squiggles

do you like to ?

meditate & study
 

study & meditate

or contemplate ?

somewhere else another reality

the hollow and the silk

parity groupings in two dimensions and the array is shown in expanded view

asea & piles

triples of perception

when looking at a sequence of symbols 3 numbers are perceptually present
00          we see 2 symbols and 1 adjacent pair and count 2 symbols from the adjacent pair
000        we see 3 symbols and 2 adjacent pairs and count 4 symbols from the adjacent pairs
0000      we see 4 symbols and 3 adjacent pairs and count 6 symbols from the adjacent pairs
00000    we see 5 symbols and 4 adjacent pairs and count 8 symbols from the adjacent pairs
000000  we see 6 symbols and 5 adjacent pairs and count 10 symbols from the adjacent pairs
0-n0s-0  we see n symbols and n-1 adjacent pairs and count 2(n-1) symbols from the adjacent pairs
so with a worked out example

how many numbers do we naturally see in  oooo  ??

oooo            4

oo oo oo      3

oooooo        6

or the numbers 12 and 13 used in musical scales

12 notes from C to B

13 notes from C to C

12 semitones from C to C

and from the triples of perception idea we see these numbers

12, 11, 22

13, 12, 24

exploring adjacency with math art

the simplest example of emergence one could possibly think of is

the zig zag motion eyes do when counting adjacent pairs in a sequence

two tw wo

spooky

sp po oo ok ky

sppooookky

sp pp po oo oo oo ok kk ky

 

 

Langford(16) spacecraft

disintegration

pattern from systems of linear congruences and a fractal sequence

 

Harshad numbers and triangle variations

 

fusc

untitled3

negabinary

[[{{ institutions of higher learning }}]] and {{ LPFs & GPFs together }}

add 1, multiply by 1, add 2, multiply by 2, etcetera

prime powers in mauve, LPFs in orange with bonus pink squares to reach the GPFs

thue morse

binary runs

curvature

Like sand poured out from a bucket so are the curves of our world .

An angel at my picnic

The ruler function: 2^a(n) divides 2n. Or, a(n) = 2-adic valuation of 2n.  ( OEIS A001511 )
1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, ...
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).   ( OEIS A002620 )
0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, ...
Some examples :
for n=2, 3, 4, 5, 6, 7, 8, 9, 10  have  1*1,  1*2,  2*2,  2*3,  3*3,  3*4,  4*4,  4*5  and  5*5

drifting plankton

untitled2

untitled

after the disco

differences = complement { subtle }

reptend128

thing~mats & distortion

 

silly choices & smooch

 

compare

disco ball

the weaver bird

Weaver birds, also known as weaver finches, get their name because of their elaborately woven nests
(the most elaborate of any birds'), though some are notable for their selective parasitic nesting habits. The nests vary in size, shape, material used, and construction techniques from species to species.
(Source: Wikipedia)
Yay, for all the wonderful bird life in nature !

admixture

transference

strange fruit & fruitfly

 

distances

transport Egyptian fractions { keep on truckin' }

5/12 = 1/4 + 1/10 + 1/15 = 1/5 + 1/6 + 1/20

Lcm( 12, 4, 10, 15, 5, 6, 20 ) = 60

25/60 = 15/60 + 6/60 + 4/60 = 12/60 + 10/60 + 3/60

60 is the smallest number with 6 representations as a sum of 2 primes:

60 = 7+53 = 13+47 = 17+43 = 19+41 = 23+37 = 29+31

grey satin

fiboprimary & melancholy

 

Octagon island and Langford's caterpillars

 

 

gravity force ~ ^", '> ;-~

The dynamics of positive integers with regards to happy numbers

A happy number example
A number such as 334 is three hundred and thirty four with digits forming the multiset { 3, 3, 4 }
We then sum the squares of these digits: 3^2 + 3^2 + 4^2 = 9 + 9 + 16 = 34
Now we do the same for 34: 3^2 + 4^2 = 9 + 16 = 25
And iterating this process:
2^2 + 5^2 =  4 + 25 = 29
2^2 + 9^2 =  4 + 81 = 85
8^2 + 5^2 =  64 + 25 = 89
8^2 + 9^2 =  64 + 81 = 145
1^2 + 4^2 + 5^2 =  1 + 16 + 25 = 42
4^2 + 2^2 =  16 + 4 = 20
2^2 + 0^2 =  4 + 0 = 4
If we keep going there is a cycle:
4^2 = 16 ;  1^2 + 6^2 = 37 ;   3^2 + 7^2 = 58 ;   5^2 + 8^2 = 89 ;   and then 145, 42, 20, 4,  as before.
If the number enters into the cycle 4, 16, 37, 58, 89, 145, 42, 20, 4 the number is called “happy”.
The only other cycle is 1, 1, 1, 1, …
An interesting question is:  how do happy numbers “enter into” the cycle ?
For example, with 334, the number enters into the cycle at 89.
This visualisation shows the dynamics of this process for numbers 1 up to 40 using distinctive colours for the digit values 0, 1, 2, …, 9 and box borders to show where the number enters into a cycle.