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Heck, there are some quite interesting subsets of \(\Bbb N\) that are closed under multiplication. Consider for example { 54, 64, 81, 96, 144, 216, 256, 324, 384, 486, 576, 729, 864, 1024, 196, 1536, 1944, 2304, 2916, 3456, 4096, 4374, 5184, 6144, 6561, 7776, 9216, 11664, 13824, 16384, 17496, 20736, 24576, 26244, 31104, 36864, 39366, 46656, 55296, 59049, 65536, 69984, 82944, 98304, … }.
That's the set of all integers that are (a) \( 2^i3^j \), (b) \( 5k\pm1\), and (c) at least \(54\).

Consider the family F that contains the subsets of \(\Bbb Q\) that are closed under multiplication. Some elements of F are well-known (\( \varnothing\), \( \Bbb Q \) itself; others less so. (Consider the set of rational numbers with denominators of the form \( 5k \pm 1\)] or the set with denominators \( 2^i3^j\) .)

\( F \setminus\{\varnothing\} \) has a very interesting lattice structure.

Mathematically disappointing 

Color-blind readers: I would appreciate your feedback on the diagrams in . Please reply here or by email to . Thanks.

New math post on my blog: The least common divisor and the greatest common multiple

Dividing a chocolate bar into any proportions:

The bar has L squares, and you want to give each of m people an integer number of squares, but the integers are not known in advance. How to break the bar into few pieces so this will always be possible?

Reid Hardison asked this months ago but Ilya Bogdanov answered with an efficient construction of the optimal partition much more recently.

Quasiperiodic bobbin lace patterns:, Veronika Irvine, Therese Biedl, and Craig S. Kaplan, via — aperiodic tilings in fiber arts.

The attached image is a photo of lace (not an illustration), braided into an Ammann–Beenker tiling pattern.

I decided that the world needed to see the Maclaurin series thrashing like a screen door in a hurricane.

Wikipedia has this animation of the Maclaurin series converging for e^x, but it emphasizes the right-hand side, where the behavior is much less bizarre, and only hints at the left-hand side where the partial sums flap back and forth like a screen door in a hurricane.

New math post on my blog: Technical devices for reducing the number of axioms

It's late. I misread a notification as “favourited your cactus”.

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