Sunday, April 30, 2023

284: Don't Square Numbers, Triangle Them

 Audio Link

Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school.   Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host.  And now, on to the math.

As I mentioned in the last episode, recently I read an unusual book, “A Fuller Explanation”, that attempts to explain the mathematical philosophy and ideas of Buckminster Fuller.  You probably know Fuller’s name due to his popularization of geodesic domes, those sphere-like structures built of a large number of connected triangles, which evenly distribute forces to make the dome self-supporting.   His unique approach to mathematics, “synergetics”, probably didn’t make enough true mathematical contributions to justify all the new terminology he created, but does look at many conventional ideas in different ways and through unusual philosophical lenses, which likely helped him to derive new and original ideas in architecture.   Today we’ll discuss another of his key points, his central focus on triangles as a key foundation of mathematics.

While most people who are conventionally educated tend to primarily think of geometry in terms of right angles and square coordinate systems, Fuller considered triangles to be a more fundamental element.   This follows directly from their essential ability to distribute forces without deforming.   Think about it for a minute:  if you apply pressure to the corner of a 4-sided figure, such as a square, its angles can be distorted to something different without changing the lengths of the sides:   for example, you can squish it into a parallelogram.   Triangles don’t have this flaw:  assuming the sides are rigid and well-fastened together, you can’t distort it at all.   This stems from the elementary geometric fact that the lengths of the three sides fully determine the triangle:  for any 3 side lengths, there is only one possible set of angles to go with them and make a triangle.  Thus, if you are building something in the real world, triangles and related 3-dimensional shapes naturally result in strong self-supporting structures.

According to one of the many legends of Fuller’s life, he first discovered the strength of this type of construction when he was a child in kindergarten.   His teacher had given the class an exercise where they were to build a small model house out of toothpicks and semi-dried peas.  At the time, Fuller’s parents had not yet discovered his need for glasses, so he was nearly blind and could not see what his classmates were doing at all.   As he recalled it, “All the other kids, the minute they were told to make structures, immediately tried to imitate houses.   I couldn’t see, so I felt.   And a triangle felt great!”  Basing his decisions on how the strength of the building felt as he was constructing it, he ended up developing a triangle-based complex of tetrahedra and octahedra.   The unusual look accompanied by the unexpected solidity and strength of his little house surprised his teachers and classmates, and many years later was patented by Fuller as the “Octet Truss”.    Fuller wasn’t the first architect to discover the strength of triangles, of course, but he put a lot more thought into extending them to useful 3-D structures than many had in the past.  

With the strong memory of such early discoveries, it’s probably not surprising that Fuller decided he wanted to build an entire mathematical system based on triangles.   He took this philosophy into all areas of mathematics that he looked at, sometimes to a bit of an extreme.   For example, a fundamental building block of algebra is looking at higher powers of numbers, starting with squaring.   But Fuller considered the operation of “triangling” much more important.  Of course this is a useful aspect of general algebra:  you might recall that the triangular numbers represent the number of units it takes to form an equilateral triangle:  start with 1 ball, add a row of 2 under it, add a row of 3 under that, etc.   So the series of triangular numbers start with  1, 3, 6, 10, and so on, representing the value ((n squared - n) / 2) for each n.   Fuller considered these numbers especially significant because they also happen to represent the set of pairwise relationships between n items.   For example, if you have 4 people and want to connect phone lines so any 2 can always talk to each other, you need 6 phone lines, the 3rd triangular number.   For 5 people you need 10, the 4th triangular number, etc.   Fuller was pleased that this abstract concept of the number of mutual relationships had such a simple geometric counterpart.   This observation helped to guide him in many of his further explorations related to constructing strong 3-dimensional patterns that could be used in construction.

So, are you convinced yet that triangles are important?   Of course these have been a fundamental element of mathematics for thousands of years, but it’s been relatively rare for people to view them and think about them exactly the way Fuller did.   That’s probably why, despite exploring areas of mathematics that had been relatively well-covered in the past, he still was able to make original and useful observations that led to significant practical results.   It’s a nice reminder that just because other people have already examined some area of mathematics, it doesn’t mean that there aren’t more fascinating discoveries waiting just under the surface to be uncovered.

And this has been your math mutation for today.



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