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A visual proof for Toeplitz Square Peg Problem of convex shapes, kungfu computer science, geometric complexity theory

Sing Kuang TanFollow

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- 1. Don’t Be a Square Man Visual Proof for Square Peg Problem with Convex Shapes Sing Kuang Tan singkuangtan@gmail.com 14 Mar 2022
- 2. Square Peg Problem • Although I am not trained in topology in pure mathematics • I am armed with knowledge in • Computational geometry • Hough transform, Fourier shape features and skeleton in Computer Vision • I think I can also tackle topology problem. Haha.
- 3. Visual Proof for Square Peg Problem for Arbitrary Convex Shapes • Square Peg Problem is also called the Inscribed Square Problem • https://en.wikipedia.org/wiki/Inscribed_square_problem • I am going to describe an visual algorithm • That will solve the Square Peg Problem for any convex shapes • By rotation and varying the length of line segments • I will show how the algorithm works for eclipse • Then for arbitrary convex shape
- 4. Research Papers • The research papers are very difficult to understand with lots of algebras • I think if my slides are not visual, nobody will want to read it • Especially those that are not mathematically inclined • I created a popular science that anyone can understand
- 5. Eclipse Take a horizontal red line with fix length Place it on top so that the ends of line touch the perimeter of eclipse
- 6. Sliding the ends of line along the perimeter clockwise
- 7. Sliding the ends of line along the perimeter clockwise
- 8. Sliding the ends of line along the perimeter clockwise
- 9. Sliding the ends of line along the perimeter clockwise
- 10. Take a vertical green line, place it on top so that the ends touch the perimeter of eclipse
- 11. Sliding the ends of line along the perimeter anti-clockwise
- 12. Sliding the ends of line along the perimeter anti-clockwise
- 13. Sliding the ends of line along the perimeter anti-clockwise
- 14. Sliding the ends of line along the perimeter anti-clockwise
- 15. Connect the ends of red and green lines with the same gradient
- 16. Connect the ends of red and green lines with the same gradient Notice that the straight lines form a parallelogram
- 17. Connect the ends of red and green lines with the same gradient At some gradient, the two lines connected by dotted lines form a rectangle We found the inscribed rectangle in the eclipse
- 18. Connect the ends of red and green lines with the same gradient Then it becomes parallelogram again
- 19. Connect the ends of red and green lines with the same gradient Then it becomes parallelogram again
- 20. Experiment with different lengths of red and green line will result in a rectangle with different aspect ratio Longer red line with result in shorter rectangle
- 21. Shorter red line with result in taller rectangle
- 22. There is a perfect red line length that will result in a square
- 23. So by rotating the red line from horizontal line to vertical line and Rotating the green line from vertical line to horizontal line Will enable us to find a inscribed rectangle
- 24. By varying the length of the lines, we will get the inscribed square
- 25. Arbitrary Convex Shape By selecting the right length for the red line and green line and rotation
- 26. By selecting the right length for the red line and green line and rotation
- 27. We will eventually find the inscribed square
- 28. We will eventually find the inscribed square
- 29. We will eventually find the inscribed square The rotation (and sliding) technique works for any convex shapes
- 30. Problems? • Can you find inscribed rectangle using my algorithm on the 2 shapes below?
- 31. Square Peg Problem Applet • https://www.math.u-bordeaux.fr/~bmatschke/squarepeg2/index.php • The link above gives us an applet to experiment with Square Peg Problem of different shapes
- 32. 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
- 33. 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/ ■ Visual Proofs for Topology ■ 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
- 34. 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