By most accounts, the NFL’s plan to expand the regular season from 16 to 18 games is a done deal. Indulge me for a moment as I take off my Bill-James-Wannabe cap and put on my dusty old Aristotle-Wannabe kausia: In addition to various practical drawbacks, moving to 18 games risks disturbing the aesthetic harmony—grounded in powerful mathematics—inherent in the 16 game season.

Analytically, it is easy to appreciate the convenience of having the season break down cleanly into 8-game halves and 4-game quarters. Powers of 2 like this are useful and aesthetically attractive: after all, we are symmetrical creatures who appreciate divisibility. But we have a possibly even more powerful aesthetic attachment to certain types of asymmetrical relationships: Mozart’s piano concertos aren’t divided into equally-sized beginnings, middles and ends. Rather, they are broken into exposition, development, and recapitulation—each progressively shorter than the last.

Similarly, the 16 game season can fairly cleanly be broken into 3 or 4 progressively shorter but more important sections. Using roughly the same proportions that Mozart would, the first 10 games (“exposition”) would set the stage and reveal who we should be paying attention to; the next 3-4 games (“development”) would be where the race for playoff positioning really begins in earnest, and the final 2-3 weeks (“recapitulation”) are where hopes are realized and hearts are broken—including the final weekend when post-season fates are settled. Now, let’s represent the season as a rectangle with sides 16 (length of the season) and 10 (length of the “exposition”), broken down into consecutively smaller squares representing each section:

^{Note: The “last” game gets the leftover space, though if the season were longer we could obviously keep going.}

At this point many of you probably know where this is going: The ratio between each square to all of the smaller pieces is roughly equal, corresponding to the “divine proportion,” which is practically ubiquitous in classical music, as well as in everything from book and movie plots to art and architecture to fractal geometry to unifying theories of “all animate and inanimate systems.” Here it is again (incredibly clumsily-sketched) in the more recognizable spiral form:

The golden ratio is represented in mathematics by the irrational constant *phi*, which is:

1.6180339887…

Which, when divided into 1 gets you:

.6180339887…

Beautiful, right? So the roughly 10/4/1/1 breakdown above is really just 16 multiplied by 1/*phi*, with the remainder multiplied by 1/*phi*, etc—9.9, 3.8, 1.4, .9—rounded to the nearest game. Whether this corresponds to your thinking about the relative significance of each portion of the season is admittedly subjective. But this is an inescapably powerful force in aesthetics (along with symmetricality and symbols of virility and fertility), and can be found in places most people would never suspect, including in professional sports. Let’s consider some anecdotal supporting evidence:

- The length of a Major League Baseball season is 162 games. Not 160, but 162. That should look familiar.
- Both NBA basketball and NHL hockey have 82-game seasons, or roughly half-
*phi*. Note 81 games would be impractical, because of need for equal number of home and road games (but bonus points if you’ve ever felt like the NBA season was exactly 1 game too long). - The “exposition” portion of a half-
*phi*season would be 50 games. The NHL and NBA All-Star breaks both take place right around game 50, or a little later, each year. - Though still solidly in between 1/2 and 2/3 of the way through the season, MLB’s “Summer Classic” usually takes place slightly earlier, around game 90 (though I might submit that the postseason crunch doesn’t really start until
*after*teams build a post-All Star record for people to talk about). - The NFL bye weeks typically end after week 10.
- Fans and even professional sports analysts are typically inclined to value “clutch” players—i.e., those who make their bones in the “Last” quadrant above—way more than a non-aesthetic analytical approach would warrant.

Etc.

So fine, say you accept this argument about how people observe sports, your next question may be: well, what’s wrong with 18 games? any number of games can be divided into phi-sized quadrants, right? Well, the answer is basically yes, it can, but it’s not pretty:

The numbers 162, 82, and 16 all share a couple of nice qualities: first they are all roughly divisible by 4, so you have nice clean quarter-seasons. Second, they each have aesthetically pleasing “exposition” periods: 100 games in MLB, 50 in the NBA and NHL, and 10 in the NFL. The “exposition” period in an 18-game season would be 11 games. Yuck! These season-lengths balance our competing aesthetic desires for the harmony of symmetry and excitement of asymmetry. We like our numbers round, but not too round. We want them dynamic, but workable.

Finally, as to why the NFL should care about vague aesthetic concerns that it takes a mathematician to identify, I can only say: I don’t think these patterns would be so pervasive in science, art, and in broader culture if they weren’t *really* important to us, whether we know it or not. Human beings are symmetrical down the middle, but as some guy in Italy noticed, golden rectangles are not only woven into our design, but into the design of the things we love. Please, NFL, don’t take that away from us.

So your argument isn’t that the Golden Ratio isn’t present in an 18-game season, but that 11 isn’t a pretty enough number?

I remain unconvinced. Especially so because European football(soccer) hasn’t exactly been held back by the use of 34- and 38-game seasons.

Also, the argument that if we see it everywhere, it must be important is circular in nature. The Golden Ratio has been known for Millennia and been used consciously by artists and designers throughout, so observing it in places where it has been consciously implemented isn’t exactly proof of it being ubiquitous.

Plus, if you’re going to look for things like that, you’re going to find them regardless of their relevance.

Couple that with the fact that for some of the examples you name (like the Mona Lisa) you have to expend some serious effort to find that golden ratio, and it doesn’t exactly fit neatly.

That’s not strictly my thinking. I think the fact that 11 isn’t “a pretty enough number” makes 18 a less attractive length in its own right, or would make it seem so over time.

True that European football doesn’t have phi-length seasons, 32 would have been the more “mathematical” choice. But really, one example of a spot where the lack of subconsciously manifested golden ratio hasn’t lead to collapse is hardly proof of its insignificance. Besides, I’m not suggesting that an 18 game season will destroy football, I’m merely saying it will make it aesthetically worse.

As to the rest, kudos to you for being more skeptical than I am—that is rare enough. But I don’t think this is your typical numerology mumbo-jumbo. First off, the fact that this ratio has been used by artists and architects intentionally for millennia isn’t the cause of the ratio’s ubiquity, it’s a symptom. That is, these people recognized its natural beauty and imitated it to make their own. More importantly, it can be found in many contexts

unintentionally— in many cases in shockingly precise fashion (such as, where the ratio is perfect out to several digits). I concede that the issue is nebulous, but I think it is delightfully so. I don’t know how familiar with the overall literature, but if you haven’t already, I encourage you to go beyond the few items that I linked, and I would be surprised if you aren’t impressed.Finally, for the mona lisa, I think your skepticism is simply unwarranted. On this count, you must simply have not looked into it very deeply: there is probably no more perfect example in all of art. It may appear subtle to the eye, as Da Vinci didn’t put tick-marks where all the boxes should be (as he did with many of his other works), but that is somewhat the beauty of it.

Thanks for commenting!

Let me say that I don’t doubt the idea that the Golden Ratio is aesthetically attractive. I do doubt its ubiquity as arising from natural causes, but that’s neither here nor there. An interesting question may be how the practice of 82- and 162-game seasons came to be, perhaps they consciously looked for a phi-sized season. I know the NFL didn’t do that consciously, as they had 14-game seasons initially.

Criticizing the Mona Lisa was probably poor form on my part.

Getting back to the point of the length of seasons, I think you dismiss the world’s most popular sport a little too easily. In fact, if we look at sports outside the USA you will usually see a total lack of phi-sized seasons. In large part because the rest of the world doesn’t share the USA’s love for divisions, instead having all teams in the league play eachother once or twice. Part of it too is a bigger focus on the regular season, as oftentimes playoffs are absent and the team that wins the season, wins the title. Which is why unbalanced schedules don’t really work for those leagues.

One more point is that while the NFL may have a 16-game season, it actually has a 17-week season. While one team’s season can be divided fairly neatly into phi-based bits, a division, conference or league season can’t be. That’d make the season structure 10/4/2/1 – which feels more aesthetically pleasing to me than 10/4/1/1.

In an 18-game season there are generally expected to be 2 bye weeks. In that case the season structure would be 12/5/2/1, and it’s actually closer to a phi-sized distribution than a 17-game season.

Sander says “In large part because the rest of the world doesn’t share the USA’s love for divisions, instead having all teams in the league play eachother once or twice.”

Perhaps you haven’t considered the Champions League, the World Cup or the Olympics.

Country-based soccer (football) leagues (e.g. Bundesliga.) have the reality of being in relatively small countries. The US leagues typically break up into leagues to save loads of cash on travel expenses and to magnify local rivalries.

I don’t think it is a nation/cultural phenomenon, just a practical one. Your thoughts five months later?

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