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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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From Peter Munn, Jul 11 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 three "0-axes" that pass through spiral point 0 and one of points 1, 2 or 3. These 0-axes are the lines along which one of the coordinates is 0.

a(n), the 3rd coordinate, is the signed distance from spiral point n to the coordinate's 0-axis, which passes through points 0 and 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 positive distance. This 3rd coordinate is the sum of the other 2. In the symmetric variant of the coordinate system, the 3rd coordinate has the opposite sense, so that the 3 coordinates sum to 0. See A345978.

We can use any 2 of the 3 coordinates to form an oblique coordinate system, in which each of the 2 coordinates specifies vectors parallel to the other coordinate's 0-axis. This means the direction of the oblique coordinate vectors depends on the choice of the other coordinate - see the illustration of coordinate pairing in the links. When both coordinates are positive, an oblique coordinate vector derived from this sequence makes a 120-degree angle with the vector derived from the other sequence; however, when A307011 and A307012 are used together, the angle is 60 degrees.

Pairing with A307012 can be viewed as follows. Let omega = -1/2 + i*sqrt(3)/2, a primitive cube root of unity. Then f(n) = a(n) + omega*A307012(n) embeds the spiral in the complex plane with spiral points 0 and 1 embedded at 0 and 1 (so that the points of the spiral embed as the Eisenstein integers, as used for A345435).

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