Chapter Text
Figure 1.1: For the purposes of this work, we define Caleb's framework of attraction as the 3D rectangular coordinate system.
Figure 1.2: Caleb's spoken coordinates x, y, and z easily match the geometry of both his framework and his understanding of his own attraction, meaning he can express himself in his own framework with relative ease.
Figure 2.1: For the purposes of this work, we define Essek's framework of attraction as the spherical coordinate system.
Figure 2.2: Though the coordinates of Essek's coordinate system may seem more complicated than Caleb's, Essek's spoken coordinates r, θ, and φ easily match the geometry of both his framework and his understanding of his own attraction, meaning he can express himself in his own framework with relative ease.
Figure 3.1: The geometry between Caleb's cube and Essek's spherical coordinate system appears incongruent. The conversion of Caleb's coordinates into Essek's coordinates looks somewhat complicated, meaning Caleb may have a hard time expressing himself using Essek's framework.
Figure 3.2: The geometry between Essek's sphere and Caleb's 3D rectangular coordinate system appears incongruent. The conversion of Essek's coordinates into Caleb's coordinates looks very complicated, meaning Essek may have a very hard time expressing himself using Caleb's framework.
Figure 4: Essek's only reference for a 'linear' coordinate is r, and Caleb's reference for coordinates does not include the letter r. Their conversation in effect reads as follows:
Caleb: So x?
Essek: ...'x'? You mean r?
Caleb: Not r? Like a line between X1 and X2.
Essek: What? So r?
Caleb: What?
Figure 5.1: In this figure are three threads of conversation. Caleb and Essek spend a while attempting to express themselves in the only ways they know how. The first thread of conversation in effect reads as follows:
Essek: ...What is 'z'?
Caleb: z is like x. z is like y. z is perpendicular to y is perpendicular to x.
Essek: 'Perpendicular' means 'not'?
Caleb: Perpendicular does not mean 'not'. ...And also z is not y is not z.
Essek: What is 'perpendicular'?
Due to some linguistic quirks, even their logical symbols don't always match up.
The second thread of conversation in effect reads as follows:
Caleb: What is 'φ'?
Essek: φ... It's like if x is zero, and then x goes to π/2.
Caleb: What's 'π'??
Essek: π is circumference divided by 2r.
Caleb: What is 'circumference'?
Essek: Circumference is 2πr, therefore π is circumference divided by 2r.
Caleb: ...
Essek tries to phrase things in a way Caleb might understand, only for them to get stuck in circular reasoning.
The third thread of conversation in effect reads as follows:
Caleb: So x is not r?
Essek: x is r if and only if r is x.
Caleb: Is r equal to x plus y plus z?
Essek: It's not because r is always a positive value.
Caleb: r squared, then? r squared is equal to all of x plus y plus z squared?
Essek: *expands Caleb's suggestion to see all its pieces*
Caleb: *asks if the parts that are not squared away are needed or unnecessary*
Caleb and Essek attempt to translate between their two disparate frameworks.
Figure 5.2: Despite the apparent differences of their frameworks, Caleb and Essek can find a point of similarity. Their conversation in effect reads as follows:
Caleb: Do you have a 'central reference point'?
Essek: Is a 'central reference point' an origin?
Both: Oh! They're the same!
Figure 5.3: Caleb and Essek learn how to translate themselves for each other. Their conversation in effect reads as follows:
Both: *agree upon an origin point and simplify their frames of reference by setting their respective polar axes (θ and z) each to 0*
Essek: r is a distance measured away from the origin.
Essek: φ is a measure of the angle r makes with respect to its 'original' position.
Caleb: What if we call r's original position where there is no φ 'x' as a fixed axis?
Essek: If x is where φ is 0, then it's -x where φ is π.
Caleb: So when φ = π/2, that's perpendicular to φ = π.
Essek: *realizes this is what Caleb meant when he said x is perpendicular to y*
Essek: φ is φ no matter where it's drawn in the angle.
Caleb, aside: Is the angle made by the x and y axis equal to φ?
Essek: That is only the case when φ = π/2.
Caleb: *names an arbitrary point using his own coordinates*
Essek: *names the same point using his own coordinates*
Both: *figure out how to convert between their coordinates using trigonometry*
Figure 5.4: Essek enjoys teasing Caleb. Caleb enjoys hearing Essek speak his language.
Figure 5.5: Essek enjoys solving Caleb puzzles. Caleb enjoys learning how Essek ticks.
Figure 6: For the purposes of this work, we represent Caleb and Essek's combined framework for understanding attraction with the cylindrical coordinate system, as it combines elements from both of their understandings. Caleb and Essek create an understanding that works for them both.