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.

1. Recreation Mathematics
Visual Proofs for Topology
Sing Kuang Tan
singkuangtan@gmail.com
11 Feb 2022

2. Prelude
• Computer science does 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 Topology in 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 a butcher 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
Analyze Radio Wave
Analyze Bomb Explosion
Analyze Robot Hand Workspace

6. Relevant to Computer Science
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 a square paper
Label the top edge and bottom edge
a
b

8. Glue edge a to 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 a to 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 edge a 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 a 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
And it becomes a mobius strip

12. Sphere
a
b
c d
a=c
b=d
Follow the above 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 sphere created
Glue edge a to edge c
Glue edge b to edge d

14. Torus
a
b
c d
a=b
c=d
Follow the above 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 torus created
Glue edge a to edge b
Glue edge c to edge d

16. Klein Bottle
a
b
Glue edge a 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 a 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
Created Klein bottle

18. Split Klein Bottle into Mobius Strip
a
b
c d
a=b
c=d

19. a1
b2
c d
a1=b1
a2=b2
c=d
a2
b1
Relabel the edges

20. a1
b2
c d
a1=b1
a2=b2
c=d
a2
b1
Cut along dotted lines

21. b2
c d
a1=b1
a2=b2
c=d
a2
b1
Separate and join edge c and d
a1

22. a1
b1
a2
b2
Reshape the separate parts.
Note that it becomes 2 mobius strips
a1=b1
a2=b2
c=d

23. Split Mobius Strip into Circular Strip
a
b
a=b
Cut along dotted line

24. Split Mobius Strip into Circular Strip
a1=b1
a2=b2
Relabel the edges
a1
b2
a2
b1

25. b2
a1=b1
a2=b2
a1
Separate the parts
a1
a2
b1

26. b2
a1=b1
a2=b2
a1
Join edge a2 and b2
a1
a2
b1

27. a1
b1
a1=b1
Reshape the object, and it becomes a circular strip

28. Expand a torus
a
b
c d
a=b
c=d
Cut along dotted lines

29. a1
b1
c d
a1=b1
a2=b2
c=d
b2
a2
Relabel the edges

30. b1
c d
a2
b2
Separate and join edge c and d
a1 a1=b1
a2=b2
c=d

31. b1
c d
a2
b2
Join edge a1 and b1
a1
a1=b1
a2=b2
c=d

32. a2
a2=b2
b2
It becomes a circular strip, a circular strip much longer than the torus

33. Poincare Conjecture (Sphere)
a
b
c d
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 rubber band around a
sphere

35. a
b
c d
a=c
b=d
Shrink the rubber band

36. a
b
c d
a=c
b=d
Shrink the rubber band

37. a
b
c d
a=c
b=d
Shrink the rubber 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 rubber band will
eventually shrink to a point

39. Poincare Conjecture (Torus)
a
b
c d
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 rubber band around a
torus, the rubber band cannot
be minimized into a point

41. a
b
c d
a=b
c=d
Wrap rubber band around a
torus, the rubber band cannot
be minimized into a point
Because the edges c, d are
separated from edges a, b

42. Double Torus
A double torus

43. Split the double loops

44. Slice parallel to the
table to get 2
halves
Merge the edges into a single point
Combine 2 tubes into 1 tube

45. Merge the edges

46. Merge the edges into single point

48. d2
a1 a2
b1
b2
c1
c2
d1 e1
e2
f1
f2
Label the edges, join a1 to a2, b1 to b2 and so on…
d2

49. d1
a1 b1 d2 f1
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 new topological objects can you create with my 2d square
diagram method?
• Leave your comments. 

51. What is the object if you connect the edges in
this way?
a1
b2
c d
a2
b1
a1=b1
a2=b2
c=d

52. Links to my papers
● https://vixra.org/author/sing_kuang_tan
● Link to my NP vs P paper
● And Discrete Markov Random Field relaxation paper

53. About Me
● My job 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 singkuangtan@gmail.com
● 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