My focus for the past few years has been on the intersection of non-halting mathematics and computer science.
The questions I've asked with prolonged contemplation include:
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How can we better understand and leverage recursive patterns?
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What's the difference between recursion and iteration?
discoveries
A Pythagorean triple example of a non-halting tiling system:
-
a
( 2 / 3 )^n
scale symmetry log function; -
a
12-by-18
parent rectangle; and -
a repeating rectangle with an
n = 0
of12-by-10
.
Printed by this alchemy repository.
OEIS contributions
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Approximate
1 7^( 1 / 2 )
with a natural number sequence. -
Approximate
1 2 * 2^( 1 / 2 )
with a natural number sequence. -
Approximate
1 10^( 1 / 2 )
with a natural number sequence. -
Approximate
1 11^( 1 / 2 )
with a natural number sequence. -
Approximate
1 2 * 3^( 1 / 2 )
with a natural number sequence. -
Approximate
1 3 * 2^( 1 / 2 )
with a natural number sequence.
Context:
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1 sqrt(#):
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1 #sqrt(2):
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1 #sqrt(3):
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1 #sqrt(4):
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#2 ~
A330395~ 5, -
#3 ~
A330396~ 7, -
#4 ~
A330397~ 9.
-
-
1 #sqrt(5):
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#2 ~
A330398, -
#3 ~
A330399, -
#4 ~
A330400.
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-
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the unique parts {
cos(#) & 0,-2,... "oscillating even"
,sin(#) & 1,-3,... "oscillating odd"
}, and -
the unique parts {
cos(#)*cos(#) & 1 "Sum"
,sin(#)*sin(#) & - "Sum"
}, rhyme with-
{
(c b) ~ 2b*x^0 2b*x^2 ...
,a ~ 2b*x^1 2b*x^3 ...
}, -
given
a^2 b^2 = c^2
and(c - b) / a = x
, which is -
context for
x^0 = 1
as well as2b = 1; 2b*x^0 = 1
;
-
-
and also rhyme with
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the A099603 Fibonacci sequence approximations of
1 sqrt5
and(1 sqrt5) / 2 = Golden ratio
, which is -
context for why the geometric series and Silver 日本 ratio are uniquely non-halting sequences.
-