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The Eisenstein integer represented by cell m is A307013(m) + A307012(m)*omega. Thus the set of Eisenstein primes is {A307013(a(n)) + A307012(a(n))*omega : n >= 2}. - Peter Munn, Jun 26 2021

The Eisenstein integer a + b*omega has norm a^2 - a*b + b^2 (see A003136). The number of Eisenstein integers of norm n is given by A004016(n).

The norms of the Eisenstein primes are given in A055664, and the number of Eisenstein primes of norm n is given in A055667.

Reid's 1910 book (still in print) is still the best reference for the Eisenstein integers and similar rings.

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 1. 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 2 has positive distance. - Peter Munn, Jul 13 2021

We can use this coordinate with the first coordinate to form an oblique coordinate system, in which each coordinate maps to an oblique coordinate vector parallel to the axis along which the other coordinate is 0. See the figure with nonperpendicular axes in the Barile link. When both of these coordinates are positive, the oblique coordinate vectors make a 60-degree angle with each other. [Made more specific by Peter Munn, Jul 19 2021]

From Peter Munn, Jul 22 2021: (Start)

The points of the spiral are equally the points of a hexagonal lattice, the points of an isometric (triangular) grid and the center points of the cells of a honeycomb (regular hexagonal tiling or grid). 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 2. 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 positive distance.

This coordinate can be paired with either of the other coordinates to form oblique coordinates as described in A307012. Alternatively, all 3 coordinates can be used together, symmetrically, as described in A345978.

There is a negated variant of the 3rd coordinate, which is the conventional sense of this coordinate for specifying (with the 2nd coordinate) the Eisenstein integers that can be the points of the spiral when it is embedded in the complex plane. See A307013.

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Cartesian coordinates (x,y) of the grid points are converted to barycentric coordinates (i,j,k) by i = x - y/sqrt(3), j = 2*y/sqrt(3), k = x + y/sqrt(3). The sequence gives i. j is given in A307016, k is given in A307017.

The sorting by polar angle affects the grid points in the shells of size A035019, starting at indices given by A038590.

For any Eisenstein integer z = u + v*w (where u and v are integers), we call u the "real" part of z and v the "w" part of z.

This sequence has connections with A316657; here we work with Eisenstein integers, there with Gaussian integers.

It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.

The sense of the spiral (clockwise/counterclockwise) and its orientation are not significant, but for the purpose of illustration, we depict a counterclockwise spiral with its first step towards the right side of the page.

Equivalence classes contain a maximum of 4 tiles. This happens when tile m's reflection about axis X is a different tile, m_x, and these 2 tiles' reflections about axis Y are 2 further tiles, m_y and m_xy, to give an equivalence class {m, m_x, m_y, m_xy}. Some equivalence classes are smaller, because a tile is its own reflection about an axis, X or Y, that passes through the center of the tile.

The Wichmann reference describes bijections from certain unique factorization domains to the hexagonal tiling. Align the spiral with the mapping so that domain identities 0 and 1 map to tiles 0 and 1 respectively. If two integers from one of the domains map to tiles in the same equivalence class, then they share the same status as units, primes or composites.

The Cartesian coordinates of the points of a counterclockwise spiral on an hexagonal grid are given by x(n) = a(n)/2 and by y(n)= A307012(n) * sqrt(3)/2.

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.

From Peter Munn, Jul 18 2021: (Start)

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.

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.

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.

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