There's something about hexagons which make them look inherently advanced. This makes sense, since they're inherently awesome, yet practical. They slot together like squares, but look more complicated despite being more efficient to build with (a given hexagon has the most area inside for the length of its sides, hence why bees use hexagonal honeycombs for storage; beeswax doesn't grow on trees). Not to mention even the name hexagon sounds a little futuristic.
The hexagon's efficiency in nature might go some way to explaining the origin of this trope. Much like smooth sleek curves and solid, practical looking blocks, there are plenty of reasons to incorporate hexagons into real world scientific and engineering projects. Most notably dome structures (since they're much easier to produce a curve with than squares), solar panels (which benefit from the aforementioned area-to-edge ratio) and hexagonal floor tiles (which look cooler than boring old rectangles). In addition, many polymers, semiconductors and other advanced materials use carbon, which is naturally inclined to form hexagonal molecules due to the way it bonds, and it is this structure that gives diamond and other carbon compounds their strength note graphite being the odd one out, since it's made up of covalently bonded "sheets" of such molecules which slide over one another).
The Beehive Barrier is a very common subtrope. See also Carbon Nanotubes. Compare Everything Is an iPod in the Future and Our Weapons Will Be Boxy in the Future. Contrast Zeerust (which this trope may well become when Science Marches On). See also: The Aesthetics of Technology.
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Anime and Manga
A recurring motif in Busou Renkin starting with the kakugane, an alchemic device transformed into a weapon by the human survival instinct, and which is shaped like a hexagon about four inches wide.
Psyren: Shiner's Hexagonal Transfer System PSI, which carves out hexagonal pieces of pretty much anything and teleports them wherever he wants.
The basic holographic fields for the Gunpla battle system in Gundam Build Fighters are hexagonal, which carries into the design of the show's title.
Films - Animated
Chicken Little in Disney's adaptation is hit in the head by a high-tech hexagon that automatically adjusts to its surroundings to make itself invisible. The hexagon turns out to be a panel from the hull of an invisible alien spaceship.
Aside from being centered on bees, Bee Movie makes a point of showing hexagons as not just a construction artifact, but ostensibly as a high-technology business organization (Honex, the primary employer of the protagonists' hive, is a member of "The Hexagon Group"), owing to the lengths the bees go through to refine their honey.
Films - Live Action
In Fantastic Voyage, the shrink ray room had hexagons all over the floor. The ship rose up on one of them once it got small enough, so that it could be shrunk one more time, and then readied for insertion into the guy's body.
In Viper, the titular high-tech supercar had an armored body/shell comprised of hexagonal tiles. Though they were normally blended together in such a way that the seams were invisible, they manifested when the vehicle suffered damage, or transformed to or from its 'less conspicuous' street car mode.
BIONICLE: In pretty much all the graphics of 2006, plus a large amount of the graphics from 2007 and 2008, there are hexagons in the background, most likely to give the images a high-tech feel.
They even changed the shape of the Matoran alphabet characters from circles to hexagons. It does make you wonder, though, why they didn't use hexagons in 2004, when the futuristic city of Metru Nui was the main setting.
Gogos has a Mega Metropolis series with a hexagonal shaped 'home' for each gogo.
'Hex Bugs' are intelligent nanobots about the size of a pen lid or marble. If the name didn't, the logo◊ says it all.
Not even just Adam, but anyone with a neural implant has a subtle or obvious hexagon indentation in their heads.
While the game's art makes a lot of use of triangles, hexagons do come in close second, being seen on armor and briefly before Adam cloaks himself to become invisible. Not only that, but if you squint at his combat vest, it is made up of minuscule hexagons.
Some of the bulletproof glass has hexagons and one building is called The Hive. Also, in the lobby outside his office Sarif has a large picture of a topless women from the back with cybernetic arms. Up close, the entire thing is covered in a film of hexagons, giving it a softer look from across the room. When inspected even closer, it becomes apparent that the hexagon grid is digital and scrolls across the image.
The logo of Mass Effect's Cerberus has a hexagonal shape and uses some materials with hexagonal surface patterns.
In Metal Gear Solid 2: Sons of Liberty, hexagons are a recurring pattern throughout the story, but it gets really extreme once you enter the final "dungeon". The Big Shell's outer buildings were hexagonal, the shells themselves form a double hexagon, some of the textures on Raiden's stealth suit exude a hexagonal pattern, the skin of arsenal gear itself is covered with a hexagonal pattern. The game case design had a hexagonal pattern on the back of it, so did the original disc. Most official supplemental materials (strategy guides, concept art, making-of materials) had either subtle or blatant hexagonal themes. In fact, since Metal Gear Solid 2, hexagonal design has been a really big thing to them.
One of the subtle hints that The Phantom Pain was really Metal Gear Solid V was the appearance of hexagonal tiles on the floor of the hospital the protagonist is escaping from.
In Persona 4, Rise's Persona Himiko is depicted as a woman with a satellite as a head, complete with hexagonal patterns.
Solatorobo: The Futzu Tower, which contains a highly advanced AI, features a lot of hexagons.
Two of the three Sonic Advance games featured high-tech, futuristic levels, both of which featured hexagons ad nauseum.
Sonic Adventure 2 uses quite a few as well. A lot the moving platforms in City Escape are hexagonal prisms.
Rise of the Reds: The Nano Shock Absorbers that can be installed on various European vehicles have a hexagon-shaped tile pattern.
StarCraft II presents these on the loading and menu screens. They also appear on some units and maps.
DoDonPachi Daifukkatsu uses these both as high-tech and as part of the bee motif of the series. They're all over the ship select screen, and each cutscene starts with a bunch of hexagons flying onto the screen and attaching to each other to form a screen. The "Expert Items" in the Xbox 360 port's Arrange B mode are also hexagonal and appear in sets of six arranged in a hexagon.
DoDonPachi Saidaioujou takes this Up to Eleven by putting hexagons everywhere — the golden star medals dropped by defeated enemies are now 6-pointed stars on a hexagonal background, the level results screen has hexagons everywhere, the bee items now appear on hexagonal stands, the Type-B helicopter now has 6 propeller blades connected in the middle by a hexagon, and the list goes on.
In Halo, a lot of Forerunner tech incorporates hexagon shapes and symbols.
In PlanetSide 2, all of the Vanu Sovereignty's equipment is covered in purple armor with hexagon patterns, to show that they are the most advanced faction. Additionally, all three empires can buy hexagonal camouflage (in neon red, blue, or purple) for their guns, armor, and vehicles.
While surprisingly absent from designs in the actual game, Vector Thrust has these plastered all over the Pause Screen and menu backgrounds. Ground targets are also denoted by a hexagon symbol on your HUD.
The 2013 edition of SimCity features hexagons in many high-tech structures such as advanced solar energy collectors, and in the "Cities of Tomorrow" expansion, futurized roads are signified by hexagonal surfaces with minor Tron Lines, in a possible prophetic move- see Real Life.
Fallout: New Vegas has an area in the futuristic Old World Blues DLC called the X-66 Hexcrete Archipelago, consisting of massive hexagonal structures made of concrete. Appropriately, they do nothing in the game other than look cool.
EVE Online will be changing the ship cloaking effect (viewable only by the player using the cloaking device) from a water-like animated texture to a hexagonal pattern in the upcoming Oceanus update.
In Star Trek: The Animated Series episode "Beyond the Farthest Star", the Enterprise crew discovers a highly advanced alien ship in orbit around a dead star. Said ship's interior structure is made up of interlocking hexagons.
In 2012, scientists at IBM's T. J. Watson research center discovered that cylinder shaped molecules known as carbon nanotubes may work as a replacement for the silicon based microchip. The tubes are capable of delivering more power at such a small size without losing any control over the electrical current that flows through them. The tubes themselves are made up of hexagonal pathways, with electrical currents constantly making a path from one end of the tube to the other by going either left or right at each corner of the hexagon.
Another hexagonal carbon molecule which has potential is graphene. In addition to being extremely strong and flexible, it's an excellent conductor of electricity. Applications include lightweight screens, semi-conductors, efficient room temperature alcohol distillation and so on.
The James Webb Space Telescope will have a mirror made out of 18 hexagonal segments.
The Solar Roadways project hopes to pave roads with durable, hexagonal solar panels that glow at night and melt snow, thus creating a road system and is safer to drive on that looks cool while producing vast amounts of energy. However, some people are skeptical about its practicality.