Posts tagged with temperature

climate estimates for 2000 years

(Source: Wikipedia)

hi-res

Why Fahrenheit is Better than Celsius

Astute reader wargut responded to yesterday’s observation about the Fahrenheit scale being affine-ish with the following incorrect assertion:

Seriously, guys, your system is bullsh~t.

It’s on.

First, the Kelvin scale is indisputably the best of {K,℉,℃} for physics. Given that ∃ a natural zero it should be reflected in the measurement system.

But Fahrenheit is the best scale for everyday use. We are not in the science lab, so all of Centigrade’s properties that are nice in chemistry class don’t matter.

Celsians brag that 0 ℃ and 100 ℃ make it easy to remember where water boils and freezes. So what? Fahrenheit makes it easy to remember the temperature of the human body and icy seawater. Or roughly the hottest day and the coldest day.

Outdoor temperatures in Indiana range from −17 ℃ on the coldest day of winter to 39 ℃ on the hottest day of summer. During the seasons I would be outdoors for more than the necessary minimum—March to November—the daily highs are between 7℃ and 29 ℃.

So most of the relevant temperature variation — the vast differences throughout all of spring, summer, and fall—are restricted to only 23 integers. (I could use decimals, if I wanted to sound like a robot.)

When I lived in ℃ places I had to pay attention to single-digit differences like 24 ℃ versus 29 ℃, wasting the first digit.

In Fahrenheit I get the basic idea with the first digit.

• "It’s in the thirties" = multiple layers and coat.
• "It’s in the nineties" = T shirt weather.

In the 70’s and 80’s I want a second sig-fig but I don’t even need 10 elements of precision. Just “upper 70’s” is enough. The first ℉ digit gives you ballpark, and the second ℉ digit gives you even more precision than you need.

In a sentence: Fahrenheit uses its digits more efficiently than Centigrade. Centigrade adopts the decimal convention but then throws away 70% of the range. Fahrenheit’s gradations are so well tuned that it only requires {0,1,2,3,4,5,6,7,8,9} × {low, medium, high}, for a cognitive savings of 7 unneeded numbers in each of 9 decades.

Celsius may be better for chemistry. Fahrenheit is better for real life.

What’s half of 100 degrees Fahrenheit?

Hint: it’s not 50 degrees Fahrenheit.

100 ℉ = 311 K, half of which is 105.5 K = −180℉

$\large \dpi{200} \bg_white \begin{matrix} 100 \, ^{\circ} \rm{F} & \longrightarrow & 311 \, \rm{K} \\ \\ && \downarrow \\ \\ -180 \, ^{\circ} \rm{F} & \longleftarrow & 155 \, ^1\!\!/\!_2 \, \rm{K} \end{matrix}$

Yup — half of 100℉ is −180℉.

The difference between the Kelvin scale ℝ⁺ and the Fahrenheit scale is like the difference between a linear scale and an affine scale.

You were taught in 6th form that `y = mx + b` is a “linear” equation, but it’s technically affine. The `+b` makes a huge difference when the mapping is iterated (like a Mandelbrot fractal) or even when it’s not, like in the temperature example above.

(The difference between affine and linear is more important in higher dimensions where `y = Mx` means `M` is a matrix and `y` & `x` vectors.)

Abstract algebraists conceive of affine algebra and manifolds like projective geometry — “relaxing the assumption” of the existence of an origin.

(Technically Fahrenheit does have a bottom just like Celsius does. But I think estadounidenses conceive of Fahrenheit being “just out there” while they conceive of Celsius being anchored by its Kelvin sea-floor. This conceptual difference is what makes Fahrenheit : Celsius :: affine : linear.)

It’s completely surprising and rad that mere linear equations can describe so many relevant, real things (examples in another post). Affine equations — that barely noticeable `+b` — do even more, without reaching into nonlinear chaos or anything trendy sounding like that.

Temperature Preference

Here’s another example of a quasimetric.  My girlfriend was arguing that winter is worse than summer.  Her reasoning was this:  if the ideal temperature is 72 Fahrenheit, plus or minus, then winter deviates much further from ideal than does summer.  In Indiana, temperatures often get down to daily highs in the 10’s, 20’s, 30’s in the winter — but they don’t get up to 110’s, 120’s, 130’s in the summer.  (And that doesn’t even take nighttime / lows into account.)

But since people do choose to live in cold places, their preferences mustn’t be symmetrical.  It must be that colder-than-ideal is not as bad as hotter-than-ideal.  Probably because you can wear a coat, but not an anti-coat!  Well, I hate wearing coats, she said.

So her preferences are more or less symmetric. But other people’s climate preferences are a quasi-metric.