[math-fun] Fwd: Rupert's algebraic mystery
The side of the largest square inscribable in the unit cube is 1.06066017177982... = 3/√2. Thus a unit cube can pass though a hole in another unit cube. The side of the largest cube inscribable in the unit tesseract can be found numerically: 1.00743475688427937609825359524... https://oeis.org/A243309 http://mathforum.org/kb/message.jspa?messageID=4637111 which is readily detected by integer lattice reduction to be a bi-quartic: In[1]:= RootApproximant[1.00743475688427937609825359524] Out[1]= Root[16 + 16 #1^2 - 7 #1^4 - 28 #1^6 + 4 #1^8 &, 3] (and thus is expressible in radicals). At G4G12, Greg Huber unveiled a mystery: Using the same trusted numerical methods (Huber & Ligocki, unpublished), he finds the largest tesseract inscribable in a 5-cube has side 1.0008394468593497886019289217565945828768033618266262653807572778501302485680777652768153338725025731471415139085497252961195484534254852511063402098761485174829601977310949912597155616952089698595643, yet this constant has so far resisted every effort (RootApproximant, FindIntegerNullVector, LatticeReduce, ...) to find its polynomial. This is really strange. I'd expect even a flawed methodology to find an algebraic maximum, albeit an erroneous one. Has anyone a clue here? --rwg
hello, about the number : 1.00083944685934978860192892175659458287680336 I will test it on my machine tonight, compared to 15.585 billion entries and some nice programs to invert a number. can we have more digits of that number ? A good question is : how much precision e have here ? I tested the number with 200 digits : nothing. also : what is the supposed degree of that algebraic number ? Could it be (for a strange reason) of very high degree ? have a nice day, Simon Plouffe
The analogous problem maximally inscribing a regular tetrahedron within a unit cube was discussed here in September 2013. The difficulty there depends on dimension modulo 4 , eg. reducing to a Hadamard matrix when n == 3 mod 4 . For n = 3,7,11 , the optimal simplex has rational coordinates. For n = 4 , optimal has quadratic algebraic. For n = 5,6 , best known lower bounds (probably optimal) have quartic. For n = 8 , BKLB (probably optimal) has rational. For n = 9 , BKLB has quintic. For n = 10 , nothing known. WFL On 4/7/16, Simon Plouffe <simon.plouffe@gmail.com> wrote:
hello,
about the number : 1.00083944685934978860192892175659458287680336
I will test it on my machine tonight, compared to 15.585 billion entries and some nice programs to invert a number.
can we have more digits of that number ?
A good question is : how much precision e have here ?
I tested the number with 200 digits : nothing.
also : what is the supposed degree of that algebraic number ?
Could it be (for a strange reason) of very high degree ?
have a nice day,
Simon Plouffe _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Simon Plouffe