Right Triangles and Wrong Triangles

This is what the Lord says: “Go down to the palace of the king of Judah and proclaim this message there: ‘Hear the word of the Lord to you, king of Judah, you who sit on David’s throne—you, your officials and your people who come through these gates.  This is what the Lord says: Do what is just and right.  Rescue from the hand of the oppressor the one who has been robbed.  Do no wrong or violence to the foreigner, the fatherless or the widow, and do not shed innocent blood in this place.  For if you are careful to carry out these commands, then kings who sit on David’s throne will come through the gates of this palace, riding in chariots and on horses, accompanied by their officials and their people.  But if you do not obey these commands, declares the Lord, I swear by myself that this palace will become a ruin.’”

  • Jeremiah 22:1-5

King Nebuchadnezzar made an image of gold, sixty cubits high and six cubits wide, and set it up on the plain of Dura in the province of Babylon.  He then summoned the satraps, prefects, governors, advisers, treasurers, judges, magistrates and all the other provincial officials to come to the dedication of the image he had set up.  So the satraps, prefects, governors, advisers, treasurers, judges, magistrates and all the other provincial officials assembled for the dedication of the image that King Nebuchadnezzar had set up, and they stood before it.
Then the herald loudly proclaimed, “Nations and peoples of every language, this is what you are commanded to do: As soon as you hear the sound of the horn, flute, zither, lyre, harp, pipe and all kinds of music, you must fall down and worship the image of gold that King Nebuchadnezzar has set up.  Whoever does not fall down and worship will immediately be thrown into a blazing furnace.”

  • Daniel 3:1-6

“Pythagoras was … a deeply religious and superstitious man.  He believed in reincarnation and the transmigration of souls, and he established a religious cult, with himself cast as a virtual messiah, in Croton, southern Italy.  His disciples lived in a collective commune, following strict behavioral and dietary rules, while studying his religious and philosophical theories.  The Pythagoreans, as his disciples were known, saw his ideas as mystical revelations…
“In geometry and mathematics he found truths that he regarded as self-evident, as if god-given, and worked out mathematical proofs that had the impact of divine revelation.
“Because these mathematical discoveries were a product of pure reasoning, Pythagoras believed they are more valuable than mere observations.  For example, the Egyptians had discovered that a triangle whose sides have ratios of 3:4:5 always has a right angle, and this was useful in practice, such as in architecture.  But Pythagoras uncovered the underlying principle behind all right-angled triangles (that the square of the hypotenuse equals the sum of the squares of the other two sides) and found it to be universally true.  This discovery was so extraordinary, and held such potential, that the Pythagoreans took it to be divine revelation.”

  • Sam Atkinson (senior editor), The Philosophy Book, Big Ideas Simply Explained

To explain the title, I was cruising through Geometry as a sophomore in high school, fifteen-years-old, now more than 50 years ago.  I was well into my second semester, and I had not gotten a question wrong.  (If you have ever done that, and with me only twice, it can be devastating to your nerves the further you go.)  Not that smart, my brain just thought in such terms and it looked natural; it looked ‘right.’  One day after class I asked the teacher that since we knew all about right triangles, what about all the wrong triangles out there?  I thought it to be a funny joke that she would appreciate.

She leaned in close to me and said, “Watch it, young man!  You have a perfect record so far, and I think your theorem for extra credit is inspiring, but do not start getting cute!”

I have often thought about what she meant by her statement.  Would she have lowered my grade for making a joke?  I knew college professors who were that much of a megalomaniac.  Was she afraid that if I started to make jokes that I might lose my focus on a perfect grade?  Or did she revere Pythagoras, as the Pythagoreans did reverently, thinking that Pythagoras made god-like revelations from above and not to be joked about?

In reading the quote above, you wonder whether Pythagoras was truly a genius or insane.  With some brilliant minds, there might be little difference, much to their discredit.  When people think they are too smart to believe in God or that they no longer need God, they might as well have drifted from brilliance to crazy.

In the two Scriptures above, we see Jeremiah prophesying to the Judah king that worshipping God, and doing that worship properly, is the only way to maintaining the kingdom.  They must do what is just and right and do no wrong to a foreigner.  Then in the second Scripture, in Daniel, we see that Nebuchadnezzar set up a golden image of himself and had everyone bow down to it.  Here was the king who God used to conquer the chosen people of Judah, to teach the Jews that they should have listened to the prophets like Jeremiah and to God Himself.  Nebuchadnezzar began to eat his own “Wheaties” as the phrase goes.  Another phrase is that Nebuchadnezzar started reading his own press clippings.  In other words, he got the big head, full of himself, as if he were god.  So, why not have everyone bow down to an idol that looked like him?

Pythagoras (570-495BC) and Nebuchadnezzar II (627-562BC) may have only overlapped in their span of life temporarily, but it is a common practice among humans to inflate the significance of our achievements.  To inflate their worth to being divine and establish yourself as someone who must be worshipped?  That crosses the line from right triangles into wrong triangles.

In my earliest army training, we built a house one afternoon, in a couple of hours, but we had spent the entire morning getting the foundation square.  How did we know it was perfectly square?  We set up batter boards and created a box using string.  Then we tied a string in each corner (3 of the 4, but then the fourth for a sanity check), making a triangle in each corner.  Once we adjusted the strings of the box so that all four triangles were 3:4:5 triangles, we knew the building would be square.  Too many builders do not take that extra time to get the strings perfect and then the corners of the rooms within the house are a bit “off.”  You are thinking an entire house in one afternoon, two hours?  How ridiculous!  But the house was a predesigned, prefab-style house, a TO structure (Theater of Operations – Army talk).  The panels were made as we were getting the foundation right.  The plumbing and electrical were laid out roughly at the same time, adjusted once the foundation was square.  Then the construction was basically like erecting a toy house when you were a child (out of logs or blocks), just throw the pieces up and nail it in place.

I just threw in that old memory to illustrate to those who have not done construction work that the 3:4:5 triangle of the Egyptians is still used today, but maybe lasers and computers can do it for you instead of string.  I still see batter boards at construction sites and I cannot help but smile.

We see something strange here with Pythagoras and Nebuchadnezzar.  These Geometry discoveries caused Pythagoras to think of himself as a god.  Nebuchadnezzar thought his conquests qualified him to be a god.  You wonder if that heady draught of fame affects each who achieve something great.  I know of only One who could withstand such a temptation, and He is God.

“In your relationships with one another, have the same mindset as Christ Jesus:
Who, being in very nature God,

    did not consider equality with God something to be used to his own advantage;
rather, he made himself nothing
    by taking the very nature of a servant,
    being made in human likeness.
And being found in appearance as a man,
    he humbled himself
    by becoming obedient to death—
        even death on a cross!
Therefore God exalted him to the highest place
    and gave him the name that is above every name,
that at the name of Jesus every knee should bow,
    in heaven and on earth and under the earth,
and every tongue acknowledge that Jesus Christ is Lord,
    to the glory of God the Father.”

  • Philippians 2:5-11

Oh, were you wondering about that theorem for extra credit that the teacher mentioned was ‘inspiring?’  You might find it silly, but a construction person might find it useful.  Let’s say you have a multi-sided room, like a tower room or a gazebo, and you wanted to make equal sized triangular roof sections, say out of glass.

In Geometry class we drew a triangle (of any kind: isosceles, equilateral, right-angled, or simply congruent – no side is the same length as another and none of the angles is the same as another or a right angle, 90 degrees).  Then we drew a line between two of the sides’ midpoints.  The smaller triangle that is formed looks just like the big triangle.  Our assignment was to prove the small triangle was “similar” to the large one.  “Similar” being a technical term that the angles were all the same, but the length of each side was different.

My theorem:  I went further.  I drew a line from each of the midpoints of each side.  I then proved that the four triangles that are formed was not only similar, but the same, same three angles and same three lengths of sides.

Now to make a triangular roof section out of glass, you either use very expensive large triangular glass panels or you make four smaller pieces that are all the same size and shape and then the extra framing adds structural strength, and replacement of a broken piece of glass is much less costly.  And if the base of each of the triangle roof sections is the same, you only make a whole lot of the same shaped piece, saving fabrication costs.

And believe me, I was not divinely inspired.  At first, I started doodling in class.  I then saw something that looked like it would work and then I did the math.  But then again, I used the same type of observations that Pythagoras made.  He just made them when no one else was making discoveries of that type, giving me a lot of rules and math that I could use, making it a lot easier.  Yet, even then, nothing god-like.

His contribution to philosophy was a switch from substance to form.  The Milesian philosophers studied substance like Thales of Miletus who thought everything was made of water and Heraclitus who thought everything was fire.  This school of philosophy focused on the four elements: earth, wind, fire, and water, arguing that everything boiled down to one or another among those elements.  But Pythagoras thought outside that box, actually measuring the box, discovering the form of the box was important and creating a type of mathematics to describe it.

Yet, calculating right triangles is good for construction and fun in the classroom, but Pythagoras did not create the triangle, he only observed what God had made.

Let us all humble ourselves before God.  The more that we learn, and the more knowledge that we might synthesize from the knowledge that we have learned, only means that we have uncovered a small tip of a corner of the universe that God created.  And God created the universe by simply speaking it into existence.

If you like these Tuesday morning essays about philosophy and other “heavy topics,” but you think you missed a few, you can use this LINK. I have set up a page off the home page for links to these Tuesday morning posts. I will continue to modify the page as I add more.

Soli Deo Gloria.  Only to God be the Glory.

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