So now we have both irrational and fractional fucks, we have all real fucks, and since we can have “twisted fuck” we can rotate any fuck through any angle and we have the entire complex fuck plane.

I love this post. Thank you.

I don’t know why Janet thought that the working conditions would be nice at the evil laboratory. It’s not exactly evil to take good care of your employees now, is it?

Fun fact: Most frogs don’t say ribbit, but one of the earliest film sound libraries included a frog that does say ribbit, and so that sound is the sound of a frog in many films and television programs, but not in nature documentaries which record their own audio.

So much of the English speaking world, far, far more broadly than the spread of that type of frog, think frogs typically say ribbit.

If you watch a nature documentary about frogs, you’ll hear a vast array of different sounds, and this map will make much more sense.

Ah, thank you. I wasn’t sure. However, I am sure that I don’t want it going the wrong way up my personals.

I find that when you sign up for lemmy, you very much underestimate the extent to which the community is going to be invested in you sticking your dick in grape.

It’s not a complaint, really, and it feels supportive, in a way, but it’s definitely not what I was expecting. I mean, the whole area of soft fruit isn’t really a theme I was considering exploring in any kind of sexual way, if you can appreciate where I’m coming from.

I take your point about the possibility but I’m still drawing a bit of a blank in the motivation department, if I’m honest.

The comment about grape seeds isn’t making making me any more enthusiastic, I can tell you. No offence meant.

Sounds painful. Like gallstones, but backwards. On the plus side, probably less scratchy. On the minus side, maybe more citric acid.

I’ll tell you why not! You hippie homeopaths are all the same! Science has scienced the evidence that there’s no evidence for homopathic medicines otter than the libido effect.

I just can’t see it ever happening.

I don’t mean any disrespect, and I don’t want to kink shame, but that kinda thing is just not my bag, baby.

Dude, I don’t mean to boast, but honestly, I think my dick is just WAY too big. Like, I would DESTROY that grape instantly if I tried. It’s not just a trick of the camera angle, it just is that big. Honestly, I don’t even need to get out a measuring tape to tell you that even with a massive grape, it’s just not going to fit.

The grapes? I can never tell when a grape wants some action. My whole life, I’ve missed every single signal. Well, that, or the grapes just don’t find me attractive, like, EVER.

I… I don’t think I can.

You’re very right.

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.

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

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

It’s everything to do with it. Read some of the other replies.

Egregious. I feel your pain.

You’re right on all three counts. It’s not always true, f(x0) has to be L, and the function has to be continuous.