAVEL ALONG PATHWAYS PRE-DEFINED IN SOME TYPE OF CRYSTAL LIKE STRUCTURE HAS ALWAYS BEEN EXPLAINED SINCE H.S.M.COXETER PUBLISHED THE FIRST BOOK ON PROJECTIVE GEOMETRY. THE IDEALS ARE VERY CLEAR. PRESSURE BREAKS DOWN INTO FANO PLANES. FANO PLANES SPLIT MASS AND ENERGY. THERE ARE ONLY SO MANY WAYS YOU CAN SPLIT THE FANO PLANE ASSEMBLIES FROM AND INTO NEW GEOMETRIES. DETAILS MUST BE LEARNED WATCHING MY PARTICLE PHYSICS VIDEOS. DETAILS ON MACS JO717 CAN BE FOUND ELSEWHERE AND YOU CAN SEE FOR YOURSELF; THIS FAST GROWING SYSTEM OF GALAXIES IS ACTUALLY SHARING LARGE PARTICLES BETWEEN GALAXIES AND BY DOING SO CREATES THE CRYSTALLINE STRUCTURE IN WHICH THEY TRAVERSE !!!!!!!!!!!!!!!!!!!!

published: 17 Oct 2012
views: 363

Jason Behrstock: Random graphs and applications to Coxeter groups

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area Erdös and Rényi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then explain applications of these results to the geometry of Coxeter groups. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry. Recording during the thematic meeting: "GAGTA-9 : Geometric, asymptotic and combinatorial group theory and applications" the September 15, 2015 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Pascal Vichi

published: 19 Oct 2015
views: 131

Visualizing the sphere and the hyperbolic plane: five projections of each

Visualizing the sphere and then the hyperbolic plane under various azimuthal projections (five of each). spherical projections: 1. orthographic projection of the sphere (at 17s), 2. gnomonic projection of the sphere (at 1m16s), 3. stereographic projection of the sphere (at 2m33s), 4. Lambert azimuthal equal-area projection of the sphere (at 3m59s), 5. azimuthal equidistant projection of the sphere (at 5m23s); hyperbolic projections: 6. Poincaré disk model of the hyperbolic plane (at 6m27s), analogous to stereographic, 7. Beltrami--Klein model of the hyperbolic plane (at 8m54s), analogous to gnomonic, 8. pseudorthographic projection of the hyperbolic plane (at 10m19s), analogous to orthographic, 9. Lambert azimuthal equal-area projection of the hyperbolic plane (at 11m20s), 10. azimuthal equidistant projection of the hyperbolic plane (at 11m55s). The spherical projections show the sphere with a dodecahedral tiling by regular pentagons; the hyperbolic projections show the hyperbolic plane with a uniform tiling by regular heptagons. The program used to compute the frames is available at http://www.madore.org/~david/programs/#prog_projections For the same projections of the Earth's surface, see: http://www.youtube.com/watch?v=LKcTbIsWS9c

published: 19 Nov 2013
views: 17316

Coxeter element

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.

Definitions

Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order.

There are many different ways to define the Coxeter number h of an irreducible root system.

A Coxeter element is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order.

The Coxeter number is the number of roots divided by the rank. The number of reflections in the Coxeter group is half the number of roots.

The Coxeter number is the order of any Coxeter element;.

If the highest root is ∑m_iα_i for simple roots α_i, then the Coxeter number is 1 + ∑m_i

The Coxeter number is the dimension of the corresponding Lie algebra is n(h + 1), where n is the rank and h is the Coxeter number.

Read more...

This page contains text from Wikipedia, the Free Encyclopedia - http://wn.com/Coxeter_element

This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.

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0:12

Odd-4 Graph, Fano Planes, and the Coxeter Graph

Odd-4 Graph, Fano Planes, and the Coxeter Graph

Odd-4 Graph, Fano Planes, and the Coxeter Graph
http://demonstrations.wolfram.com/Odd4GraphFanoPlanesAndTheCoxeterGraph/ The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. The odd-4 graph is constructed by taking triplets from {1, 2, 3, 4, 5, 6, 7}, then connecting the triplets that share no values. Ignoring the colored edges, the entire 35-vertex graph is the odd-4 graph. The odd-3 graph is the Petersen graph, and the od... Contributed by: Ed Pegg Jr
3:36

Dimensiones by Alexandre Rodichevski

Dimensiones by Alexandre Rodichevski

Dimensiones by Alexandre Rodichevski
(en) Dimensions by Alexandre Rodichevski The dimension is the number of degrees of freedom available for movements in a space. The world we live in has three spatial dimensions. In mathematics spaces with any number of dimensions, even infinite, are used. (it) Dimensioni di Alexandre Rodichevski La dimensione è il numero di gradi di libertà disponibili per i movimenti in uno spazio. Nel mondo in cui viviamo disponiamo di tre dimensioni spaziali. In matematica vengono utilizzati gli spazi con qualsiasi numero di dimensioni, anche infinito. (ru) Измерения Александра Родишевского Измерение -- это число степеней свободы перемещения в пространстве. В мире, в котором мы живем, имеются три пространственных измерения. Математические теории используют пространства с любым число измерений, даже бесконечным. (lt) Dimensiones (gr) Διαστάσεις (en) The following images are used to create this work. License type is indicated in parentheses. "Suprematist Composition: White on White" by Kazimir Malevich (public domain) "Suprematist Painting: Eight Red Rectangles" by Kazimir Malevich (public domain) "Spatial Construction n. 12" by Aleksandr Rodchenko (CC BY-SA 3.0) "Monologue" by Elisha Ben Yitzhak (CC BY-SA 3.0) "Number line" by Andeggs (public domain) "Five point stencil illustration" by Oleg Alexandrov (public domain) "The unknot" by Jim.belk (public domain) "A trefoil knot" by Jim.belk (public domain) "A figure-eight knot" by Jim.belk (public domain) "A cinquefoil knot" by Jim.belk (public domain) "A three-twist (5_2) knot" by Jim.belk (public domain) "A stevedore knot" by Jim.belk (public domain) "A 6 2 knot" by Jim.belk (public domain) "A 6 3 knot" by Jim.belk (public domain) "Cartesian coordinate system" by K. Bolino (public domain) "A finite set of points in two-dimensional Euclidean space" by Jean-Luc W (CC BY-SA 3.0) "Sphere closed path" by Luca Antonelli (CC BY-SA 3.0) "Lorentzian wormhole" by AllenMcC (CC BY-SA 3.0) "3D coordinate system" by Sakurambo (CC BY-SA 3.0) "Illustration of a piecewise linear function" by Oleg Alexandrov (public domain) "Golden rectangles in an icosahedron" by Fropuff, Mysid (public domain) "Illustration of Bicubic interpolation on a dataset" by Berland (public domain) "4-cube Petrie projecton in 3D" by Jgmoxness (CC BY-SA 3.0) "Hydrogen Density Plots for n up to 4" by PoorLeno (public domain) "6-simplex family graph in A6 Coxeter plane" by Tomruen (public domain) "6-orthoplexl family graph in B6 Coxeter plane" by Tomruen (public domain) "6-orthoplexl family graph in B6 Coxeter plane" by Tomruen (public domain) "6-demicube family polytopes in Coxeter planes symmetry" by Tomruen (public domain) "2_21 polytope family polytopes in Coxeter plane symmetry" by Tomruen (public domain) "1_22 polytope family polytopes in Coxeter plane symmetry" by Tomruen (public domain) "Calabi-Yau quintic" by Andrew J. Hanson (CC BY-SA 3.0) "CMS Higgs-event" by Lucas Taylor (CC BY-SA 3.0) "Schwarzschild wormhole which bridges two different universes" by Alain r (CC BY-SA 3.0) "Messier 82" by NASA (public domain) The images of this video are subject of the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0). See http://creativecommons.org/licenses/by-sa/4.0/ Music © Alexandre Rodichevski http://www.morfoedro.it
0:21

The Vieta Mapping for the Coxeter Group A_2

The Vieta Mapping for the Coxeter Group A_2

The Vieta Mapping for the Coxeter Group A_2
http://demonstrations.wolfram.com/TheVietaMappingForTheCoxeterGroupA_2 The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. This Demonstration shows the real part of the Vieta mapping, which is important in singularity theory. The domain is the plane x+y+z=0 containing the green hexagon; it has been rotated around the origin to make it horizontal for the purpose of the Demon ... Contributed by: Andrzej Kozlowski Audio created with WolframTones: http://tones.wolfram.com
0:49

chilli plane spotting

chilli plane spotting

chilli plane spotting
via YouTube Capture
7:59

Coxeter discusses the math behind Escher's circle limit

Coxeter discusses the math behind Escher's circle limit

Coxeter discusses the math behind Escher's circle limit
HSM Coxeter talks about the Poincare disk model of hyperbolic geometry. He also talks about complex inversions. Clip from the "fantastic wrld of Escher"
26:25

UHG52: Petrie polygons of a polyhedron

UHG52: Petrie polygons of a polyhedron

UHG52: Petrie polygons of a polyhedron
John Flinders Petrie was the son of the famed Egyptologist Flinders Petrie, and a good friend of Donald Coxeter. He discovered lovely polygonal paths on polytopes or polyhedra, that in the case of the Platonic solids have remarkable properties when we project these solids orthogonally onto particular planes. We also expand our view of the Platonic solids by considering more general tesselations of a surface, including the three famous regular tesselations of the plane. The Schlafli symbol of a tesselation is also introduced. These will have important consequences when we extend this thinking to the hyperbolic situation.
38:54

Lecture 1 . Coxeter Groups (Federico Ardila)

Lecture 1 . Coxeter Groups (Federico Ardila)

Lecture 1 . Coxeter Groups (Federico Ardila)
Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogota, Colombia) Spring 2008 Partially supported by: US National Science Foundation CAREER Award DMS-0956178 US National Science Foundation Grant DMS-0801075 SFSU-Colombia Combinatorics Initiative.
1:20

MACS JO717 THE CRYSTALLINE GALAXY CLUSTER

MACS JO717 THE CRYSTALLINE GALAXY CLUSTER

MACS JO717 THE CRYSTALLINE GALAXY CLUSTER
THE SECRET OF MACS JO717 , HOW IT ALLOWS MASSIVE GALAXIES TO TRAVEL ALONG PATHWAYS PRE-DEFINED IN SOME TYPE OF CRYSTAL LIKE STRUCTURE HAS ALWAYS BEEN EXPLAINED SINCE H.S.M.COXETER PUBLISHED THE FIRST BOOK ON PROJECTIVE GEOMETRY. THE IDEALS ARE VERY CLEAR. PRESSURE BREAKS DOWN INTO FANO PLANES. FANO PLANES SPLIT MASS AND ENERGY. THERE ARE ONLY SO MANY WAYS YOU CAN SPLIT THE FANO PLANE ASSEMBLIES FROM AND INTO NEW GEOMETRIES. DETAILS MUST BE LEARNED WATCHING MY PARTICLE PHYSICS VIDEOS. DETAILS ON MACS JO717 CAN BE FOUND ELSEWHERE AND YOU CAN SEE FOR YOURSELF; THIS FAST GROWING SYSTEM OF GALAXIES IS ACTUALLY SHARING LARGE PARTICLES BETWEEN GALAXIES AND BY DOING SO CREATES THE CRYSTALLINE STRUCTURE IN WHICH THEY TRAVERSE !!!!!!!!!!!!!!!!!!!!
58:23

Jason Behrstock: Random graphs and applications to Coxeter groups

Jason Behrstock: Random graphs and applications to Coxeter groups

Jason Behrstock: Random graphs and applications to Coxeter groups
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area Erdös and Rényi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then explain applications of these results to the geometry of Coxeter groups. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry. Recording during the thematic meeting: "GAGTA-9 : Geometric, asymptotic and combinatorial group theory and applications" the September 15, 2015 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Pascal Vichi
12:47

Visualizing the sphere and the hyperbolic plane: five projections of each

Visualizing the sphere and the hyperbolic plane: five projections of each

Visualizing the sphere and the hyperbolic plane: five projections of each
Visualizing the sphere and then the hyperbolic plane under various azimuthal projections (five of each). spherical projections: 1. orthographic projection of the sphere (at 17s), 2. gnomonic projection of the sphere (at 1m16s), 3. stereographic projection of the sphere (at 2m33s), 4. Lambert azimuthal equal-area projection of the sphere (at 3m59s), 5. azimuthal equidistant projection of the sphere (at 5m23s); hyperbolic projections: 6. Poincaré disk model of the hyperbolic plane (at 6m27s), analogous to stereographic, 7. Beltrami--Klein model of the hyperbolic plane (at 8m54s), analogous to gnomonic, 8. pseudorthographic projection of the hyperbolic plane (at 10m19s), analogous to orthographic, 9. Lambert azimuthal equal-area projection of the hyperbolic plane (at 11m20s), 10. azimuthal equidistant projection of the hyperbolic plane (at 11m55s). The spherical projections show the sphere with a dodecahedral tiling by regular pentagons; the hyperbolic projections show the hyperbolic plane with a uniform tiling by regular heptagons. The program used to compute the frames is available at http://www.madore.org/~david/programs/#prog_projections For the same projections of the Earth's surface, see: http://www.youtube.com/watch?v=LKcTbIsWS9c

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Send this Playlist by SMS Email this Page Play All in FS Show More Related Videos

Send this Playlist by SMS Email this Page Play All in FS Show More Related Videos

Odd-4 Graph, Fano Planes, and the Coxeter Graph
http://demonstrations.wolfram.com/Odd4GraphFanoPlanesAndTheCoxeterGraph/ The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. The odd-4 graph is constructed by taking triplets from {1, 2, 3, 4, 5, 6, 7}, then connecting the triplets that share no values. Ignoring the colored edges, the entire 35-vertex graph is the odd-4 graph. The odd-3 graph is the Petersen graph, and the od... Contributed by: Ed Pegg Jr
Dimensiones by Alexandre Rodichevski
(en) Dimensions by Alexandre Rodichevski The dimension is the number of degrees of freedom available for movements in a space. The world we live in has three spatial dimensions. In mathematics spaces with any number of dimensions, even infinite, are used. (it) Dimensioni di Alexandre Rodichevski La dimensione è il numero di gradi di libertà disponibili per i movimenti in uno spazio. Nel mondo in cui viviamo disponiamo di tre dimensioni spaziali. In matematica vengono utilizzati gli spazi con qualsiasi numero di dimensioni, anche infinito. (ru) Измерения Александра Родишевского Измерение -- это число степеней свободы перемещения в пространстве. В мире, в котором мы живем, имеются три пространственных измерения. Математические теории используют пространства с любым число измерений, даже бе...
The Vieta Mapping for the Coxeter Group A_2
http://demonstrations.wolfram.com/TheVietaMappingForTheCoxeterGroupA_2 The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. This Demonstration shows the real part of the Vieta mapping, which is important in singularity theory. The domain is the plane x+y+z=0 containing the green hexagon; it has been rotated around the origin to make it horizontal for the purpose of the Demon ... Contributed by: Andrzej Kozlowski Audio created with WolframTones: http://tones.wolfram.com
chilli plane spotting
via YouTube Capture
Coxeter discusses the math behind Escher's circle limit
HSM Coxeter talks about the Poincare disk model of hyperbolic geometry. He also talks about complex inversions. Clip from the "fantastic wrld of Escher"
UHG52: Petrie polygons of a polyhedron
John Flinders Petrie was the son of the famed Egyptologist Flinders Petrie, and a good friend of Donald Coxeter. He discovered lovely polygonal paths on polytopes or polyhedra, that in the case of the Platonic solids have remarkable properties when we project these solids orthogonally onto particular planes. We also expand our view of the Platonic solids by considering more general tesselations of a surface, including the three famous regular tesselations of the plane. The Schlafli symbol of a tesselation is also introduced. These will have important consequences when we extend this thinking to the hyperbolic situation.
Lecture 1 . Coxeter Groups (Federico Ardila)
Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogota, Colombia) Spring 2008 Partially supported by: US National Science Foundation CAREER Award DMS-0956178 US National Science Foundation Grant DMS-0801075 SFSU-Colombia Combinatorics Initiative.
MACS JO717 THE CRYSTALLINE GALAXY CLUSTER
THE SECRET OF MACS JO717 , HOW IT ALLOWS MASSIVE GALAXIES TO TRAVEL ALONG PATHWAYS PRE-DEFINED IN SOME TYPE OF CRYSTAL LIKE STRUCTURE HAS ALWAYS BEEN EXPLAINED SINCE H.S.M.COXETER PUBLISHED THE FIRST BOOK ON PROJECTIVE GEOMETRY. THE IDEALS ARE VERY CLEAR. PRESSURE BREAKS DOWN INTO FANO PLANES. FANO PLANES SPLIT MASS AND ENERGY. THERE ARE ONLY SO MANY WAYS YOU CAN SPLIT THE FANO PLANE ASSEMBLIES FROM AND INTO NEW GEOMETRIES. DETAILS MUST BE LEARNED WATCHING MY PARTICLE PHYSICS VIDEOS. DETAILS ON MACS JO717 CAN BE FOUND ELSEWHERE AND YOU CAN SEE FOR YOURSELF; THIS FAST GROWING SYSTEM OF GALAXIES IS ACTUALLY SHARING LARGE PARTICLES BETWEEN GALAXIES AND BY DOING SO CREATES THE CRYSTALLINE STRUCTURE IN WHICH THEY TRAVERSE !!!!!!!!!!!!!!!!!!!!
Jason Behrstock: Random graphs and applications to Coxeter groups
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area Erdös and Rényi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then explain applications of these res...
Visualizing the sphere and the hyperbolic plane: five projections of each
Visualizing the sphere and then the hyperbolic plane under various azimuthal projections (five of each). spherical projections: 1. orthographic projection of the sphere (at 17s), 2. gnomonic projection of the sphere (at 1m16s), 3. stereographic projection of the sphere (at 2m33s), 4. Lambert azimuthal equal-area projection of the sphere (at 3m59s), 5. azimuthal equidistant projection of the sphere (at 5m23s); hyperbolic projections: 6. Poincaré disk model of the hyperbolic plane (at 6m27s), analogous to stereographic, 7. Beltrami--Klein model of the hyperbolic plane (at 8m54s), analogous to gnomonic, 8. pseudorthographic projection of the hyperbolic plane (at 10m19s), analogous to orthographic, 9. Lambert azimuthal equal-area projection of the hyperbolic plane (at 11m20s), 10. ...

Odd-4 Graph, Fano Planes, and the Coxeter Graph

Odd-4 Graph, Fano Planes, and the Coxeter Graph

Order: Reorder
Duration: 0:12
Updated: 11 Jul 2009
views: 710

videos

http://demonstrations.wolfram.com/Odd4GraphFanoPlanesAndTheCoxeterGraph/ The Wolfram Demonstrations Project contains thousands of free interactive visualizatio...

http://demonstrations.wolfram.com/Odd4GraphFanoPlanesAndTheCoxeterGraph/ The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. The odd-4 graph is constructed by taking triplets from {1, 2, 3, 4, 5, 6, 7}, then connecting the triplets that share no values. Ignoring the colored edges, the entire 35-vertex graph is the odd-4 graph. The odd-3 graph is the Petersen graph, and the od... Contributed by: Ed Pegg Jr

wn.com/Odd 4 Graph, Fano Planes, And The Coxeter Graph

http://demonstrations.wolfram.com/Odd4GraphFanoPlanesAndTheCoxeterGraph/ The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. The odd-4 graph is constructed by taking triplets from {1, 2, 3, 4, 5, 6, 7}, then connecting the triplets that share no values. Ignoring the colored edges, the entire 35-vertex graph is the odd-4 graph. The odd-3 graph is the Petersen graph, and the od... Contributed by: Ed Pegg Jr

published: 11 Jul 2009
views: 710

Dimensiones by Alexandre Rodichevski

Dimensiones by Alexandre Rodichevski

Order: Reorder
Duration: 3:36
Updated: 05 Apr 2014
views: 156

videos

(en) Dimensions by Alexandre Rodichevski The dimension is the number of degrees of freedom available for movements in a space. The world we live in has three sp...

(en) Dimensions by Alexandre Rodichevski The dimension is the number of degrees of freedom available for movements in a space. The world we live in has three spatial dimensions. In mathematics spaces with any number of dimensions, even infinite, are used. (it) Dimensioni di Alexandre Rodichevski La dimensione è il numero di gradi di libertà disponibili per i movimenti in uno spazio. Nel mondo in cui viviamo disponiamo di tre dimensioni spaziali. In matematica vengono utilizzati gli spazi con qualsiasi numero di dimensioni, anche infinito. (ru) Измерения Александра Родишевского Измерение -- это число степеней свободы перемещения в пространстве. В мире, в котором мы живем, имеются три пространственных измерения. Математические теории используют пространства с любым число измерений, даже бесконечным. (lt) Dimensiones (gr) Διαστάσεις (en) The following images are used to create this work. License type is indicated in parentheses. "Suprematist Composition: White on White" by Kazimir Malevich (public domain) "Suprematist Painting: Eight Red Rectangles" by Kazimir Malevich (public domain) "Spatial Construction n. 12" by Aleksandr Rodchenko (CC BY-SA 3.0) "Monologue" by Elisha Ben Yitzhak (CC BY-SA 3.0) "Number line" by Andeggs (public domain) "Five point stencil illustration" by Oleg Alexandrov (public domain) "The unknot" by Jim.belk (public domain) "A trefoil knot" by Jim.belk (public domain) "A figure-eight knot" by Jim.belk (public domain) "A cinquefoil knot" by Jim.belk (public domain) "A three-twist (5_2) knot" by Jim.belk (public domain) "A stevedore knot" by Jim.belk (public domain) "A 6 2 knot" by Jim.belk (public domain) "A 6 3 knot" by Jim.belk (public domain) "Cartesian coordinate system" by K. Bolino (public domain) "A finite set of points in two-dimensional Euclidean space" by Jean-Luc W (CC BY-SA 3.0) "Sphere closed path" by Luca Antonelli (CC BY-SA 3.0) "Lorentzian wormhole" by AllenMcC (CC BY-SA 3.0) "3D coordinate system" by Sakurambo (CC BY-SA 3.0) "Illustration of a piecewise linear function" by Oleg Alexandrov (public domain) "Golden rectangles in an icosahedron" by Fropuff, Mysid (public domain) "Illustration of Bicubic interpolation on a dataset" by Berland (public domain) "4-cube Petrie projecton in 3D" by Jgmoxness (CC BY-SA 3.0) "Hydrogen Density Plots for n up to 4" by PoorLeno (public domain) "6-simplex family graph in A6 Coxeter plane" by Tomruen (public domain) "6-orthoplexl family graph in B6 Coxeter plane" by Tomruen (public domain) "6-orthoplexl family graph in B6 Coxeter plane" by Tomruen (public domain) "6-demicube family polytopes in Coxeter planes symmetry" by Tomruen (public domain) "2_21 polytope family polytopes in Coxeter plane symmetry" by Tomruen (public domain) "1_22 polytope family polytopes in Coxeter plane symmetry" by Tomruen (public domain) "Calabi-Yau quintic" by Andrew J. Hanson (CC BY-SA 3.0) "CMS Higgs-event" by Lucas Taylor (CC BY-SA 3.0) "Schwarzschild wormhole which bridges two different universes" by Alain r (CC BY-SA 3.0) "Messier 82" by NASA (public domain) The images of this video are subject of the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0). See http://creativecommons.org/licenses/by-sa/4.0/ Music © Alexandre Rodichevski http://www.morfoedro.it

wn.com/Dimensiones By Alexandre Rodichevski

(en) Dimensions by Alexandre Rodichevski The dimension is the number of degrees of freedom available for movements in a space. The world we live in has three spatial dimensions. In mathematics spaces with any number of dimensions, even infinite, are used. (it) Dimensioni di Alexandre Rodichevski La dimensione è il numero di gradi di libertà disponibili per i movimenti in uno spazio. Nel mondo in cui viviamo disponiamo di tre dimensioni spaziali. In matematica vengono utilizzati gli spazi con qualsiasi numero di dimensioni, anche infinito. (ru) Измерения Александра Родишевского Измерение -- это число степеней свободы перемещения в пространстве. В мире, в котором мы живем, имеются три пространственных измерения. Математические теории используют пространства с любым число измерений, даже бесконечным. (lt) Dimensiones (gr) Διαστάσεις (en) The following images are used to create this work. License type is indicated in parentheses. "Suprematist Composition: White on White" by Kazimir Malevich (public domain) "Suprematist Painting: Eight Red Rectangles" by Kazimir Malevich (public domain) "Spatial Construction n. 12" by Aleksandr Rodchenko (CC BY-SA 3.0) "Monologue" by Elisha Ben Yitzhak (CC BY-SA 3.0) "Number line" by Andeggs (public domain) "Five point stencil illustration" by Oleg Alexandrov (public domain) "The unknot" by Jim.belk (public domain) "A trefoil knot" by Jim.belk (public domain) "A figure-eight knot" by Jim.belk (public domain) "A cinquefoil knot" by Jim.belk (public domain) "A three-twist (5_2) knot" by Jim.belk (public domain) "A stevedore knot" by Jim.belk (public domain) "A 6 2 knot" by Jim.belk (public domain) "A 6 3 knot" by Jim.belk (public domain) "Cartesian coordinate system" by K. Bolino (public domain) "A finite set of points in two-dimensional Euclidean space" by Jean-Luc W (CC BY-SA 3.0) "Sphere closed path" by Luca Antonelli (CC BY-SA 3.0) "Lorentzian wormhole" by AllenMcC (CC BY-SA 3.0) "3D coordinate system" by Sakurambo (CC BY-SA 3.0) "Illustration of a piecewise linear function" by Oleg Alexandrov (public domain) "Golden rectangles in an icosahedron" by Fropuff, Mysid (public domain) "Illustration of Bicubic interpolation on a dataset" by Berland (public domain) "4-cube Petrie projecton in 3D" by Jgmoxness (CC BY-SA 3.0) "Hydrogen Density Plots for n up to 4" by PoorLeno (public domain) "6-simplex family graph in A6 Coxeter plane" by Tomruen (public domain) "6-orthoplexl family graph in B6 Coxeter plane" by Tomruen (public domain) "6-orthoplexl family graph in B6 Coxeter plane" by Tomruen (public domain) "6-demicube family polytopes in Coxeter planes symmetry" by Tomruen (public domain) "2_21 polytope family polytopes in Coxeter plane symmetry" by Tomruen (public domain) "1_22 polytope family polytopes in Coxeter plane symmetry" by Tomruen (public domain) "Calabi-Yau quintic" by Andrew J. Hanson (CC BY-SA 3.0) "CMS Higgs-event" by Lucas Taylor (CC BY-SA 3.0) "Schwarzschild wormhole which bridges two different universes" by Alain r (CC BY-SA 3.0) "Messier 82" by NASA (public domain) The images of this video are subject of the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0). See http://creativecommons.org/licenses/by-sa/4.0/ Music © Alexandre Rodichevski http://www.morfoedro.it

published: 05 Apr 2014
views: 156

The Vieta Mapping for the Coxeter Group A_2

The Vieta Mapping for the Coxeter Group A_2

Order: Reorder
Duration: 0:21
Updated: 12 Nov 2012
views: 99

videos

http://demonstrations.wolfram.com/TheVietaMappingForTheCoxeterGroupA_2 The Wolfram Demonstrations Project contains thousands of free interactive visualizations...

http://demonstrations.wolfram.com/TheVietaMappingForTheCoxeterGroupA_2 The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. This Demonstration shows the real part of the Vieta mapping, which is important in singularity theory. The domain is the plane x+y+z=0 containing the green hexagon; it has been rotated around the origin to make it horizontal for the purpose of the Demon ... Contributed by: Andrzej Kozlowski Audio created with WolframTones: http://tones.wolfram.com

wn.com/The Vieta Mapping For The Coxeter Group A 2

http://demonstrations.wolfram.com/TheVietaMappingForTheCoxeterGroupA_2 The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. This Demonstration shows the real part of the Vieta mapping, which is important in singularity theory. The domain is the plane x+y+z=0 containing the green hexagon; it has been rotated around the origin to make it horizontal for the purpose of the Demon ... Contributed by: Andrzej Kozlowski Audio created with WolframTones: http://tones.wolfram.com

published: 12 Nov 2012
views: 99

chilli plane spotting

chilli plane spotting

Order: Reorder
Duration: 0:49
Updated: 08 Aug 2013
views: 38

videos

via YouTube Capture

via YouTube Capture

wn.com/Chilli Plane Spotting

via YouTube Capture

published: 08 Aug 2013
views: 38

Coxeter discusses the math behind Escher's circle limit

Coxeter discusses the math behind Escher's circle limit

Order: Reorder
Duration: 7:59
Updated: 27 Dec 2010
views: 14343

videos

HSM Coxeter talks about the Poincare disk model of hyperbolic geometry. He also talks about complex inversions. Clip from the "fantastic wrld of Escher"

HSM Coxeter talks about the Poincare disk model of hyperbolic geometry. He also talks about complex inversions. Clip from the "fantastic wrld of Escher"

wn.com/Coxeter Discusses The Math Behind Escher's Circle Limit

HSM Coxeter talks about the Poincare disk model of hyperbolic geometry. He also talks about complex inversions. Clip from the "fantastic wrld of Escher"

published: 27 Dec 2010
views: 14343

UHG52: Petrie polygons of a polyhedron

UHG52: Petrie polygons of a polyhedron

Order: Reorder
Duration: 26:25
Updated: 29 Nov 2015
views: 1391

videos

John Flinders Petrie was the son of the famed Egyptologist Flinders Petrie, and a good friend of Donald Coxeter. He discovered lovely polygonal paths on polytop...

John Flinders Petrie was the son of the famed Egyptologist Flinders Petrie, and a good friend of Donald Coxeter. He discovered lovely polygonal paths on polytopes or polyhedra, that in the case of the Platonic solids have remarkable properties when we project these solids orthogonally onto particular planes. We also expand our view of the Platonic solids by considering more general tesselations of a surface, including the three famous regular tesselations of the plane. The Schlafli symbol of a tesselation is also introduced. These will have important consequences when we extend this thinking to the hyperbolic situation.

wn.com/Uhg52 Petrie Polygons Of A Polyhedron

John Flinders Petrie was the son of the famed Egyptologist Flinders Petrie, and a good friend of Donald Coxeter. He discovered lovely polygonal paths on polytopes or polyhedra, that in the case of the Platonic solids have remarkable properties when we project these solids orthogonally onto particular planes. We also expand our view of the Platonic solids by considering more general tesselations of a surface, including the three famous regular tesselations of the plane. The Schlafli symbol of a tesselation is also introduced. These will have important consequences when we extend this thinking to the hyperbolic situation.

published: 29 Nov 2015
views: 1391

Lecture 1 . Coxeter Groups (Federico Ardila)

Lecture 1 . Coxeter Groups (Federico Ardila)

Order: Reorder
Duration: 38:54
Updated: 07 Sep 2012
views: 2604

videos

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogot ...

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogota, Colombia) Spring 2008 Partially supported by: US National Science Foundation CAREER Award DMS-0956178 US National Science Foundation Grant DMS-0801075 SFSU-Colombia Combinatorics Initiative.

wn.com/Lecture 1 . Coxeter Groups (Federico Ardila)

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogota, Colombia) Spring 2008 Partially supported by: US National Science Foundation CAREER Award DMS-0956178 US National Science Foundation Grant DMS-0801075 SFSU-Colombia Combinatorics Initiative.

published: 07 Sep 2012
views: 2604

MACS JO717 THE CRYSTALLINE GALAXY CLUSTER

MACS JO717 THE CRYSTALLINE GALAXY CLUSTER

Order: Reorder
Duration: 1:20
Updated: 17 Oct 2012
views: 363

videos

THE SECRET OF MACS JO717 , HOW IT ALLOWS MASSIVE GALAXIES TO TRAVEL ALONG PATHWAYS PRE-DEFINED IN SOME TYPE OF CRYSTAL LIKE STRUCTURE HAS ALWAYS BEEN EXPLAINED ...

THE SECRET OF MACS JO717 , HOW IT ALLOWS MASSIVE GALAXIES TO TRAVEL ALONG PATHWAYS PRE-DEFINED IN SOME TYPE OF CRYSTAL LIKE STRUCTURE HAS ALWAYS BEEN EXPLAINED SINCE H.S.M.COXETER PUBLISHED THE FIRST BOOK ON PROJECTIVE GEOMETRY. THE IDEALS ARE VERY CLEAR. PRESSURE BREAKS DOWN INTO FANO PLANES. FANO PLANES SPLIT MASS AND ENERGY. THERE ARE ONLY SO MANY WAYS YOU CAN SPLIT THE FANO PLANE ASSEMBLIES FROM AND INTO NEW GEOMETRIES. DETAILS MUST BE LEARNED WATCHING MY PARTICLE PHYSICS VIDEOS. DETAILS ON MACS JO717 CAN BE FOUND ELSEWHERE AND YOU CAN SEE FOR YOURSELF; THIS FAST GROWING SYSTEM OF GALAXIES IS ACTUALLY SHARING LARGE PARTICLES BETWEEN GALAXIES AND BY DOING SO CREATES THE CRYSTALLINE STRUCTURE IN WHICH THEY TRAVERSE !!!!!!!!!!!!!!!!!!!!

wn.com/Macs Jo717 The Crystalline Galaxy Cluster

THE SECRET OF MACS JO717 , HOW IT ALLOWS MASSIVE GALAXIES TO TRAVEL ALONG PATHWAYS PRE-DEFINED IN SOME TYPE OF CRYSTAL LIKE STRUCTURE HAS ALWAYS BEEN EXPLAINED SINCE H.S.M.COXETER PUBLISHED THE FIRST BOOK ON PROJECTIVE GEOMETRY. THE IDEALS ARE VERY CLEAR. PRESSURE BREAKS DOWN INTO FANO PLANES. FANO PLANES SPLIT MASS AND ENERGY. THERE ARE ONLY SO MANY WAYS YOU CAN SPLIT THE FANO PLANE ASSEMBLIES FROM AND INTO NEW GEOMETRIES. DETAILS MUST BE LEARNED WATCHING MY PARTICLE PHYSICS VIDEOS. DETAILS ON MACS JO717 CAN BE FOUND ELSEWHERE AND YOU CAN SEE FOR YOURSELF; THIS FAST GROWING SYSTEM OF GALAXIES IS ACTUALLY SHARING LARGE PARTICLES BETWEEN GALAXIES AND BY DOING SO CREATES THE CRYSTALLINE STRUCTURE IN WHICH THEY TRAVERSE !!!!!!!!!!!!!!!!!!!!

published: 17 Oct 2012
views: 363

Jason Behrstock: Random graphs and applications to Coxeter groups

Jason Behrstock: Random graphs and applications to Coxeter groups

Order: Reorder
Duration: 58:23
Updated: 19 Oct 2015
views: 131

videos

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its f...

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area Erdös and Rényi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then explain applications of these results to the geometry of Coxeter groups. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry. Recording during the thematic meeting: "GAGTA-9 : Geometric, asymptotic and combinatorial group theory and applications" the September 15, 2015 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Pascal Vichi

wn.com/Jason Behrstock Random Graphs And Applications To Coxeter Groups

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area Erdös and Rényi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then explain applications of these results to the geometry of Coxeter groups. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry. Recording during the thematic meeting: "GAGTA-9 : Geometric, asymptotic and combinatorial group theory and applications" the September 15, 2015 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Pascal Vichi

published: 19 Oct 2015
views: 131

Visualizing the sphere and the hyperbolic plane: five projections of each

Visualizing the sphere and the hyperbolic plane: five projections of each

Order: Reorder
Duration: 12:47
Updated: 19 Nov 2013
views: 17316

videos

Visualizing the sphere and then the hyperbolic plane under various azimuthal projections (five of each). spherical projections: 1. orthographic projection of ...

Visualizing the sphere and then the hyperbolic plane under various azimuthal projections (five of each). spherical projections: 1. orthographic projection of the sphere (at 17s), 2. gnomonic projection of the sphere (at 1m16s), 3. stereographic projection of the sphere (at 2m33s), 4. Lambert azimuthal equal-area projection of the sphere (at 3m59s), 5. azimuthal equidistant projection of the sphere (at 5m23s); hyperbolic projections: 6. Poincaré disk model of the hyperbolic plane (at 6m27s), analogous to stereographic, 7. Beltrami--Klein model of the hyperbolic plane (at 8m54s), analogous to gnomonic, 8. pseudorthographic projection of the hyperbolic plane (at 10m19s), analogous to orthographic, 9. Lambert azimuthal equal-area projection of the hyperbolic plane (at 11m20s), 10. azimuthal equidistant projection of the hyperbolic plane (at 11m55s). The spherical projections show the sphere with a dodecahedral tiling by regular pentagons; the hyperbolic projections show the hyperbolic plane with a uniform tiling by regular heptagons. The program used to compute the frames is available at http://www.madore.org/~david/programs/#prog_projections For the same projections of the Earth's surface, see: http://www.youtube.com/watch?v=LKcTbIsWS9c

wn.com/Visualizing The Sphere And The Hyperbolic Plane Five Projections Of Each

Visualizing the sphere and then the hyperbolic plane under various azimuthal projections (five of each). spherical projections: 1. orthographic projection of the sphere (at 17s), 2. gnomonic projection of the sphere (at 1m16s), 3. stereographic projection of the sphere (at 2m33s), 4. Lambert azimuthal equal-area projection of the sphere (at 3m59s), 5. azimuthal equidistant projection of the sphere (at 5m23s); hyperbolic projections: 6. Poincaré disk model of the hyperbolic plane (at 6m27s), analogous to stereographic, 7. Beltrami--Klein model of the hyperbolic plane (at 8m54s), analogous to gnomonic, 8. pseudorthographic projection of the hyperbolic plane (at 10m19s), analogous to orthographic, 9. Lambert azimuthal equal-area projection of the hyperbolic plane (at 11m20s), 10. azimuthal equidistant projection of the hyperbolic plane (at 11m55s). The spherical projections show the sphere with a dodecahedral tiling by regular pentagons; the hyperbolic projections show the hyperbolic plane with a uniform tiling by regular heptagons. The program used to compute the frames is available at http://www.madore.org/~david/programs/#prog_projections For the same projections of the Earth's surface, see: http://www.youtube.com/watch?v=LKcTbIsWS9c

published: 19 Nov 2013
views: 17316

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Lecture 3 . Coxeter Groups (Federico Ardila)
Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogota, Colombia) Spring 2008 Partially supported by: US National Science Foundation CAREER Award DMS-0956178 US National Science Foundation Grant DMS-0801075 SFSU-Colombia Combinatorics Initiative.
FPG3: Axioms, Desargue, The Fano Plane
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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...
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Cédric Bonnafé: Calogero-Moser cellular characters : the smooth case
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area Using the representation theory of Cherednik algebra at t= 0, we define a family of "Calogero-Moser cellular characters" for any complex reflection group W. Whenever W is a Coxeter group, we conjecture that they coincide with the "Kazhdan-Lusztig cellular characters". We shall give some evidences for this conjecture. Our main result is that, whenever the associated Calogero-Moser space is s...

Lecture 3 . Coxeter Groups (Federico Ardila)

Lecture 3 . Coxeter Groups (Federico Ardila)

Order: Reorder
Duration: 54:02
Updated: 07 Sep 2012
views: 796

videos

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogot ...

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogota, Colombia) Spring 2008 Partially supported by: US National Science Foundation CAREER Award DMS-0956178 US National Science Foundation Grant DMS-0801075 SFSU-Colombia Combinatorics Initiative.

wn.com/Lecture 3 . Coxeter Groups (Federico Ardila)

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogota, Colombia) Spring 2008 Partially supported by: US National Science Foundation CAREER Award DMS-0956178 US National Science Foundation Grant DMS-0801075 SFSU-Colombia Combinatorics Initiative.

published: 07 Sep 2012
views: 796

FPG3: Axioms, Desargue, The Fano Plane

FPG3: Axioms, Desargue, The Fano Plane

Order: Reorder
Duration: 59:42
Updated: 23 May 2015
views: 490

videos

In this lecture we describe Coexeter's six axioms for the projective plane. We take Desargue's Theorem as one of the axioms, however we also take a detailed loo...

In this lecture we describe Coexeter's six axioms for the projective plane. We take Desargue's Theorem as one of the axioms, however we also take a detailed look at the converse of Desargue's Theorem, and the interesting way it is proved, using Desargue's Configuration. After setting out the axioms, we show several results which can be inferred from them. Comparing these results to the initial axioms makes it clear that points and lines are dual objects to one another, with respect to the projective plane. As a concrete example, we also discuss the Fano Plane, and explain how the corresponding incidence structure can be visualized as a network (a Levi graph). In fact, the structure turns out to be the famous Heawood Graph. Projective geometry is more basic and important than Euclidean geometry, because it uses less assumptions, and in concerned with statements which remain true for a much wider range of different geometric setups. In fact, with this algebraic approach, we do not even define a metric. We shall see how such ideas, as well as those of polarity, harmony and conic curves arise as natural consequences of our small set of initial axioms. I created this video with the YouTube Video Editor (http://www.youtube.com/editor)

wn.com/Fpg3 Axioms, Desargue, The Fano Plane

In this lecture we describe Coexeter's six axioms for the projective plane. We take Desargue's Theorem as one of the axioms, however we also take a detailed look at the converse of Desargue's Theorem, and the interesting way it is proved, using Desargue's Configuration. After setting out the axioms, we show several results which can be inferred from them. Comparing these results to the initial axioms makes it clear that points and lines are dual objects to one another, with respect to the projective plane. As a concrete example, we also discuss the Fano Plane, and explain how the corresponding incidence structure can be visualized as a network (a Levi graph). In fact, the structure turns out to be the famous Heawood Graph. Projective geometry is more basic and important than Euclidean geometry, because it uses less assumptions, and in concerned with statements which remain true for a much wider range of different geometric setups. In fact, with this algebraic approach, we do not even define a metric. We shall see how such ideas, as well as those of polarity, harmony and conic curves arise as natural consequences of our small set of initial axioms. I created this video with the YouTube Video Editor (http://www.youtube.com/editor)

published: 23 May 2015
views: 490

Universal Hyperbolic Geometry 0: Introduction

Universal Hyperbolic Geometry 0: Introduction

Order: Reorder
Duration: 23:13
Updated: 13 Apr 2011
views: 32236

videos

Hyperbolic geometry, in this new series, is made simpler, more logical, more general and... more beautiful! The new approach will be called `Universal Hyperboli...

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. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

wn.com/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. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

published: 13 Apr 2011
views: 32236

Hyperbolic Geometry is Projective Relativistic Geometry (full lecture)

Hyperbolic Geometry is Projective Relativistic Geometry (full lecture)

Order: Reorder
Duration: 51:17
Updated: 04 Mar 2012
views: 7735

videos

This is the full lecture of a seminar on a new way of thinking about Hyperbolic Geometry, basically viewing it as relativistic geometry projectivized, that I ga...

This is the full lecture of a seminar on a new way of thinking about Hyperbolic Geometry, basically viewing it as relativistic geometry projectivized, that I gave a few years ago at UNSW. We discuss three dimensional relativistic space and its quadratic/bilinear form, particularly the upper sheet of the two sheeted hyperbola, which is a Riemannian manifold under the induced metric from the Lorentzian quadratic form x^2+y^2-z^2. Two views give the Poincare and Klein models (both due to Beltrami!) and it is the latter that we extend to include the entire projective plane. We discuss some related projective geometry, harmonic ranges and cross ratio, introduce the main metrical notions of quadrance and spread, illustrate with diagrams of circles, connect with classical notions of distance and angle, and state the main laws of trigonometry as well as some important secondary results, such as the Parallax theorem. We explain the Null perspective theorem and related null trigonometric results, the 48/64 theorem and finish with the remarkable Jumping Jack theorem---my personal favourite theorem in mathematics. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

wn.com/Hyperbolic Geometry Is Projective Relativistic Geometry (Full Lecture)

This is the full lecture of a seminar on a new way of thinking about Hyperbolic Geometry, basically viewing it as relativistic geometry projectivized, that I gave a few years ago at UNSW. We discuss three dimensional relativistic space and its quadratic/bilinear form, particularly the upper sheet of the two sheeted hyperbola, which is a Riemannian manifold under the induced metric from the Lorentzian quadratic form x^2+y^2-z^2. Two views give the Poincare and Klein models (both due to Beltrami!) and it is the latter that we extend to include the entire projective plane. We discuss some related projective geometry, harmonic ranges and cross ratio, introduce the main metrical notions of quadrance and spread, illustrate with diagrams of circles, connect with classical notions of distance and angle, and state the main laws of trigonometry as well as some important secondary results, such as the Parallax theorem. We explain the Null perspective theorem and related null trigonometric results, the 48/64 theorem and finish with the remarkable Jumping Jack theorem---my personal favourite theorem in mathematics. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

published: 04 Mar 2012
views: 7735

Topology For Beginners: Topology Meets Hyperbolic Geometry

Topology For Beginners: Topology Meets Hyperbolic Geometry

Order: Reorder
Duration: 25:48
Updated: 22 Jan 2014
views: 3956

videos

https://sites.google.com/site/richardsouthwell254/home In this video I will briefly introduce hyperbolic geometry and continue discussing the relationship betw...

https://sites.google.com/site/richardsouthwell254/home In this video I will briefly introduce hyperbolic geometry and continue discussing the relationship between curvature, tilings, non-euclidean geometry and surfaces with boundary. Hyperbolic geometry is introduced using computer simulations of the Beltrami-Poincare disk, and Hyperbolic paper. In particular, I argue that surfaces with different genus (number of wholes) can have different types of geometry and hence admit different topologies. The Gauss-Bonnet formula is described at the end to concrete these ideas. Resources The Shape of Space (book by Jeffrey R. Weeks) Non Euclidean Geometry & Hyperbolic Social Networks (by Richard Southwell) https://www.youtube.com/watch?v=qmaTIO4avM4 Universal Hyperbolic Geometry (by Norman Wildberger) http://www.youtube.com/watch?v=EvP8VtyhzXs Wolfram Demonstrations Project: Tiling the Hyperbolic Plane with Regular Polygons (by Gašper Zadnik) http://demonstrations.wolfram.com/TilingTheHyperbolicPlaneWithRegularPolygons/ Wolfram Demonstrations Project: Lines through Points in the Poincaré Disk (by Brett Champion) http://demonstrations.wolfram.com/LinesThroughPointsInThePoincareDisk/

wn.com/Topology For Beginners Topology Meets Hyperbolic Geometry

https://sites.google.com/site/richardsouthwell254/home In this video I will briefly introduce hyperbolic geometry and continue discussing the relationship between curvature, tilings, non-euclidean geometry and surfaces with boundary. Hyperbolic geometry is introduced using computer simulations of the Beltrami-Poincare disk, and Hyperbolic paper. In particular, I argue that surfaces with different genus (number of wholes) can have different types of geometry and hence admit different topologies. The Gauss-Bonnet formula is described at the end to concrete these ideas. Resources The Shape of Space (book by Jeffrey R. Weeks) Non Euclidean Geometry & Hyperbolic Social Networks (by Richard Southwell) https://www.youtube.com/watch?v=qmaTIO4avM4 Universal Hyperbolic Geometry (by Norman Wildberger) http://www.youtube.com/watch?v=EvP8VtyhzXs Wolfram Demonstrations Project: Tiling the Hyperbolic Plane with Regular Polygons (by Gašper Zadnik) http://demonstrations.wolfram.com/TilingTheHyperbolicPlaneWithRegularPolygons/ Wolfram Demonstrations Project: Lines through Points in the Poincaré Disk (by Brett Champion) http://demonstrations.wolfram.com/LinesThroughPointsInThePoincareDisk/

published: 22 Jan 2014
views: 3956

8 Inversion in Unit Circle

8 Inversion in Unit Circle

Order: Reorder
Duration: 21:07
Updated: 14 Jul 2013
views: 809

videos

An important map of (most of) the complex plane to itself is considered.

An important map of (most of) the complex plane to itself is considered.

wn.com/8 Inversion In Unit Circle

An important map of (most of) the complex plane to itself is considered.

published: 14 Jul 2013
views: 809

Cédric Bonnafé: Calogero-Moser cellular characters : the smooth case

Cédric Bonnafé: Calogero-Moser cellular characters : the smooth case

Order: Reorder
Duration: 65:18
Updated: 01 Jun 2015
views: 80

videos

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its f...

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area Using the representation theory of Cherednik algebra at t= 0, we define a family of "Calogero-Moser cellular characters" for any complex reflection group W. Whenever W is a Coxeter group, we conjecture that they coincide with the "Kazhdan-Lusztig cellular characters". We shall give some evidences for this conjecture. Our main result is that, whenever the associated Calogero-Moser space is smooth, then all the Calogero-Moser cellular characters are irreducible. This implies in particular that our conjecture holds in type A and for some particular choices of the parameters in type B. Recording during the thematic meeting: «Algebraic and finite groups, geometry and representations. 50 years of Chevalley seminar» the September 23, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent

wn.com/Cédric Bonnafé Calogero Moser Cellular Characters The Smooth Case

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area Using the representation theory of Cherednik algebra at t= 0, we define a family of "Calogero-Moser cellular characters" for any complex reflection group W. Whenever W is a Coxeter group, we conjecture that they coincide with the "Kazhdan-Lusztig cellular characters". We shall give some evidences for this conjecture. Our main result is that, whenever the associated Calogero-Moser space is smooth, then all the Calogero-Moser cellular characters are irreducible. This implies in particular that our conjecture holds in type A and for some particular choices of the parameters in type B. Recording during the thematic meeting: «Algebraic and finite groups, geometry and representations. 50 years of Chevalley seminar» the September 23, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent

published: 01 Jun 2015
views: 80

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0:12

Odd-4 Graph, Fano Planes, and the Coxeter Graph

http://demonstrations.wolfram.com/Odd4GraphFanoPlanesAndTheCoxeterGraph/ The Wolfram Demo...

published: 11 Jul 2009

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Odd-4 Graph, Fano Planes, and the Coxeter Graph

Odd-4 Graph, Fano Planes, and the Coxeter Graph

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published: 11 Jul 2009
views: 710

http://demonstrations.wolfram.com/Odd4GraphFanoPlanesAndTheCoxeterGraph/ The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. The odd-4 graph is constructed by taking triplets from {1, 2, 3, 4, 5, 6, 7}, then connecting the triplets that share no values. Ignoring the colored edges, the entire 35-vertex graph is the odd-4 graph. The odd-3 graph is the Petersen graph, and the od... Contributed by: Ed Pegg Jr

3:36

Dimensiones by Alexandre Rodichevski

(en) Dimensions by Alexandre Rodichevski The dimension is the number of degrees of freedom...

published: 05 Apr 2014

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Dimensiones by Alexandre Rodichevski

Dimensiones by Alexandre Rodichevski

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published: 05 Apr 2014
views: 156

(en) Dimensions by Alexandre Rodichevski The dimension is the number of degrees of freedom available for movements in a space. The world we live in has three spatial dimensions. In mathematics spaces with any number of dimensions, even infinite, are used. (it) Dimensioni di Alexandre Rodichevski La dimensione è il numero di gradi di libertà disponibili per i movimenti in uno spazio. Nel mondo in cui viviamo disponiamo di tre dimensioni spaziali. In matematica vengono utilizzati gli spazi con qualsiasi numero di dimensioni, anche infinito. (ru) Измерения Александра Родишевского Измерение -- это число степеней свободы перемещения в пространстве. В мире, в котором мы живем, имеются три пространственных измерения. Математические теории используют пространства с любым число измерений, даже бесконечным. (lt) Dimensiones (gr) Διαστάσεις (en) The following images are used to create this work. License type is indicated in parentheses. "Suprematist Composition: White on White" by Kazimir Malevich (public domain) "Suprematist Painting: Eight Red Rectangles" by Kazimir Malevich (public domain) "Spatial Construction n. 12" by Aleksandr Rodchenko (CC BY-SA 3.0) "Monologue" by Elisha Ben Yitzhak (CC BY-SA 3.0) "Number line" by Andeggs (public domain) "Five point stencil illustration" by Oleg Alexandrov (public domain) "The unknot" by Jim.belk (public domain) "A trefoil knot" by Jim.belk (public domain) "A figure-eight knot" by Jim.belk (public domain) "A cinquefoil knot" by Jim.belk (public domain) "A three-twist (5_2) knot" by Jim.belk (public domain) "A stevedore knot" by Jim.belk (public domain) "A 6 2 knot" by Jim.belk (public domain) "A 6 3 knot" by Jim.belk (public domain) "Cartesian coordinate system" by K. Bolino (public domain) "A finite set of points in two-dimensional Euclidean space" by Jean-Luc W (CC BY-SA 3.0) "Sphere closed path" by Luca Antonelli (CC BY-SA 3.0) "Lorentzian wormhole" by AllenMcC (CC BY-SA 3.0) "3D coordinate system" by Sakurambo (CC BY-SA 3.0) "Illustration of a piecewise linear function" by Oleg Alexandrov (public domain) "Golden rectangles in an icosahedron" by Fropuff, Mysid (public domain) "Illustration of Bicubic interpolation on a dataset" by Berland (public domain) "4-cube Petrie projecton in 3D" by Jgmoxness (CC BY-SA 3.0) "Hydrogen Density Plots for n up to 4" by PoorLeno (public domain) "6-simplex family graph in A6 Coxeter plane" by Tomruen (public domain) "6-orthoplexl family graph in B6 Coxeter plane" by Tomruen (public domain) "6-orthoplexl family graph in B6 Coxeter plane" by Tomruen (public domain) "6-demicube family polytopes in Coxeter planes symmetry" by Tomruen (public domain) "2_21 polytope family polytopes in Coxeter plane symmetry" by Tomruen (public domain) "1_22 polytope family polytopes in Coxeter plane symmetry" by Tomruen (public domain) "Calabi-Yau quintic" by Andrew J. Hanson (CC BY-SA 3.0) "CMS Higgs-event" by Lucas Taylor (CC BY-SA 3.0) "Schwarzschild wormhole which bridges two different universes" by Alain r (CC BY-SA 3.0) "Messier 82" by NASA (public domain) The images of this video are subject of the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0). See http://creativecommons.org/licenses/by-sa/4.0/ Music © Alexandre Rodichevski http://www.morfoedro.it

0:21

The Vieta Mapping for the Coxeter Group A_2

http://demonstrations.wolfram.com/TheVietaMappingForTheCoxeterGroupA_2 The Wolfram Demons...

published: 12 Nov 2012

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The Vieta Mapping for the Coxeter Group A_2

The Vieta Mapping for the Coxeter Group A_2

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published: 12 Nov 2012
views: 99

http://demonstrations.wolfram.com/TheVietaMappingForTheCoxeterGroupA_2 The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. This Demonstration shows the real part of the Vieta mapping, which is important in singularity theory. The domain is the plane x+y+z=0 containing the green hexagon; it has been rotated around the origin to make it horizontal for the purpose of the Demon ... Contributed by: Andrzej Kozlowski Audio created with WolframTones: http://tones.wolfram.com

0:49

chilli plane spotting

via YouTube Capture

published: 08 Aug 2013

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chilli plane spotting

chilli plane spotting

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published: 08 Aug 2013
views: 38

via YouTube Capture

7:59

Coxeter discusses the math behind Escher's circle limit

HSM Coxeter talks about the Poincare disk model of hyperbolic geometry. He also talks abou...

published: 27 Dec 2010

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Coxeter discusses the math behind Escher's circle limit

Coxeter discusses the math behind Escher's circle limit

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published: 27 Dec 2010
views: 14343

HSM Coxeter talks about the Poincare disk model of hyperbolic geometry. He also talks about complex inversions. Clip from the "fantastic wrld of Escher"

26:25

UHG52: Petrie polygons of a polyhedron

John Flinders Petrie was the son of the famed Egyptologist Flinders Petrie, and a good fri...

published: 29 Nov 2015

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UHG52: Petrie polygons of a polyhedron

UHG52: Petrie polygons of a polyhedron

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published: 29 Nov 2015
views: 1391

John Flinders Petrie was the son of the famed Egyptologist Flinders Petrie, and a good friend of Donald Coxeter. He discovered lovely polygonal paths on polytopes or polyhedra, that in the case of the Platonic solids have remarkable properties when we project these solids orthogonally onto particular planes. We also expand our view of the Platonic solids by considering more general tesselations of a surface, including the three famous regular tesselations of the plane. The Schlafli symbol of a tesselation is also introduced. These will have important consequences when we extend this thinking to the hyperbolic situation.

38:54

Lecture 1 . Coxeter Groups (Federico Ardila)

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco...

published: 07 Sep 2012

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Lecture 1 . Coxeter Groups (Federico Ardila)

Lecture 1 . Coxeter Groups (Federico Ardila)

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published: 07 Sep 2012
views: 2604

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogota, Colombia) Spring 2008 Partially supported by: US National Science Foundation CAREER Award DMS-0956178 US National Science Foundation Grant DMS-0801075 SFSU-Colombia Combinatorics Initiative.

1:20

MACS JO717 THE CRYSTALLINE GALAXY CLUSTER

THE SECRET OF MACS JO717 , HOW IT ALLOWS MASSIVE GALAXIES TO TRAVEL ALONG PATHWAYS PRE-DEF...

published: 17 Oct 2012

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MACS JO717 THE CRYSTALLINE GALAXY CLUSTER

MACS JO717 THE CRYSTALLINE GALAXY CLUSTER

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published: 17 Oct 2012
views: 363

THE SECRET OF MACS JO717 , HOW IT ALLOWS MASSIVE GALAXIES TO TRAVEL ALONG PATHWAYS PRE-DEFINED IN SOME TYPE OF CRYSTAL LIKE STRUCTURE HAS ALWAYS BEEN EXPLAINED SINCE H.S.M.COXETER PUBLISHED THE FIRST BOOK ON PROJECTIVE GEOMETRY. THE IDEALS ARE VERY CLEAR. PRESSURE BREAKS DOWN INTO FANO PLANES. FANO PLANES SPLIT MASS AND ENERGY. THERE ARE ONLY SO MANY WAYS YOU CAN SPLIT THE FANO PLANE ASSEMBLIES FROM AND INTO NEW GEOMETRIES. DETAILS MUST BE LEARNED WATCHING MY PARTICLE PHYSICS VIDEOS. DETAILS ON MACS JO717 CAN BE FOUND ELSEWHERE AND YOU CAN SEE FOR YOURSELF; THIS FAST GROWING SYSTEM OF GALAXIES IS ACTUALLY SHARING LARGE PARTICLES BETWEEN GALAXIES AND BY DOING SO CREATES THE CRYSTALLINE STRUCTURE IN WHICH THEY TRAVERSE !!!!!!!!!!!!!!!!!!!!

58:23

Jason Behrstock: Random graphs and applications to Coxeter groups

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Ma...

published: 19 Oct 2015

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Jason Behrstock: Random graphs and applications to Coxeter groups

Jason Behrstock: Random graphs and applications to Coxeter groups

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published: 19 Oct 2015
views: 131

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area Erdös and Rényi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then explain applications of these results to the geometry of Coxeter groups. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry. Recording during the thematic meeting: "GAGTA-9 : Geometric, asymptotic and combinatorial group theory and applications" the September 15, 2015 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Pascal Vichi

12:47

Visualizing the sphere and the hyperbolic plane: five projections of each

Visualizing the sphere and then the hyperbolic plane under various azimuthal projections (...

published: 19 Nov 2013

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Visualizing the sphere and the hyperbolic plane: five projections of each

Visualizing the sphere and the hyperbolic plane: five projections of each

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published: 19 Nov 2013
views: 17316

Visualizing the sphere and then the hyperbolic plane under various azimuthal projections (five of each). spherical projections: 1. orthographic projection of the sphere (at 17s), 2. gnomonic projection of the sphere (at 1m16s), 3. stereographic projection of the sphere (at 2m33s), 4. Lambert azimuthal equal-area projection of the sphere (at 3m59s), 5. azimuthal equidistant projection of the sphere (at 5m23s); hyperbolic projections: 6. Poincaré disk model of the hyperbolic plane (at 6m27s), analogous to stereographic, 7. Beltrami--Klein model of the hyperbolic plane (at 8m54s), analogous to gnomonic, 8. pseudorthographic projection of the hyperbolic plane (at 10m19s), analogous to orthographic, 9. Lambert azimuthal equal-area projection of the hyperbolic plane (at 11m20s), 10. azimuthal equidistant projection of the hyperbolic plane (at 11m55s). The spherical projections show the sphere with a dodecahedral tiling by regular pentagons; the hyperbolic projections show the hyperbolic plane with a uniform tiling by regular heptagons. The program used to compute the frames is available at http://www.madore.org/~david/programs/#prog_projections For the same projections of the Earth's surface, see: http://www.youtube.com/watch?v=LKcTbIsWS9c

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54:02

Lecture 3 . Coxeter Groups (Federico Ardila)

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco...

published: 07 Sep 2012

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Lecture 3 . Coxeter Groups (Federico Ardila)

Lecture 3 . Coxeter Groups (Federico Ardila)

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published: 07 Sep 2012
views: 796

Lecture notes at: http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html San Francisco State University (San Francisco, USA) Universidad de Los Andes (Bogota, Colombia) Spring 2008 Partially supported by: US National Science Foundation CAREER Award DMS-0956178 US National Science Foundation Grant DMS-0801075 SFSU-Colombia Combinatorics Initiative.

59:42

FPG3: Axioms, Desargue, The Fano Plane

In this lecture we describe Coexeter's six axioms for the projective plane. We take Desarg...

published: 23 May 2015

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FPG3: Axioms, Desargue, The Fano Plane

FPG3: Axioms, Desargue, The Fano Plane

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published: 23 May 2015
views: 490

In this lecture we describe Coexeter's six axioms for the projective plane. We take Desargue's Theorem as one of the axioms, however we also take a detailed look at the converse of Desargue's Theorem, and the interesting way it is proved, using Desargue's Configuration. After setting out the axioms, we show several results which can be inferred from them. Comparing these results to the initial axioms makes it clear that points and lines are dual objects to one another, with respect to the projective plane. As a concrete example, we also discuss the Fano Plane, and explain how the corresponding incidence structure can be visualized as a network (a Levi graph). In fact, the structure turns out to be the famous Heawood Graph. Projective geometry is more basic and important than Euclidean geometry, because it uses less assumptions, and in concerned with statements which remain true for a much wider range of different geometric setups. In fact, with this algebraic approach, we do not even define a metric. We shall see how such ideas, as well as those of polarity, harmony and conic curves arise as natural consequences of our small set of initial axioms. I created this video with the YouTube Video Editor (http://www.youtube.com/editor)

23:13

Universal Hyperbolic Geometry 0: Introduction

Hyperbolic geometry, in this new series, is made simpler, more logical, more general and.....

published: 13 Apr 2011

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Universal Hyperbolic Geometry 0: Introduction

Universal Hyperbolic Geometry 0: Introduction

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published: 13 Apr 2011
views: 32236

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. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

51:17

Hyperbolic Geometry is Projective Relativistic Geometry (full lecture)

This is the full lecture of a seminar on a new way of thinking about Hyperbolic Geometry, ...

published: 04 Mar 2012

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Hyperbolic Geometry is Projective Relativistic Geometry (full lecture)

Hyperbolic Geometry is Projective Relativistic Geometry (full lecture)

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published: 04 Mar 2012
views: 7735

This is the full lecture of a seminar on a new way of thinking about Hyperbolic Geometry, basically viewing it as relativistic geometry projectivized, that I gave a few years ago at UNSW. We discuss three dimensional relativistic space and its quadratic/bilinear form, particularly the upper sheet of the two sheeted hyperbola, which is a Riemannian manifold under the induced metric from the Lorentzian quadratic form x^2+y^2-z^2. Two views give the Poincare and Klein models (both due to Beltrami!) and it is the latter that we extend to include the entire projective plane. We discuss some related projective geometry, harmonic ranges and cross ratio, introduce the main metrical notions of quadrance and spread, illustrate with diagrams of circles, connect with classical notions of distance and angle, and state the main laws of trigonometry as well as some important secondary results, such as the Parallax theorem. We explain the Null perspective theorem and related null trigonometric results, the 48/64 theorem and finish with the remarkable Jumping Jack theorem---my personal favourite theorem in mathematics. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

25:48

Topology For Beginners: Topology Meets Hyperbolic Geometry

https://sites.google.com/site/richardsouthwell254/home In this video I will briefly intro...

published: 22 Jan 2014

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Topology For Beginners: Topology Meets Hyperbolic Geometry

Topology For Beginners: Topology Meets Hyperbolic Geometry

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published: 22 Jan 2014
views: 3956

https://sites.google.com/site/richardsouthwell254/home In this video I will briefly introduce hyperbolic geometry and continue discussing the relationship between curvature, tilings, non-euclidean geometry and surfaces with boundary. Hyperbolic geometry is introduced using computer simulations of the Beltrami-Poincare disk, and Hyperbolic paper. In particular, I argue that surfaces with different genus (number of wholes) can have different types of geometry and hence admit different topologies. The Gauss-Bonnet formula is described at the end to concrete these ideas. Resources The Shape of Space (book by Jeffrey R. Weeks) Non Euclidean Geometry & Hyperbolic Social Networks (by Richard Southwell) https://www.youtube.com/watch?v=qmaTIO4avM4 Universal Hyperbolic Geometry (by Norman Wildberger) http://www.youtube.com/watch?v=EvP8VtyhzXs Wolfram Demonstrations Project: Tiling the Hyperbolic Plane with Regular Polygons (by Gašper Zadnik) http://demonstrations.wolfram.com/TilingTheHyperbolicPlaneWithRegularPolygons/ Wolfram Demonstrations Project: Lines through Points in the Poincaré Disk (by Brett Champion) http://demonstrations.wolfram.com/LinesThroughPointsInThePoincareDisk/

21:07

8 Inversion in Unit Circle

An important map of (most of) the complex plane to itself is considered.

published: 14 Jul 2013

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8 Inversion in Unit Circle

8 Inversion in Unit Circle

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published: 14 Jul 2013
views: 809

An important map of (most of) the complex plane to itself is considered.

65:18

Cédric Bonnafé: Calogero-Moser cellular characters : the smooth case

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Ma...

published: 01 Jun 2015

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Cédric Bonnafé: Calogero-Moser cellular characters : the smooth case

Cédric Bonnafé: Calogero-Moser cellular characters : the smooth case

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published: 01 Jun 2015
views: 80

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, bibliographies, Mathematics Subject Classification - Multi-criteria search by author, title, tags, mathematical area Using the representation theory of Cherednik algebra at t= 0, we define a family of "Calogero-Moser cellular characters" for any complex reflection group W. Whenever W is a Coxeter group, we conjecture that they coincide with the "Kazhdan-Lusztig cellular characters". We shall give some evidences for this conjecture. Our main result is that, whenever the associated Calogero-Moser space is smooth, then all the Calogero-Moser cellular characters are irreducible. This implies in particular that our conjecture holds in type A and for some particular choices of the parameters in type B. Recording during the thematic meeting: «Algebraic and finite groups, geometry and representations. 50 years of Chevalley seminar» the September 23, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent

Lecture 3 . Coxeter Groups (Federico Ardila)

Lecture 3 . Coxeter Groups (Federico Ardila)...

FPG3: Axioms, Desargue, The Fano Plane

FPG3: Axioms, Desargue, The Fano Plane...

Universal Hyperbolic Geometry 0: Introduction

Universal Hyperbolic Geometry 0: Introduction...

Hyperbolic Geometry is Projective Relativistic Geometry (ful

Hyperbolic Geometry is Projective Relativistic Geo...

Topology For Beginners: Topology Meets Hyperbolic Geometry

Topology For Beginners: Topology Meets Hyperbolic ...

8 Inversion in Unit Circle

8 Inversion in Unit Circle...

Cédric Bonnafé: Calogero-Moser cellular characters : the smo

Cédric Bonnafé: Calogero-Moser cellular characters...

photo: Creative Commons

Two iPhones

iPhone 8 Leaks, New iPhone 7 Features Reported

Edit Inquisitr 09 Jul 2016

New leaks are reportedly showing details of the iPhone 8 while new features of the iPhone 7 are coming to the surface, as well. According to two separate reports by Forbes, the iPhone 8 and the iPhone 7 leaks show amazing new features coming to the latest models, expected to be released in 2017 and late 2016 respectively ... Leaks about features on the iPhone 7 and the iPhone 8 have made their way onto the Internet ... [Image....

photo: AP / Anjum Naveed, File

FILE - In this Monday, Aug. 2, 2010 file photo, Pakistan's humanitarian leader Abdul Sattar Edhi collects donations at a roadside in Peshawar, Pakistan. The family of Pakistan's legendary philanthropist Abdul Sattar Edhi says he has died at a hospital in Karachi. (AP Photo/Anjum Naveed, File)

Pakistan's 'Angel of Mercy' Edhi Dies at 88

Edit WorldNews.com 08 Jul 2016

Pakistan's legendary philanthropist Abdul Sattar Edhi, known as the “Angel of Mercy' who devoted his life to the poor and the destitute, died on Friday at a hospital in Karachi following a prolonged illness, The Associated Press reports ... We have lost a great servant of humanity ... This loss is irreparable for the people of Pakistan."...

photo: AP / Jason Patinkin

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Edit CNN 09 Jul 2016

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photo: AP / Ben Curtis

Serena Williams of the U.S raises the trophy after beating Angelique Kerber of Germany in the women's singles final on day thirteen of the Wimbledon Tennis Championships in London, Saturday, July 9, 2016.

Wimbledon 2016: Serena Williams beats Angelique Kerber to win 22nd Grand Slam

Edit BBC News 09 Jul 2016

Williams won her first Wimbledon title in 2002 ... More to follow. ....

photo: AP

Anton Chekhov in Winter at Moscow North River Port

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Edit Deccan Chronicle 09 Jul 2016

Dallas. A newlywed and a Navy veteran of three tours in Iraq were among the five Dallas police officers killed in a sniper ambush on Thursday night. The victims were identified on Friday by officials, relatives and the media. Seven other police officers and two civilians were wounded in the shooting at the end of a protest over this week’s killing of two black men by police in Baton Rouge, Louisiana and a suburb of St ... ....

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