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User:Juan Barajas Martin

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I am a geologist working in computing, and like to be an amateur for some mathematics, as all those related to spatial geometry. I have been working for computation of the minimum surface areas of polycube sets of volume n as a function of n, and found an infinite infinite sequence of polycubes with minimal surface area. This is built between pairs of sucesive polycubes with cubic shape and edge values (e) and (e+1), meaning polycube volumes e**3 and (e+1)**3 and minimum surface areas of 6*(e**2).

Initial cubic shape polycube: edge = e; volume = e**3; minimum surface area = 6*(e**2);

Target cubic shape polycube: edge = e+1; volume = (e+1)**3; minimum surface area = 6*((e+1)**2);

The target polycube is built by the addition ot 3 succesive orthogonal layers of unit cubes to the faces of the Initial polycube, the number of unit cubes added in each layer take sucessive values e**2, e*(e+1) and (e+1)**2, and every layer is fully completed before starting the following layer. This is so because the first unit cube added on every layer will add a surface area of 4 to the previous poilycube, greater than keeping the new unit cube in the current layer, where we ensure that surface area will increase by 2 or not increase at all.

Every layer is constructed as an Ulam spiral,, starting by the center and until the completion of the layer. Except for the first cube placed on every layer which will increase the surface area of the previous polycube by 4, we can count 2 surface area increments for the unit cubes placed at the positions given by the function of the Quarter-squares + 1 sequence (i.e.OEIS A033638 = A002620 + 1), which set the locations of right angle turns in Ulam square spiral, ( Donald S. McDonald (don.mcdonald(AT)paradise.net.nz), Jan 09 2003).