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Over several hundred years, a tide-locked, but otherwise Earth-like, planet orbiting a red dwarf is terraformed. The terminator is a very habitable zone, in terms of temperature, atmosphere, etc. Chemical composition is (now) almost identical with Earth's.

Given the star always appearing near the horizon, would plants grow diagonally? They'd almost all orient toward the sunny side of the planet, no? So walking through a forest would look very different than here, even if everything else was the same?

Though of course it wouldn't be the same. What Earth flora might survive there, in the absence of the day/night cycle? I'm going to handwave 'modified' plants, so this doesn't need to be utterly feasible, but what would you expect to do least-poorly in such an environment?

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I don't think there is a complete list, but tomatoes do not generally like continuous light. Otherwise most plants seem to be okay with it, but continuous light might be detrimental to the health of animals (and humans) that have a diurnal cycle.

Plants that photosynthesize will certainly grow towards the light. This phenomenon is called phototropism. But gravity has an effect on plant growth as well (this is called gravitropism), so stem-building plants will at first not grow horizontally but lean towards the light.

We might assume that as the plants adapt to the directional light, the effect of gravity on growth direction might decrease, causing the plants to grow horizontally towards the low standing sun. But if all plants grow along the ground, most of them will continuously "stand" (or lie) in the shadow of other plants closer to the sun, so growing away from the ground might in fact be an advantage. I therefore think that most plants will continue to grow against gravity, especially as some of the sunlight will be refracted by the atmosphere and come from other directions as well.

There are also many plants that prefer shade to direct sunlight and those will not grow towards the sun at all.

Animals and humans will develop what is called a "free running activity cycle" under continuous light and loose their 24 hour diurnal cycle. I once lived above the polar circle in Russia, and many people there worked during what the clock said was night. Animals show similar behavior.

I've already linked to an article about detrimental health effects of constant light. Its authors assume that constant light will reduce the life span, so people and animals on your planet might not live as long as on Earth due to accelerated aging processes.

Social changes coming with continuous light are already apparent on Earth. Before the advent of candles, the night was dark and people slept. Since we have cheap electric light, people can be awake and active at all hours, and many are. There are joggers with headlamps in the park, adolescents playing computer games until the early morning, young adults partying all weekend without pause, workers working in night shifts, and so on. The day-night-cycle still influences the lives of most of us, but under continuous sunlight we might assume that at any given time of the "day" there will always be one third of the population asleep and two thirds awake, with people tuning their cycles to their jobs or their friends and family. There might be a more active part of the "day", because it might be more efficient to synchronize certain activities. For example, a school might want to have all kids there at the same time. Other activities might be desynchronized intentionally, with shops and leisure activities opening when people don't have to work.

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