Persist List

Multiplicative persistence is a complex term but a simple concept. Take a number, multiply its digits, repeat. Sooner or later the result is a single digit:

25 → 2 x 5 = 10 → 1 x 0 = 0 (mp=2)
39 → 3 x 9 = 27 → 2 x 7 = 14 → 1 x 4 = 4 (mp=3)

So 25 has a multiplicative persistence of 2 and 39 a multiplicative persistence of 3. Each is the smallest number with that m.p. in base-10. Further records are set by these numbers:

77 → 49 → 36 → 18 → 8 (mp=4)
679 → 378 → 168 → 48 → 32 → 6 (mp=5)
6788 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=6)
68889 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=7)
2677889 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=8)
26888999 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=9)
3778888999 → 438939648 → 4478976 → 338688 → 27648 → 2688 → 768 → 336 → 54 → 20 → 0 (mp=10)

Now here’s base-9:

25[b=9] → 11 → 1 (mp=2)
38[b=9] → 26 → 13 → 3 (mp=3)
57[b=9] → 38 → 26 → 13 → 3 (mp=4)
477[b=9] → 237 → 46 → 26 → 13 → 3 (mp=5)
45788[b=9] → 13255 → 176 → 46 → 26 → 13 → 3 (mp=6)
2577777[b=9] → 275484 → 13255 → 176 → 46 → 26 → 13 → 3 (mp=7)

And base-11:

26[b=11] → 11 → 1 (mp=2)
3A[b=11] → 28 → 15 → 5 (mp=3)
69[b=11] → 4A → 37 → 1A → A (=10b=10) (mp=4)
269[b=11] → 99 → 74 → 26 → 11 → 1 (mp=5)
3579[b=11] → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=6)
26778[b=11] → 3597 → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=7)
47788A[b=11] → 86277 → 3597 → 78A → 46A → 1A9 → 82 → 15 → 5 (mp=8)
67899AAA[b=11] → 143A9869 → 299596 → 2A954 → 2783 → 286 → 88 → 59 → 41 → 4 (mp=9)
77777889999[b=11] → 2AA174996A → 143A9869 → 299596 → 2A954 → 2783 → 286 → 88 → 59 → 41 → 4 (mp=10)

I was also interested in the narcissism of multiplicative persistence. That is, are any numbers equal to the sum of the numbers created while calculating their multiplicative persistence? Yes:

86 = (8 x 6 = 48) + (4 x 8 = 32) + (3 x 2 = 6)

I haven’t found any more in base-10 (apart from the trivial 0 to 9) and can’t prove that this is the only one. Base-9 offers this:

78[b=9] = 62 + 13 + 3

I can’t find any at all in base-11, but here are base-12 and base-27:

57[b=12] = 2B + 1A + A
A8[b=12] = 68 + 40 + 0

4[23][b=27] = 3B + 16 + 6
7[24][b=27] = 66 + 19 + 9
A[18][b=27] = 6[18] + 40 + 0
[26][24][b=27] = [23]3 + 2F + 13 + 3
[26][23][26][b=27] = [21]8[23] + 583 + 4C + 1[21] + [21]

But the richest base I’ve found so far is base-108, with fourteen narcissistic multiplicative-persistence sums:

4[92][b=108] = 3[44] + 1[24] + [24]
5[63][b=108] = 2[99] + 1[90] + [90]
7[96][b=108] = 6[24] + 1[36] + [36]
A[72][b=108] = 6[72] + 40 + 0
[19][81][b=108] = E[27] + 3[54] + 1[54] + [54]
[26][96][b=108] = [23]C + 2[60] + 1C + C
[35][81][b=108] = [26][27] + 6[54] + 30 + 0
[37][55][b=108] = [18][91] + F[18] + 2[54] + 10 + 0
[73][60][b=108] = [40][60] + [22][24] + 4[96] + 3[60] + 1[72] + [72]
[107][66][b=108] = [65][42] + [25][30] + 6[102] + 5[72] + 3[36] + 10 + 0
[71][84][b=108] = [55][24] + C[24] + 2[72] + 1[36] + [36]
[107][99][b=108] = [98]9 + 8[18] + 1[36] + [36]
5[92][96][b=108] = 3[84][96] + 280 + 0
8[107][100][b=108] = 7[36][64] + 1[41][36] + D[72] + 8[72] + 5[36] + 1[72] + [72]


Update (10/ii/14): The best now is base-180 with eighteen multiplicative-persistence sums.

5[105][b=180] = 2[165] + 1[150] + [150]
7[118][b=180] = 4[106] + 2[64] + [128]
7[160][b=180] = 6[40] + 1[60] + [60]
8[108][b=180] = 4[144] + 3[36] + [108]
A[120][b=180] = 6[120] + 40 + 0 (s=5)
[19][135][b=180] = E[45] + 3[90] + 1[90] + [90]
[21][108][b=180] = C[108] + 7[36] + 1[72] + [72]
[26][160][b=180] = [23][20] + 2[100] + 1[20] + [20]
[31][98][b=180] = [16][158] + E8 + [112]
[35][135][b=180] = [26][45] + 6[90] + 30 + 0 (s=10)
[44][96][b=180] = [23][84] + A[132] + 7[60] + 2[60] + [120]
[71][140][b=180] = [55][40] + C[40] + 2[120] + 1[60] + [60]
[73][100][b=180] = [40][100] + [22][40] + 4[160] + 3[100] + 1[120] + [120]
[107][110][b=180] = [65][70] + [25][50] + 6[170] + 5[120] + 3[60] + 10 + 0
[107][165][b=180] = [98]F + 8[30] + 1[60] + [60] (s=15)
[172][132][b=180] = [126][24] + [16][144] + C[144] + 9[108] + 5[72] + 20 + 0
5[173][145][b=180] = 3[156][145] + 2[17]0 + 0
E[170][120][b=180] = 8[146][120] + 4[58][120] + [154][120] + [102][120] + [68]0 + 0

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Ghosts in the Cathedral

Front cover of The Neutrino Hunters by Ray JayawardhanaThe Neutrino Hunters: The Chase for the Ghost Particle and the Secrets of the Universe, Ray Jayawardhana (Oneworld 2013)

An easy read on a difficult topic: Ray Jayawardhana takes some complicated ideas and makes them a pleasure to absorb. Humans have only recently discovered neutrinos, but neutrinos have always known us from the inside:

…about a hundred trillion neutrinos produced in the nuclear furnace at the Sun’s core pass through your body every second of the day and night, yet they do no harm and leave no trace. During your entire lifetime, perhaps one neutrino will interact with an atom in your body. Neutrinos travel right through the Earth unhindered, like bullets cutting through a fog. (ch. 1, “The Hunt Heats Up”, pg. 9)

In a way, “ghost particle” is a misnomer: to neutrinos, we are the ghosts, because they pass through all solid matter almost as though it’s not there:

Neutrinos are elementary particles, just like electrons that buzz around atomic nuclei or quarks that combine to make protons and neutrons. They are fundamental building blocks of matter, but they don’t remain trapped inside atoms. Also unlike their subatomic cousins, neutrinos carry no electric charge, have a tiny mass and hardly ever interact with other particles. A typical neutrino can travel through a light-year’s worth of lead without interacting with any atoms. (ch. 1, pg. 7)

That’s a lot of lead, but a little of neutrino. With a different ratio – a lot less matter and a lot more neutrino – it’s possible to detect them on earth. Because so many are passing through the earth at any moment, a large piece of matter watched for long enough will eventually catch a ghost. So neutrino-hunters sink optical sensors into the transparent ice of the Antarctic and fill huge tanks with carbon tetrachloride or water. Then they wait:

Every once in a while, a solar neutrino would collide with an electron in the water and propel it forward, like a billiard ball that’s hit head-on. The fast-moving electron would create an electromagnetic “wake”, or cone of light, along its path. The resulting pale blue radiation is called “Cherenkov radiation”, after the Russian physicist Pavel Cherenkov, who investigated the phenomenon. Phototubes lining the inside walls of the tank would register each light flash and reveal an electron’s interaction with a neutrino. The Kamiokande provided two extra bits of information to researchers: from the direction of the light cone scientists would infer the direction of the incoming neutrino and from its intensity they could determine the neutrino’s energy. (ch. 4, “Sun Underground”, pg. 95)

That’s a description of a neutrino-hunt in “3,000 tons of pure water” in a mine “150 miles west of Tokyo”: big brains around the world are obsessed with the “little neutral one”. That’s what “neutrino” means in Italian, because the particle was named by the physicist Enrico Fermi (1901-54) after the original proposal, “neutron”, was taken over by another, and much bigger, particle with no electric charge. Fermi was one of the greatest physicists of all time and oversaw the first “controlled nuclear chain reaction” at the University of Chicago in 1942. That is, he helped build the first nuclear reactor. Like the sun, reactors are rich sources of neutrinos and because neutrinos pass easily through any form of shielding, a reactor can’t be hidden from a neutrino-detector. Nor can a supernova: one of the most interesting sections of the book discusses the way exploding stars flood the universe with a lot of light and a lot more neutrinos:

Alex Friedland of the Los Alamos National Laboratory explained that a supernova is in essence a “neutrino bomb”, since the explosion releases a truly staggering number – some 10^58, or ten billion trillion trillion trillion trillion – of these particles. … In fact, the energy emitted in the form of neutrinos within a few seconds is several hundred times what the Sun emits in the form of photons over its entire lifetime of nearly 10 billion years. What’s more, during the supernova explosion, 99 percent of the precursor star’s gravitational binding energy goes into the neutrinos of all flavors, while barely half a percent appears as visible light. (ch. 6, “Exploding Star”, pg. 125)

That light is remarkably bright, but it can be blocked by interstellar dust. The neutrinos can’t, so they’re a way to detect supernovae that are otherwise invisible. However, Supernova 1987A was highly visible: a lot of photons were captured by a lot of telescopes when it flared in the Large Magellanic Cloud. Nearly four hours before that, a few neutrino-detectors had captured far fewer neutrinos:

Detecting a grand total of two dozen particles may not sound like much to crow about. But the significance of these two dozen neutrino events is underlined by the fact that they have been the subject of hundreds of scientific papers over the years. Supernova 1987A was the first time that we had observed neutrinos coming from an astronomical source other than the Sun. (ch. 6, pg. 124)

The timing of the two dozen was very important: it came before the visible explosion and “meant that astrophysicists like Bahcall and his colleagues were right about what happened during a supernova explosion” (pg. 123). That’s John Bahcall (1931-2005), an American who wanted to be a rabbi but ended up a physicist after taking a science course during his philosophy degree at Berkeley. He had predicted how many solar neutrinos his colleague Raymond Davis (1914-2006) should detect interacting with atoms in a giant tank of “dry-cleaning fluid”, as carbon tetrachloride is also known. But Davis found “only a third as many as Bahcall’s model calculation predicted” (ch. 4, pg. 90). Was Davis missing some? Was Bahcall’s model wrong? The answer would take decades to arrive, as Davis refined his apparatus and Bahcall re-checked his calculations. This book is about several kinds of interaction: between neutrinos and atoms, between theory and experiment, between mathematics and matter. Neutrinos were predicted with maths before they were detected in matter. The Austrian physicist Wolfgang Pauli (1900-58) produced the prediction; Davis and others did the detecting.

The Super-Kamiokande neutrino-cathedral

The Super-Kamiokande neutrino-cathedral (click for larger image)

Pauli was famously witty; another big brain in the book, the Englishman Paul Dirac (1902-84), was famously taciturn. Big brains are often strange ones too. That’s part of why they’re attracted to the very strange world of atomic physics. Jayawardhana also discusses the Italian physicist Ettore Majorana (1906-?1938), who disappeared at the age of thirty-two, and his colleague Bruno Pontecorvo (1913-93), who defected to the Soviet Union. Neutrinos are fascinating and so are the humans who have hunted for them. So is the history that surrounded them. Quantum physics was convulsing science at the same time as communism and Nazism were convulsing Europe. As the Danish physicist Niels Bohr (1885-1962) said: “Anyone who is not shocked by quantum theory has not understood it.” Modern physicists have been called a new priesthood, devoted to lofty and remote ideas incomprehensible and irrelevant to ordinary people. But ordinary people fund the devices the priests build to pursue their ideas with. And some of the neutrino-detectors pictured here are as huge and awe-inspiring as cathedrals. Some might say they’re as futile as cathedrals too. But if understanding the universe isn’t enough in itself, there may be practical uses for neutrinos on the way. At present, we have to communicate over the earth’s surface; a beam of neutrinos can travel right through the earth.

The universe is also a dangerous place: some scientists theorized that the neutrino deficit in Ray Davis’s experiments meant the sun was about to go nova. It wasn’t, but neutrinos may help the human race spot other dangers and exploit new opportunities. We still know only a fraction of what’s out there and the ghost particle is a messenger from the heart not only of supernovae and the sun, but also of the earth itself. There’s radioactivity deep in the earth, so there are neutrinos streaming upward. As methods of detecting them get better, we’ll understand the interior of the earth better. But Jayawardhana doesn’t discuss another possibility: that we might even discover advanced life down there, living under huge pressures at very high temperatures, as Arthur C. Clarke suggested in his short-story “The Fires Within” (1949).

Clarke also suggested that life could exist inside the sun. There’s presently no way of testing his ideas, but neutrinos may carry even more secrets than standard science has guessed. Either way, I think Clarke would have enjoyed this book and perhaps Jayawardhana, who’s of Sri Lankan origin, was influenced by him. Jayawardhana’s writing certainly reminds me of Clarke’s writing. It’s clear, enthusiastic and a pleasure to read, wearing its learning lightly and carrying you easily over vast stretches of space and time. The Neutrino Hunters is an excellent introduction to the hunters, the hunted and the history, with a good glossary and index too.


Previously pre-posted (please peruse):

Think Ink – Review of 50 Quantum Physics Ideas You Really Need to Know

World Wide Watchmen

I prefer to self-issue books in a library. It’s quicker and more convenient. And you feel okay about borrowing books suggestive of sordid and socially unacceptable tastes. For example, who would want to hand a copy of Watch You Bleed to a live librarian?

Well, I wouldn’t mind. I find it amusing to be mistaken for a G’n’R fan, just as I find it amusing to be mistaken for a Guardian-reader. But there are limits, so I’m grateful for self-issue when I borrow, say, a biography of Martin Amis or that book about The Simpsons. The trouble is, nowadays we have to be more dubious about self-issue than we used to be. It’s all on computer and it isn’t just librarians who might be scanning the record of books you borrow. No, you also have to ask yourself: What will the NSA, GCHQ and MOSSAD think?

With this in mind, I’d like to put it clearly on record: I got that book out last year for research purposes only. Nothing more. I am not – repeat not – a fan of Iron Maiden. The same applies to that other book this year. I got it out for research purposes only, I swear. Inter alia, I had a hypothesis to confirm. I am not – repeat not – a fan of his.

Front cover of Iron Maiden: On Board Flight 666

And was the hypothesis confirmed? Yes, thanks for asking, it was.


As for Big Numbers, Moore asserted: “It is the most advanced comic work I’ve ever done in terms of the storytelling.” — Magic Words: The Extraordinary Life of Alan Moore, Lance Parkin, pg. 266 (Aurum 2013)


Elsewhere other-posted:

Ex-term-in-ate!

She Say, She Sigh, She Sow

“Those who view mathematical science, not merely as a vast body of abstract and immutable truths, whose intrinsic beauty, symmetry and logical completeness, when regarded in their connection together as a whole, entitle them to a prominent place in the interest of all profound and logical minds, but as possessing a yet deeper interest for the human race, when it is remembered that this science constitutes the language through which alone we can adequately express the great facts of the natural world, and those unceasing changes of mutual relationship which, visibly or invisibly, consciously or unconsciously to our immediate physical perceptions, are interminably going on in the agencies of the creation we live amidst: those who thus think on mathematical truth as the instrument through which the weak mind of man can most effectively read his Creator’s works, will regard with especial interest all that can tend to facilitate the translation of its principles into explicit practical forms.” — Ada Lovelace (née Byron) (1815-52).

Six Six Nix

4 x 3 = 13. A mistake? Not in base-9, where 13 = 1×9^1 + 3 = 12 in base-10. This means that 13 is a sum-product number in base-9: first add its digits, then multiply them, then multiply the digit-sum by the digit-product: (1+3) x (1×3) = 13[9]. There are four more sum-product numbers in this base:

2086[9] = 17 x 116 = (2 + 8 + 6) x (2 x 8 x 6) = 1536[10] = 16 x 96
281876[9] = 35 x 7333 = (2 + 8 + 1 + 8 + 7 + 6) x (2 x 8 x 1 x 8 x 7 x 6) = 172032[10] = 32 x 5376
724856[9] = 35 x 20383 = (7 + 2 + 4 + 8 + 5 + 6) x (7 x 2 x 4 x 8 x 5 x 6) = 430080[10] = 32 x 13440
7487248[9] = 44 x 162582 = (7 + 4 + 8 + 7 + 2 + 4 + 8) x (7 x 4 x 8 x 7 x 2 x 4 x 8) = 4014080[10] = 40 x 100352

And that’s the lot, apart from the trivial 0 = (0) x (0) and 1 = (1) x (1), which are true in all bases.

What about base-10?

135 = 9 x 15 = (1 + 3 + 5) x (1 x 3 x 5)
144 = 9 x 16 = (1 + 4 + 4) x (1 x 4 x 4)
1088 = 17 x 64 = (1 + 8 + 8) x (1 x 8 x 8)

1088 is missing from the list at Wikipedia and the Encyclopedia of Integer Sequences, but I like the look of it, so I’m including it here. Base-11 has five sum-product numbers:

419[11] = 13 x 33 = (4 + 1 + 9) x (4 x 1 x 9) = 504[10] = 14 x 36
253[11] = [10] x 28 = (2 + 5 + 3) x (2 x 5 x 3) = 300[10] = 10 x 30
2189[11] = 19 x 121 = (2 + 1 + 8 + 9) x (2 x 1 x 8 x 9) = 2880[10] = 20 x 144
7634[11] = 19 x 419 = (7 + 6 + 3 + 4) x (7 x 6 x 3 x 4) = 10080[10] = 20 x 504
82974[11] = 28 x 3036 = (8 + 2 + 9 + 7 + 4) x (8 x 2 x 9 x 7 x 4) = 120960[10] = 30 x 4032

But the record for bases below 50 is set by 7:

22[7] = 4 x 4 = (2 + 2) x (2 x 2) = 16[10] = 4 x 4
505[7] = 13 x 34 = (5 + 5) x (5 x 5) = 250[10] = 10 x 25
242[7] = 11 x 22 = (2 + 4 + 2) x (2 x 4 x 2) = 128[10] = 8 x 16
1254[7] = 15 x 55 = (1 + 2 + 5 + 4) x (1 x 2 x 5 x 4) = 480[10] = 12 x 40
2343[7] = 15 x 132 = (2 + 3 + 4 + 3) x (2 x 3 x 4 x 3) = 864[10] = 12 x 72
116655[7] = 33 x 2424 = (1 + 1 + 6 + 6 + 5 + 5) x (1 x 1 x 6 x 6 x 5 x 5) = 21600[10] = 24 x 900
346236[7] = 33 x 10362 = (3 + 4 + 6 + 2 + 3 + 6) x (3 x 4 x 6 x 2 x 3 x 6) = 62208[10] = 24 x 2592
424644[7] = 33 x 11646 = (4 + 2 + 4 + 6 + 4 + 4) x (4 x 2 x 4 x 6 x 4 x 4) = 73728[10] = 24 x 3072

And base-6? Six Nix. There are no sum-product numbers unique to that base (to the best of my far-from-infallible knowledge). Here is the full list for base-3 to base-50 (not counting 0 and 1 as sum-product numbers):

5 in base-11 4 in base-21 3 in base-31 2 in base-41
4 in base-12 5 in base-22 1 in base-32 3 in base-42
0 in base-3 3 in base-13 4 in base-23 3 in base-33 4 in base-43
2 in base-4 3 in base-14 2 in base-24 4 in base-34 5 in base-44
1 in base-5 2 in base-15 3 in base-25 2 in base-35 6 in base-45
0 in base-6 2 in base-16 6 in base-26 2 in base-36 7 in base-46
8 in base-7 6 in base-17 0 in base-27 3 in base-37 3 in base-47
1 in base-8 5 in base-18 1 in base-28 3 in base-38 7 in base-48
5 in base-9 7 in base-19 0 in base-29 1 in base-39 5 in base-49
3 in base-10 3 in base-20 2 in base-30 2 in base-40 3 in base-50

Performativizing Papyrocentricity #18

Papyrocentric Performativity Presents:

Der ÜbergmenschDougal Haston: The Philosophy of Risk, Jeff Connors (Canongate Books 2002)

Book with Bite Steve Backshall’s Most Poisonous Creatures, Steve Backshall (New Holland 2013)

The Politics of PretenceMo Mowlam: The Biography, Julia Langdon (Little, Brown 2000)

Guns’n’GladioliA Light That Never Goes Out: The Enduring Saga of the Smiths, Tony Fletcher (Windmill Books 2013) (posted @ Overlord of the Über-Feral)

Think Ink50 Quantum Physics Ideas You Really Need to Know, Joanne Baker (Quercus 2013) (posted @ O.o.t.Ü.-F.)


Or Read a Review at Random: RaRaR

Guns’n’Gladioli

Front cover of A Light That Never Goes Out by Tony FletcherA Light That Never Goes Out: The Enduring Saga of the Smiths, Tony Fletcher (Windmill Books 2013)

Coke, booze, earsplitting volume. Not a combination you associate with the Smiths. But it was there, as you’ll learn from this book. Towards the end, they were almost turning into Guns’n’Gladioli. Morrissey, of course, was the odd one out: he wasn’t battering his brain-cells with drink and drugs on their final American tour. But back home his Lichtmusik was also lout-music: the Smiths didn’t just appeal to bedsit miserabilists in rain-hammered humdrum towns. No, they appealed to some football hooligans too, including a Chelsea fan who didn’t mind being asked, “You still wanking off over that miserable northern poof?” as he travelled north by train to do battle with Manchester United and Manchester City, who also supplied hoolifans to the Smiths (pp. 509-10). So did football clubs in Glasgow and Edinburgh. The Smiths are easy to caricature, but the caricatures don’t capture their complexity.

Tony Fletcher does capture it: the band, their music, their fans, friends, producers, studio-engineers and record-labels. He’s definitely a Guardianista, but his prose is plodding rather than painful and he does a good job of putting the poof and his partners into context. The 1980s is one important part of that context. So are Irish Catholicism and Manchester. When you look at pictures of the Smiths, you can see two clear divisions. One of them separates the singer, guitarist and drummer from the bassist: the dark-haired, bushy-browed, strong-faced Morrissey, Johnny Marr and Andy Rourke clearly belong to one race and the light-haired, lesser-browed, milder-faced Mike Joyce to another. They’re Irish and he’s English: the British Isles are rich in language and rich in biology too. But Morrissey’s height and handsomeness also separate him from Marr, Rourke and Joyce, like his polysyllabic name. Both must be related to his intelligence, his creativity and his ability to turn himself into the Pope of Mope and become much more famous than any of the other three. Fletcher doesn’t talk about this biology – as I said, he’s a Guardianista – but it’s implicit in his descriptions of Irish settlement in Manchester and of Morrissey’s genius.

Mozza with some flora and fauna

Mozza with some flora and fauna

Is that too strong a word? Maybe. Morrissey is certainly the interesting and original one in this book and it ends with his story only just beginning. You can feel the tug of his later career throughout the book: it’s not discussed, but you know it’s there. But Fletcher isn’t concentrating on Morrissey and doesn’t seem very interested in Carry On and Brit-film in the 1960s, so he’s less good on what might be called the Smythos: the world created by Morrissey in his lyrics and interviews. Morrissey’s influences are better explained in Simon Goddard’s Mozipedia (2009), which isn’t just about the New York Dolls, the Cockney Rejects and vegetarianism. It has also entries for everyone from Hawtrey and Housman to Williams and Wilde by way of Sandy Shaw, Shelagh Delaney and Jobriath. No-one will ever devote an encyclopaedia to Marr like that: music doesn’t have as much meaning and metaphor in it. It has emotion and beauty instead and Fletcher is good at describing how Marr created a lot of both on albums like Meat Is Murder and Strangeways Here We Come.

Front cover of Mozipedia by Simon Goddard

Front cover of Mozipedia by Simon Goddard

I’ve never liked him much, though. I like what he did with the guitar and in the studio, but I don’t like what he did to his body and mind. Or what he put on his body: he didn’t have Mozza’s way with weeds either. In the photos, you can clearly see Morrissey’s narcissism and Marr’s weediness. It’s no surprise that Marr smoked a lot of marijuana, preferred working at night and didn’t eat properly. But he’s weedy in more ways than the physical: there’s also a photo of him with Billy Bragg, the committed socialist behind Red Wedge. This was a collective of musicians and bands who wanted to make the world a better place by fighting Fatcher, fascism and free speech with their fantastic music. Morrissey had his lefty opinions too, but he didn’t like collectives and he didn’t scorn just Margaret Thatcher and the Queen: Bob Geldof and Live Aid got the sharp side of his tongue too. Which is good. Mozza is worshipped by Guardianistas, but he’s not a Guardianista himself.

Or not wholly. The hive-mind hasn’t been able to hum him fully into line, unlike Marr and Bragg. As for Rourke and Joyce: their politics don’t matter and the most interesting thing one of them does in this book is get stung by a sting-ray (pp. 539-40). They were competent musicians, but they weren’t essential to the Smiths. Joyce is most important for causing trouble, not for strumming his bass: first there was the heroin addiction, then the 21st-century court-case in which he sued for more money and earnt Morrissey’s undying enmity. Fletcher barely mentions the court-case and ends the book in the 1980s, with the Smiths exhausted, antagonistic and unfulfilled. They never achieved their full potential and though few bands do, few bands have had more to offer than the Smiths. The Beatles were one and managed to offer it from the nearby northern city of Liverpool. They were Irish Catholic too. But, like the Smiths, they achieved success in England, not Ireland. That’s important and the younger band captured it in their name. “Smiths” is an Anglo-Saxon word with ancient roots and difficult phonetics. It seems simple, but it isn’t. Rather like light.

Hex Appeal

A polyiamond is a shape consisting of equilateral triangles joined edge-to-edge. There is one moniamond, consisting of one equilateral triangle, and one diamond, consisting of two. After that, there are one triamond, three tetriamonds, four pentiamonds and twelve hexiamonds. The most famous hexiamond is known as the sphinx, because it’s reminiscent of the Great Sphinx of Giza:

sphinx_hexiamond

It’s famous because it is the only known pentagonal rep-tile, or shape that can be divided completely into smaller copies of itself. You can divide a sphinx into either four copies of itself or nine copies, like this (please open images in a new window if they fail to animate):

sphinx4

sphinx9

So far, no other pentagonal rep-tile has been discovered. Unless you count this double-triangle as a pentagon:

double_triangle_rep-tile

It has five sides, five vertices and is divisible into sixteen copies of itself. But one of the vertices sits on one of the sides, so it’s not a normal pentagon. Some might argue that this vertex divides the side into two, making the shape a hexagon. I would appeal to these ancient definitions: a point is “that which has no part” and a line is “a length without breadth” (see Neuclid on the Block). The vertex is a partless point on the breadthless line of the side, which isn’t altered by it.

But, unlike the sphinx, the double-triangle has two internal areas, not one. It can be completely drawn with five continuous lines uniting five unique points, but it definitely isn’t a normal pentagon. Even less normal are two more rep-tiles that can be drawn with five continuous lines uniting five unique points: the fish that can be created from three equilateral triangles and the fish that can be created from four isosceles right triangles:

equilateral_triangle_fish_rep-tile

right_triangle_fish_rep-tile

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Front cover of 50 Quantum Physics Ideas You Really Need to Know by Joanne Baker50 Quantum Physics Ideas You Really Need to Know, Joanne Baker (Quercus 2013)

A very good introduction to a very difficult subject. A very superficial introduction too, because it doesn’t use proper mathematics. If it did, I’d be lost: like most people’s, my maths is far too weak for me to understand quantum physics. Here’s one of the side-quotes that help make this book such an interesting read: “We must be clear that when it comes to atoms, language can be used only as in poetry.”

That’s by the Jewish-Danish physicist Niels Bohr (1885-1962). It applies to quantum physics in general. Without the full maths, you’re peering through a frost-covered window into a sweetshop, you’re not inside sampling the wares. But even without the full maths, the concepts and ideas in this book are still difficult and challenging, from the early puzzles thrown up by the ultra-violet catastrophe to the ingenious experiments that have proved particle-wave duality and action at a distance.

But there’s a paradox here.

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