3. Initial conditions. Two populations makes it obvious that you'll need two initial conditions.

Seemed to go okay. Managed to also bring in exponential of matrices which was fun!

Just guessing here, but to render the eleven tiles would need a finer grid on the cube. I think I can see how to see each tile on a 3×3, but not smaller. So to render a spherical tiled knot on a n×n×n grid you'd need a 3n×3n×3n sized cube.

Yeah, Adobe editions would be very useful.

I read that as saying that I can't read books bought from bookshop.org on my kobo.

It'd be nice if I could as this sounds like a great way to support local bookshops with the convenience of ebooks.

The bookshop.org FAQ says:

"Can I read my ebooks on my Kindle, Kobo, etc.?
• Ebooks from Bookshop.org must be read on either our Apple or Android app, or via a web browser, with the exception of DRM-free titles that can be downloaded and transferred to your reader app or device of choice."

Exactly, so the binomial polynomials are the discrete analogues of the divided powers.

I shall have to break my "no video" rule as this is one of those topics where I Have Opinions.

Define the nth binomial polynomial as b_n(x) = n(n+1)(n+2)...(n+k-1)/k! then the summation operator on sequences takes b_n(x) to b_{n+1}(x).

Possibly easier to see in reverse: that the difference operator takes b_n to b_{n-1}

Neither of which I want associated with the very deep and profound sense of wonder that I get when - after a lot of time and effort - I truly understand some piece of mathematics.

Apologies for not being clear.

I'm objecting to the (over)use of the *word* beauty because (to me) it conveys a superficialness together with a sense of "if you don't see it yourself, you'll never be part of the in-crowd".

I don't think that there's one replacement word, rather we use a more accurate word for each circumstance.

Incidentally, it's not a child-like appreciation of beauty that I'm objecting to, but a superficial one.

I'd've bought one ... if I didn't already own a copy.

(Which, as it is half term, I am now reading)

I just feel that the word "beauty" conveys something one either sees or doesn't and that if you've had to work at it - as in your beethoven example - then there'll be a more pertinent word that expresses what it is that you experience.