From what I can find there's no real formal term I can use when it comes to the design tendency of verticality. I use the term 'verticality' here to refer to environments that spread their content with an inclination to a vertical plane as opposed to a horizontal one, stacking areas on top of each other in a harmonious, interconnected fashion. These environments often tend to be busy, convoluted, dense and internally complex. Verticality is a design principle I enjoy for its atmosphere and aesthetic. In this short written piece I'm going to focus mainly on the use of this principle in games but I'll touch on film and art as well.
I'm a big fan of verticality in regards to level design. With an overlapping and convoluted environment to work with you can pack it with secrets and discoveries around every other corner. It encourages a player to feel constantly engaged with the game world and eager to explore around other pathways and sidedoors. I enjoy plenty of white space in environment design as well. I often find that verticality is enhanced by a contrast between the dense and the empty (for instance, a self-contained world surrounded by aether). A dense vertical environment is also a good way to introduce short, snappy bursts of variety in environment and colour palette. Rather than having a gradual fade from one landscape style to the next a player can walk through a door, cross a bridge or climb some stairs and find themselves in a completely different looking area without it feeling jarring or absurd.
As an example, games like Dark Souls, Ico and Spec Ops: The Line have used verticality effectively in varying ways for design and atmosphere. Ico and Dark Souls share similarities in that the player wanders through levels, making progress and unlocking shortcuts through to earlier areas and crossroads. The player constantly finds themselves traversing previously unreachable pathways through areas they have already passed through. The player finds that the progress they make elsewhere affects their ability to make their way through other areas. I really enjoy this kind of organically progressive, interlinked level design. Metroidvania style games are a good example of this in action.
Dark Souls also uses verticality as a means to pack a lot of content into a relatively small space. The twisting, double-back pathways, hidden corridors, long elevators into the sky or down into the dark and the interlinked, stacked areas come together to create a dense but potent game world. Observant players can also look from many points in the game and pick out familiar areas and landmarks they have visited before or will visit in the future. It's quite poignant in a very simple and effective way.
Spec Ops : The Line uses verticality as an abstraction. Over the course of the game the player constantly finds themselves descending into yawning, impossible pits and this reflects a lot of the psychological aspects of the game's theme, lending towards an unsettling surreal atmosphere that reflects the psyche of the player character. The city of Dubai, as viewed from the top of skyscrapers, is shown to fade down into darkness such that the ground is not visible, surrounded by sheer, shifting cliffs of sand as if it is being slowly sucked and dragged down into the earth.
Verticality can also keep a player engaged and thinking as they travel, depending on how treacherous and precarious the level and environment design is. To avoid slipping off narrow pathways and stairways into the abyss requires constant attention. The player can't afford to zone out like they can when there is plenty of space to move and a lot of flat distance to run. Pack those pathways with wandering enemies, traps and secrets and the player is going to have very little 'white space time'. It keeps the player thinking, acting and on-their-toes with their focus in the game world.
I used the term 'white space time' there. It's something I've come to use when referring to the time a player uses to travel between points of interest within the game world with little to interrupt or waylay them. A good example of this is galloping through the forbidden land in Shadow of the Colossus, or flying between points in a game like World of Warcraft. Some games are based entirely around 'white space time', such as flight sims, space exploration sims and Euro Truck Simulator. I often find myself enjoying a good white space game. It's relaxing listening to podcasts and music while using something like the Google Earth flight simulator to explore around different places, skimming across the landscape. White space games are even more engaging to me if there are things to see and discover tucked away in little corners and valleys of the game map. I've spent a lot of hours just exploring the world of Shadow of the Colossus. There's a lot of detail and strange little areas hidden in the overworld.
In general I'm more inclined to be a fan of open world game design rather than level based. Open world design makes a game environment feel more consistent and 'real' to me, a place I can take my time, explore and wander through rather than being pulled along by a predetermined set of events, which can sometimes make some games and their worlds feel a little like shallow filmsets in comparison. Open world design also allows different parts of the world to be affected by the player in different ways as they progress through the game, like being inside a giant clockwork machine. Pulling a hidden lever opens a door or activates an elevator in some other place in the world, and the player can visit that area later at their leisure. A character tells you of something they lost, and hours later you can find it and can return to where they are to give it back. It lends a feeling of intricate interconnectedness and depth.
However, a twisted, vertical open world (with all its inherent strengths and weaknesses) needs to have clearly communicated ways through which the player is guided along the correct paths. A game without well designed guidance tools runs the risk of getting their players lost, stuck and frustrated. How a designer finds a solution and integrates it is entirely dependent on the game. It could be through arrows and directions on the HUD or game map, conversations with NPCs, locked doors and inaccessible pathways or guiding clues in the environment. Giving the player a faster way to move through cleared areas quickly is important too lest they get frustrated with slogging back and forth. A good amount of travel time is important as it gives a world weight and consistency, but too much can be detrimental. Dark Souls for example fulfills this need with warping, various shortcuts and elevators.
I'm also very drawn to verticality on an aesthetic level. I feel that it gives a world, environment or piece of architecture weight, depth and a hint of surreality when you can look down and see the sky, as you wander a dead city hanging suspended in the air or hanging like a birdcage from the arm of an ancient, hollow giant. Ghibli movies often use beautiful, effective vertical environments and sequences. Laputa has a good example of verticality in the flying castles, but even the mining village at the beginning of the film is a series of little towns, rusted factories and industrial areas clinging to the edges of sheer, massive cliffs where you never see the bottom of the ravines, traced with spindly little traintracks. The bath house from Spirited Away also has a very vertical, stacked, dense and convoluted design, surrounded to the horizon by a plane of dream logic and shifting distances. Ghibli fantasy stories in general often take place around small, densely packed but conceptually 'large' worlds, such as Howl's Castle and the old house in Totoro.
I find that vertical worlds, environments and architectural designs often give me a feeling of vertigo-inducing awe, wanderlust and a mildly surreal sense of the uncanny in the impossible. It lends itself to the creation of complex, interconnected and engaging game worlds. A well designed vertical world reminds me a little of the design structure of a park or campus, traced with paths branching off into the distance with something to discover around every corner.