# The Rule of Plato Project

## What is The Rule of Plato?

I am doing the The Rule of Plato project. This project is all about comparing the rules of Plato and Pythagorean and generating Pythagoras triples. The formula for the rule of Plato is that for an even number, m, three legs of a right triangle are given by: m, m squared divided by 4 then minus 1 and m squared divided by four then plus 1. The rule of Pythagoras says that for an odd number, m, three legs of a right triangle are given by: m, m squared minus 1 divided by 2 and m squared plus 1 divided by 2. Then I analyze the two formulas to see how if they are alike or different.

## The Rule of Plato

So with the formula: For an even number, m, three legs of a right triangle are given by: m, m squared divided by 4 then minus 1 and m squared divided by four then plus 1. Now I am going to generate 10 Pythagoras triples using this formula. For my first Pythagorean triple, my even number is 64. m=64. 64 squared=4096. 4096/4=1024. 1024-1=1023. Then 64 squared=4096. 4096/4=1024. 1024+1=1025. 64, 1023, 1025 are Pythagorean triples.

Another even number is 4. m=4. 4 squared=16. 16/4=4. 4-1=3. Then 4 squared=16. 16/4=4. 4+1=5. That means 4, 3, 5, are Pythagorean triples.

The next even number I am going to use 10. m=10. 10 squared=100. 100/4=25. 25-1=24. Next 10 squared=100. 100/4=25. 25+1=26. 10, 24, 26, are Pythagorean triples.

Another even number I am going to use is 14. m=14. 14 squared=256. 256/4=64. 64-1=63. 14 squared=256. 256=64. 64+1=65. 14, 63, 65, are Pythagorean triple.

The even number I am going to use is 84. m=84. 84 squared 7056. 7056/4=1764. 1764-1=1763. 84 squared=-7056. 7056/4=1764. 1764+1=1765. 84, 1763, 1765, are Pythagorean triples.

More Pythagorean triples will be listed below:

6, 8,10

8, 15, 17

12, 37, 39

16, 63, 65

20, 99, 101

I noticed that when you use the even 2, you can not use it because 2 squared=4. 4/4=1. 1-1 would be 0 so you can not use 2 as an even number.

## The Rule of Pythagoras

The rule of Pythagoras says that for an odd number, m, three legs of a right triangle are given by: m, m squared minus 1 divided by 2 and m squared plus 1 divided by 2. So for the first odd number I will use, it will be 7. m=7. 7 squared=49. 49-1=48. 48/2=24. 7 squared=49. 49+1=50. 50/2=25. 7, 24, 25.

Another odd number is 13. m=13. 13 squared=169. 169-1=168. 168/2=84. 13 squared=169. 169+1=170. 170/2=85. 13, 84, 85.

11. m=11. 11 squared=121. 121-1=120. 120/2=60. 11 squared=121. 121+1=122. 122/2=61. 11, 60, 61.

9. m=9. 9 squared=81. 81-1=80. 80/2=40. 9 squared=81. 81+1=82. 82/2=41. 9, 40, 41.

3. m=3. 3 squared=9. 9-1=8. 8/2=4. 3 squared=9. 9+1=10. 10/2=5. 3, 4, 5.

(Interesting. When I was using 4 on the rule of Plato formula I got the same answer when I using 3 with the rule of Pythagoras.)

More Pythagorean triples are listed below:

5, 12, 13

15, 112, 113

17, 144, 145

19, 180, 181

21, 220, 221

## Analyze Time

After generating 10 Pythagorean triples using both formulas I was noticing that for the Rule of Plato, the last two numbers in the Pythagorean triple they are two apart from each other. So when I was generating Pythagorean triples using the Rule of Plato, I only needed to know the second number in the Pythagorean triples. But when I was generating Pythagorean triples with using the Rule of Pythagoras the last two numbers are one apart.