cp's OEIS Frontend

A345978 Third coordinate of the points of a counterclockwise spiral on an hexagonal grid in a symmetric redundant hexagonal coordinate system.

%I A345978 #14 Jul 25 2021 02:15:49
%S A345978 0,-1,-1,0,1,1,0,-1,-2,-2,-2,-1,0,1,2,2,2,1,0,-1,-2,-3,-3,-3,-3,-2,-1,
%T A345978 0,1,2,3,3,3,3,2,1,0,-1,-2,-3,-4,-4,-4,-4,-4,-3,-2,-1,0,1,2,3,4,4,4,4,
%U A345978 4,3,2,1,0,-1,-2,-3,-4,-5,-5,-5,-5,-5,-5,-4,-3,-2,-1,0,1,2
%N A345978 Third coordinate of the points of a counterclockwise spiral on an hexagonal grid in a symmetric redundant hexagonal coordinate system.
%C A345978 This is a negated version of A307013 with the advantage of symmetry, i.e., A307011(n) + A307012(n) + a(n) = 0. The mutual angles of the 3 coordinate axes then are 120 or 240 degrees.
%C A345978 From _Peter Munn_, Jul 18 2021: (Start)
%C A345978 The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0. a(n) is the signed distance from spiral point n to the axis that passes through point 3. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 1 has negative distance.
%C A345978 The coordinates may be used in 2 ways. Firstly, any 2 of the 3 coordinates can be paired as oblique coordinates, which entails mapping each coordinate to a vector that is parallel to the line along which the other coordinate is 0 (described further in A307012). Alternatively, each of the 3 coordinates is mapped to a vector perpendicular to the line along which the coordinate is 0, then the sum of the vectors is divided by the square root of 3.
%C A345978 This coordinate system has been used for more than half a century. See the extract from Moffatt, Pearsall and Wulff included in the linked Princeton MAE page (which refers to a 4th coordinate, making it a 3D system). "Cube coordinates" appears to be a currently popular term for the system in some information technology communities. This refers to the useful isometric view of the cubic cells from a 3 dimensional lattice that are indexed by 3 coordinates that sum to zero.
%C A345978 (End)
%D A345978 William G Moffatt, George W Pearsall and John Wulff, The Structure and Properties of Materials Volume I: Structure, Wiley, 1964.
%H A345978 Margherita Barile, <a href="https://mathworld.wolfram.com/ObliqueCoordinates.html">Oblique Coordinates</a>, entry in Eric Weisstein's World of Mathematics.
%H A345978 HandWiki, <a href="https://handwiki.org/wiki/Hexagonal_lattice">Hexagonal Lattice</a>.
%H A345978 Princeton University Mechanical & Aerospace Engineering, The Structure of Solids, <a href="https://www.princeton.edu/~maelabs/mae324/03/03mae_21.htm">The Hexagonal Lattice</a>.
%H A345978 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Signed_distance_function">Signed distance function</a>.
%F A345978 a(n) = -A307013(n) = -(A307011(n) + A307012(n)).
%Y A345978 Cf. A307011, A307012, A307013.
%K A345978 sign
%O A345978 0,9
%A A345978 _Hugo Pfoertner_, Jul 15 2021
%E A345978 Name revised by _Peter Munn_, Jul 22 2021