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Visual topology
This document provides a visual proof technique for topology problems using 2D diagrams to represent 2D surfaces in 3D worlds. It demonstrates how to construct common topological objects like the circular strip, Mobius strip, sphere, torus, and Klein bottle. It also discusses applications in fields like mechanical design, radio waves, and robotics. The author believes this visual proof technique can prove all topology problems and encourages sharing his work to spread his ideas.
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Visual topology
- 1. Recreation Mathematics Visual Proofsfor Topology Sing Kuang Tan [email protected] 11 Feb 2022
- 2. Prelude • Computer sciencedoes not look like real mathematics • Programming looks stupid (LOL), does not look like mathematics • Those who do topology win prizes. LOL • Topology looks like real mathematics • Today I going to show you what real mathematics that I know • Using Visual Proofs to prove well known Topology problems • I believe my visual proof technique can prove all Topology problems
- 3. 2D Surface Topologyin 3D world • I going to prove Topologies made up of 2D surfaces in 3D world • All these Topologies can be projected on a 2D diagram and solve using the 2D diagram
- 4. What do abutcher and a 3D graphic designer have in common? Butcher 3D Graphic Designer They experimented with Topology!!!
- 5. Applications https://www.youtube.com/watch?v=xN9hTo3iR6A Design Mechanical Parts AnalyzeRadio Wave Analyze Bomb Explosion Analyze Robot Hand Workspace
- 6. Relevant to ComputerScience Images of a car viewing at different angles lie on a manifold Transform a boy image to a girl image gradually lies on a manifold
- 7. Circular Strip Take asquare paper Label the top edge and bottom edge a b
- 8. Glue edge ato edge b Note that the edge directions matter a b a=b The equation above represent edge a glued to edge b a b Glued these 2 edges together
- 9. Glue edge ato edge b Note that the edge directions matter a b a=b The equation above represent edge a glued to edge b And it becomes a circular strip
- 10. Mobius Strip a b Glue edgea to edge b Note that edge b direction is opposite of circular strip a=b The equation above represent edge a glued to edge b a b Glued these 2 edges together
- 11. a b Glue edge ato edge b Note that edge b direction is opposite of circular strip a=b The equation above represent edge a glued to edge b And it becomes a mobius strip
- 12. Sphere a b c d a=c b=d Follow theabove steps to create sphere Glue edge a to edge c Glue edge b to edge d
- 13. Sphere a b c d a=c b=d A spherecreated Glue edge a to edge c Glue edge b to edge d
- 14. Torus a b c d a=b c=d Follow theabove steps to create torus Glue edge a to edge b Glue edge c to edge d
- 15. Torus a b c d a=b c=d A toruscreated Glue edge a to edge b Glue edge c to edge d
- 16. Klein Bottle a b Glue edgea to edge b Glue edge c to edge d Note that edge b direction is opposite of edge a c d a=b c=d The equation above represent edge a glued to edge b And edge c is glued to edge d Glue edge c and d Extend and rotate Extend and rotate Poke through itself Flip edge b inside out and glue to edge a 1 2 3 4 5 5 steps to create a Klein bottle
- 17. a b Glue edge ato edge b Glue edge c to edge d Note that edge b direction is opposite of edge a c d a=b c=d The equation above represent edge a glued to edge b And edge c is glued to edge d Created Klein bottle
- 18. Split Klein Bottleinto Mobius Strip a b c d a=b c=d
- 19.
- 20. a1 b2 c d a1=b1 a2=b2 c=d a2 b1 Cut alongdotted lines
- 21. b2 c d a1=b1 a2=b2 c=d a2 b1 Separate andjoin edge c and d a1
- 22. a1 b1 a2 b2 Reshape the separateparts. Note that it becomes 2 mobius strips a1=b1 a2=b2 c=d
- 23. Split Mobius Stripinto Circular Strip a b a=b Cut along dotted line
- 24. Split Mobius Stripinto Circular Strip a1=b1 a2=b2 Relabel the edges a1 b2 a2 b1
- 25.
- 26. b2 a1=b1 a2=b2 a1 Join edge a2and b2 a1 a2 b1
- 27. a1 b1 a1=b1 Reshape the object,and it becomes a circular strip
- 28. Expand a torus a b cd a=b c=d Cut along dotted lines
- 29.
- 30. b1 c d a2 b2 Separate andjoin edge c and d a1 a1=b1 a2=b2 c=d
- 31. b1 c d a2 b2 Join edgea1 and b1 a1 a1=b1 a2=b2 c=d
- 32. a2 a2=b2 b2 It becomes acircular strip, a circular strip much longer than the torus
- 33. Poincare Conjecture (Sphere) a b cd a=c b=d Wrap rubber band around a sphere, the rubber band will minimized into a point
- 34. a b c d a=c b=d Wrap rubberband around a sphere
- 35. a b c d a=c b=d Shrink therubber band
- 36. a b c d a=c b=d Shrink therubber band
- 37. a b c d a=c b=d Shrink therubber band Because the edges are connected, the rubber band will shrink to a small circle
- 38. a b c d a=c b=d The rubberband will eventually shrink to a point
- 39. Poincare Conjecture (Torus) a b cd a=b c=d Wrap rubber band around a torus, the rubber band cannot be minimized into a point
- 40. a b c d a=b c=d Wrap rubberband around a torus, the rubber band cannot be minimized into a point
- 41. a b c d a=b c=d Wrap rubberband around a torus, the rubber band cannot be minimized into a point Because the edges c, d are separated from edges a, b
- 42.
- 43. Split the doubleloops
- 44. Slice parallel tothe table to get 2 halves Merge the edges into a single point Combine 2 tubes into 1 tube
- 45.
- 46. Merge the edgesinto single point
- 48. d2 a1 a2 b1 b2 c1 c2 d1 e1 e2 f1 f2 Labelthe edges, join a1 to a2, b1 to b2 and so on… d2
- 49. d1 a1 b1 d2f1 b2 e1 a2 c1 e2 f2 c2 a1=a2 b1=b2 c1=c2 d1=d2 e1=e2 f1=f2 Join the edges using the rules above, And you will get the double torus This is the double torus in canonical form in 2D diagram
- 50. • What newtopological objects can you create with my 2d square diagram method? • Leave your comments.
- 51. What is theobject if you connect the edges in this way? a1 b2 c d a2 b1 a1=b1 a2=b2 c=d
- 52. Links to mypapers ● https://vixra.org/author/sing_kuang_tan ● Link to my NP vs P paper ● And Discrete Markov Random Field relaxation paper
- 53. About Me ● Myjob uses Machine Learning to solve problems ○ Like my posts or slides in LinkedIn, Twitter or Slideshare ○ Follow me on LinkedIn ■ https://www.linkedin.com/in/sing-kuang-tan-b189279/ ○ Follow me on Twitter ■ https://twitter.com/Tan_Sing_Kuang ○ Send me comments through these links ● Look at my Slideshare slides ○ https://www.slideshare.net/SingKuangTan ○ https://slideplayer.com/user/21705658/ ■ Implement Data Structure Fast with Python ■ Discrete Markov Random Field Relaxation ■ NP vs P Proof using Discrete Finite Automata ■ Use Inductive or Deductive Logic to solve NP vs P? ■ Kung Fu Computer Science, Clique Problem: Step by Step ■ Beyond Shannon, Sipser and Razborov; Solve Clique Problem like an Electronic Engineer ■ A weird Soviet method to partially solve the Perebor Problems ■ 8 trends in Hang Seng Index ■ 4 types of Mathematical Proofs ■ How I prove NP vs P ○ Follow me on Slideshare
- 54. Share my links ●I am a Small Person with Big Dreams ○ Please help me to repost my links to other platforms so that I can spread my ideas to the rest of the world ● 我人小，但因梦想而伟大。 ○ 请帮我的文件链接传发到其他平台，让我的思想能传遍天下。 ● Comments? Send to [email protected] ● Link to my paper NP vs P paper ○ https://www.slideshare.net/SingKuangTan/brief-np-vspexplain-249524831 ○ Prove Np not equal P using Markov Random Field and Boolean Algebra Simplification ○ https://vixra.org/abs/2105.0181 ○ Other link ■ https://www.slideshare.net/SingKuangTan