Si hortum in bibliotheca habes, deerit nihil. – Cicero (106-43 BC), *Epistulae ad Familiares*, Liber IX, Epistula IV

• “If you have a garden and a library, you lack for nothing.” — Cicero, *Letters to Friends*, Book 9, Letter 4

Si hortum in bibliotheca habes, deerit nihil. – Cicero (106-43 BC), *Epistulae ad Familiares*, Liber IX, Epistula IV

• “If you have a garden and a library, you lack for nothing.” — Cicero, *Letters to Friends*, Book 9, Letter 4

What’s the next number in this sequence of integers?

5, 14, 19, 23, 28, 32, 37, 41, 46, 50, 55... (A227793 at the OEIS)

It shouldn’t be hard to work out that it’s 64 — the sum-of-digits of *n* is divisible by 5, i.e., digsum(n) mod 5 = 0. Now try summing the numbers in that sequence:

5 + 14 = 19

19 + 19 = 38

38 + 23 = 61

61 + 28 = 89

89 + 32 = 121

121 + 37 = 158

158 + 41 = 199

199 + 46 = 245

[...]

Here are the cumulative sums as another sequence:

5, 19, 38, 61, 89, 121, 158, 199, 245, 295, 350, 414, 483, 556, 634, 716, 803, 894, 990, 1094, 1203, 1316, 1434, 1556, 1683, 1814, 1950, 2090, 2235, 2389, 2548, 2711, 2879, 3051, 3228, 3409, 3595, 3785, 3980, 4183, 4391, 4603, 4820, 5041, 5267, 5497, 5732, 5976, 6225...

And there’s that cumulative-sum sequence represented as a spiral:

Spiral for cumulative sum of *n* where digsum(*n*) mod 5 = 0

You can see how the spiral is created by following 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E… from the center:

ZYXWVU

GFEDCT

H432BS

I501AR

J6789Q

KLMNOP

What about other values for the cumulative sums of digsum(*n*) mod *m* = 0? Here’s *m* = 2,3,4,5,6,7:

Spiral for cumulative sum of *n* where digsum(*n*) mod 2 = 0

s1 = 2, 4, 6, 8, 11, 13, 15, 17, 19, 20, 22…

s2 = 2, 6, 12, 20, 31, 44, 59, 76, 95, 115… (cumulative sum of s1)

sum of digsum(*n*) mod 3 = 0

s1 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33…

s2 = 3, 9, 18, 30, 45, 63, 84, 108, 135, 165…

sum of digsum(*n*) mod 4 = 0

s1 = 4, 8, 13, 17, 22, 26, 31, 35, 39, 40, 44…

s2 = 4, 12, 25, 42, 64, 90, 121, 156, 195, 235…

sum of digsum(*n*) mod 5 = 0

s1 = 5, 14, 19, 23, 28, 32, 37, 41, 46, 50, 55…

s2 = 5, 19, 38, 61, 89, 121, 158, 199, 245, 295…

sum of digsum(*n*) mod 6 = 0

s1 = 6, 15, 24, 33, 39, 42, 48, 51, 57, 60, 66…

s2 = 6, 21, 45, 78, 117, 159, 207, 258, 315, 375…

sum of digsum(*n*) mod 7 = 0

s1 = 7, 16, 25, 34, 43, 52, 59, 61, 68, 70, 77…

s2 = 7, 23, 48, 82, 125, 177, 236, 297, 365, 435…

The spiral for *m* = 2 is strange, but the spirals are similar after that. Until *m* = 8, when something strange happens again:

sum of digsum(*n*) mod 8 = 0

s1 = 8, 17, 26, 35, 44, 53, 62, 71, 79, 80, 88…

s2 = 8, 25, 51, 86, 130, 183, 245, 316, 395, 475…

Then the spirals return to normal for *m* = 9, 10:

sum of digsum(*n*) mod 9 = 0

s1 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99…

s2 = 9, 27, 54, 90, 135, 189, 252, 324, 405, 495…

sum of digsum(*n*) mod 10 = 0

s1 = 19, 28, 37, 46, 55, 64, 73, 82, 91, 109, 118…

s2 = 19, 47, 84, 130, 185, 249, 322, 404, 495, 604…

Here’s an animated gif of *m* = 8 at higher and higher resolution:

sum of digsum(*n*) mod 8 = 0 (animated gif)

You might think this strange behavior is dependant on the base in which the dig-sum is calculated. It isn’t. Here’s an animated gif for other bases in which the mod-8 spiral behaves strangely:

sum of digsum(*n*) mod 8 = 0 in base *b* = 5, 6, 7, 9, 11, 12, 13 (animated gif)

But the mod-8 spiral stops behaving strangely when the spiral is like this, as a diamond:

W

XIV

YJ8HU

ZK927GT

LA3016FS

MB45ER

NCDQ

OP

Now the mod-8 spiral looks like this:

sum of digsum(*n*) mod 8 = 0 (diamond spiral)

But the mod-4 and mod-9 spirals look like this:

sum of digsum(*n*) mod 4 = 0 (diamond spiral)

sum of digsum(*n*) mod 9 = 0 (diamond spiral)

You can also construct the spirals as a triangle, like this:

U

VCT

WD2CS

XE301AR

YF456789Q

ZGHIJKLMNOP

Here’s the beginning of the mod-5 triangular spiral:

sum of digsum(*n*) mod 5 = 0 (triangular spiral) (open in new window for full size)

And the beginning of the mod-8 triangular spiral:

sum of digsum(*n*) mod 8 = 0 (triangular spiral) (open in new window for full size)

The mod-8 spiral is behaving strangely again. So the strangeness is partly an artefact of the way the spirals are constructed.

**Post-Performative Post-Scriptum**

“Spiral Artefact”, the title of this incendiary intervention, is of course a tip-of-the-hat to core Black-Sabbath track “Spiral Architect”, off core Black-Sabbath album *Sabbath Bloody Sabbath*, issued in core Black-Sabbath success-period of 1973.

“If this work is about hell,” he says, “it’s not only about hell in terms of content. It’s also about hell in terms of its hellishness in terms of production.” — maximally maverick artist Jake Chapman describes how he and his brother Dinos made the transgressive sculpture *Hell* (2000), as quoted in Simon Garfield’s *In Miniature: How Small Things Illuminate the World* (2018)

**Elsewhere Other-Accessible**

• Ex-term-in-nate! — incendiarily interrogating issues around “in terms of”…

• *All* O.o.t.Ü.-F. posts interrogating issues around “in terms of”…

**Peri-Performative Post-Scriptum…**

Yes, this was an über-ideal quote for posting on the *23rd* in terms of the month… But I was so taken with it that I couldn’t delay any longer. And anyway: it *is* the 23rd of the months in base 11. (I.e., 21_{11} = 2 * 11 + 1 = 22 + 1 = 23.)

It caught my eye, it caught my eye,

That fluttering flake of fallen sky.

It rode the wind as cars bored by

And did not die:

And *shall* not die,

That fluttering flake of fallen sky.

**Post-Performative Post-Scriptum**

A poem written months ago about a briefly glimpsed blue butterfly flying along — and over — a busy road. I don’t know the species, but *Polyommatus icarus* seems a reasonable guess.

**Elsewhere Other-Engageable**

• Guise and Molls — review of *Front cover of Octopus: The Ocean’s Intelligent Invertebrate: A Natural History* (2010)

• Magna Mater Marina — review of *The Illustrated World Encyclopedia of Marine Fish and Sea Creatures* (2007)

Here’s a famous paradox (or a variant of it at least):

• There are two errers in this sentence.

The only visible error is the misspelt “errers”. But if the sentence claims to have two errors while having only one, that is another error and there are two errors after all.

Now for another variant. I’m not sure if I’ve thought this up for myself, but try this sentence:

• There are three errors in this sentence.

There are no visible errors in the sentence. Therefore it has one error: the claim that it has three errors when there is in fact no error. But if it has one error, it’s in error to claim that it has three errors. Therefore the sentence has two errors. And if it has two errors, again it’s in error, because it claims to have three errors while having only two. Therefore it has three errors after all.

The same reasoning can be applied to any integral number of errors:

• There are five errors in this sentence.

• There are 719 errors in this sentence.

• There are 1,000,000 errors in this sentence.

• There are 1,000,000,000 errors in this sentence.

No matter how large the number of errors, the sentence becomes true instantly, because each time the sentence makes a false claim, it makes another error. But those “times of error” don’t take place in time, any more than this equation does:

• 2 = 1 + 1/2 + 1/4 + 1/8 + 1/16…

So I think these sentences are instantly true:

• There are infinitely many errors in this sentence.

• There are ∞ errors in this sentence.

But there are infinitely many infinities. Ordinary infinity, the infinity of 1,2,3…, is called ℵ_{0} or aleph-zero. It’s a countable infinity. Above that comes ℵ_{1}, an uncountable infinity. So does this sentence instantly become true?

• There are ℵ_{1} errors in this sentence.

I’m not sure. But I think I can argue for the validity of sentences claiming fractional or irrational number of errors:

• There is 1.5 errors in this sentence.

• There are π errors in this sentence.

Let’s have a look at “There is 1.5 errors in this sentence”. There are no visible errors, so there’s one error: the claim that sentence contains 1.5 errors. So now there seems to be another error: the sentence has one error but claims to have 1.5 errors. But does it therefore have two errors? No, because if it has two errors, it’s still in error and has three errors. And that generates another error and another and another, and so on for ever. The sentence becomes unstoppably and infinitely false.

So let’s go back to the point at which the sentence contains one error. Now, the difference between 1 error and 1.5 errors is small — less than a full error. So how big is the error of claiming to have 1.5 errors when having 1 error? Well, it’s obviously 0.5 of an error. So the sentence contains 1.5 errors after all.

Now for “There are π errors in this sentence”. There are no visible errors, so there’s one error: the claim that the sentence contains π errors. Therefore it contains one error. But it claims to have π errors, so it has another error. And if it has 2 errors and claims to have π errors, it has another and third error. But if it has three errors and claims to have π error, it’s still in error. But only slightly — it’s now committing a small amount of an error. How much? It can only be 0.14159265… of an error. Therefore it’s committing 3.14159265… = π errors and is a true sentence.

Now try:

• There is -1 error in this sentence.

What is a negative error? A truth. So I think that sentence is valid too. But I can’t think of how to use *i*, or the square root of -1, in a sentence like that.

Uncle, whose inventive brains

Kept evolving aeroplanes,

Fell from an enormous height

Upon my garden lawn last night.

Flying is a fatal sport:

Uncle wrecked the tennis court. — Harry Graham (1874-1936)

**Peri-Performative Post-Scriptum**

*Pteric* means “of or like a wing”; *ptosis* meant “fall, falling” in ancient Greek and is now used in medicine to mean “drooping of the eyelid; sagging or lowering of an organ”, etc.

**Peri-Performative Post-Scriptum**

“I Like Aix” corely references “I Like Ike”, a slogan for Dwight “Ike” Eisenhower’s presidential campaign in the 1950s. *Aix galericulata* means “crested aix”, the word αἴξ, *aix*, being used by Aristotle for an unknown variety of water-bird. In Greek, it would have been pronounced something like “aye-ks”, which is what I’ve used in the title of this incendiary intervention. But “ay-ks” is probably better in modern English.

“I am sick to death of people saying that we’ve made 11 albums that sound exactly the same. In fact we’ve made 12 albums that sound exactly the same.” — Angus Young of AC/DC

**Elsewhere other-accessible**

• Bon and Off — a rogue review at Papyrocentric Performativity of *Two Sides to Every Glory: AC/DC: The Complete Biography*