Generated in Excel 2010 by Sheet1!C3=MOD(SUM(B1:B2,A2),MAX($A$1,2)) Filled to 2624400 cells A1=3
Animated in Excel 2010 by C3=INDIRECT("sheet1!"&ADDRESS((ROW()-3)*$A$1+3,(COLUMN()-3)*$A$1+3)) Filled to 291600 cells A1=A2 A2=IF(B2,3,A3) A3=A2/B1 B1=3^(1/80)
No interpolation...
I'm seeing now I forgot to FLOOR() the scale coordinates. Turns out ADDRESS() hard casts float to int. Who knew? That explains the visual glitches.
Generated in Excel 2010 by Sheet1!C3=MOD(SUM(B1:B2,A2),MAX($A$1,2)) Filled to 2624400 cells A1=3
Animated in Excel 2010 by C3=INDIRECT("sheet1!"&ADDRESS((ROW()-3)*$A$1+3,(COLUMN()-3)*$A$1+3)) Filled to 291600 cells A1=A2 A2=IF(B2,3,A3) A3=A2/B1 B1=3^(1/80)
No interpolation...
Sometimes you want to play Lite Brite.
This is the dihedral group with six elements again. The green, yellow and blue structure resembles a [mod 2 addition] structure because green, yellow, and blue represent the 3 reflections of an equilateral triangle - and reflections are elements of order 2. Red and orange represent the two rotations of the same triangle.
steven wolfram generated sierpinski's gasket with a cellular automaton. the rule of action for this automaton is logical xor. I notice loxigal xor is equivalent to addition modulo 2. I make a bunch of art, mostly in addition modulo 2 or 3.
I think to myself, modular addition is the operation of cyclic groups. Why restrict myself to cyclic groups? why restrict myself to abelian groups even? Just because Excel's architecture assumes cell contents are real numbers? Pah. I write a formula to iterate wolfram's sierpinski rule using arbitrarily defined group operation. B11=INDIRECT(ADDRESS(A11+3,B10+3)). This will compute B11 as group operation A11⊕B10, where ⊕ is designated by the cayley table whose upper left corner is C3. Since B10 is above B11 and A10 is to B11's left, the pattern formed by ⊕ cascades down and to the right.
Here's a sierpinski gasket "modulo" [the dihedral group of order 6]. The group D6 is generated by the actions of [horizontal reflection] and [120° rotation] on a plane.
This one is the quaternion group. The trick is finding the right initial conditions to showcase the weirdness of the group. Here my top row repeats i j k i j k i j k ...
In the closeup you can see the Cayley table I painstakingly transcribed from wikipedia. I have to code each element of the group as an integer! 0 represents 1, and 1 represents -1... we stay silly XD
steven wolfram generated sierpinski's gasket with a cellular automaton. the rule of action for this automaton is logical xor. I notice loxigal xor is equivalent to addition modulo 2. I make a bunch of art, mostly in addition modulo 2 or 3.
I think to myself, modular addition is the operation of cyclic groups. Why restrict myself to cyclic groups? why restrict myself to abelian groups even? Just because Excel's architecture assumes cell contents are real numbers? Pah. I write a formula to iterate wolfram's sierpinski rule using arbitrarily defined group operation. B11=INDIRECT(ADDRESS(A11+3,B10+3)). This will compute B11 as group operation A11⊕B10, where ⊕ is designated by the cayley table whose upper left corner is C3. Since B10 is above B11 and A10 is to B11's left, the pattern formed by ⊕ cascades down and to the right.
Here's a sierpinski gasket "modulo" [the dihedral group of order 6]. The group D6 is generated by the actions of [horizontal reflection] and [120° rotation] on a plane.
Well this post is pretty seriffic :)
hihi! just found your blog and really love fractals, so i'm having a great time exploring it. i noticed your minecraft build and i thought it was very neat because extremely early on in the game's lifespan (in the first couple weeks of it existing at all) people were building a LOT of fractals, particularly menger sponges as the game's grid system is particularly well-suited for that, but they built a lot of others as well, and i think it's very cool that this practice is still alive and well 15+ years later!
hehe glad you enjoy! I didn't get into minecraft until 2014 so I missed that era. It is well suited to fractals! Here's my attempt to build an actual working cellular automaton XD
Rule 150 fractal, Minecraft, September 2017
Sierpiński gasket with Fibonacci initial conditions, modulo 4.
Left edge: 1,1,2,3,1,0,1,1... Top edge: 1,3,0,3,3,2,1,3...
A simple mspaint fractal. To construct the next tier, fan out three of the current tier, rotated 180°, 90° and 0° respectively. The first tier is the T block▗▙ from Tetris.
When this fractal is twice as long, it contains three times as much stuff! Therefore it is log₂(3)-dimensional.
"Dandelions" 512x768
I wrote a formula to deploy programmable periodic initial conditions, specifically dashed lines:
B2=IF(MOD(COLUMN(B2)+ROW(B2)-$B$1-4,$C$1+$D$1)<$C$1,1,0)
Here I used a period of 8 with C1=3 on, D1=5 off. Like this:
The cellular automaton growing out of these conditions is my old standby with 7 parents: C3=MOD(A1+A2+A3+B1+B3+C1+C2,2)
Since this is the same automaton as used in headache, this render has the same self-symmetry type as headache.
in some sense, this fractal's generator is f(f(x)), where f(x) generates a menger carpet
