## Today is Phi Day -- at least, it ought to be

**Jan 6, 2015**

By now, most everyone is aware of Pi Day, celebrating the famous mathematical constant
\(\pi \approx 3.14159\)
on ^{3}⁄_{14}. On this day each year, students and math enthusiasts eat pie and engage in light-hearted
\(\pi\)
-related activities. Then there is e Day, a day for commemorating the equally-important-if-somewhat-less-famous constant
\(e \approx 2.71828\)
on ^{2}⁄_{7}. The activities are similar, though it’s less clear-cut what food one should eat (a high school teacher of mine insisted that the proper e Day food is waffles, evocative of the Cartesian coordinate system). Even events like Pi Approximation Day (^{22}⁄_{7} in day/month format), and Mole Day (6:02 ^{10}⁄_{23}) have gained enough momentum to warrant dedicated results from both Google and Bing.

So when is Phi Day?

\(\phi \approx 1.61803\) , the Golden Ratio, is arguably the 3rd-most “pop-famous” mathematical constant after \(\pi\) and \(e\) . It has enough geek-cred status that you can buy shirts with hundreds of its digits on the front, arranged into the shape of the \(\phi\) character. So you’d think Phi Day, whenever it is, must surely be marked on numberphiles’ calendars worldwide.

As it turns out, there are major disagreements over the date on which Phi Day should be held, and this lack on consensus has kind of killed the whole thing before it ever got started. The surprisingly long list of candidate dates has fractured the community such that a critical mass of participation on a single day has never been reached.

What are these different Phi Day camps?

### The digits division

Following the precedent set by Pi Day on ^{3}⁄_{14} and e Day on ^{2}⁄_{7}, the most straightforward choice for Phi Day is ** ^{1}⁄_{6}**, based on the leading decimal digits of the constant. And indeed, there is some support for this date: see here, here, and here.

Interestingly, the more popular digit-based candidate seems to be ** ^{6}⁄_{18}**. Some basic web searches turn up a number of results, e.g. here, here, here, here, here, and here. The standard explanation for this choice is that it matches the first digits

*after*the decimal point, or that it matches the leading digits of the reciprocal form \(\Phi = 1/\phi \approx 0.61803\) .

### The fraction faction

There is another group which argues that picking the date of Phi Day based on decimal digits is passé and should be avoided. Maybe that’s good enough for Pi Day and e Day, they say, but with Phi Day we have an opportunity for something more meaningful, man!!

The idea uniting this faction is to pick Phi Day as the date which divides the year into two parts, where the parts are in proportion matching the Golden Ratio.

This is actually a neat idea, but sadly even this camp shatters into further subdivisions.

Taking the year to be 365 days, one gets closest to a division in the ratio of
\(\phi\)
by taking the 226th day of the year (226/(365 - 226) is about 1.626), which is ** ^{8}⁄_{14}**. See here and here for advocacy of this date.

But what about leap years? In those years there are 366 days, of which 8/14 is the 227th. In this case the previous day, ^{8}⁄_{13}, is actually a better pick, as 226/(366 - 226) = 1.614 comes closer to the desired ratio than 227/(366 - 227) = 1.633.

Thus ^{8}⁄_{14} is rejected by some in favor of the more robust alternative where the shorter division is put first. ** ^{5}⁄_{19}** is the 139th day of standard years, and the 140th day of leap years. Happily, this ends up being the ideal pick for both cases: (365 - 139)/139 = 1.626 and (366 - 140)/140 = 1.614. See here for advocacy of this date.

Finally, this prominent site catapults the navel-gazing to stunning new heights by condemning the Gregorian calendar itself as too mainstream, declaring that the Phi Day year-division should be measured starting from a more “natural” delimitation of the annum, namely the vernal equinox.

But wait, doesn’t the Spring equinox occur on different dates in the Northern and Southern Hemisphere? Indeed, this site goes so far as to advocate for *two separate Phi Days*: ** ^{10}⁄_{31}** in the Northern Hemisphere,

**in the Southern Hemisphere (where the dates are computed the same way as**

^{5}⁄_{6}^{8}⁄

_{14}, but starting from the respective equinoxes).

### Keep it simple

My stance is that Phi Day belongs on the obvious choice: ** ^{1}⁄_{6}**. There is strong precedent in Pi Day and e Day, it’s easy to remember, and there are no worries about leap years or hemispheres. It’s accessible and easy to explain to anyone.

^{6}⁄_{18} strikes me as an arbitrary choice to ignore digits, or to prefer the secondary, reciprocal form. I just don’t see what’s gained by that choice. A further disadvantage is that many schools (in the US, at least) are already out for summer break by this time, so this date can’t really be used as a fun day for students.

And all of the year-fraction dates can be dismissed outright, if you ask me. Yes, explaining their position within the year provides a potential teaching opportunity, but that computation (let alone the overall explanation) is just totally inaccessible to most people. I had to break out a bit of code and lean on a date/time library just to verify these dates, and I assume most others would need to do the same. When there is such a high hurdle to even figure out what date the event is on, there’s no way it will ever gain traction outside of fanatics.

The ultimate goal of these silly “math holidays” is to provide a fun excuse for enthusiasts to celebrate, but also to create a friendly and inviting opportunity for *non*-enthusiasts to learn a little bit and join the fun without feeling intimidated. ^{1}⁄_{6} seems like the clear choice.

Happy Phi Day!