Triangles

This is an essay about symbolism, poetry, truth, and most of all a triangle. It rambles a bit and wanders more than many of my posts. We’re going to have to take a bit of a detour through mathematics, and consider a common equation as we make this journey. Sometimes, we forget some very basic things which are very important, and other times we try to insist on the importance of very unimportant things. There won’t be a test later, but you may want to pay attention nonetheless.

Think about triangles, and what you know about them. I’m guessing that most folks came up with three sides, quite a few that the three inside angles add up to 180 degrees, some folks may have remembered that all sides are equal if all angles are equal, and those who work with math in their daily lives may have recalled a grab bag of interesting facts about sine, cosine, and tangent. I bet a lot of folks also thought of “a^2+b^2=c^2” the long lasting Pythagorean Theorem. It’s that last one I want to talk about, and travel alongside here for a bit.

For starters, let’s say that I walk up to a child of say 6 years old, who has a very advanced grasp of conversational English, but no education in formal mathematics. If I were to say to them “a^2+b^2=c^2” I would likely be greeted by either confusion, or by questions about what I meant. We can see that there is not any kind of inherent truth or meaning to this statement, only to the idea that it stands for. If we wish to talk about this idea with someone who lacks any modern geometry, we would need to alter our terms.

So let’s go back to something else, sadly I don’t have access to any record of how Pythagoras stated the theory for his proofs in ancient Greek, but a fairly standard English version is "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides." Well, that is better, but I’d wager that I still get an odd look from my hypothetical six year old. Let’s really engage our imaginations and pretend that this child is willing and able to sit patiently as I revise my statement further. I would need to provide an explanation for:

• area of the square: “if you made a square out of pennies, the area would be how many pennies it took to make the whole square, not just the edges”
• square built upon: “If we draw a square so that two of its corners are the same place as the corners of our triangle, and its other edges go away from the triangle” 
• hypotenuse : “the longest of the three edges of this kind of triangle”
• right triangle: “A triangle where one corner is perfectly straight on both sides like the corner of a door or a piece of paper”

Now, I’m giving my child credit for a fairly robust vocabulary here, and even so I’m making other cultural assumptions, such as the prevalence of rectangular doors. Even so our explanation now becomes:

“The number of pennies it took to make the whole square, not just the edges, of a square that has two of its corners at the same place as the corners of a triangle, and it’s other edges going away from the triangle, along the longest of the three edges of a triangle where one corner is perfectly straight on both sides like the corner of a door or a piece of paper is equal to the number of pennies used to make both the squares that we draw off of other two sides of the triangle.”

Whew. That is bulky, ineloquent, difficult to remember and unlikely to stick in the mind of the child even if they do grasp the concept. They might understand, but be unable to pass the idea on to others as they trip over trying to remember how to give the explanation. Now, let’s bring a little bit of drawing into this lesson, instead of telling the child everything merely by verbal expansion. Now I draw this figure:

(Possibly even using pennies on a table to make a 3,4,5 triangle) And explain the above lesson, pointing out each of the parts as I go, and once I have done so, I give the popular reduction of “a^2+b^2=c^2”. (Probably explaining that “^2” means making the square with sides equal to the line we are looking at, and that each letter can stand for a number of pennies) Now we have a shared language, a shared base of imagery, and I suspect the child is much more likely to remember the image and the quick version, making him more able to recall the ideas and explain them to his friends. Even if he abandons my asides about pennies or paper, he can pick new words of his own and transmit the truth about the triangle to others, and do so without error.

Now let me digress for a moment into a symbolic description of the thwarting of an assassin:

Devout Aaron stood watch upon the wall fifteen cubits above the cold and silent desert sands, his keen eyes searching for any signs of those who might wish to enter the fertile gardens beyond the walls and harm his master. He thought he heard an odd noise within the brush along the inside of the wall and narrowed his watch lamp to a beam, sharp as a spear to cast about for danger. Guided by the spirit of the Lord, he turned his lamp to look within the walls, and his spear of light revealed the feet of a figure some eight cubits from his tower pressed to the base of the wall, and the light piercing the darkness also pierced the veils of deception, giving Aaron knowledge that this figure was here to slay his master. Driven by his duty and his faith, he leapt from the tower and flew along the line of the lantern’s beam as if it were a well cobbled road, and despite flying twenty cubits through the air, he landed miraculously unharmed at the criminal’s feet and engaged him, calling out for other guards. The deceitful adder managed to strike Aaron a blow which would have slain a less righteous man, but then the other guards arrived to take the night viper into custody to be brought before the judges.

I made that up, to avoid any possible arguments about poor translation or corrupted text. What is this passage about? It is about a devoted guard who takes his task seriously, and is willing to place himself in harm’s way to do what he knows is right. It documents what the author considers to be minor manifestations of divine aid and intervention, and seeks to teach a moral about the need for faithful service. Oh, it also has bad math. The description establishes a clear triangle (Vertical wall, base of wall, line of light along the wall) which it claims has edges of 8, 15, and 20. An actual right triangle with the wall edges of 8 and 15 would have a hypotenuse of 17. 3 cubits (a bit over a meter and a half) is too large an error to explain away as the real world roughing up of the abstract perfect math. The author (me) is clearly not providing an accurate account of this triangle. But I was not trying to. My account of the distances was not intended to educate the reader on how to measure, it was to provide context for understanding the events.

Aaron is on a tall wall, around 7.5 meters tall, making it clear that this is a mighty city with considerable wealth to have such walls, maybe it really was this high, maybe this is just my symbolic shorthand to let the reader know I am speaking of a mighty city without having to provide a long list of details, much as many folk tales introduce the powerful, wealthy land owner by calling him a King.

Then the figure is seen about 4 meters down the wall from the guard post. Perhaps this is intended to show a sense of clear danger; our hero was guided by God to spot this assassin in the final moments before he would have passed out of sight, or launched a strike at our hero. This distance establishes a certain nick of time element, a foreboding of immediate danger the reader can share.

Finally the great leap of our hero, not content to simply let the height of the wall speak for itself, I point out that the guard had jumped an impressive distance which anyone familiar with heights or with falls would know usually leads to broken limbs, or for skilled jumpers landing in a ball exposed to anyone nearby. That our hero leaps this distance and engages the assassin in combat clearly shows the depth of his dedication to his duty, and the favor that follows him because of it.

One can also see that I have called the assassin a literal snake at least once, and yet most any reader would know at once that this element is symbolic to evoke his character. I’ve spoken before about how the elements of symbolic writing change over time, but I wanted to come back to this idea. Sometimes the best way to teach a lesson is to do so with less precision. Let’s go back to my triangle above and the use of pennies. Clearly the pennies are imperfect and will not make either our square or our triangle come out with precision, and yet they can convey the idea of that precision in a way more complete and easy to grasp than the more accurate use of just abstract and precise words. Likewise in my tale, I can convey the elements of tension and danger more fully by placing a quick and simple statement of scale with more tension than if I placed more mathematically accurate distances into the narrative.I n both cases the truth of what I am saying is preserved and intact… but the truth of what I am not saying is not warranted, and is not material. I am not seeking to educate my reader on the nature of right triangles, so what my text may happen to say about them may or may not happen to be true.

When we read poetry, we know that what we see is symbolic, and we undertake effort to unravel the symbolism before we claim to know what the poem is telling us, and do not complain that a given tyrant was not actually known to ever wear a velvet glove. Why would we treat Scripture with any less respect? To think that if God felt that he needed to reveal the age of the world as Divine Fact of importance to our salvation, he would elect to do so by giving us a long genealogy and a few statements about individual life spans rather than give us a lesson on the age of the earth seems somewhat suspect. To claim that God is only allowed to use symbolic statements and literary flourishes when it suits the way I want to read the Bible is the height of arrogance. There is a reason that the Holy Spirit is with us, and that the Risen Lord wants a personal relationship with each of us. We are not supposed to have a static relationship with dead ink, but a dynamic relationship with a living God.

I said that this was most of all about a triangle, and it is. The triangle of Father, Son, and Holy Ghost… I can’t tell you how to construct the triangle, I can’t tell you the length of one side or the angle of any corner, but the idea of the Triangle is important, and helps us to remember that we have One God, even if we know him from Three Sides. We can "know" that the Triangle is God, and that the Edges are Father, Son, and Holy Ghost, each part of God, but each distinct. If we can get that idea right, we can share it, even if we change some of the words.