cjwatson: (Default)
Happy Christmas and happy Hanukkah (they coincide this year)! This is not especially festive except that I was reminded of it at Midnight Mass while counting verses of a rather long litany.

When I was a teenager, I invented what as far as I know is an original method of finger-counting; at least I was unaware of having based it on anything else and I haven't seen it used anywhere since. No, please stop backing away slowly, I promise I'm not dangerous. I don't remember exactly why I bothered, but it may have had something to do with being a cellist and therefore occasionally having to count off long rests in a reasonably discreet way that was harder to lose track of than just counting in my head. I still sometimes use it in similar circumstances.

My method goes as follows:
  • Begin by counting off the three segments of each finger on your left hand by touching the palmward side of them with your left thumb: 1, 2, 3 for the tip, middle, base of your index finger, 4, 5, 6 for your middle finger, and so on. This takes you to 12.
  • Touch your left palm with your left thumb for 13.
  • Now return to your left fingers as before, but this time touch the backs of the segments: 14, 15, 16 for the tip, middle, base of your index finger, and so on. This takes you to 25 on a single hand.
  • If you need more, use your right hand in the same way as a 25s place. The upper limit is therefore 625.
I was certainly aware of binary and hexadecimal bases by that time and reasonably fluent in both, and I thought of finger binary independently, but converting between bases can take a bit of thought, and the point was to be easily usable in situations where I didn't have much spare brainpower available, for example when in the middle of an orchestral concert with lots of other stuff going on. I basically wanted to be able to delegate the job of counting to a simple motor task and be reasonably sure of getting it right.

This method has several nice properties:
  • 25 is very decimal-friendly, at least at smallish values. It's pretty rare to have to "manually" count higher than 100, and that's just 0 on the left hand and 4 on the right. Most numbers one is likely to need to count to come out easily. Wikipedia tells me that there are Asian systems that use finger segments in a similar way to reach 12 on each hand, but that's not as decimal-friendly.
  • Only involves small movements, mostly within the natural crook of your hand. You can quite easily count this way in a context where other methods would be awkward or gauche, and probably nobody will notice.
  • Reasonably useful upper limit with a single hand. (As a cellist I was usually holding a bow with my right hand, but during rests my left hand was free.)
It is, I suspect, not at all useful for communication: distinguishing between two different sides of a finger is quite easy by touch but probably not by sight.

Am I weird? Is anyone aware of a previous base-25 system like this? Feel free to only answer the second of those questions.

cjwatson: (Default)
On our date a couple of weeks ago, I found myself explaining to [livejournal.com profile] ghoti the basics of why P vs. NP is an interesting question. (Clearly, we have the best romantic conversations.) I'd like to explain this at a bit more length and to more people. You'll have to care at least a little bit about maths to find this interesting, but I hope I've managed to explain it clearly enough that it doesn't require any specialist knowledge. Note that I'm not actually a complexity researcher, just an interested person with some relevant background.

the essence of hard problems )

This post is part of my December days series. Please prompt me!
cjwatson: (Default)
Dear Lazyweb:

[livejournal.com profile] ghoti and I were trying to explain a strange shape to [livejournal.com profile] xanna and Jacob in the hope that they knew the proper geometric name for it. This is the shape of the containers that the ice lollies we like come in (old but still accurate picture, search for "Jubbly"). It's the original tetrahedron shape used by Tetra Pak, now called the Tetra Classic (indeed the pack is so labelled). It's not a regular tetrahedron, though: it's constructed from four isosceles triangles. It's most easily constructed using this net (apologies for the dodgy quality of my Inkscape use):



Fold it such that edges A meet; in doing so edges B will also meet, along with edges C.

MathWorld reckons that it's a special case of an isosceles tetrahedron, but that only requires opposite pairs of the tetrahedral edges with the same length as A to have the same length, whereas in fact four edges have the same length in this shape.

Does anyone know if there's a proper name for this polyhedron? Doubly-isosceles tetrahedron or something?

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