Ratios and Geometry

Rectangle Reactor

Continued Fractions, Square Cuts, and the Golden-Only Spiral Story

Verified Verified Verified False

Ratio

55/34

[1; 1, 1, 1, 1, 1, 1, 2]

Golden Case

Fibonacci approximation to the golden-ratio dissection

Quarter-circle guides are available for the golden-special case. They are not a generic logarithmic spiral proof.

Convergents

i	q_i	p_i / q_i	decimal	error
1	1	1/1	1.0	0.61764706
2	1	2/1	2.0	-0.38235294
3	1	3/2	1.5	0.11764706
4	1	5/3	1.66666667	-0.04901961
5	1	8/5	1.6	0.01764706
6	1	13/8	1.625	-0.00735294
7	1	21/13	1.61538462	0.00226244
8	2	55/34	1.61764706	0.0

Claim Status

Verified Verified Verified False

Running the Euclidean algorithm on positive integers yields the finite continued fraction of a/b.
Rectangle-to-squares dissection follows the same quotient sequence as the Euclidean algorithm.
The logarithmic-spiral story belongs to the special golden-rectangle case, not to every Euclidean rectangle dissection.
Generic inward-spiral or logarithmic-spiral claims for arbitrary gear or rectangle models are not valid in general.

Fun Fact

phi

The golden spiral story depends on self-similarity. Arbitrary Euclidean rectangle cuts do not automatically inherit it.

Related Labs

Sources