I am assuming you've read the previous articles on energy, elements, and the elemental triplicities. If you haven't, I recommend starting there and coming back.
I may make a second pass and clean up these articles, but I'm currently trying to just get SOMETHING out there. Gemini energy is a form of deep, non-linear thought.
In the world of the intellectual, there are many systems. Languages, logics, and models. These entities are composed of rules, exceptions, and connections. As one will find learning mathematics, laws have corallaries which embed them in the surrounding theory. Most important pieces within a system are connected like cogs turning and impinging upon other cogs. Together, there is a motion throughout a mental construct like this. If the logic that binds the ideas is consistent, the motion is synchronized in one direction. This is the property of all useful tools. 2+2=4 is not correct because it exists in the external world, but because it is internally consistent, and that internal consistency maps to consistencies we may observe in the external world. We may name a clump of minerals "one stone", and do the same three more times. We may then form groups of two, and we may use our reasoning to oberve the truth of the equation. This natural number addition is a most-useful model for counting up groups of discrete objects.
But what of other systems? Let's take a map, the style with cities and roads on it. This is spatial data, and by its very nature is non-linear. The sense of proximity that we might have for numbers of similar magnitude only grows marginally more complex when we apply it to two dimensions. And yet, it's trivial for the human brain to imagine data in 2D. Rote memorization of nine items is far more difficult than envisioning a 3x3 grid and placing each item in one cell. This is the trick of the "method of loci. It can work in 3D as well(a 3x3x3 matrix of 27 cells. for instance). Set aside the parlor trick, and you will a powerful mode of thought. Few engage in it consciously, but it's at the heart of answering queries like "how do you get from my house to the bank?"
In a dynamic system, there are feedback loops. As element A exerts force onto element B, element B exerts force back onto element A. Someone insults you, so you slap them. They slap you back harder. You punch them. They tackle you. A positive feedback loop escalates with further interaction, usually approaching some threshold. In the case of a fight, a third party interrupts, or else one party is neutralized and unable to continue the interaction. A system of two elements is easily read in a linear fashion. Let us move past linearity.
Consider a makeshift mobile. A coat hander, and on each side another coat hanger. From the four corners of the bottom coat hangers, four gems dangle from string. If you were to dangle one ornament from your hand, you'd notice a pendulum effect, a negative feedback loop where the ornament swings back and forth in a shrinking arc that settles at an equilibrium. The entire mobile holds this property. Should you tug on one ornament connected to the mobile, you can obserbe motion on all the others. The one ornament sharing the same coat hanger is effected by first-order interaction, and the ornaments on the other coat hanger are effected by second-order interaction. However, each ornament's pendulum motion pulls on the three coat hangers of the mobile. There's a feedback loop between 7 entities(three coat hangers, four pendulums). There's some variables, such as the length of the ornaments' strings, the weight of the ornaments, and which ornament you tug on, and how hard you tug on it that effect how the system will behave as it returns to rest.
Let us consider a baseball game. At its coarsest, there are three entities: two teams and an audience. Which team will win? Of course, the team with the best skill/experience/coordination will be advantaged. So too, the team with the most vocal support from the audience. The away team will suffer a minor disadvantage. This might sound like Sun Tzu's art of war a little bit. We can go deeper than that, though. Zoom in, and you may consider individual players and members of the audience. Particular players may have particular performance, loved ones present or absent in the audience, and may have unique reactions to being the away/home team. If I knew anything of sports, I might drill into the anatomy of the different roles on a baseball team and how individual players effected the performance of the team overall. The point here isn't to understand sports. The point here is to ask "how would you determine which side holds the advantage"? Analysis of competition lends aries energy to this situation. Note that it is not in the spirit of gemini energy to compete, but merely to understand.
Very-well, let's say we've dug through and thoroughly analyzed baseball and observed enough games to hone our model to determine advantage. As you read this, you might be thinking "oh, this reminds me of XYZ" as with Sun Tzu's art of war. You might think "this reminds me of football, or soccer, or League of Legends". Your human brain contains a powerful pattern recognition computer. Given time, it will begin to find connections between things that were only trivial truths of one system to form laws or theories that govern multiple systems. Newton's third law can be felt broadly in many, many systems.
Generally, this intersystem analysis is subconscious. An article I can't find from the Angry GM's D&D blog mentions that one of the best ways to improve your DMing skill is by doing literally anything else that is mentally stimulating. Reading fiction grants a library of tropes and plot elements to pluck from, and even more importantly, he notes that visiting the zoo and watching how dolphins swim gave him insight about how D&D combat is run. He didn't consciously say "I'm going to find some common ground between this dolphin and D&D". Instead, he crammed novel and interesting information into his brain and let it digest. As it digested, the pattern between the two systems emerged, allowing him to gain better understanding of D&D.
While Sagittarius energy possesses a curiosity that compells it to seek out and ingest new experiences such as visiting a zoo for their own sake, the act of deeply digesting said experiences falls exclusively within the realm of Gemini energy. Gemini takes in a system whole, but dissolves away the skin and meat to find the bones that structure the system. Comparing bones of one system to another is the very act of intersystem analyis I mean to highlight here.
In the above paragraph, I describe digesting systems and comparing bones. But what of the rest of them? The trivialities? Perhaps, in your analysis of past major league baseball teams, you notice that all the baseball players you see before a certain date are white. A thing then happens in your mind. You ask, "why?" This is an essential aspect of Gemini energy. Sagittarius may see a horizon and say "I want to expand that", and thus pursue novel things for the sake of novelty. Gemini, however, is locked with questions. In pursuit of this baseball trivia, you come to learn about the US civil rights movements. In the process, you may gain a new appreciation for existing knowledge of the movements, or tease out new patterns related to discrimination that you can find alive in today's society. A Sagittarian scholar of baseball may be able to cite by memory a list of first black major league baseball players without blinking, but via Gemini energy you are able to probe the connections and implications.
Bringing back the example of the Angry GM, knowledge of the dolphin was trivia. It was unimported and unconnected. That is, until Gemini energy digested it and teased out patterns.
Coherency is the connective tissue which binds all useful forms of thought. Even the intentual employment of incoherency is bound by coherency. However, the dark side of Gemini energy is incoherent, purposeless pondering. From entering this wandering non-linear analysis, the dark dGemini fails to commit themself to a single coherent course of action. In the worst instances, one starts a train of thought and abandons it without culmination many times in a row. Physical deterioration of the health of the brain may encourage this loss of attention span. Likewise, anxiety may push someone to think faster than is coherently possible. Conspiracy theorists are addicted to the rush of that "woah!" feeling to the extent that their beliefs abandon coherency. A conspiracy theorists's worldview can vary dramatically from one day to the next, the only common thread being the intense feelings thinking about it confers. In the most severe cases, this form of dark Gemini energy leads people to take horse dewormer for unrelated diseases, to buy into pump-and-dump scams, and to make themselves pawns for power brokers who construct propaganda and radicalization pipelines categering to their psychological vulnerability.
There is another shadow of Gemini energy, which is its total absence. When a Gemini reads the ingredients on the back of a pack of gum, they are left with questions. What the heck is xanthan gum? Why are there so many huge polysllable words? What role does each ingredient play in the process of making gum, or for the consumer's experience of chewwing the gum? Also, how is gum made? With one's Gemini energy sapped, one finishes reading the list and says "Well, that was certainly boring." They will never ask any questions, and they will unlock any new mysteries. Without Gemini energy, the mind is a sandy soil in which no seeds may survive to become plants, and which may bare no intellectual fruits.
Pardon my rambling. Writing about air signs seems to bring that out of me. Revisiting Gemini energy, there is a curiosity that asks questions. This is the heart of the socratic method. There is a deep digestion of ideas that leads one to notice similarities and make connections between seemingly unrelated concepts. While Sagittarius and Gemini are both hoarders of ideas, Gemini breaks those ideas into building blocks capable of readily building new ideas. In this way, Gemini is an archaeologist, analyst, and an inventor.