What are the odds that 3 'random' points on a sphere will form an acute triangle? 

6:57 "Out of nowhere." If there's one thing I've learned in maths and physics, it's that mathematical constants (not physics ones like c or α, afaik) show up in the most unexpected places… until you realize that most of those "unexpected places" have one thing in common: a circle (or sphere) can easily pop up if you play with the numbers a bit.

Oh hell yeah a nerdy zach star video! Oh how i have missed these

I discovered, quite a few years ago, that the bisector of the larger of the two non-right-angles in a 1-2-√5 triangle cuts the longer orthogonal side of that triangle at a distance of 1 / φ from the right angle. I'm sure many people had already discovered it, even centuries before me, but it was still satisfying. This problem reminded me of it.

hell yeah! Another Zach Star video!

You should look at the construction of a pentagon. When you draw the 5-pointed star using the points of the corners of the pentagon the golden ratio shows up for the line segments that form the 5-pointed star. Also, the ratio of the side of the outer pentagon to the inner pentagon of the 5-pointed star is equal to phi squared. Also, the value of sin(54deg)=phi/2. The values of trig functions for angles of 18deg, 36deg, 54 deg and 72deg have phi buried in the expression. Have fun with it.

“Bro, here’s why you’re a fvcking m0ron!” “Now, do you like interesting math puzzles that utilize the golden ratio?” Zach Star in a nutshell lol.

Does Bertrand's paradox play a role here given that the selection method for the 3rd point hasn't been specified further?

3:54 Really cool fact! I think you can link this to the 3b1b video on surface area of the sphere, specifically the animation that breaks apart the sphere and rearranges it into a cylinder

You didn't justify why does only green area results in acute triangles.

Man your math videos are really good. Would love more of these.

Alternatively, use compound angles to write P(theta)=(sqrt(5)/2)sin(theta/2-phi)-1/2 (where the angle phi is an exercise), with maximum (sqrt(5)-1)/2.

Can you do a tier list of the hardest type of mathematics to learn that have real life applications? OR, or, what about a tier list about the most challenging fields of ingeniering? or mabye something more like, rocket science vs architecture or stuff like that. Thanks for your videos.

welcome back, zach

I wish Zach is my Math teacher

Sometimes I forget this guy is a genius

Omg..., I have been waiting for your video:)))

Man that's awesome. We can never get enough of these nerdy videos

Where do you get puzzles like this from?

Hey Zach, I’ve been interested in Aerospace engineering and I was wondering if you could recommend any specific RU-vidrs that give a similar experience as you, like giving their experience through college and such? Thank you so much😁

A curiosity I've had for a while is: how would a second time dimension work? Like a second spacial dimension allows us to move along two axes and occupy a higher level of space. It allows us to combine coordinates in axes to occupy different positions in the 2D/3D space. But how would that even work for time? Like would you be able to age in two directions? How would that affect processes that depend on time, or entropy? What would that even mean for measurements such as velocity? Or would time just be measured as the distance from O on the axes? Super weird. I can't even begin to think of what that would mean, unlike with spacial dimensions, for which it's at least possible to intuit some of it based off of flat land observations and comparisons.

more discussion of the golden ratio and speculation about how it is fundamentally related to this problem would have been interesting to add, give another 2-3 mins to the video

Hey im kinda confused if theta 126.9 degrees and the centre is in in green area doesn't that mean that some parts of green are red area as the triangle won't be acute?

your explanation and animation are surperb, can you telle how you do your animation?

Would be curious about how this problem changes when the triangle is drawn on the surface of the sphere (using great arcs) instead of inside it.

Golden ratio is the fabric of the universe beyond metaphysics

Chef's kiss Phi and pi are two sneaky bois

Sometimes I forget this is a math channel and I always think you are gonna butt in with a goofy punchline

Can you make videos on these topics. Architecture Systems engineering Biotechnology Quantum Computing/Engineering

6:44, outstanding video, but you said ‘odds’ when you meant probability. It’s in the title too. Odds is the ratio of winners to losers. Probability is the ration of winners to the total. Hope it helps!

Just for the record: is this angles of a 3D triangle, i.e. tunneling through the sphere? Or non-euclidean triangles along the surface of the sphere?

Arctan? Yes.

How is the 3rd point chosen 'randomly'? If it was chosen by picking from uniform distributions of latitudes and longitudes then this would not work.

So why do many video titles start with an unnecessary 'so' at the moment?

I watch these videos without really taking anything in to feel smarter about myself.

Burgundy

U have to make a sketch like "If god was an engineer." where an atheist and god have a conversation in afterlife

👍 👍 👍 👍 👍

real

Because it shows up everywhere? Lol

👌

Hey Zach, I was curious about your opinion. I would think that the laws of probability are independent of time. If this were the case, then if you could time travel you could never get back to your future. For example, if you traveled 10 years back in time and then traveled back to your future, all the events based on probability would be "re-rolled". Another example is say you memorized the lottery numbers that won in last weeks lottery and then traveled back in time. The odds of winning this particular lottery is, say, 1 in 350 million. Since the balls "have no memory" it would still be 1 in 350 million chance that those numbers would come up. There's no guarantee that those numbers would come up again. I don't know, it's just something I thought about, what do you think?

I read the title seven times... Tried to understand the joke or gag, but then I realized... Lol

I'm not happy with the explanation from 3:05 onwards. You can't just say "these bits aren't red so they're green". I have no confidence that ALL the points in that section are valid

I picked a random video, it was Zach Star, he's a cute so I win

0:23 yeah the triangle is very cute I WIN!!!

I don't get it shouldn't you just put those points as close to each other as possible

So your saying I can unlock tusk 4, got it

The people in these comments are far smarter than me holy

5:10 I disagree, it should be written as: 4pi*r*z- pi*R2d2 I cannot believe you didn’t write it that way, very disappointing.

You do make interesting videos, but i find it hard to stay focused because of the many little pauses you make while speaking. Some of them may be a good thing, but for me it's too many. Just a suggestion for improving, but maybe it's just a me thing and the way you record you videos is fine 🤷

Can I use this to defeat the United States president?

"promosm"

Boring.