The Forbidden Triangle

Exercise

This is the Forbidden Triangle:

Two sides are specified and no angle is given a specific value. It is only known that no angle is the same and neither angle is 90 degrees or 30 degrees or 60 degrees. What is the value of the missing side?

Facts

With no angles specified, it is said that the missing side can have any value between 1 and 29 exclusive.

If the 15 line and the 14 line were exactly aligned, a stick 1 unit long would complete the puzzle. If the 15 and 14 line were laid end-to-end, a stick 29 units long would complete the puzzle. Any stick could solve this triangle but the fact of the matter remains that it was drawn on paper just so, and if you had a perfectly accurate ruler you could measure the sides and find the most-right answer: 13.

Thoughts and Opinions

The Forbidden Triangle is one of my favorite Geometrical ideas. You will notice if you read my following explanation that, except for this sentence, I will not mention “Math” as the “Math Answer” is overly general and not helpful. That answer is given in the first remarks in the Facts section. If that first answer is perfectly acceptable to you then you may move on from the Forbidden Triangle back to more comforting levels of rigor on the topic of triangles.

My last answer as a matter of fact is specific and it is right. There is a difference between right and correct, or in this case most-correct and most-right. Please be mindful that I say “most-right” and not  “mostly right.” You may even be right on a topic without being correct, but that is an idea for another day. People are all too ready to throw around the words “impossible” and “trivial” when they are unable or unwilling to accept the truth of the most-right.

Consider a simpler triangle:

Everybody agrees that, if the angles are all equal (an “equilateral” triangle in some circles), there is one correct answer and it is 3.

When you go to build the triangle, you will find that these matters of fact are emergent from the act of making imagination reality. If I take 3 sticks each 3 feet long and lay them as pictured, I will find that the angles are all exactly equal and they are all exactly 60 degrees. You didn’t need a rule book to tell you that, these are just matters of fact on the “3, 3, 3” triangle.

Consider another triangle. This one is taught extensively in schools:

A sensible person immediately knows the answer is 5 for the missing side of this humble triangle. They say that the angle joining the sides 3 and 4 is a 90 degree joint (a “right angle”). When these conditions are met, the missing side is always bigger than any of the other two individually. There is a formula for these things but as before, I will find that a stick 3 feet long and a stick 4 feet long arranged according to these rules will complete a perfect triangle with a 5 foot stick joining them.

These triangle sketches are representations of the laying of these sticks. I use these pictures to stand in for the actual Physical act of building these things. It is labor of the imagination through building, or making art into reality, that reveals the most-right answer. Not at all coincidentally, the most-right answers are also the most beautiful.

Consider this next triangle:

I have taken the liberty to write in the size of each stick used to build this triangle. This one is 3 times larger than the so-called “3, 4, 5 triangle” we considered last. From my picture, you might even be able to tell that I am building a tent large enough to stand in. I need it to be 12 feet high with enough space, say 9 feet, to walk side-to-side comfortably. In order to build my tent, I need material at least 15 feet long to make sure rain does not get in. This is a very big tent, but I have very big needs for shelter and safety.

This isn’t my only option. Pictured here is another triangle I built to suit my needs:

You see, the 12 foot stick I use to support my tent could be flanked by two other sticks any size. I choose the sizes of 5 feet and 13 feet because they are beautiful. People make choices based on this criterion in life so often that we take it for granted. On the topic of triangles however, people too often forget this is an option at all. Is this an insult to the triangles, or an insult to our instructors who teach them? Or is it a self-handicap meant to ensure our survival? Are people comforted more by the most-correct over the most-right because their lives depend on it?

Finish building my tent and you will see why the most-right answer for the Forbidden Triangle is indeed 13.

Feel free to email me with your thoughts and opinions on this matter at andrew@ahogan.org.

 

The article above is dedicated to Eric and Will. Thank you for your patience and thoughtful discussions.