Curious Cuneus

A triangle subdivided into three more triangles, each of which is subdivided too...

A triangle in which dots represent the centers of the triangles into which it is repeatedly subdivided

A close-up of the central-dot triangle

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Death and the Midden

Front cover of The Mitchell Beazley Pocket Guide to Mushrooms and Toadstools by David N. PeglerThe Mitchell Beazley Pocket Guide to Mushrooms and Toadstools, David N. Pegler (1982)

A little gem of a book in a consistently excellent natural history series. Rather like its subject, it’s an example of something very rich and rewarding that’s growing quietly in a neglected niche. Representational art, banished from the academies and galleries over the past century, has survived in natural history illustration. When I think of contemporary art that’s moved or delighted me I often think of men like Richard Lewington, illustrator of Field Guide to the Dragonflies of Britain and Europe, and Ralph Thompson, who illustrated Gerald Durrell’s books about animal-collecting in Africa and South America. David N. Pegler’s art is more realistic and detailed than Thompson’s and he may be an even better draughtsman. But if you think he has less scope for quirkiness and humor, with non-animal, let alone non-mammalian, subjects, you’d be wrong. Each of the fungi illustrated here is a finely detailed, delicately tinted portrait in miniature and in situ, often accompanied by the dried leaves or bark or pine-needles of the spot in which Pegler presumably found it. And one of the pleasures of looking through the book is uncovering the unique and often witty touches Pelger has added to some of the portraits. For example, there’s the beetle crawling towards two specimens of Tricholoma portenosum – ‘so good to eat the French call it “Marvellous Tricholoma” (Tricholome merveilleux)’ – and the crumpled sweet-wrapper lying near three Agaricus xanthodermus, the Yellow-staining mushroom found in or on “Parks, roadsides and wasteland”.

But Pegler usually lets the fungi speak for themselves in their bewildering variety of voices from their startlingly wide range of habitats: there are fungi that specialize in sand, marsh, burnt ground, and dung, as well as the more familiar dead wood and leaf-litter. As so often, the English-speaking world still has a lot to learn from the French: where many Brits or Americans are familiar with two or three edible species, the French are familiar with dozens. The Italians, on the other hand, knew a lot about another kind of mushroom during the Renaissance: the poisonous varieties whose symbols – black-skull-on-white-background for “dangerous” and white-skull-on-black-background for “deadly” – add a regular macabre frisson to Pegler’s drawings.

Inner pages of Mushrooms and Toadstools

One of the deadliest fungi, the Destroying Angel (Amanita virosa), is one of the most beautiful too, like an evil young witch out of Grimms’ Fairy Tales: it’s pure white, slender-stemmed, and with lacy clinging veils, but it reveals its true nature by its “heavy soporific smell”. “Do not mistake for Agaricus silvicola”, Pegler warns (the Latin adjective silvicola, meaning “wood-dwelling”, only exists in the feminine form). One of the ways to avoid mistaking the two is that A. silvicola, the Wood mushroom, “smells of aniseed”. Fungi can delight, or revolt, the nose as well as the eye: there’s the Coconut-scented milk-cap (Lactarius glyciosmus) and the Geranium-scented russula (Russula fellea) on the delightful side, and the Nitrous mycena (Mycena leptocephala), “often smell[ing] of nitric acid”, and the Stinking parasol (Lepiota cristata), with its “unpleasant rubbery smell”, on the revolting.

Unless it can assist identification like that, Pegler doesn’t usually say much about any particular fungus, because he’s writing mainly for identification and has to cram hundreds of species into a pocket-sized space. But each species must have its own unique ecological story and Pegler has managed to make his drawings portraits from the wild and not just mycological mug-shots. And each is accompanied by an illustration of its spores, as a further aid to identification and further invitation for the browsing eye. Spores, like fungi themselves, come in many different shapes and sizes. All of which makes this book my favorite in the Mitchell Beazley series. Every book is worth owning or looking at, but the Pocket Guide to Butterflies, for example, has no artistic charm or whimsy. The butterflies are drawn strictly and severely for identification, with nothing accompanying them: no plants, no landscapes, and no jeux d’esprit. And European butterflies don’t come in many varieties or colors: although they often have hidden charms, most of them are frumpish and dowdy when set beside their glittering, gleaming, multi-spectacular cousins from the tropics.

That isn’t true of European fungi, as Pegler demonstrates: both they and their spores come in all shapes, sizes, and patterns. And all colors too. The Hygrocybe genus gleams with reds, yellows, and lilacs, and the species there look much more like magic mushrooms than the genuine article: the unassuming little Liberty Cap, Psilocybe semilanceata, which can open the doors of perception to a world of wonder. Fungi can drive you mad, kill you, or delight your palate, eye, and intellect, and this book captures their richness and variety better than any other I’ve come across. Art, natural history, and culinary guide: it’s all here and The Mitchell Beazley Pocket Guide to Mushrooms and Toadstools is, in its quiet way, a much greater example of European high culture than anything the modern Turner Prize has produced.

Back to Bases

(N.B. I am not a mathematician and often make stupid mistakes in my recreational maths. Caveat lector.)

101 isn’t a number, it’s a label for a number. In fact, it’s a label for infinitely many numbers. In base 2, 1012 = 5; in base 3, 1013 = 10; 1014 = 17; 1015 = 26; and so on, for ever. In some bases, like 2 and 4, the number labelled 101 is prime. In other bases, it isn’t. But it is always a palindrome: that is, it’s the same read forward and back. But 101, the number itself, is a palindrome in only two bases: base 10 and base 100.1 Note that 100 = 101-1: with the exception of 2, all integers, or whole numbers, are palindromic in at least one base, the base that is one less than the integer itself. So 3 = 112; 4 = 113; 5 = 114; 101 = 11100; and so on.

Less trivial is the question of which integers set progressive records for palindromicity, or for the number of palindromes they create in bases less than the integers themselves. You might guess that the bigger the integer, the more palindromes it will create, but it isn’t as simple as that. Here is 10 represented in bases 2 through 9:

10102 | 1013* | 224* | 205 | 146 | 137 | 128 | 119*

10 is a palindrome in bases 3, 4, and 9. Now here is 30 represented in bases 2 through 29 (note that a number between square brackets represents a single digit in that base):2

111102 | 10103 | 1324 | 1105 | 506 | 427 | 368 | 339* | 30 | 2811 | 2612 | 2413 | 2214* | 2015 | 1[14]16 | 1[13]17 | 1[12]18 | 1[11]19 | 1[10]20 | 1921 | 1822 | 1723 | 1624 | 1525 | 1426 | 1327 | 1228
| 1129*

30, despite being three times bigger than 10, creates only three palindromes too: in bases 9, 14, and 29. Here is a graph showing the number of palindromes for each number from 3 to 100 (prime numbers are in red):

Graph of palindromes in various bases for n=3 to 100

The number of palindromes a number has is related to the number of factors, or divisors, it has. A prime number has only one factor,  itself (and 1), so primes tend to be less palindromic than composite numbers. Even large primes can have only one palindrome, in the base b=n-1 (55,440 has 119 factors and 61 palindromes; 65,381 has one factor and one palindrome, 1165380). Here is a graph showing the number of factors for each number from 3 to 100:

Graph of number of factors for n = 3 to 100

And here is an animated gif combining the two previous images:

Animated gif of number of palindromes and factors, n=3 to 100

Here is a graph indicating where palindromes appear when n, along the x-axis, is represented in the bases b=2 to n-1, along the y-axis:

Graph showing where palindromes occur in various bases for n = 3 to 1000

The red line are the palindromes in base b=n-1, which is “11” for every n>2. The lines below it arise because every sufficiently large n with divisor d can be represented in the form d·n1 + d. For example, 8 = 2·3 + 2, so 8 in base 3 = 223; 18 = 3·5 + 3, so 18 = 335; 32 = 4.7 + 4, so 32 = 447; 391 = 17·22 + 17, so 391 = [17][17]22.

And here, finally, is a table showing integers that set progressive records for palindromicity (p = number of palindromes, f = total number of factors, prime and non-prime):

n Prime Factors p f    n Prime Factors p f
3 3 1 1    2,520 23·32·5·7 25 47
5 5 2 1    3,600 24·32·52 26 44
10 2·5 3 3    5,040 24·32·5·7 30 59
21 3·7 4 3    7,560 23·33·5·7 32 63
36 22·32 5 8    9,240 23·3·5·7·11 35 63
60 22·3·5 6 11    10,080 25·32·5·7 36 71
80 24·5 7 9    12,600 23·32·52·7 38 71
120 23·3·5 8 15    15,120 24·33·5·7 40 79
180 22·32·5 9 17    18,480 24·3·5·7·11 43 79
252 22·32·7 11 17    25,200 24·32·52·7 47 89
300 22·3·52 13 17    27,720 23·32·5·7·11 49 95
720 24·32·5 16 29    36,960 25·3·5·7·11 50 95
1,080 23·33·5 17 31    41,580 22·33·5·7·11 51 95
1,440 25·32·5 18 35    45,360 24·34·5·7 52 99
1,680 24·3·5·7 20 39    50,400 25·32·52·7 54 107
2,160 24·33·5 21 39    55,440 24·32·5·7·11 61 119

Notes

1. That is, it’s only a palindrome in two bases less than 101. In higher bases, “101” is a single digit, so is trivially a palindrome (as the numbers 1 through 9 are in base 10).

2. In base b, there are b digits, including 0. So base 2 has two digits, 0 and 1; base 10 has ten digits, 0-9; base 16 has sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

Monteverde’s Mathemorphosis

Here are some fractals based on the Angelo di Monteverde or Angelo della Resurrezione carved by the Italian sculptor Giulio Monteverde (1837-1917) for the monumental cemetery of Staglieno in Genoa.

A fractal based on an angel carved by Giulio Monteverde

A fractal based on an angel carved by Giulio Monteverde

A fractal based on an angel carved by Giulio Monteverde

A fractal based on an angel carved by Giulio Monteverde

A fractal based on an angel carved by Giulio Monteverde

A fractal based on an angel carved by Giulio Monteverde

A fractal based on an angel carved by Giulio Monteverde

Jug is the Drug

In “The Gems of Rebbuqqa”, I interrogated notions around the concept of priestesses who permanently juggle three giant eye-like gems, a ruby, a sapphire, and an emerald, atop a sandstone altar. In “The Schismatarch” (forthcoming), I will interrogate notions around the concept of a Himalayan sect that believes this universe is one of three juggled by a god called Nganāma. Each of these universes contains a smaller Nganāma who juggles three dwarf universes; et sic ad infinitum. Moreover, the Nganāma juggling our universe sits in a larger universe, one of three juggled by a giant Nganāma in a larger universe still, which is one of three on a higher plane; et sic ad infinitum. The cosmology of the Nganāma-sect is fractal: ut supra, sic infra: as above, so below. Here are some animated gifs inspired by these two stories and based on juggled eye-gem fractals.

A fractal of three juggled blue eyes

 A fractal comprising three juggled eyes

A fractal of three sets of three juggled eyes

A fractal comprising three sets of three sets (sic) of three juggled eyes

A fractal based on juggled eyeballs

A fractal of three juggled eyes, in front of each of which three more eyes are juggled

A fractal of 27 juggled eyes