JoeBartholomew.com Rectangles, Table 2

The Golden Rectangle and Sequences of Rectangles
with Similar Properties (continued)

Tables	Return
	Table 1. First Sequence of Rectangles Table 3. Thrid Sequence of Rectangles

Table 2. Second Sequence of Rectangles
	Golden Rectangle	1st Order	2nd Order	3rd Order
Ratio Formula	b/(a–b) = a/b	b/(a–b/2) = a/b	b/(a–b/4) = a/b	b/(a–b/8) = a/b
Solution, a/b	(1+√5)/2	(1+√17)/4	(1+√65)/8	(1+√257)/16
Approximate value	1.61803	1.28078	1.13278	1.06445
Number sequence	F(i) = F(i-1) + F(i-2)	F(i) = F(i-1)/2 + F(i-2)	F(i) = F(i-1)/4 + F(i-2)	F(i) = F(i-1)/8 + F(i-2)
Unit squares to construct rectangle (1)	1/2	1/4	1/8	1/16
Squares to remove (2)	1	1/2	1/4	1/8
	4th Order	5th Order	6th Order	7th Order
Ratio Formula	b/(a–b/16) = a/b	b/(a–b/32) = a/b	b/(a–b/64) = a/b	b/(a–b/128) = a/b
Solution, a/b	(1+√1025)/32	(1+√4097)/64	(1+√16385)/128	(1+√65537)/256
Approximate value	1.03174	1.01575	1.00784	1.00391
Number sequence	F(i) = F(i-1)/16 + F(i-2)	F(i) = F(i-1)/32 + F(i-2)	F(i) = F(i-1)/64 + F(i-2)	F(i) = F(i-1)/128 + F(i-2)
Unit squares to construct rectangle (1)	1/32	1/64	1/128	1/256
Squares to remove (2)	1/16	1/32	1/64	1/128

(1) Unit squares to construct rectangle provide the length of the long side when the short side is one. The length is the value in the table plus the diagonal from one corner to the opposite corner of the unit squares. See Construction, Second Sequence.
(2) Squares to remove are the fractions of a unit square to remove when constructing new, progressively smaller rectangles with the same proportions as the first. See Spirals, Second Sequence.

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