OBSERVATION (ambiguity of icon usage)

Notice that icons are ambiguous: they can be notifying you of a STATUS (as for the muted/unmuted icon in a Hangouts meeting), or offering you an OPTION (as for the show/don’t show icon in the gmail login screen). So, when you are in a Hangouts meeting and see that your mic has a slash through it, it means you are MUTED, but when logging in to gmail and see that the eye icon is slashed, it means that your password will be VISIBLE. So, the rule seems to be: when you are static, the icon shows your status, but when you are dynamic, the icon shows your option.
keywords: confusion, cyberspace, Information Technology (IT), user-friendliness, convenience

complementarity (of logarithms and two squares)

The process of obtaining the product of two numbers via logarithms and the process of proving that the product of two circular numbers is itself also circular are complementary. The former is thick at the ends, and thin in the middle, while the latter is thin at the two ends and thick in the middle, the middle portion of both being of the same nature: separate, or split, the components and then perform an operation on those components.

Here are the schemas of the two processes.

process 1
Take each of the two items separately to the logarithmic domain.
Compute their sum.
Take that sum (via exponentiation) back to the original venue.

process 2
Remain at the original venue.
Split each item into its (canonical) complex-conjugate factorization.
Multiply, separately, the two first factors together, and the two second factors together.
Note that the result is a complex-conjugate factorization.

circular: being representable as the sum of two squares
thin: being infinitesimal, or zero

keywords: Mathematics, Number Theory, Calculus, Proof Theory, similarity of processes