2D: If you draw a perhaps wobbly circle shape (loop) on the ground, it has an inside that you can colour in. If your loop is elastic, it can contract to be all in a tiny heap. Topologists call this “simply connected”.

3D: The water on your bath is also simply connected. Your elastic loop, whatever its shape, can shrink back down to tiny.

2D: The surface of your tennis ball is simply connected because any elastic loop on its surface can shrink to nothing, but the surface of your ring donut isn’t, because you could cut your elastic and wrap it arround the donut and it couldn’t shrink because the donut would stop it. Ants living on the surface of the donut might not immediately realise it wasn’t simply connected because they’d never drawn a big enough loop to find out that it couldn’t be shrunk.

3D: The solid donut is also not simply connected, because the ring could contain an elastic band that goes all the way around the ring and back to the start, and it couldn’t shrink to nothing because it would have to leave the donut.

2-Manifolds: a 2-manifold is some kind of surface that doesn’t have an edge and when you look up close it looks like it’s flat-ish. You could make it by sticking lots of tiny sheets of rubber flat to each other but there’s not allowed to be an edge. The simplest 2-manifolds are an infinite plane, the surface of a ball and the surface of a donut. The small ones are called closed. The technical reason for that is to do with not having any edges but still being finite, but you can think of closed to mean finite.

Manifolds may not be as the srrm: If you live in a 2-manifold you might not immediately realise that it’s ball surface and you might not realise it’s a donut surface. If you have a computer game from yesteryear where when you go off the top of the screen you come back on at the same angle and position on the bottom of the screen, and the same for left and right, that’s actually got the same layout as the surface of a donut. To help you see that, imagine your screen was triple widescreen and made of rubber. Roll it up to glue the top to the bottom and then glue the two ends of the tube to each other. You haven’t changed the game play at all but now you can see it’s the surface of a donut shape.

3-manifolds: anything that looks like 3D space up close is a 3-manifold. The simplest 3-manifolds are an ordinary infinite 3D space, a 3-sphere, which is like the 3D version of the surface of a ball, but it’s hard to imagine the 4D ball it’s wrapped around, and the 3D version of the computer game.

The universe: It looks simply connected, but we can’t see that directly, because maybe there’s a very long loop we haven’t gone on yet that gets back where you started without being shrinkable. This is hard to imagine, but it could be like being in the 3D version of the computer game where there’s a long loop that can’t shrink because it goes through one side of the screen and comes out the other before coming back. It can’t be shrink at all, especially not to nothing. The universe is a 3-manifold.

The Poincare conjecture says that every simply connected “closed” (finite) 3-manifold is essentially the same as the 3-sphere. If ALL your loops shrink, no matter how big, and the universe is finite and has no end wall, then it’s the 3 sphere.

Mathematicians have been trying to prove that it’s true for a long long time, and there was a 1M USD prize for proving it that this guy turned down. The prize was largely unnecessary because lots of mathematicians were trying to prove it anyway because it’s so famous and enticing.

And 3 to 25 bunches is between 30 and 250 apples gone bad, so that A few bad apples could spoill half of the apples, and there you have the problem with the police force, especially of the USA where officially it’s not a crime to drive a car whilst having dark skin tone but you can definitely still be summarily executed for it.

Congratulations: civilisation has arrived in your home. We’ll talk no more of what happened before.

That at least has the weight of tradition behind it.

Wait, you do mean a stove kettle, don’t you? You don’t mean just a normal vegetables pot? You can’t mean that, surely?

This is why Trump is still in the race.

This is the kindest thing. Thank you so much.

<3

If only we could get them to be less afraid of everything that they’re unfamiliar with, or familiar with everything they’re afraid of, maybe they would lash out in fear less.

I’m on board with this proposal.

That’s path connectedness. Convex shapes are ones in which any two points can be joined by a straight line internal to the shape.

I deduce that your phone never randomly locks while you’re using it as a satnav. Lucky you.

Nice dick, bro, but I should emphasise that this can’t be just anyone’s mouth and you should check that the other person is enthusiastic about it all before beginning.

But it also finishes on a Friday, so 4/5, not 4/7.

A crescent isn’t a convex shape though, and ironically you left out the word ‘straight’ in your rule about convex.

I’d also like to note that I agree with Abortion if it’s simply removing a dead foetus. I see that as less of an abortion and more of removing a dead foetus that’s already dead.

Yes, that’s not abortion, that’s just D&C after the foetus died.

As for abortions in case of the pregnancy might kill the woman, those are exceedingly rare as is, but should still be allowed.

But it’s the same D&C procedure whether the foetus was deliberately terminated or it died of natural causes, and a doctor performing a D&C is at real and genuine risk of losing their career and going to prison, and a woman in need of D&C is at risk of losing her fertility or her life from sepsis or other complications. Criminalizing healthcare kills women. Doctors need a lot of good luck keeping their job in a red state if they agree to perform a D&C. Don’t rely on facts winning a case when anti-abortion rhetoric is one of the two main ways republicans have got and kept power.

It’s clear that you think God loves unborn children. Do you think God loves women who stillbirth? They are at genuine risk of jail time because their baby died. Do you think God loves children who were raped? Do you think God wants rape victims to bear their rapists children? Do you believe that God always chooses the baby over the mother, no matter the cost or the consequences? Do you think God doesn’t like it when we intervene in matters of life and death?

Do you, and this is the central question, and the only one that matters really, believe that God wants the government to put doctors and women in prison whenever an abortion is performed, and do you think God wants the increased maternal death rates amongst otherwise healthy women who have miscarried that these new policies are creating? Before you answer, please realise that early miscarriage is far, far more common and far far less admitted in public than people who have never admitted to having a miscarriage with other women in a safe environment realise. Are miscarriages hateful to God too? Is it the woman’s fault, and should she go to prison if she can’t prove her miscarriage was natural?

The second question is this: does God care more about putting women and doctors in prison after a foetus dies more than God cares about feeding the poor and the homeless? More than looking after foreigners? More than loving your neighbour and your enemy? Is it more important to love God or to love money?

Because what I see in america is a lot of people justifying being super condemnatory about women and immigrants and hating on helping, all in the name of religion, and I just don’t see much of “love God and love your neighbour”. No wonder young people are leaving the american church in droves. What’s all that hate got that might draw them in? Maybe if it had more for them than the promise of jailing women and doctors, they might see the point, but a lot of church folks can’t get past the jailing of women and doctors and the condemnation of sinners they judge to be particularly scandalously sinful, unless of course they’re republicans or church leaders, in which case denial that anything went wrong. Not forgiveness, that would require an admission of guilt. Just minimisation and cover up.

I remember stories about Jesus getting cross with the temple being full of money and the religious leaders acting pious while making up hard and harsh rules for everyone else to follow (not so much themselves). Is it a relevant story for today? Isn’t it really sad and far from following commandments if the church gets into the judging others and condemnation business? I don’t think it was ever supposed to be about big theatres, big money, politics, professional music and plenty of prison time for plenty of sinners. Hadn’t it better get back very urgently to what Jesus actually said to do?

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Yeah, Nikki Haley and her muslim-hating “finish them” rhetoric. Fucking republicans.

Can’t help wondering what you’re deleting here.

It’s an imperialist semi-democratic republic. Up until Jan 6th it had a long run of peaceful transfer of power. It’s not a dictatorship yet. Trump promises to make it one. I didn’t think any of that would be unknown to you?