# The Rule of Plato Project

### November 18, 2014 By: Hagen Shook

## What is The Rule of Plato?

__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__

**even**__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.__

**odd**## The Rule of Plato

__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.__

**even**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

__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