The Modest 3-4-5 Triangle

I was watching a deck being built. That’s how this all started. If you remember any geometry, you might remember the 3-4-5 triangle. It turns out that if the ratios of the sides of a triangle are 3:4:5, it will be a right triangle. In other words, the sides can be 3,4,5 inches, 3,4,5 feet, 3,4,5 furlongs, 6,8,10 furlongs, 6,8,10 feet, or any multiple. I mean, it could even be a 6 inch dachshund, an 8 inch dachshund and a 10 inch dachshund. If this is true, then one of its internal angles will be exactly 90 degrees. How can this be? If you take three anythings, whose lengths are related by 3:4:5, and put them together to form a triangle they will automatically make a perfect right triangle, perfect! In other words, one of the angles will be 90 degrees, exactly 90 degrees! What the heck do 3,4,5 and 90 degrees have to do with each other, I ask you?

How did this triangle get so smart? I will not attempt to answer for I will be lured into a world with which I do not want to associate. - The Occult. As a matter of fact, I don’t WANT to know how this can be. I just know it is, and it’s as close to a religion as a guy can get. As a matter of fact, the secret society The Pythagoreans, who believed in the sanctity of whole numbers must have revered the 3:4:5 triangle. But don’t get them started on the 1:1:? triangle. That bad boy has a third side that’s not a whole number. Now, a 1:1:? triangle is half of a square with the ? side the diagonal of the square. A square is a beautiful thing, being all square and all. Matter of fact, the Pathagorean motto was, “Be There AND be Square.” A guy, a Pathagorean, was killed accidentally on purpose after discovering this ugly not a whole number number inside the Holy Square and then blabbing all over Greece that the number wasn’t even rational. It was The Ugly of Uglies. It was the square root of two. It was supposed to be a secret. I think that sealed his fate.

Want to hear a cool way of using the 3-4-5 triangle? No? Too bad. Suppose you’re building a deck (yes, THAT deck) on the back of your house. You make a rectangular area by taking three 2 x 10’s toenailing them to a ledger board on the side of the house and then to each other. Now, how can you make them square? They wobble all over making every sort of parallelogram except the one you want, a nice rectangle. You can’t use a little dinky carpenter’s square- the sides aren’t long enough to give you an accurate set. Here’s what you do. Mark one joist 3 feet from the corner, mark the adjacent one 4 feet from the corner. Then take a 2 x 4, drive a nail in one end, measure 5 feet and drive another nail in. Now drive one of the nails clear through and into one of the joists at the 3 foot mark. Now, wiggle the whole thing until the second nail strikes the second joist exactly at the 4 foot mark. What do you have? A 3-4-5 triangle. That means the angle between the two joists is exactly 90 degrees and the three joists defining the perimeter of the deck is now a nice rectangle! I LOVE that.

I know smarty-pants topologists call this child’s play, but it doesn’t prevent me from being mesmerized by it all. Just the thought that this is true throughout the Euclidean Cosmos is awesome. Now, you might think it depends on us using the base 10 number system, supposedly originating from the fact that we have 10 fingers. Well, that’s not exactly true. As long as you use a base 6 or above, the ratio will still look like 3:4:5. Any lower than that and you’re in trouble because of carrying and the numbers look odd. From this, it follows that we are NOT the only intelligent life in the Universe and they all have more than 5 fingers! Sagan and Drake! Jealous? Q E freekin’ D.