There are clear 2-D geometric interpretations of
Pell numbers, one of which involves spirals of squares described here.
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:
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 numbersand 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.
Template
Link to large (5741 x 2378) black and white version, interpolated
horizontally so that it is centered along the odd dimension direction.
Compound spirals
Logical combinations (here and exclusive OR of the
colors) of two pairs of these 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:
Similarity tilings and tesselations
Similarity tilings, and square tilings using the above combinations as motifs:
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