A188635 Continued fraction expansion of length/width of metagolden rectangle.

2, 10, 2, 40, 10, 2, 2, 1, 14, 1, 12, 3, 2, 1, 3, 9, 2, 12, 3, 1, 5, 1, 51, 29, 1, 3, 2, 35, 1, 27, 3, 75, 5, 1, 3, 2, 36, 1, 5, 1, 1, 3, 1, 2, 40, 1, 2, 7, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 8, 11, 6, 1, 2, 1, 3, 1, 2, 7, 1, 9, 1, 1, 9, 2, 1, 2, 1, 2, 7, 2, 34, 9, 13, 1, 2, 2, 24, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 1, 2, 4, 11, 3, 1, 3, 1, 6, 1, 5, 20, 2, 1, 1, 4, 10

Offset: 0

Author

Clark Kimberling, Apr 06 2011

Keywords

• nonn
• cofr

Comments

The metagolden rectangle is introduced here as a rectangle such that if a golden rectangle is removed from one end, the remaining rectangle is metagolden. That is, using mu to represent the metagolden ratio, mu - phi = 1/mu, so that when the phi X 1 golden rectangle is removed, the remaining rectangle is 1/mu X 1, the same metagolden shape as the original mu X 1 rectangle. - Geoffrey Caveney, Apr 20 2014

(Recall that a rectangle is golden if the removal of a square from one end leaves a rectangle having the same shape as the original.)

To generalize, suppose that a rectangle R of length L and width W is to be partitioned into a set of squares. One way to do this matches the ordinary continued fraction of the shape (defined as L/W). Explicitly, if L/W = [a(0), a(1), a(2), ...] (finite or infinite), then a(0) squares are removed in step 1, then a(1) squares in step 2, then a(2) squares, and so on. An example is the golden rectangle, of shape [1,1,1,...].

To generalize from squares to rectangles all having a common shape s >= 1, in such a manner that a(n) of these rectangles are removed at step k, the original rectangle must have shape given by the continued fraction [s*a(0), s*a(1), s*a(2), ...]. This result follows easily from the Euclidean algorithm for two positive real numbers (as in the paper cited below on golden triangles, generalized to match continued fractions). Here, the terms of the continued fraction need not be integers, and by Seidel's theorem of 1846, the continued fraction converges (e.g., Roberts, pages 97-105). We shall refer to the geometric partitioning of the original rectangle as the continued fraction partition (matching the continued fraction [s*a(0), s*a(1), s*a(2), ...]).

Thus, the constant at A136319 is the shape of a rectangle whose continued fraction partition is an infinite collection of golden rectangles.

References

• Joe Roberts, Elementary Number Theory: A Problem Oriented Approach, MIT Press, Cambridge, Massachusetts, 1977.

Links

• D. Huylebrouck, The Meta-Golden Ratio Chi, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture.
• Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.
• Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165-171.

Crossrefs

Cf. A136319 (decimal expansion), A188636, A188637.

Programs

Mathematica

t=(1+5^(1/2))/2; r=(t+(t^2+4)^(1/2))/2
FullSimplify[r]
N[r, 130]
RealDigits[N[r, 130]][[1]]
ContinuedFraction[r,120]
(*End.  Following is another approach.*)
r = (1 + 5^(1/2))/2;
FromContinuedFraction[{r, {r}}]
FullSimplify[%]
N[%, 150]
RealDigits[%]  (*A136319*)
ContinuedFraction[%%, 120] (*A188635*)

Formula

A188635 is the continued fraction of the number given by the decimal expansion at A136319. It is also the continued fraction [r,r,r,...], where r=(golden ratio).

Extensions

Offset changed by Andrew Howroyd, Jul 08 2024