[math-fun] Galois-conjugate fractals
Does anyone know of two fractals of respective fractal dimension log alpha and log beta such that alpha and beta are conjugate algebraic numbers? I'd be curious to see whether they have any family resemblance owing to their common algebraic origin. Jim Propp
This may well be trivial, since I haven't thought about it more than a few seconds. But let Q^ denote the least set of reals containing Q+ and closed under both multiplication and exponentiation. I.e., Q^ is the union of all Q_n, n >= 0, where * Q_0 := Q+ * Q_(n+1) = (Q_n)^(Q_n) * (Q_n)^(Q_n), n >= 1. where for any subsets X and Y of R+, X op Y := {x op y | x in X and Y in Y} for op in {*,^}. Question: Is Q^ closed under addition? --Dan
My guess would be no. Specifically, I guess that 2^(2^(1/2)) + 3^(2^(1/2)) is not in Q^.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Sunday, January 18, 2015 6:01 PM To: math-fun Subject: [math-fun] Least set of reals containing Q+ and closed under both multiplication and exponentiation
This may well be trivial, since I haven't thought about it more than a few seconds.
But let Q^ denote the least set of reals containing Q+ and closed under both multiplication and exponentiation.
I.e., Q^ is the union of all Q_n, n >= 0, where
* Q_0 := Q+
* Q_(n+1) = (Q_n)^(Q_n) * (Q_n)^(Q_n), n >= 1.
where for any subsets X and Y of R+,
X op Y := {x op y | x in X and Y in Y}
for op in {*,^}.
Question: Is Q^ closed under addition?
--Dan
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participants (3)
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Dan Asimov -
David Wilson -
James Propp