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Numbered daze: shift-shuffling by 4 or 9 to stay caput or to bend towards the sun

tivoli font

fountain at Tivoli

Our days here are numbered. Not that we know what that number is or where we're going next, but j officially resigned from her job last week so there's nothing keeping us "caput," as the saying goes .... which in this case is caput mundi.

dozen-titted woman

woman with 13 breasts & a fern skirt (Tivoli)

If you stop to think about it, everyone's days everywhere are numbered.

ruins of a roof

'ruins of roof' or 'roof of ruins'?

I used to like positing this riddle:

«There's a man headed towards a field & when he gets there he knows he will die. Why?»

... not because i thought the answer was interesting (the guy jumped from a plane & his parachute wouldn't open) but i just like the idea of a man headed towards a field knowing he will die.

bare-ass & green water

headless but not buttless (Tivoli)

I am not sure where this blog post will end up. I haven't written a blogject for a few weeks ... since our Balkan trip. Since then we've been laying low mostly, enduring the heat of Rome, taking small little weekend trips to nearby places that we've been meaning to go, like Pompeii. And other places that end in i, like Tivoli, Chianti & Capri (where these photos were taken). And reading books by the likes of Bergson, Cixous, Delillo, Vladislaviċ, Gass & Gleick. Books & places i may or may not get around to talking about here.


turtle at Tivoli

Since our days are numbered & we don't know that number is or where we are going next, we might just disappear for a spell & resurface elsewhere. But for now we are here in Rome, with a UK IP address, watching the olympics.


a driveway somewhere in Tuscany

The other day Christian Peet wrote a post about Ark Codex & numbers, amongst other things, in which he posited a question to me. In his chapbook The Nines (for which i made the cover), he uses a line-shuffling constraint that, given a number of lines or topics, shifts the old last to the new first, the old first to the new second, the old second-to-last to the new third, the old second to the new fourth, etc. He runs through what this is like for poems or sestinas containing 3-9 lines & makes the observation that after a certain number of iterations, depending on the number, the pattern cycles back to its original order. For 3 lines, the pattern repeats after 3 iterations, for 4 lines the pattern also repeats after 3 iterations, then for 5 lines it repeats after 5 iterations & here's where you might just say «, etc.» ... but it's not that easy. For the first 9, the sequence goes like this (where this number is the number of iterations before it cycles back to its original state): 1, 2, 3, 3, 5, 6, 4, 4, 9. Xtian stops at 9 (the number he used to constrain the topical shuffling in The Nines). But this pattern had me intrigued. The fact that the pattern doesn't just shuffle into a chaotic stream of numbers is quite astonishing.

sunflowers in tuscany

sunflowers in Tuscany

Before i was into words, i was into numbers. Not that i'd say it was my "first love," but they came easy for me. And i could never resist trying to solve things or figuring out patterns in numbers. When i had to pick a major (at UC Santa Cruz) as an undergraduate, i picked math just because i'd already taken enough math classes to fulfill the requirement (even in high school i was taking AP & college level classes at the local community college in the foothills, just for the hell of it). All i had to do was a thesis, which i did on Fibonacci numbers & phyllotaxis (the spiraling patterns of plants). What intrigued me most was not that plants exhibited whorling patterns in sequential Fibonacci numbers (that most everyone is probably familiar with, if not, google it or count for yourself the number of spirals in either direction on a pineapple, artichoke, pinecone or sunflower), or that the ratio of these successive Fibonacci numbers approaches the so-called Golden Ratio as you get higher in the sequence, but why .... Ends up if you use the Golden Ratio to order leafs around the stem of a plant, this ratio optimizes the leaves such that they never overlap (thus maximizing sunlight & photosynthesis). So by using successive Fibonacci numbers, plants are optimizing the sunlight that reaches their leaves. (And strangely, as i am writing this, with iTunes on shuffle, Jeff Buckley & Elizabeth Fraser are singing: «all flowers in time, bend towards the sun».)

tivoli gardens

Optimizing sunlight ... that's the reason in a nutshell, whether the plants know it or not. I go into more here, in relation to MS 408, Roman artichokes & Ark Codex. But back to the sequence of numbers Xtian derailed/prompted me with (a 9-fold pattern which he credits to Ted Berrigan & Reuben Hersh). To satisfy my own curiosity, i wrote out the next 2 using this shuffle-shift. This is what happens with 10 (using 0 as the 10th number).

1 2 3 4 5 6 7 8 9 0
0 1 9 2 8 3 7 4 6 5
5 0 6 1 4 9 7 2 3 8
8 5 3 0 2 6 7 1 9 4
4 8 9 5 1 3 7 0 6 2
2 4 6 8 0 9 7 5 3 1

... at which point, after 6 iterations, the sequence cycles back to the beginning. Interesting to note though, is what happens in the last row, with all the even numbers on the left & odd numbers to the right, increasing towards the middle. Also interesting is that although all the other numbers shuffle around, there is always a 7 in the 7th column (& similarly for the 5th column when you have 7 lines/topics). If you look carefully you'll start to notice other odd patterns, like look at the 3rd & 9th columns in the above, and how otherwise besides the 3rd & 9th columns there is a linear down-shifting going on between the other columns. You could probably follow any of these sub-patterns to some sort of insight.

etruscan ruin

looking out into a cypress forest from some Etruscan ruin

And here it's worked out for the number 11 (for which i will use an E):

1 2 3 4 5 6 7 8 9 0 E
E 1 0 2 9 3 8 4 7 5 6
6 E 5 1 7 0 4 2 8 9 3
3 6 9 E 8 5 2 1 4 7 0
0 3 7 6 4 9 1 E 2 8 5
5 0 8 3 2 7 E 6 1 4 9
9 5 4 0 1 8 6 3 E 2 7
7 9 2 5 E 4 3 0 6 1 8
8 7 1 9 6 2 0 5 3 E 4
4 8 E 7 3 1 5 9 0 6 2
2 4 6 8 0 E 9 7 5 3 1

... so it repeats after 11 iterations. (You can also use sequences of letters to do this if you don't like numbers, for example for the 9-lettered ARTICHOKE, you'd get these 9 iterations before it repeats:

R I H K E O C T A ).

So, in summary, the sequence of numbers this shuffling operation generates is: 1, 2, 3, 3, 5, 6, 4, 4, 9, 6, 11, ... If you are good at math, you wouldn't need to write out the sequences this far, if at all. Just by virtue of the operation (shifting one & then shuffling alternating from the beginning & end of the last stack) you should be able to generalize the pattern with an equation. You'd expect modular arithmetic to be at play here (a sort of arithmetic for whole integers where remainders are what counts, rather than the decimal, or as wikipedia puts it, for sequences of numbers (like the 12-hour clock) that 'wrap around'). And you'd also expect the number 2 (or 1/2) to be involved since the operation of shuffling from either end is like cutting a deck in half. Ends up (this is the part where i do some hand-waving), the equation that generalizes this sequence is:

2m = ±1 mod (2n + 1)

where n is the number of lines/topics in the sequence & m is how many iterations the pattern gets shuffled before it repeats. Feel better now, knowing that, Mr. Peet? Interestingly, the math here is similar to the math used to describe card shuffling. So if you wanted to cheat at cards, you could take a rigged deck & shuffle it (albeit, shuffle perfectly, after pulling the bottom card to the top) 12 times & you'd end up with the same exact order you started with (at which point deal the cards) .... where 12 is obtained by plugging n = 52 in the above equation (lot easier than doing this by hand with 52 variables!), or if you'd rather, you can see the whole 'A003558 ' sequence written out here, or graphed even:


What's interesting is that if you had a deck of 50 cards, you'd have to shuffle 50 times before it gets back to the original state, and for 51 cards you'd have to shuffle 51 times, and for 53 cards you'd have to shuffle 53 times, so there's something special about the number 52 that perhaps early developers of cards games figured out. Also interesting is that there are 26 letters in the English alphabet, or 52 if you consider miniscule & majuscule forms. Or why in my last post it bothered me to list my 50 literary pillars, and i felt inclined to list 52. Though there i said it had to do with my recent obsession with the number 4.

roofless tuscan house

roofless house somewhere in the Chianti region

Most of my life until now i haven't show much love for the number 4 .... i've always liked 3 or 13, but never thought about 4 much. But lately i've been mulling over the number 4, specifically in regards to the narrative structure of the text i'm working on, which for now i'm calling 'The Raft Manifest,' but likely i will call something else as i'm almost finished with the first book & they (a mixed litter of feral children & dogs) haven't even gotten around to building this raft (just like how in the Ark Codex they built the ark, but never set sail in it). The way i'm building the narrative structure is in cycles of 4. There will be 4 books, and 4 chapters to each book, and each of these chapters will be divided into 4 sections, and 4 sub-sections each with 4 paragraphs, each with 4 sentences .... or at least i am using this 4-folded structure to loosely organize my thinking. It's not a rigid constraint so much as a way to construct an iterative canon, based on event cycles that happen in 4s... the seasons, cardinal directions, dimensions (including time), the elements, Maxwell's equations, the Carnot 4-stroke engine, the 4 stages of competence in psychology, the 4 stages of enlightenment in Buddhism, the 4 stages of metamorphosis, etc. 4 is also the first number that is not a Fibonacci number, so it is not built iteratively on the numbers before it except to say it is 22 (i.e. the first squared number, unless you count 11 = 1).

Tivoli headless

That's about all i have to say about the Raft Manifest for now, or anything for that matter. As usual I set out to write a post about Tivoli & Chianti & Pompeii & Bergson & Delillo & Gass, etc. & ended up blogging about something else entirely. Go figure.

Tivoli Tree



>> Un-numbered wordatlas: what to tell your grandchildren when you don't have any, another tangential appeal to the imagination & why i quit twitter

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