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