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A373537
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Decimal expansion of the Euclidean length of the shortest minimum-link polygonal chains joining all the vertices of the cube [0,1]^3.
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0
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1, 1, 1, 0, 5, 2, 5, 1, 1, 2, 3, 0, 6, 5, 3, 3, 1, 7, 9, 7, 3, 5, 9, 1, 7, 1, 1, 2, 1, 5, 2, 4, 1, 9, 5, 1, 2, 7, 9, 3, 9, 2, 0, 9, 8, 0, 9, 9, 1, 9, 1, 7, 3, 4, 3, 8, 5, 9, 0, 0, 5, 5, 1, 8, 2, 1, 6, 5, 5, 0, 6, 1, 1, 2, 7, 2, 8, 5, 2, 4, 2, 1, 8, 3, 1, 7, 3
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OFFSET
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2,5
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COMMENTS
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It has been proved that it is not possible to join the 8 vertices of a cube with a polygonal chain that has fewer than 6 edges (see Links, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, Theorem 2.2).
Here we consider the additional constraint of minimizing the total (Euclidean) length of the minimum-link polygonal chains (which consist of exactly 6 line segments connected at their endpoints) that join all the vertices of the cube [0,1] X [0,1] X [0,1].
A solution to the above-stated problem is provided by the 6-link polygonal chain (0,0,1)-(0,0,0)-(1+(x+2+sqrt(2))/(2*sqrt(2)(x+sqrt(2))),1+(x+2+sqrt(2))/(2*sqrt(2)(x+sqrt(2))),0)-(1/2,1/2,1+x/sqrt(2))-(- (x+2+sqrt(2))/(2*sqrt(2)(x+sqrt(2))),1+(x+2+sqrt(2))/(2*sqrt(2)(x+sqrt(2))),0)-(1,0,0)-(1,0,1), where x = (1/2)*sqrt((2/3)^(2/3)*((9+sqrt(177)))^(1/3) - 4*(2/(27+3*sqrt(177)))^(1/3)) + (1/2)*sqrt(4*(2/(27+3*sqrt(177)))^(1/3) - (2/3)^(2/3)*(9+sqrt(177))^(1/3) + 4*sqrt(2/((2/3)^(2/3)*(9+sqrt(177))^(1/3) - 4*(2/(27+3*sqrt(177)))^(1/3)))) = 1.597920933550032074764705350780465558827883608091828573735862154752648...
The total (Euclidean) length of the mentioned polygonal chain is about 11.105251123 and this value cannot be beaten by any other 6-link polygonal chain covering all the vertices belonging to the set {0,1} X {0,1} X {0,1} (a nice proof was posted on MathOverflow on June 5, 2024 by a new user, DR.LL, whose profile was subsequently deleted for unknown reasons).
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LINKS
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FORMULA
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Equals 2*(1+1/sqrt(2)+((2+sqrt(2)*x)/2)*(1/x+sqrt(1+1/x^2))), where x = (1/2)*sqrt((2/3)^(2/3)*((9+sqrt(177)))^(1/3) - 4*(2/(27+3*sqrt(177)))^(1/3)) + (1/2)*sqrt(4*(2/(27+3*sqrt(177)))^(1/3) - (2/3)^(2/3)*(9+sqrt(177))^(1/3) + 4*sqrt(2/((2/3)^(2/3)*(9+sqrt(177))^(1/3) - 4*(2/(27+3*sqrt(177)))^(1/3)))) = 1.59792093355003207476470...
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EXAMPLE
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11.10525112306533179735917112152419512793920980991917343859...
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PROG
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(PARI) my(x=solve(x=1.5, 1.7, 4-8*x^2-4*x^4+x^8)); 2 + sqrt(2) + (sqrt(1 + 1/x^2) + 1/x) * (2 + sqrt(2)*x) \\ Hugo Pfoertner, Jun 10 2024
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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