h a l f b a k e r yThe word "How?" springs to mind at this point.
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These dice are shaped like ordinary d6 (cubic, six sided dice).
However, on each face - instead of a simple number or value in 'pips' - there is a short expression which evaluates to the number required.
Each expression is chosen to optimise some combination of elegance, brevity and obscurity,
while still giving the exact integral result.
Constants used may include:
e : 2.71828... Euler's number
pi: ratio of a circle's circumference to its diameter : 3.14159265...
gamma : Euler's constant : 0.57721566...
i : the square root of -1
phi : the golden ratio : 1.61803...
For example, the 'one' side could use the expression : -e^ (i * pi) (shown in neat mathematical notation; that isn't possible here)
I have a couple of ideas for other sides, although not great options for all of them. For example I can get five by reorganising the equation which gives the golden ratio (although this uses the number two twice, and one once, which isn't ideal).
I'm open to suggestions.
I'd like to avoid using floor and ceiling operators as far as possible, although they could each be used once in total.
Similarly, I'm not entirely averse to using a trigonometric function, provided it can be formed into a relatively short formula.
- The issue is that you need numbers to define the angle, and trying to avoid those leads to equations like 1/cos(pi/floor(pi))
I like the idea, and think it's similar to the kind of thing on this mathematician's clock.
https://www.reddit....thematicians_clock/
[zen_tom, May 09 2024]
[link]
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You can't make a solid integer out of an irrational without referencing either a few well-known formulae or just adding an integer minus the same irrational, such as pi+1-pi. A die doesn't' have much space for a formula. |
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//You can't make a solid integer out of an irrational without referencing either a few well-known formulae or just adding an integer minus the same irrational, such as pi+1-pi.// |
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You mean, apart from the various other options I covered? ...And perhaps other methods I don't know about.
I don't mind enormously if the formula is well-known to mathematicians, or a derivative of one. |
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//A die doesn't' have much space for a formula.// |
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This is why I /do/ care about it being short. |
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Examples, a1?
Well, I gave you three already, for one and five, and one more you can calculate yourself.
The first was explicit, and generates an integral value using two irrational constants and the imaginary unit - it's actually quite famous because of its surprising nature, to the point I might not use it directly. |
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The second example (for five) is pretty straightforward to figure out for yourself, and the third formula (given at the very end of the idea) is something you might enjoy doing in your head, if you were so inclined. |
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There's a very simple way of generating six - maybe too simple; it may need additional obfuscation. |
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This idea is actually something I've considered baking, so I don't particularly want to give everything away. I'm kind of hoping for suggestions for some of the values. |
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I think limits over sums and products provide a nice way to arrive at some values, and I'd imagine there's some neat geometrical ways of defining integers that would come out of known Pythagorean triples. So that's 3,4,5 covered one would imagine. |
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Also, have posted link to a clock with a similar idea - no idea how accurate the formulae are on that one. |
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Given that -1/phi = 1-phi (sometimes labelled with the Greek "psi"), you could do quite a few sides with just phi and/or psi & inverses & squares & such. |
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