Pell spirals

There are clear 2-D geometric interpretations of Pell numbers, one of which involves spirals of squares described here.

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Compound spirals
Similarity tilings and tesselations

The Pell sequence

The Pell numbers are a sequences of integers that can be defined by a recurrence relation similar to that for the Fibonacci numbers. Starting with 0 and 1, each Pell number is the sum of twice the previous Pell number and the Pell number before that -- for example 0 + 2*1 = 2 and 29 + 2*70 = 169:

0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741... (sequence A000129 in OEIS).

The sequence grow exponentially, and the ratio of pairs of adjacent terms approaches the silver ratio. So, for example, a good approximation of the square root of 2 is 2378/985 - 1. There are a delightfully wide range of generating functions and relationships between this sequence and other analysis topics; see Pell numbers and A000129 for examples.

A 2-D graphical representation of the sequence uses a rectangle composed of squares with sides the length of Pell numbers. The dimensions of each rectangular boundary is the sum of twice the largest and second largest enclosed square, in the same way that the recurrence relation defines the one dimensional sequence. Coloring one of the two sets of squares in the same direction about the center results in two interlocked spirals, equivalent up to a pi rotation:

The key feature of this geometry is that the side of each integral square component is matched by three adjacent integral squares (169 = 70 + 29 + 70), and if the end squares are removed the rectangle remaining has the same properties (but not the exact ratio of sides) with a pi/2 rotation.

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Link to large (5741 x 2378) black and white version, interpolated horizontally so that it is centered along the odd dimension direction.

Compound Pell spirals

Logical combinations (here and exclusive OR of the colors) of two pairs of Pell spirals, one rotated by pi/2, gives a compound set of spirals at the overlapping center:

Logical combination (XOR) after mirroring one of the pairs gives a similarity tiling symmetric about the diagonals:

Logical combinations of black and white Pell spirals and their rotations and mirrors. These are similarity tilings as the whole pattern is replicated at smaller scales, ad infinitum:

Pell spirals, compounded with rotation and mirror

Similarity tilings and tesselations

Similarity tilings, and square tilings using the compound Pell spiral (see above) combinations as motifs:

The center of this similarity tiling is an inverted (or rotated) scaled copy of the whole, and corners of each quadrant are inverted scaled copies of the quadrant.

The pattern is composed of a single "dart" shape derived from a compound Pell spiral, with a fractal but finite length boundary. It is a binary fractal.

The pattern is composed of an array of similarity tilings, each of which is formed by a compound Pell spiral. These are interspersed with gray "background" squares.

I like how it gives the impression of overlapping black and white squares. This effect was the inspiration for the motif of "Corners tilings"

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