La Mathematica dell’Arte: let’s think about length

320px-Sparkle_water_drop-otherPrime: You know, magnitudes are great.

Naught: What are you saying all of a sudden?

Prime: This idea that things have magnitudes is just marvelous.

Naught: You want me to ask “why,” right?

Prime: ”Why,” you ask? Well, how else would you compare my desk with your desk?

Naught: I don’t know, how about by their shapes?

Prime: Try to be more precise; you can’t really say that my desk is more desk-shaped than yours.

Naught: By their lengths, then.

Prime: Lovely! Length is a magnitude that is so convenient when comparing two things.

Naught: Yeah, I’d say so too.

Prime: The natural way to determine length is by measuring the distance between its initial and end points, perhaps by using a ruler. But let’s think outside of the box. What could be another way to compare the heights of our desks?

Naught: Oh boy. Let’s see… comparing the length of the shadows they cast?

Prime: That’s not very accurate, as it’ll depend on the time of the day and the angles. I want you even more outside the box.

Naught: Even more outside… How about the time that a drop of water will take for it to travel from the top to the bottom along its leg.

Prime: I love it! Let’s call that…“aquatic length” [1]. Assuming that gravitational acceleration and cohesive forces are the same in both of our universes – although it is very inefficient and the water drop may never reach the bottom – that is a great way to compare the height of two things. Analytical chemists use paper chromatography, which uses a similar principle – but instead of measuring time, they measure the length each pigments rise.

Naught: What are you saying?

Prime: Never mind, I just wanted to point out that your idea isn’t so out there. But back to mathematics! Say your desk and my desk have about the same height but say your desk and my desk have “aquatic lengths” of 40 and 20

Naught: Sure, that sounds reasonable., respectively, for some reason.  Then, we can conclude that my desk is “aquatically” shorter than yours.

Prime: But we can measure “aquatic length” for anything, say that of that window and get  20. Then although my desk is shorter than the window in the conventional length, they “aquatically” have the same length.

Naught: Oh, I think that’s pretty cool. So something that is taller in centimeters could be “aquatically” shorter?

Prime: Great observation! This relates to my research as well. The measurement of length in the geometry I work with has the name “rational length” and it is the greatest common divisor (gcd) of the numerator and the denominator of the slope of a line, and we make sure that this measurement exists by only having line segments with rational slope (i.e. the slopes can be represented by a fraction). [2]

Naught: Remind me what gcd is?

Prime: It is the largest integer that divides both numbers. So if the slopes are 2/4, 6/9, then the rational lengths are 2 and 3, respectively.

Naught: So a line with slopes 2/4 and 6/8 both have rational length of 2?

Prime: Exactly! They have the same “length,” just like how my desk and the window

both had the same “aquatic length.” I restrict myself to only work with objects that have a rational length of 1. Then neat things start appearing!

[1] “Aquatic length” is not a real measurement, and is probably impractical too. But the idea is that there exists different measurements which may have the same length in one measurement but not in the other.

[2] This is not completely accurate, as rational length is the gcd of the vectors of the line segments. The representation of rational numbers is not unique; 1/2, 2/4 and 3/6 represent the same slope, but if they are actually vectors – [1,2], [2, 4] and [3, 6] – the rational lengths are 1, 2 and 3, respectively.

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