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What if the rotation of the Earth about its own axis was in the direction North-South (instead of East-West) while the rotation about the sun remained as it is now? What would the consequences be other than "time zones" (meaning exposure to sunlight) would be fixed in horizontal sections/frames? I guess the main differences would be for countries that are thin in an either North-South sense, like Cuba, or in an East-West sense, like Chile. Is there a reason for why the rotation is in an East-West sense?

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    $\begingroup$ Aside: the planet Uranus has it's axis of rotation roughly in the plane of the ecliptic, so there is a physical model of this notion near by. That said, the notion of "time zones" doesn't make much sense in a case where most of the planet is "above the [ant]arctic circles" and experiences "all day" illumination of darkness for nearly half the year. $\endgroup$ – dmckee Nov 4 at 17:59
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Don't worry about individual countries - there won't be any.

The earth's axis of rotation is close to perpendicular to the orbital plane around the sun (offset by 23 degrees). This means that the poles are always cold, and the equator is always warm. Except for the arctic/antarctic circles, sunlight falls everywhere on the planet every day.

If the earth's axis of rotation is parallel to its orbital plane, this changes everything. As the earth orbits the sun, you'll have one pole pointed directly at the sun, and 6 months later, the other pole will be pointed directly at the sun. Half of the world will be in complete darkness, and the other half will be burned by constant sunlight, before gradually swapping to be the other way around over the course of half a year.

A day/night cycle will take an entire year, not 24 hours. Life would likely have evolved very differently on a planet like this, so the concerns about time zones and country-level effects aren't what you should be worried about. Humans may not have evolved at all on a planet with such harsh, prolonged, and inescapable extremes.

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