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it's subtle, but I might be finally starting to gain a sense of self. I'm still getting used to it.

I've been depersonalized and spaced out pretty much all of my life.

It doesn’t have to be a regular geometric shape. Any conceivable surface, no matter how awkward and lumpy — unless you have some infinitely rough mathematical object or other weird thing that doesn’t exist in real life — will permit the definition of geodesic curves.

But also, how much detail you account for is inherently a subjective decision tied to the level of precision you need for your specific application. Are you going to consider every whitecap and speck of foam when you describe your path? That depends! If your goal is to steer an ocean liner in a roughly straight line without hitting any landmasses, then probably not!

If you’re a satellite looking for sea-level defects due to gravitational anomalies, then you might need a little more precision.

Also, you can define a geodesic for any surface, the concept isn’t specific to spheres.

The question asks for an approximate answer, so you can approximate the earth as a sphere. Your path will be roughly correct, since the earth’s deviation from being a sphere is extremely small, on the order of a few kilometers.

define a space for the surface of a sphere with fewer than 3 dimensions.

latitude and longitude

such a map is guaranteed to not be a bijection

just put the poles wherever it's convenient

idk, to me this is kinda semantics, whether or not we say a 2d surface in 3d space is 2d or 3d.

just assume "straight line" refers to a geodesic on the surface, then you can use whatever coordinates you want.

A "straight line on a curved surface" is called a geodesic. It's the path a tiny car would drive on the surface if you didn't move the steering wheel. Generally it's the shortest path between two points on the surface, although that definition gets iffy for long paths that start to loop back — obviously, going 90% of the way around the equator is longer than going 10% in the other direction, even though both paths would be geodesics. I prefer the car explanation.

a vote in our current system means "I approve of this alternative over the others"

Almost, but not quite.

What a vote actually means is "I prefer the result I expect from this vote over the result I expect from the other vote."

If withholding your vote this time means they offer more next time, that's a result you might want. On the other hand, if you always vote blue as long as you slightly prefer the democrats, you give the democrats permission to move to the right, slightly behind the republicans, indefinitely.

in practice, though, they'll always find some way to gin up votes without offering substance. that's their whole job.

Political power is wielded by those who hold office, thanks to the people who got them into office. When that is a handful of billionaires and will connected political operatives, that power is used to their benefit. To the extent that thanks is owed to small dollar working and lower class people, to thousands of door knockers and surrogates, that power will be used to their benefit.

This is not true in the slightest.

Why would it be? The Democrats only need to be slightly less bad than the Republicans, and your vote is guaranteed.

And it requires that we acknowledge that we approve of what the American Left has done more than the alternative.

1. Democrats are not the American Left.

2. What have the Democrats actually done?

The policies of the Democratic party are bought by corporate interests. The material interests of a corporation run directly counter to those of a worker. Workers want higher wages, companies want lower. Workers want public healthcare, insurance companies want private. Workers want less war, the MIC wants more. Workers want lower housing costs, hedge funds and landlords want higher. Who do the Democrats side with? What direction is America headed in?

Is this a democracy or a good-cop bad-cop routine?

Please elect me haha

The existence of a sincere, conscientious left makes them insecure in their worldview

I looked at the guy's instagram and saw this picture

I had no idea caves could form fucking funnel shapes on the ceiling.

Caption says the location is a cenote in Tulum, Mexico

Edit, wikipedia article for those funnel things https://en.wikipedia.org/wiki/Hells_Bells_(cave_formations)

It's a self-fulfilling prophecy because the closer I get to people the more anxious I feel, which makes me clam up and become less fun to be around.