Juegos topológicos

Posts Tagged ‘Windows Mixed Reality

screen_1920x1080_2018 -12-03_19-52-43.png

The Császár polyhedron is the first polyhedral realization of a torus with 7 vertices, without diagonals and without self-intersections (Császár, 1949), with 21 edges and 14 triangular faces.

The 1-skeleton is the complete graph on 7 vertices. So we can manipulate it to get the graph of the Csaszar’s polyhedron. One can construct this model of the torus as the quoatient of a square, and build step by step the construction. There is a last option to say Csaszar in high voice (or Csaszar with faces) to get the polyhedron automatically on the scene, thanks to the Speech Recognition System of Neotrie.  One can enlarge the figure, and visit it from the inside, fly throught the hole of the torus and see its interior too.

screen_1920x1080_2018 -12-06_14-52-18

screen_1920x1080_2018 -12-06_14-33-47.png

screen_1920x1080_2018 -12-06_14-56-21

screen_1920x1080_2018 -12-03_19-21-54.png

screen_1920x1080_2018 -12-03_19-35-16.png

Demonstrations of Neotrie in Paris

Last week we visited the “Palais de la Découverte” and  the “Cité des Sciences” in Paris, invited by Guillaume Reuiller.

During two days we were testing Neotrie VR and planning future collaborations with members and responsibles of Universcience, Science Ouverte, teachers of GIPTIP, and “Comité de Culture Mathématique de l’Institut Henry Poincaré”. It was a motivating and great experience! I would like to thank them here for their interest on Neotrie VR and its future applications and uses.

The activity of the Császár’s polyhedron was one of the VR experiences started by Roger Mansuy (second of the right in the next picture). We  have completed this in this post.



  1. J. Bokowski and A. Eggert: Toutes les réalisations du tore de Moebius avec sept sommets, Topologie Struct. 17 (1991), 59-78.
  2. A. Császár: A polyhedron without diagonals, Acta Sci. Math., Szeged 13 (1949-1950), 140-142.
  3. F.H. Lutz,  Császár’s TorusElectronic Geometry Models: 2001.02.069.



Pulsa sobre imagen para acceder a web del evento.

One can now visit the famous Imaginary exhibition in Virtual Reality inside Neotrie VR!

Neotrie VR allows to visualize and interact with any 3D object in Virtual Reality. You can play with your favourite math figures and interact with them as never before.


Furthermore, the experience of flying through these surfaces is incredible:

Want to create your own gallery of VR math objects for education, research or public exhibitions? Visit our webpage:


Estadísticas del blog

  • 823.844 visitas

Síguenos en

Google translator

Geometry in Virtual Reality

AR Platónic Solids

Disfruta de un montón de figuras en realidad aumentada.

App Surface Projection

A new app to play topology. Get it for free filling

Sierpinski carpet project

Juego alicatado con hilos.


Premiados en 2012, 2013, 2014, 2015, 2017


Escribe tu dirección de correo electrónico para suscribirte a este blog, y recibir notificaciones de nuevos mensajes por correo.

Únete a otros 81 seguidores

Actualizaciones de Twitter