Ratios and Geometry

Quadratic Surd Loop

Periodic Continued Fractions for sqrt(n)

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[4; (2, 1, 3, 1, 2, 8)]

a0

4

Period

6

Pell Hint

170, 39

ia_ip_i / q_idecimal
1 4 4/1 4.0
2 2 9/2 4.5
3 1 13/3 4.33333333
4 3 48/11 4.36363636
5 1 61/14 4.35714286
6 2 170/39 4.35897436
7 8 1421/326 4.35889571
8 2 3012/691 4.35890014
9 1 4433/1017 4.35889872
10 3 16311/3742 4.35889898
11 1 20744/4759 4.35889893
12 2 57799/13260 4.35889894
13 8 483136/110839 4.35889894
14 2 1024071/234938 4.35889894
15 1 1507207/345777 4.35889894
16 3 5545692/1272269 4.35889894

Claim Status

Rigour stays visible

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  • Running the Euclidean algorithm on positive integers yields the finite continued fraction of a/b.
  • The logarithmic-spiral story belongs to the special golden-rectangle case, not to every Euclidean rectangle dissection.

Fun Fact

phi

Special, Not Generic

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