Peyton Manning was a bit more difficult than the others, mostly because there was so much information about exactly how many points he scored and yards he threw for in which games (detailed in Asperger's-like fashion), and a little less information about his personality. I still pulled out the bits I felt best represented him specifically (as opposed to how many yards he got in each game blah blah blah), so it's a smaller amount of formulas and info, but still representative of him as a success story and as a celebrity.

At the top is alpha greater than beta. These in the field of statistics uses alpha and beta to represent type I (false positive) and type II (false negative) errors, respectively. Peyton was not known for missing many good opportunities, so his false negative rate (beta) is super low. However, he was known for throwing a fair number of interceptions (false positives), so his alpha number is high. Overall he seems throughout his career to be heavily biased toward action instead of inaction, which is probably why he was so successful. He doesn't let good opportunities go by (though he screws up a few times in the process, he moves on).

Over on the right side I have an homage to how far he was willing to go for his career. He ended up having to leave the Colts after serious spinal injuries/trouble required neck surgery, which inhibited the range of motion in his throwing arm for long enough that he had to be let go. Then after getting the full spinal fusion surgery for his neck based on an MRI, he was able to return to football with the Broncos. The energy difference between nuclear spin states is what generates the signal used to generate an MRI image (that's the formula), and similar to how lower energy states can be "flipped" to higher energy states before "relaxing" back to the lower energy state in MRI, he had to undergo a difficult transition before returning to what he loved.

On the left the three acceleration formulas (one for horizontal acceleration, one for vertical acceleration, and one for Coriolis apparent acceleration) reference his legendary accuracy in increasing detail and to increasing degree. The first (horizontal) one accounts for air drag on a projectile, which is a kind of standard accuracy that every quarterback must think about. The second (vertical) one accounts for both air drag and changing gravity due to widely varying heights above the center of earth's gravity (as if he were launching the football into the upper atmosphere and back). The third one shows the apparent acceleration in three dimensions (Northward, Eastward, and up toward the sky) due to the Coriolis effect, which has to do with appreciating that the football field isn't actually flat but is laid out over the curved surface of a sphere (the earth). For instance, if you were to launch a rocket directly in the north direction from the equator, it would totally miss the North Pole, appearing to swerve Eastward as it traveled north, because the earth's rotation must be taken into account. I remember reading somewhere that the Coriolis effect (which would only be relevant if the throws were made in the north or south directions but not for eastward or westward throws) could potentially seemingly alter a football's course by a measurable but unnoticeable distance (something like half an inch) over the course of the entire field. The facetious reference is the idea that his throws are so accurate that he has to take the Coriolis effect into account, like a sniper.

This is perhaps where I should bring in the "Standard of Perfection: Anas platyrhynchos" bit. Despite being known for his ability to get passes to their exact target at the exact right time, he was also known for throwing ugly passes with wobbly spirals. In football parlance, such ugly throws are called "ducks" (probably because they're goofy-looking and not elegant like a perfect football spiral is). The (American) Standard of Perfection is actually a standard for the breeding of various poultry (kind of like dog competitions except for ducks, turkeys, geese, etc--yeah apparently that actually exists), and the Latin but after it is the scientific species name for the Mallard duck. So basically I'm implying that although his throws are ducks, they're perfect ducks, because they win games. He was once called out during a press conference for throwing ducks (to the effect of "what do you have to say about the claims that you throw ducks?"), and he famously retorted that he's thrown a lot of yards and touchdown ducks. He's doubling down on his choices here, famously being a very hard worker and being focused on winning, purely and simply, without concerns about any superfluous qualities to his football. He's basically saying "yeah, my throws are ugly, but they win games". The 1/2(1+a^2/b^2)cos(theta) thing beside it is the spin-to-wobble ratio, which is an important aerodynamic quantity (his presumably being low).

Another formulas related to his focus are the two beneath those three formulas for acceleration. The first one is the (tangent of the) apparent elevation angle for a receiver looking to catch a pass. It's constantly increasing over time (this is true in real life actually--even as the ball falls down toward the catcher from its peak, its elevation angle still increases from the perspective of the catcher), and nothing in the environment effects this constant change. I'm likening this to his constant focus on increasing his skills over time and his doing so regardless of his external environment. He was known to inspire his teammates and even rivals with his diligence, and in their eyes he was always improving even though he never later had a season as good as his peak in 2004. He is a complicated character though, and despite his seriousness about improving and winning is known to not be humorless. The lower formula ( 1/2(1-cos(e)) ) is the proportion of the visible moon that can be seen based on the elongation angle between the sun, moon, and earth, and it's referencing an incident back from his time in college at Tennessee when he mooned a teammate in the locker room while his physical therapist was occupied looking at his foot. (When she looked back up at him, she got a face full of ass, and ended up settling with the university for a buttload (har har, couldn't help it) of money.) So despite his seriousness he also had a humorous side (despite that incident attracting a fair amount of media controversy well after the fact, and his legendary football stats have outlived that controversy in the public imagination anyway).

The other formula on the right side is a description of the Reynolds number of an American football, which is about 200,000. It has to do with more aerodynamic properties of its flight. Reynolds numbers are very cool, and have to do with other interesting physics principles like the Magnus effect, but I'd say this formula can be kept or dropped based on how you feel.

The series of number fractions added up are a numerological reading of his jersey number (18 for all his years with the Colts and Broncos, or his whole professional career). In mathematical circles, a number whose divisors sum up to more than the number itself is called an "abundant" number, and a number whose nearly complete list of divisors sums up to exactly equal to the number itself is called "semi-perfect" (if the full set of divisors adds up to exactly the number itself, it's called a "perfect" number: 6 is an example because 1, 2, and 3 all divide evenly into 6, and 1+2+3=6). Peyton Manning has an abundance of NFL records to find under his name and jersey number, and he's a nearly perfect quarterback (perhaps just his duck-like throws and relatively high number of interceptions thrown are his weaker points). There's also another reference hidden in his jersey: the "Ar" is the chemical symbol for Argon (element number 18), which is a "noble" gas (Peyton is known for his admirable and upstanding nature as well as philanthropy). Additionally, the argon symbol is shown in "core electron" notation (with the brackets), which shows that he doesn't have any internal weaknesses and is pretty hard core, not reacting much to external pressure (somewhat like the quote at the bottom that you picked out from earlier).

You can see the formula I provided for that quote at the very bottom. P is the standard symbol for pressure, and the right side is the measure of surprisal (roughly describable as the mathematical measure of how surprising some piece of information turns out to be, given all the other values it could have had--for more, read up on Claude Shannon, the guy, basically a genius, who essentially came up with the entire field of information theory on his own). The right side is how surprising an event e is given the entire set of events (E) that could have happened, and the formula shows pressure being proportional to that surprise (when you don't know what the hell you're doing). It goes a bit deeper too, as there are two ways to reduce the surprise of an event: either work to make the one you're expecting even more likely, or work to reduce the total number of other things that could happen. Peyton was almost always ahead of the game due to his relentless effort. But the other side also held up too: he got visibly spooked for the rest of the game when his center accidentally snapped the ball early during the very first play of Super Bowl 48 and it flew over his shoulder and into his own end zone for a safety. Seriously, look up www.youtube.com/watch?v=AvtemlwTr_Q and see what happened. He didn't play very well for the rest of the game and they lost that one to the Seahawks. I think the pressure got to him on that one, haha.

The diagram I'm proudest of for this project on Peyton Manning is the perhaps inexplicable diagram on the right. It's a diagram of a quincunx (just look it up--it's kinda like pachinko), which is usually used to demonstrate the statistical concept of regression to the mean (pretty relevant concept here too, as it unusually *doesn't* apply much to Peyton--most athletes have an unbelievable season (like Peyton did in 2004) and end up on the cover of Sports Illustrated, but then fail to perform well afterwards. Why though? Because that unusually good season was the result of random chance artificially boosting their apparent performance and isn't an accurate reflection of their inherent skill. Not so for Peyton though--he continued to do well because he's just that good.). But here I'm using it to illustrate the idea of a statistical outlier, which is normally defined as a data point more than 1.5 interquartile range's worth of space above the third quartile (that's basically a rigorous statistical way of saying "it's way out of the range of the rest of the data"). You can see that his data point (the marble that's off to the right and is supposed to have an 18 written in it to represent him) is way out of the rest of the range of the marbles (other players), which are roughly normally distributed. So not only is he far enough out on the high skill end to be an outlier, he's literally outside the box, seemingly impossible because you put all the marbles in the top of the box to see them randomly distributed. It's a way of showing both how unusually good he was (2004 season widely regarded as the best season of any quarterback ever) and also how outside the box he is for a typical star quarterback, as he's apparently a very low-key and high-diligence guy who doesn't carry around an outsized ego with him.