Yellow Submarine

Free, one-sided, and fixed polyknights
There are three common ways of distinguishing polyominoes and polyknights
free polyknights are distinct when none is a rigid transformation ( translation, rotation, reflection or glide reflection ) of another ( pieces that can be picked up and flipped over ).  one-sided polyknights are distinct when none is a translation or rotation of another  ( pieces that cannot be flipped over ).
fixed polyknights are distinct when none is a translation of another  ( pieces that can be neither flipped nor rotated ).  Yellow submarine shows polyknights of various types with 2 cells.

Chessboard Polyominoes

Chessboard Polyominoes with n squares. ( A001933 )
2, 1, 4, 7, ...

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

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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

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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