What could cause a culture to change their writing style from left to right similar to English to vertically like traditional Chinese?
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Usually it happens because the authority / authorities in charge of these matters decides it to be so.
Hangul would be a prime example of this.
And when Korea was occupied in the 1910's, Japanese was made the official language, another example of authority say-so.
It might be easier to use a different format in a particular technology or the other language is used for trade.
Trade languages can spread and there can be a simpler way to write things down, which can then spread as an idea--which will then have to taught to others. This means that whoever controls knowledge and learning will have the power to do this.
My example is the switch from Roman Numerals to Arabic. In this case the change took hundreds of years. It was introduced in round about the 900s (earlier in some places, but for all practical purposes...900s) and monks that were into math thought it was really neat--plus it took up less room in their ledgers. They wrote to each other and it became a fad, however, for the common folk and most people, it was difficult. Roman numerals follow the pattern of the abacus, which is what most folks used for math--each symbol representing a unit of a certain number. Arabic was thought to be confusing enough (after all, a decimal point or a missing place could change the value entirely) and was even outlawed in the 12 & 13th centuries in some places.
The system was already available and part of the culture when the printing press was invented. This is where tech comes in. Roman numbers were much longer, and Arabic was much easier to use with movable type. One of the complaints about the Arabic system was the similarity between the form of numbers, which, when people were writing them down, led to misunderstandings. The standardized print helped with that, and spread the format more consistently.
This is old tech, but my point is that if a language is more compatible with the available tech, an earlier form may be abandoned by and large because of it.
It took a very long time for this conversion to happen with numbers,but the world is faster paced today, particularly when it comes to language changes.
The way I can see it happening is this: a small country gets free tech from a country with a different language system. It's not easily compatible with their language as written, so people figure out a different way to use it. It might be a poor country that didn't have much of an education system in the first place, and they want to take advantage of what they have, but maybe they don't have many computer programmers--and those they have aren't interested in converting all the computers or writing a program to accommodate their language. Instead, it's just easier to adjust how it's written.
Now, that's just an example of how that could work--there's lots of other ways you could do it--even by ramping the tech back.
The formula is thus:
1) Another system of language or way of thinking is available
2) The new tech or lack of tech or shift in writing tech makes the current/old system more difficult to use than the other style.
3) People begin using the new tech and teaching the new system of language.
Slow changes and authority say-so. An example would be the Japanese system:
All this mess has been cleaned to the modern format by multiple authority say-so, often ratifying the most common practice at the time. That bold part is really important because the change often goes to multiple directions and the authority say-so is a way to keep it together.
It would be really hard for English to become vertical because the letters have evolved to such that they are best red horizontally. The verticality of Chinese is because the ink and sleeves are not compatible.
The idea for a real-life case:
The change would probably be in a language or an adoption of a completely new writing system. Horizontality is wide-spread and the current technology creates a lock-in for it. The best realistic reason I can invent for the change to occur is that machines get their own writing system. Not all robots have a printing machine, so the system is to be such that it is easy and fast for them. The change would probably be also in the language. People would then adopt this system by using the both systems in parallel, but slowly use less and less the old system, so that at some point the new system has replaced the old one.
Basically variant of MichaelK answer: Religion requires you to change your writing style and everyone follows such religion.
Newly introduced script is easy to understand by people using current style of writing. Example: Jan Hus and his change to Czech Alphabet
Why did Chinese do just exactly the opposite? As I recall, a major scientific journal published that it would do so, not for the sake of being different, but to better handle the inclusion of math formulas and tables.
The earliest widely known Chinese publication using horizontal alignment was the magazine Science (科學). Its first issue in January 1915 explained the (then) unusual format:
This magazine is printed so that it goes sideways from the top left, and is marked with Western punctuation. This is to make the insertion of mathematical, physical and chemical formulae convenient, not for the sake of novelty-hunting. We ask our readers to excuse us.
So, more generally, the new layout is better for the medium or better able to handle complex mixed content.
Languages evolve indeed, but that kind of change requires a willful reform. The only way to have that change into people's habit is to have the state enforce it.
Logistically (assuming there is no popular opposition), doing it is pretty straightforward: change the school, change the official papers. Providing the rest of society follows, it will be considered perfectly normal in a matter of decades.
The main problem is of course to get the private text providers to follow. Depending on your government:
Now why would a ruler/ruling class/society want to do that?
A despotic ruler may want to complicate life for the elderly. That's not the best way to do it, but it may come with other bothering rules.
If under control by Chinese (or other vertical-writer) authority, the government may have to adopt the ruler's language. The original language may survive through the occupation. But once the occupant is gone and there's a country to rebuild with little money... changing the layout of everything is costly. If everybody is now used to Chinese layout, and we get back to the former writing, we can all be happy!
A new service that uses some vertical writing starts being used by everyone. The vertical features becomes popular and is also more adapted to smartphones. This sounds dumb but smaller features have already changed our language #chirper.
Educational standards as set by the government or government-related bodies.
Assume a free country where people can mostly write as they like. If public schools (and any private schools which are considered equivalent) adopt a single, unified system for grading purposes, after a few years any other system will appear uneducated, or possibly just old-fashioned. Either way, parents who want their children to do well in school will adapt the new system.
There are examples of this in the real world. The Chinese language developed an independent writing system that is written vertically, so many of the surrounding younger cultures also write vertically, even if their writing system comes from somewhere else. The Manchu and Mongol scripts are derived from central Asian alphabets that were originally written horizontally, but now they're vertical.
Conversely, books in mainland China are now normally printed horizontally (left to right), and non-decorative handwriting is also mostly horizontal. This change happened in the 20th century, when this layout was seen as more modern, and made it easier to include words from western languages or mathematical expressions. For the same reason, many Japanese books on science are also written horizontally, even though most narrative texts keep the vertical layout.
So here are two options why a language might change:
Look at us. 50 years ago, everybody was writing in cursive with pens. 30 years ago everybody was using a typewriter with its fixed-width characters. 10 years ago everybody was using computers and word processors. Nowadays everybody is typing emojis in their phones.
Our own writing system is being (re)defined by the tools we use to write. Of particular interest are the typesetting and linotype technologies.
See also how technological artefacts such as fixed-width character matrices made writing systems to adapt to them in weird forms such as double-width characters.
So cursive versus fixed-width characters are not as radical a change as horizontal-to-vertical, but it's plausible that your culture has a typesetting technology, or material, which can only handle verticality. Maybe there is an influx of foreign machine vertical typesetters. Maybe the paper-like material works better / survives longer when placed vertically. Whatever the reason, the practicality/cost of using the new tech can take over a pre-existing writing system, even at the cost of changing it.
One thing that has different impact on vertical writing is gravity. If a culture starts writing on fabric and writing is used mostly for public labels (e.g. on important buildings), it makes sense to write from top to bottom. This allows you to write something and hang it like a flag while preserving its readability (horizontal writing would depend more on wind). For a short vertical text, you also need just one point of support.
You can also fit more (or bigger) symbols without wrapping on all things which are more tall than they are wide - e.g. a door, an obelisk.
One more plus is that on some surfaces (like rock) it's easier to make straight vertical lines and to verify that they really are straight.
A Sudden Genetic Shift
The answers here are a good place to start.
There is a decent amount of evidence that how we write is due to the tools we use and the fact that the majority of the population is right handed.
So if a demon cursed the population, or mad scientist unleashed a virus, that caused righties and lefties to switch, that would be a reason to change the direction we write. Add to that a shift in our optical muscles change to make vertical tracking faster and less tiring. Then you have a population that would be physically inclined to prefer righting top to bottom, and right to left.
An archaeologist once explained to me why older writing (hieroglyphics, Hebrew) generally goes from right to left.
If you are writing with a hammer and chisel, doing it from right-to-left or top-to-bottom means you can see what you just wrote.
When you are writing with one hand, with a stick or a pen, you need to write from left-to-right to see what you just wrote, and with ink, to stop your hand from smearing the writing.
Obviously for left-handers it is the other way round. Ancient Egyptians wrote in both directions, maybe because of left-handedness, and you can tell which way to read it by the way the birds are facing.
So the direction is dictated by the method. In a future where nobody writes with their hands, you could go in any direction you like, but then you need the birds to show your readers the way.
Gotta stop those [insert national group here] from being able to read my communications.
If you think I'm kidding just look at what happened in England with the street signs. War is a great excuse for everything stupid.
A change in writing style is most easily explained by a change in language. If, perhaps due to some virus, the prevalence of deafness in the population massively increased, culture might change to one like Matha's Vineyard in the 18th to mid 20th centuries, where everyone, hearing and Deaf, used a sign language. Sign languages are not commonly written, but when they are, they are written vertically, in Sutton SignWriting. As the use of sign languages became more prevalent, they would probably be written more.
Spoken languages would not disappear, of course, but as people became used to writing vertically, they might start writing their spoken languages that way too.
Before WWII Gothic Fraktur was one of most popular typefaces in Germany and all around the Europe. But after Hitler adopted it for his ideology and then lost the WWII it was widely abandoned as part of anti-hitler ideology. And it happened mostly organically – it had disappeared even in countries where there were no law forbidding it as people started perceiving it as a "nazi" typeface, while previously it was just one of historical ones originating from medieval calligraphy.
Other example: shifts between Cyrillic and Latin scripts in slavic languages. It was often connected with religious and national identity, with cyrillic being connected with pro-russian politics and orthodox church and latin being pro-western and catholic. There are even langiages like belarussian which had been written in Latin script when the region was part of Polish-lithuanian Commonwealth then moved to Cyrillic after becoming part of Soviet Union (I oversimplify it but you see the point).
Conclusion: if your country exists in a context where either the abandoned way of writing is distinctive for an enemy or the new one is taken from ally the ideology is enough to justify the change.