the habitable zone of that surface could start at about 0.99au from the sun (where water might boil), and end at around 1.004au from the sun (where water might freeze)
this means that the distance towards/away from the sun between boiling and freezing would be 2.0944e+7km, around 522.6 times the circumference of the earth
and this habitable band would stretch into a ring around the sun, with a total surface area of around 7.019e+22 km2
this is an image of that shape to scale. The circle at the center is the sun, with a radius of 696,340km:
it is important to note that "habitable" in this case is being defined by the possibility of liquid water.
at the hot inner edge of the ring, the temperature is 100°c, and at the cold outer edge, the temperature is 0°c.
the coldest permanently inhabited settlement on earth is Oymyakon, Russia, with an average temperature reaching -50°c.
the hottest inhabited place on earth is Dallol, Ethiopia, with an average mean temperature of 34.4c.
assuming the temperature changes linearly with distance, and a human-safe range of -50°c to 35°c, this leaves 85° inhabitable "distance"
the lowest 50° of this area would be outside of the previously established habitable zone, as temperatures there range from -50°c to 0°c and the previous zone ranges from 0°c to 100°c.
this new ring would have an inner edge 0.9991 au from the sun, and an outer edge 1.011 au from the sun, with each edge 1.78e+6km apart.
this new distance is 44.4167 times the circumference of the earth, as opposed to the original ring's 522.6 times the circumference of the earth.
...thats a pretty extremely smaller distance so I feel I must have done something wrong by this point.
such a world would be physically impossible anyway, so worrying too much about the specific physical distances required for human life isnt very important to me.
also, I dont really believe people can live in a place that is consistently -50°c. I think they can live in Oymyakon because the temperature still goes up to like 60-80f in the summer.
this world would have no day/night cycle, no seasons, and no years.
the sun is always in the same direction, and sitting on the horizon.
shadows will always point away from the sun, creating cold areas in permanent night "behind" mountains.
the world might also experience an endless wind, since air closer to the sun would be heated and rise.
cold air from further away would flow in to fill the gap, so air near the surface would always flow towards the sun.
at a certain elevation, this would switch and the air would always be flowing away from the sun.
if people were to live in a world like this, they would probably have fundamentally different concepts of directions and time.
on Earth, we define North as towards the North pole, South as towards the South pole, and East/West as being around the equator.
on the plane world, the sun and shadows would provite a constant reference for direction, and so directions might be defined as in/out/left/right
"in" being the direction towards the sun, "out" being away from the sun, and left/right being to one's left and right when facing the sun.
in such a world, saying something is "to my right" could be a shortened version of saying "if I were facing in, it would be right of me"
and without specifying "my" right, universal left and right are used similarly to East and West on earth.
I find it unlikely that people living on such a world would easily determine the true shape of their world.
the distances and areas involved are simply so massive that it would be very difficult to ever travel far enough left or right to actually measure the world's curve around the sun.
to their knowledge, the common conception of the world would probably be more like this:
the world is more or less a rectangular shape, cold at one edge (outer edge) and hot on the opposite edge (inner edge). The two other edges have never been found.
places further in get hotter until you can't travel in any more, places further out get colder until you can't travel out any more, and left and right go straight for a long distance.
people would probably relate In to heat and light, as things get hotter the further in you are and the light comes from that way.
similarly, Out might be considered related to the concepts of darkness, shadows, and the cold.
I think the languages of such a world would likely name In and Out with names based on this conception.
Alternatively, the names of these directions could be based on the permanent wind of this world, which always travels In.
it seems pretty reasonable that they would call In "towards the light" or "with the wind" and Out "towards the shadow" or "against the wind"
There is a problem with this though, in that if the world is really 40 - 500 times the earth's circumference deep, nobody would ever travel far enough to actually relate In to heat or Out to cold.
Unless you live on the very extreme edges, you are simply surrounded by habitable land in all directions further than you can explore.
I think this setting would be more interesting if the habitable distance were more like 7,000 km.
if the laws of physics were applied here, this small scale would require some really extreme temperature differences and the wind would be pretty strong.
if the laws of physics aren't applied, the whole math of the distance to the sun and radius of the world-ring becomes kind of irrelevant.
I think it would be more interesting to just say the world has earthlike biomes, a constant but survivable wind, and measures around 7000km In to Out.
as for how long the world is left to right, it's just Way Too Bigtm
civilizations at one point along the ring might never contact civilizations a few degrees away.
space travel might be impossible, as there is no way to orbit the infinite plane world.
to keep something up, you would need to constantly provide at least some thrust upward.
despite this, it might not be completely improbable for people to eventually determine the approximate shape of the world.
this could mostly be motivated by an interest in shortening travel times.
since the world is a ring, travel to somewhere to the left or right could be made shorter by going In.
ideally, the shortest distance would be achieved by travelling a truly straight line towards the destination, at an angle Inward rather than straight left or right.
it would be difficult to prove that going In is actually faster, since the scale is still impractically large and going Inward requires travelling over hotter terrain the more In you go.
but if someone were to create an extraordinarily long and verifyably straight object, aligned perfectly left and right, and which casts a shadow,
it might be possible to measure a minute difference in angle from the shadow at one and and the other to determine that this straight left/right line is actually curved.
from this, you could theorize that the habitable world is a ring, and approximate the size of that world.
Since the wind on this world is always travelling In, it might only rain Inward of a body of water.
anything Out of the area's main water source would be comparatively arid, though rain might not be impossible there.
since the air high enough up travels the opposite direction, it might be possible for water to reach that height, travel some way Out, and then fall.
the people would likely have a name for this rare rain in the arid Outland, probably related to the word for Out.
it would be incredibly difficult to measure time in a world with no seasons, years, or day/night cycle.
people would still need to sleep, but theres no universal way to measure durations unless you have some naturally observable constant to compare them to.
because of this, there would be no nocturnal or diurnal animals- everything just sleeps when it has to, and even among humans people can sleep on completely offset schedules.
since Outer is colder, at a certain point the moisture in the air freezes, and could then be brought Inward with the wind in the form of cold fog.
the time-keeping problem could be avoided by having a naturally periodic source of this fog, too far Out to be observed.
if the fog is consistent and extreme enough, it could even block out the sun enough to create a day/night cycle.
units of time could then be derived from the durations of day and night, and people could be more or less synchronized.
the further one travels In, the shorter the nights and longer the days, as the fog dissipates as it travels.
at a certain point Inward, it is always day and never sees fog, and at a certain point Outward, it is always night and permanently in fog.
even if the fog is not on top of you, light from the sun can't travel through it so it remains night until the fog travels In enough to dissipate.
places Outward would experience fog, then an extended dark night, while places Inward would experience fog, which then dissipates directly back into daylight.
if the fog were too frequent it would be night forever at a certain distance Outward, even if the fog were not constant.
it would be difficult to see stars in the sky at night, mostly because they do not exist, but also because the fog would obscure the sky for a certain portion at the beginning of each night.
There would be no horizon on this world, and any viewpoint is only limited by the nearest vertical barrier in any given direction.
when standing on something tall enough, you can see all the way to either the fog, the sun, or something just as tall.
if a place is at higher altitude than the fog, it would experience perpetual daylight. Any gradual landforms would push the fog up with them, though, so this only happens with sheer edges.
plant and animal life would be quite different under these conditions.
plants which use the sun's light would naturally grow to have large surfaces facing permanently In, rather than being somewhat random like on earth.
the reproduction of these plant species would also be problematic, as any seeds sown too closely Outward of the parent plant will be forever in its shadow and die, while any sown Inward would eventually starve their parent.
since the latter problem takes longer, and spreading seeds with the wind is easier than against it, plants would likely tend to propagate Inward.
Similarly to rain, it could be possible for seeds to sometimes travel back Out with the higher reversed wind, but less frequently.
plant density would gradually increase as one travels Inward, and then suddenly drop off as the environment eventually becomes too hot or dry.
some species would still grow large vertical structures like trees, to avoid the shadows of other plants Inward of them.
instead of grass, which is arranged somewhat vertically, the surface of this world might be covered in a vine-like plant which always grows Inward.
this species might have somewhat regularly spaced and sized leaves growing upward, so as to avoid any leaf casting a shadow on the next Outward leaf.
kinda like --/--/---/--/--/---/-- shaped. Landscapes covered in that would be pretty.