hi-res

Posts tagged with **temperature**

A list of quasimetrics from real life:

- one-way street
- temperature preferences
- ratchet
- running distance around the track
- magnitude
- downside risk in a financial portfolio
- Facebook EdgeRank
- progress
- time
- I love you, but you like me back. Ouch. (“He loves me in his own way”, however, is not on a 1-D scale against “like” — so a functional to a quasimetric would pervert the meaning.)

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.

*Hint: it’s not 50 degrees Fahrenheit.*

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

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.

A list of quasimetrics from real life:

- one-way street
- temperature preferences
- ratchet
- running distance around the track
- magnitude
- downside risk in a financial portfolio
- Facebook EdgeRank
- progress
- DNA sequences
- time

arXiv papers on quasimetrics (to read):

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**.