http://i.imgur.com/7I5ZR.png -Binary Pyramid
http://i.imgur.com/s27CS.png -Binary Staircase
They both seem to be fractal, and I was curious if there were any other concepts I could be made aware of regarding these renderings.
The second one (http://i.imgur.com/s27CS.png) is obviously a representation of the natural numbers (1,2,3, etc.) in binary notation (one number to each column), with red representing the digit 1 and blue the digit 0. The first is a similar pattern, but it doesn’t seem to me to map so neatly to a specific mathematical object.
The pyramid can be read as the ordered sequence of signed binary numbers. Each level increases the bit depth, and the numbers increase from left to right. The numbers read from top to bottom. Blue is 0, red is 1.
E.g. at level 3, using two’s complement, you’d have RBB going down on the left, which = 100 = -4, then -3, -2, -1, 0, 1, 2, and finally 3 (BRR) in the rightmost location.
On edit, I’d say one’s complement is maybe better for the symmetry of the structure (-3,-2,-1,-0,0,1,2,3 on level 3).
Giles-
I see that thank you! So you basically count down from the top of each column, and the totals of each column form a linear line of integers from 1 to 127.
panamajack-
I looked it up, and the ‘one’s complement binary system’ just means if you invert any bit it becomes the negative equivalent. So ‘10’ (two) becomes ‘01’ (negative two).
So this time, red represents ‘0’ and blue represents ‘1’. From the top of the pyramid, each horizontal level represents a linear sequence of integers centered around the number zero.
…(0,0)
…(-1,0,0,1)
…(-2,-1,0,0,1,2)
…(-3,-2,-1,0,0,1,2,3)
…(-4,-3,-2,-1,0,0,1,2,3,4)
(-5,-4,-3,-2,-1,0,0,1,2,3,4,5)
Is this correct?
That’s correct. Although the usual convention, would have red be 1 and blue be 0. Then the sequence for, say, four bits is ( 1000, 1001, 1010, 1011, 1100, … 1111, 0000, 0001, 0010, 0011 … 0111). This has the advantage that the positive numbers (starting with +0 = 0000) correspond to the typical unsigned/positive numbers. For a visual interpretation of this idea, note that if you squared off all the steps in your staircase with blue bricks (and included just one solid blue column at the front), it would be usable as the right half of one level of the pyramid.
The math still works encoding it the other way; it’s just not the most common implementation.
Let me correct myself (my earlier response was rushed and I can’t decipher it atm sadly)…
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Take each horizontal level (there are 6 total) as a chunk.
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Red = 1, Blue = 0
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In each vertical one-button column in a level, convert the red/blue sequence into decimal (going from the top down to get the binary number).
The numbers on the left half of the pyramid are a positive linear sequence of numbers, while the right side is its ‘bit-flipped’ mirror image…
For example, look at the first column of the third level from the top. It goes RBB, which equals ‘100’ in binary and ‘4’ when converted to decimal. Then look at the last column of the same level. It goes BRR, which is ‘001’ in binary and ‘(-4)’ in decimal (because it is the bit-flipped version of 4).
The entire pattern of the pyramid is thus:
…(1,-1)
…(2, 3,-3,-2)
…(4, 5, 6, 7,-7,-6,-5,-4)
(8, 9,10,11,12,13,14,15,-15,-14,-13,-12,-11,-10,-9,-8)
(16 through 31, -31 through -16)
(32 through 63, -63 through -32)
You can count directly from 1-63 by reading the left half of the pyramid ‘like a book’, and you can count from -1 to -63 by reading the right half of the pyramid like a book in Hebrew.
Please correct me if I’m wrong (but I hope I’m right this time!)
You might start with the Koch snowflake.
I think that whatever else these patterns are, they aren’t fractal. You can’t build a accurate physical model of a fractal object. I don’t even see any way that the patterns are low-level approximations of fractal items in the sense that the typical picture of the Mandelbrot set is a low-level approximation of a fractal object.
Here’s what I was thinking for the staircase:
Look at the staircase, and see the blue legos as negative space (like the ‘holes’ in the Sierpinski triangle). Then zoom in to any of the six level chunks, and ignore the top red line. You will see it is a mini version of the whole (and one degree granier). I agree that this is not pure self-similarity, as it requires selective chopping, but at least the chopping method is uniform.
So I guess in the purest sense, the staircase is not a fractal, but it does have a certain form of ‘self-similarity with a simple rule applied to each level’, and I do not know a term for that.