This Means RaWaR

The Overlord of the Über-Feral says: Welcome to my bijou bloguette. You can scroll down to sample more or simply:

• Read a Writerization at Random: RaWaR


• O.o.t.Ü.-F.: More Maverick than a Monkey-Munching* Mingrelian Myrmecologist Marinated in Mescaline…

• ¿And What Doth It Mean To Be Flesh?

მათემატიკა მსოფლიოს მეფე


*Der Muntsch ist Etwas, das überwunden werden soll.

Perfect Performative Pairing

Salt and celery, cheese and chocolate, yams and yoghurt — some things just taste better together. But that’s true of much more than foods and flavors. As a keyly — and corely — committed member of the anti-racist community, I’m proud and passionate to report that it’s also true of ideology and “in terms of”:

Unsurprisingly for a 200-year-old institution, the Guardian has not always got it right in terms of race coverage. — From slavery to BLM: the ups and downs of 200 years of Guardian race reporting, The Guardian, 6v21

For me, anti-racism just wouldn’t be the maximally moral movement that it is without a steady seasoning of “in terms of”. They’re a perfect performative pairing in an atrabiliously imperfect world.


Elsewhere other-engageable…

Ex-term-in-nate! — interrogating issues around “in terms of”
All O.o.t.Ü.-F. posts interrogating issues around “in terms of”…

More Mythical Mathicality

In a prev-previous post, I looked at this interesting fractal image on the front cover of a Ray Bradbury book:

Cover of Ray Bradbury’s I Sing the Body Electric (1969)

It seems obvious that the image is created from photographs: only the body of the centaur is drawn by hand. And here’s my attempt at extending the fractality of the image:

Further fractality for the centaur

Elsewhere other-accessible

Mythical Mathical — Man-Horse! — the pre-previous post about the fractal centaur

Knostrils

• εἰ πάντα τὰ ὄντα καπνὸς γένοιτο, ῥῖνες ἂν διαγνοῖεν. — Ἡράκλειτος ὁ Ἐφέσιος

• • Si toutes choses devenaient fumée, on connaîtrait par les narines. — Héraclite d’Ephèse

• • • If all things were turned to smoke, the nostrils would tell them apart. — Heraclitus of Ephesos, quoted in Aristotle’s De sensu, 5, 443a 23

Think Inc #2

In a pre-previous post called “Think Inc”, I looked at the fractals created by a point first jumping halfway towards the vertex of a square, then using a set of increments to decide which vertex to jump towards next. For example, if the inc-set was [0, 1, 3], the point would jump next towards the same vertex, v[i]+0, or the vertex immediately clockwise, v[i]+1, or the vertex immediately anti-clockwise, v[i]+3. And it would trace all possible routes using that inc-set. Then I added refinements to the process like giving the point extra jumping-targets half-way along each side.

Here are some more variations on the inc-set theme using two and three extra jumping-targets along each side of the square. First of all, try two extra jumping-targets along each side and a set of three increments:

inc = 0, 1, 6


inc = 0, 2, 6


inc = 0, 2, 8


inc = 0, 3, 6


inc = 0, 3, 9


inc = 0, 4, 8


inc = 0, 5, 6


inc = 0, 5, 7


inc = 1, 6, 11


inc = 2, 6, 10


inc = 3, 6, 9


Now try two extra jumping-targets along each side and a set of four increments:

inc = 0, 1, 6, 11


inc = 0, 2, 8, 10


inc = 0, 3, 7, 9


inc = 0, 4, 8, 10


inc = 0, 5, 6, 7


inc = 0, 5, 7, 8


inc = 1, 6, 7, 9


inc = 1, 4, 6, 11


inc = 1, 5, 7, 11


inc = 2, 4, 8, 10


inc = 3, 5, 7, 9


And finally, three extra jumping-targets along each side and a set of three increments:

inc = 0, 3, 13


inc = 0, 4, 8


inc = 0, 4, 12


inc = 0, 5, 11

inc = 0, 6, 9


inc = 0, 7, 9


Previously Pre-Posted

Think Inc — an earlier look at inc-set fractals

Vacancy Vanquished

We never sighted the slightest suggestion of life all the way to Vancouver, twelve days of chilly boredom, though there was a certain impressiveness in the very dreariness and desolation. There was a hint of the curious horror that emptiness always evokes, whether it is a space of starless night or a bleak and barren waste of land. The one exception is the Sahara Desert where, for some reason that I cannot name, the suggestion is not in the least of vacancy and barrenness, but rather of some subtle and secret spring of life. — The Confessions of Aleister Crowley: An Autohagiography (1929), ch. 57


Previously Pre-Posted…

Leech Unleashed
Crowley on Crystals

Mythical Mathical — Man-Horse!

Cover of Ray Bradbury’s I Sing the Body Electric (1969), published by Corgi in 1972

That’s a striking cover — and more than that. The blog where I found the cover says this: “This very odd cover clearly features a heavily rouged glam rock centaur with a rather natty feather-cut hairstyle flexing his biceps, his forearms transmogrifying into miniature bicep flexing glam rock figures. I think I’m slowly losing the plot here.” Losing the plot? No, losing the mathematicality in the mythical. The artist has started to make the centaur into a fractal. Or rather, the artist has started to make more explicit what is already there in the human body. As I wrote in “Fingering the Frigit”:

Fingers are fractal. Where a tree has a trunk, branches and twigs, a human being has a torso, arms and fingers. And human beings move in fractal ways. We use our legs to move large distances, then reach out with our arms over smaller distances, then move our fingers over smaller distances still. We’re fractal beings, inside and out, brains and blood-vessels, fingers and toes.

Sliv and Let Tri

Fluvius, planus et altus, in quo et agnus ambulet et elephas natet,” wrote Pope Gregory the Great (540-604). “There’s a river, wide and deep, where a lamb may wade and an elephant swim.” He was talking about the Word of God, but you can easily apply his words to mathematics. However, in the river of mathematics, the very shallow and the very deep are often a single step apart.

Here’s a good example. Take the integer 2. How many different ways can it be represented as an sum of separate integers? Easy. First of all it can be represented as itself: 2 = 2. Next, it can be represented as 2 = 1 + 1. And that’s it. There are two partitions of 2, as mathematicians say:

2 = 2 = 1+1 (p=2)


Now try 3, 4, 5, 6:

3 = 3 = 1+2 = 1+1+1 (p=3)
4 = 4 = 1+3 = 2+2 = 1+1+2 = 1+1+1+1 (p=5)
5 = 5 = 1+4 = 2+3 = 1+1+3 = 1+2+2 = 1+1+1+2 = 1+1+1+1+1 (p=7)
6 = 6 = 1+5 = 2+4 = 3+3 = 1+1+4 = 1+2+3 = 2+2+2 = 1+1+1+3 = 1+1+2+2 = 1+1+1+1+2 = 1+1+1+1+1+1 (p=11)


So the partitions of 2, 3, 4, 5, 6 are 2, 3, 5, 7, 11. That’s interesting — the partition-counts are the prime numbers in sequence. So you might conjecture that p(7) = 13 and p(8) = 17. Alas, you’d be wrong. Here are the partitions of n = 1..10:

1 = 1 (p=1)
2 = 2 = 1+1 (p=2)
3 = 3 = 1+2 = 1+1+1 (p=3)
4 = 4 = 1+3 = 2+2 = 1+1+2 = 1+1+1+1 (p=5)
5 = 5 = 1+4 = 2+3 = 1+1+3 = 1+2+2 = 1+1+1+2 = 1+1+1+1+1 (p=7)
6 = 6 = 1+5 = 2+4 = 3+3 = 1+1+4 = 1+2+3 = 2+2+2 = 1+1+1+3 = 1+1+2+2 = 1+1+1+1+2 = 1+1+1+1+1+1 (p=11)
7 = 7 = 1+6 = 2+5 = 3+4 = 1+1+5 = 1+2+4 = 1+3+3 = 2+2+3 = 1+1+1+4 = 1+1+2+3 = 1+2+2+2 = 1+1+1+1+3 = 1+1+1+2+2 = 1+1+1+1+1+2 = 1+1+1+1+1+1+1 (p=15)
8 = 8 = 1+7 = 2+6 = 3+5 = 4+4 = 1+1+6 = 1+2+5 = 1+3+4 = 2+2+4 = 2+3+3 = 1+1+1+5 = 1+1+2+4 = 1+1+3+3 = 1+2+2+3 = 2+2+2+2 = 1+1+1+1+4 = 1+1+1+2+3 = 1+1+2+2+2 = 1+1+1+1+1+3 = 1+1+1+1+2+2 = 1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1 (p=22)
9 = 9 = 1+8 = 2+7 = 3+6 = 4+5 = 1+1+7 = 1+2+6 = 1+3+5 = 1+4+4 = 2+2+5 = 2+3+4 = 3+3+3 = 1+1+1+6 = 1+1+2+5 = 1+1+3+4 = 1+2+2+4 = 1+2+3+3 = 2+2+2+3 = 1+1+1+1+5 = 1+1+1+2+4 = 1+1+1+3+3 = 1+1+2+2+3 = 1+2+2+2+2 = 1+1+1+1+1+4 = 1+1+1+1+2+3 = 1+1+1+2+2+2 = 1+1+1+1+1+1+3 = 1+1+1+1+1+2+2 = 1+1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1+1 (p=30)
10 = 10 = 1+9 = 2+8 = 3+7 = 4+6 = 5+5 = 1+1+8 = 1+2+7 = 1+3+6 = 1+4+5 = 2+2+6 = 2+3+5 = 2+4+4 = 3+3+4 = 1+1+1+7 = 1+1+2+6 = 1+1+3+5 = 1+1+4+4 = 1+2+2+5 = 1+2+3+4 = 1+3+3+3 = 2+2+2+4 = 2+2+3+3 = 1+1+1+1+6 = 1+1+1+2+5 = 1+1+1+3+4 = 1+1+2+2+4 = 1+1+2+3+3 = 1+2+2+2+3 = 2+2+2+2+2 = 1+1+1+1+1+5 = 1+1+1+1+2+4 = 1+1+1+1+3+3 = 1+1+1+2+2+3 = 1+1+2+2+2+2 = 1+1+1+1+1+1+4 = 1+1+1+1+1+2+3 = 1+1+1+1+2+2+2 = 1+1+1+1+1+1+1+3 = 1+1+1+1+1+1+2+2 = 1+1+1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1+1+1 (p=42)


It’s very simple to understand what a partition is, but very difficult to say how many partitions, p(n), a particular number will have. Here’s a partition: 11 = 4 + 3 + 2 + 2. But what is p(11)? Is there a formula for the sequence of p(n)?

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 3118 5, 37338, 44583, 53174, 63261... (A000041 at the OEIS)

Yes, there is a formula, but it is very difficult to understand the Partition function that supplies it. So that part of the river of mathematics is very deep. But a step away the river of mathematics is very shallow. Here’s another question: If you multiply the numbers in a partition of n, what’s the largest possible product? Try using the partitions of 5:

4 = 1 * 4
6 = 2 * 3
3 = 1 * 1 * 3
4 = 1 * 2 * 2
2 = 1 * 1 * 1 * 2
1 = 1 * 1 * 1 * 1 * 1

The largest product is 6 = 2 * 3. So the answer is easy for n = 5, but I assumed that as n got bigger, the largest product got more interesting, using a subtler and subtler mix of prime factors. I was wrong. You don’t have to struggle to find a formula for what you might call the maximum multiplicity of the partitions of n:

1 = 1 (n=1)
2 = 2 (n=2)
3 = 3 (n=3)
4 = 2 * 2 (n=4)
6 = 2 * 3 (n=5)
9 = 3 * 3 (n=6)
12 = 2 * 2 * 3 (n=7)
18 = 2 * 3 * 3 (n=8)
27 = 3 * 3 * 3 (n=9)
36 = 2 * 2 * 3 * 3 (n=10)
54 = 2 * 3 * 3 * 3 (n=11)
81 = 3 * 3 * 3 * 3 (n=12)
108 = 2 * 2 * 3 * 3 * 3 (n=13)
162 = 2 * 3 * 3 * 3 * 3 162(n=14)
243 = 3 * 3 * 3 * 3 * 3 (n=15)
324 = 2 * 2 * 3 * 3 * 3 * 3 (n=16)
486 = 2 * 3 * 3 * 3 * 3 * 3 (n=17)
729 = 3 * 3 * 3 * 3 * 3 * 3 (n=18)


It’s easy to see why the greatest prime factor is always 3. If you use 5 or 7 as a factor, the product can always be beaten by splitting the 5 into 2*3 or the 7 into 2*2*3:

15 = 3 * 5 < 18 = 3 * 2*3 (n=8)
14 = 2 * 7 < 24 = 2 * 2*2*3 (n=9)
35 = 5 * 7 < 72 = 2*3 * 2*2*3 (n=12)

And if you’re using 7 → 2*2*3 as factors, you can convert them to 1*3*3, then add the 1 to another factor to make a bigger product still:

14 = 2 * 7 < 24 = 2 * 2*2*3 < 27 = 3 * 3 * 3 (n=9)
35 = 5 * 7 < 72 = 2*3 * 2*2*3 < 81 = 3 * 3 * 3 * 3 (n=12)


Post-Performative Post-Scriptum

The title of this post is, of course, a paronomasia on core Beatles album Live and Let Die (1954). But what does it mean? Well, if you think of the partitions of n as slivers of n, then you sliv n to find its partitions:

9 = 9 = 1+8 = 2+7 = 3+6 = 4+5 = 1+1+7 = 1+2+6 = 1+3+5 = 1+4+4 = 2+2+5 = 2+3+4 = 3+3+3 = 1+1+1+6 = 1+1+2+5 = 1+1+3+4 = 1+2+2+4 = 1+2+3+3 = 2+2+2+3 = 1+1+1+1+5 = 1+1+1+2+4 = 1+1+1+3+3 = 1+1+2+2+3 = 1+2+2+2+2 = 1+1+1+1+1+4 = 1+1+1+1+2+3 = 1+1+1+2+2+2 = 1+1+1+1+1+1+3 = 1+1+1+1+1+2+2 = 1+1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1+1 (p=30)

And when you find the greatest product among those partitions, you let 3 or “tri” work its multiplicative magic. So you “Sliv and Let Tri”:

8 = 1 * 8
14 = 2 * 7
18 = 3 * 6
20 = 4 * 5
7 = 1 * 1 * 7
12 = 1 * 2 * 6
15 = 1 * 3 * 5
16 = 1 * 4 * 4
20 = 2 * 2 * 5
24 = 2 * 3 * 4
27 = 3 * 3 * 3 ←
6 = 1 * 1 * 1 * 6
10 = 1 * 1 * 2 * 5
12 = 1 * 1 * 3 * 4
16 = 1 * 2 * 2 * 4
12 = 1 * 2 * 3 * 3
24 = 2 * 2 * 2 * 3
5 = 1 * 1 * 1 * 1 * 5
8 = 1 * 1 * 1 * 2 * 4
9 = 1 * 1 * 1 * 3 * 3
12 = 1 * 1 * 2 * 2 * 3
16 = 1 * 2 * 2 * 2 * 2
4 = 1 * 1 * 1 * 1 * 1 * 4
6 = 1 * 1 * 1 * 1 * 2 * 3
8 = 1 * 1 * 1 * 2 * 2 * 2
3 = 1 * 1 * 1 * 1 * 1 * 1 * 3
4 = 1 * 1 * 1 * 1 * 1 * 2 * 2
2 = 1 * 1 * 1 * 1 * 1 * 1 * 1 * 2
1 = 1 * 1 * 1 * 1 * 1 * 1 * 1 * 1 * 1