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"Archimedean Polychora" are a subcategory of "Uniform Polychora" aka "Uniform 4-Polytopes". I believe there are 47 of them in 4-D, depending on how you count.

I would say it would be much easier than predicting the number of a roll of dice and maybe a bit harder than predicting the trajectory of a thrown ball. There definitely is a chaotic component to the unstable equilibrium, so the longer you let it go the less predictable it would be.

This thesis has a full analytic solution worked out. There's an interesting graph of the angular momentum components on page 24.

They are not indeterminate or unpredictable. As long as you know the moments of inertia and the initial rotational velocities you can predict the flipping period by solving Euler's equations.

They only have one dihedral angle (by duality, since Archimedean solids have a single edge length). Did you mean a different kind of angle?

Every permutation can be written as a product of transpositions. If the product is of an even number of transpositions, then it is an even permutation, and vice versa.

Suppose the numbers are x and y, with x < y. Choose your favorite probability distribution supported on the whole real line, let's say N(0,1), and sample a random value z from it. If the number you see is greater than or equal to z, guess it is the larger of the two. Using the law of total...

What do you mean by 'correct'? The convex regular polytopes have a simple recursive definition (for any dimension), and we can figure out their properties from there. In Euclidean space there are: 1 in 0D, 1 in 1D, infinitely many in 2D, 5 in 3D, 6 in 4D, and 3 in each dimension 5 and higher...

Ok, but that definitely feels like a bit of bait-and-switch. It isn't that classical theorems are intuitionist theorems, it's that there are ways of fiddling with the statements of classical theorems to get related theorems that are shared. But intuitionist theorems are literally classical...

Yup. :) Sounds like Platonism to me. In essence, do mathematical entities exist in reality or not? When I talk about an idealized triangle or the number 1, am I referring to something 'out there'? If they exist in reality, where? How can I determine their properties while sitting at home...

Do they each think that? I might agree that the 'think they're trying to prove the wrong theorems' could be true from a more restrictive view of less restrictive reasoning, but why the other way? I think you'd be hard-pressed to find a classical mathematician who thinks that about intuitionist...

Fair enough. BTW, I'm not sure if you saw, but I started a post in the math forum and was hoping to see your thoughts on the philosophy of math.

We go through cycles of posters valiantly trying to get untermensche to understand the basics of logical argument before eventually running out of patience and giving up. It's been years, and trust me, you can lead this horse to water, but you can't make him think.

Here's a thread for discussing the philosophies of mathematics, their merits and faults. Constructivism, formalism, logicism, platonism, etc. What do you think? IMO, at the surface level, the majority of working mathematicians think of their day-to-day mathematics platonistically, with the...

The calculation for sampling without replacement is not too difficult,and it's good to see how much that changes the answer. I get 1 - C(19999947,20000)/C(20000000,20000) = 0.0516452 for the probability without replacement and 1 - (19999947/20000000)20000 = 0.0516201 for the probability with...

Very easily, it all freezes really well if sealed tightly. You can even freeze them already formed (pre or post bake) for a quick prep meal, just wrap them so they don't stick together. The filling is already cooked, so all you really need to do is to finish the crust and warm it through. I may...

He he

In a way, kind of, but in a sense it's also worse than just a single quantum 'god of the gaps' argument. You need two distinct steps, at least, and it's the second one that's harder to plausibly satisfy. Discreteness is forbidden (even at arbitrarily small lattice sizes) in the standard model...

JFC, if you aren't able to read, then we're done here.

Luckily for all of us, physicists are much smarter than you are, and they actually thought through the consequences of the different possibilities. They figured out that the potential discretized spacetimes imply a violation of Lorentz invariance, which in turn implies consequence that would be...