A Mathematical Explanation Why Valentine's Day Might Not Suck

We have yet to ring in the new year, but in anticipation of other disappointing holidays to come (seriously, isn't New Years set up to be anticlimactic?), let's consider Valentine's Day. Like New Years, it's meant to be a special occasion. Valentine's Day has the added pressure to create a particularly romantic day with your partner, a sure sign that reality may not meet expectations. Or maybe you shun the cliché chocolate and roses, but then the pressure of anti-Valentine's Day can be just as great! You have to wear sweatpants so hard while you care so little because it's all just a commercial holiday anyway, right?

Today in lab, my coworkers and I discovered what might be a seasonal proof to warm your icy, disappointed heart this February 14th. Just coming off the winter solstice, we were reflecting on how day length does not grow or shrink evenly throughout the year. Rather, following a sine curve, the day length barely budges around the solstices in June and December, while it rockets upward in spring or down in fall, at the equinoxes.

Just take a look at the sweet graph I drew to explain this. The horizontal axis corresponds to the day of the year, or time. The slope of the sine curve reflects the rate at which day length changes. A steeper portion of the line corresponds to times when daylight shrinks or expands at a faster clip from day to day. The fastest rate of change occurs at the autumnal equinox (AE) and the vernal equinox (VE). The peak and trough, at the summer solstice (SS) and winter solstice (WS) are flatter, meaning the day length doesn't change as quickly. Having just experienced the winter solstice a week ago, we are, sadly, at the bottom of the graph.

Given the continuous nature of the increases in daylight, when will we subjectively experience a slow thawing from winter darkness? By the vernal equinox, the days will be as long as nights and we will clearly have sensed the earlier sunrises and later sunsets. That is three months from now though, or about 90 days. Maybe halfway to the vernal equinox, 45 days from now, we will remark to our coworkers and spouse "Hey! It's staying lighter out now, isn't it?"

45 days from today is February 15th. A little rounding in either direction and I will make the bold prediction that come Valentine's Day, February 14th, you may be alone, your partner may have disappointed you, boxed chocolate may still taste terrible and roses may still have their thorns, but you'll probably wake up that day and think, "Well, at least it's not as dark as it has been." Comforting, right? An extra hour and a half of daylight can make up for so many of life's disappointments.

(The graph on the bottom is what we would expect day length changes to look like if it was a constant increase or decrease every day of the year, simply reversing at the solstices. The brackets on the main graph show how roughly equal periods of time (distance in the horizontal direction) leads to small changes in day length near the solstices, but big swings near the equinoxes.)