Author Archives: Krilling for Company

Know Your Limaçons

by in Book Reviews, Fractals, Geometry, Mathematics and tagged , , , , , , , , , , , , , , , , , , , | Leave a comment

Front cover of The Penguin Dictionary of Curious and Interesting Geometry by David WellsThe Penguin Dictionary of Curious and Interesting Geometry, David Wells (1991)

Mathematics is an ocean in which a child can paddle and an elephant can swim. Or a whale, indeed. This book, a sequel to Wells’ excellent Penguin Dictionary of Curious and Interesting Mathematics, is suitable for both paddlers and plungers. Plumbers, even, because you can dive into some very deep mathematics here.

Far too deep for me, I have to admit, but I can wade a little way into the shallows and enjoy looking further out at what I don’t understand, because the advantage of geometry over number theory is that it can appeal to the eye even when it baffles the brain. If this book is more expensive than its prequel, that’s because it needs to be. It’s a paperback, but a large one, to accommodate the illustrations.

Fortunately, plenty of them appeal to the eye without baffling the brain, like the absurdly simple yet mindstretching Koch snowflake. Take a triangle and divide each side into thirds. Erect another triangle on each middle third. Take each new line of the shape and do the same: divide into thirds, erect another triangle on the middle third. Then repeat. And repeat. For ever.

A Koch snowflake (from Wikipedia)

A Koch snowflake (from Wikipedia)

The result is a shape with a finite area enclosed by an infinite perimeter, and it is in fact a very early example of a fractal. Early in this case means it was invented in 1907, but many of the other beautiful shapes and theorems in this book stretch back much further: through Étienne Pascal and his oddly organic limaçon (which looks like a kidney) to the ancient Greeks and beyond. Some, on the other hand, are very modern, and this book was out-of-date on the day it was printed. Despite the thousands of years devoted by mathematicians to shapes and the relationship between them, new discoveries are being made all the time. Knots have probably been tied by human beings for as long as human beings have existed, but we’ve only now started to classify them properly and even find new uses for them in biology and physics.

Which is not to say knots are not included here, because they are. But even the older geometry Wells looks at would be enough to keep amateur and recreational mathematicians happy for years, proving, re-creating, and generalizing as they work their way through variations on all manner of trigonomic, topological, and tessellatory themes.

Previously pre-posted (please peruse):

• Poulet’s Propeller — discussion of Wells’ Penguin Dictionary of Curious and Interesting Numbers (1986)

Persecution Complex

by in Geometry, Mathematics, Pursuit Curves and tagged , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Imagine four mice sitting on the corners of a square. Each mouse begins to run towards its clockwise neighbour. What happens? This:

Four mice chasing each other

Four mice chasing each other

The mice spiral to the centre and meet, creating what are called pursuit curves. Now imagine eight mice on a square, four sitting on the corners, four sitting on the midpoints of the sides. Each mouse begins to run towards its clockwise neighbour. Now what happens? This:

Eight mice chasing each other

Eight mice chasing each other

But what happens if each of the eight mice begins to run towards its neighbour-but-one? Or its neighbour-but-two? And so on. The curves begin to get more complex:

square+midpoint+2

(Please open the following image in a new window if it fails to animate.)

square+midpoint+3

You can also make the mice run at different speeds or towards neighbours displaced by different amounts. As these variables change, so do the patterns traced by the mice:

• Continue reading Persecution Complexified

Poulet’s Propeller

by in Biology, History, Magistra Mundi, Mathematics, Prime numbers, Psychology and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

The Penguin Dictionary of Curious and Interesting Numbers (1986) is one of my favourite books. It’s a fascinating mixture of math, mathecdote and math-joke:

2·618 0333…

The square of φ, the golden ratio, and the only positive number such that √n = n-1. (pg. 45)

6

Kepler discussed the 6-fold symmetry of snowflakes, and attempted to explain it by considering the close packing of spheres in a hexagonal array. (pg. 69)

39

This appears to be the first uninteresting number, which of course makes it an especially interesting number, because it is the smallest number to have the property of being uninteresting.

It is therefore also the first number to be simultaneously interesting and uninteresting. (pg. 120)

David Wells, who wrote the Dictionary, “had the rare distinction of being a Cambridge scholar in mathematics and failing his degree”. He must be the mathematical equivalent of the astronomer Patrick Moore: a popularizer responsible for opening many minds and inspiring many careers. He’s also written books on geometry and mathematical puzzles. But not everyone appreciates his efforts. This is a sideswipe in a review of William Hartston’s The Book of Numbers:

Thankfully, this book is more concerned with facts than mathematics. Anyone wanting to learn more about [π] or the Fibonacci sequence should turn to the Penguin Dictionary of Curious and Interesting Numbers, a volume which none but propeller-heads will find either curious or interesting. (Review in The Independent, 18th December 1997)

Continue reading: Poulet’s Propeller…

Narcissarithmetic #2

by in Base Behavior, Digit-sums, Mathematics, Narcissistic numbers, Powers and tagged , , , , , , , , , , , , , , , , , , , , , | Leave a comment

It’s easy to find patterns like these in base ten:

81 = (8 + 1)^2 = 9^2 = 81

512 = (5 + 1 + 2)^3 = 8^3 = 512
4913 = (4 + 9 + 1 + 3)^3 = 17^3 = 4913
5832 = (5 + 8 + 3 + 2)^3 = 18^3 = 5832
17576 = (1 + 7 + 5 + 7 + 6)^3 = 26^3 = 17576
19683 = (1 + 9 + 6 + 8 + 3)^3 = 27^3 = 19683

2401 = (2 + 4 + 0 + 1)^4 = 7^4 = 2401
234256 = (2 + 3 + 4 + 2 + 5 + 6)^4 = 22^4 = 234256
390625 = (3 + 9 + 0 + 6 + 2 + 5)^4 = 25^4 = 390625
614656 = (6 + 1 + 4 + 6 + 5 + 6)^4 = 28^4 = 614656
1679616 = (1 + 6 + 7 + 9 + 6 + 1 + 6)^4 = 36^4 = 1679616

17210368 = (1 + 7 + 2 + 1 + 0 + 3 + 6 + 8)^5 = 28^5 = 17210368
52521875 = (5 + 2 + 5 + 2 + 1 + 8 + 7 + 5)^5 = 35^5 = 52521875
60466176 = (6 + 0 + 4 + 6 + 6 + 1 + 7 + 6)^5 = 36^5 = 60466176
205962976 = (2 + 0 + 5 + 9 + 6 + 2 + 9 + 7 + 6)^5 = 46^5 = 205962976

1215766545905692880100000000000000000000 = (1 + 2 + 1 + 5 + 7 + 6 + 6 + 5 + 4 + 5 + 9 + 0 + 5 + 6 + 9 + 2 + 8 + 8 + 0 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0)^20 = 90^20 = 1215766545905692880100000000000000000000

Patterns like this are much rarer:

914457600 = (9 x 1 x 4 x 4 x 5 x 7 x 6)^2 = 30240^2 = 914457600

3657830400 = (3 x 6 x 5 x 7 x 8 x 3 x 4)^2 = 60480^2 = 3657830400

I haven’t found a cube like that in base ten, but base six supplies them:

2212 = (2 x 2 x 1 x 2)^3 = 12^3 = 2212 (b=6) = 8^3 = 512 (b=10)
325000 = (3 x 2 x 5)^3 = 50^3 = 325000 (b=6) = 30^3 = 27000 (b=10)
411412 = (4 x 1 x 1 x 4 x 1 x 2)^3 = 52^3 = 411412 (b=6) = 32^3 = 32768 (b=10)

And base nine supplies a fourth and fifth power:

31400 = (3 x 1 x 4)^4 = 13^4 = 31400 (b=9) = 12^4 = 20736 (b=10)
11600 = (1 x 1 x 6)^5 = 6^5 = 11600 (b=9) = 6^5 = 7776 (b=10)

Then base ten is rich in patterns like these:

81 = (8^1 + 1^1) x (8 + 1) = 9 x 9 = 81

133 = (1^2 + 3^2 + 3^2) x (1 + 3 + 3) = 19 x 7 = 133
315 = (3^2 + 1^2 + 5^2) x (3 + 1 + 5) = 35 x 9 = 315
803 = (8^2 + 0^2 + 3^2) x (8 + 0 + 3) = 73 x 11 = 803
1148 = (1^2 + 1^2 + 4^2 + 8^2) x (1 + 1 + 4 + 8) = 82 x 14 = 1148
1547 = (1^2 + 5^2 + 4^2 + 7^2) x (1 + 5 + 4 + 7) = 91 x 17 = 1547
2196 = (2^2 + 1^2 + 9^2 + 6^2) x (2 + 1 + 9 + 6) = 122 x 18 = 2196

1215 = (1^3 + 2^3 + 1^3 + 5^3) x (1 + 2 + 1 + 5) = 135 x 9 = 1215
3700 = (3^3 + 7^3 + 0^3 + 0^3) x (3 + 7 + 0 + 0) = 370 x 10 = 3700
11680 = (1^3 + 1^3 + 6^3 + 8^3 + 0^3) x (1 + 1 + 6 + 8 + 0) = 730 x 16 = 11680
13608 = (1^3 + 3^3 + 6^3 + 0^3 + 8^3) x (1 + 3 + 6 + 0 + 8) = 756 x 18 = 13608
87949 = (8^3 + 7^3 + 9^3 + 4^3 + 9^3) x (8 + 7 + 9 + 4 + 9) = 2377 x 37 = 87949

182380 = (1^4 + 8^4 + 2^4 + 3^4 + 8^4 + 0^4) x (1 + 8 + 2 + 3 + 8 + 0) = 8290 x 22 = 182380
444992 = (4^4 + 4^4 + 4^4 + 9^4 + 9^4 + 2^4) x (4 + 4 + 4 + 9 + 9 + 2) = 13906 x 32 = 444992

41500 = (4^5 + 1^5 + 5^5 + 0^5 + 0^5) x (4 + 1 + 5 + 0 + 0) = 4150 x 10 = 41500
3508936 = (3^5 + 5^5 + 0^5 + 8^5 + 9^5 + 3^5 + 6^5) x (3 + 5 + 0 + 8 + 9 + 3 + 6) = 103204 x 34 = 3508936
3828816 = (3^5 + 8^5 + 2^5 + 8^5 + 8^5 + 1^5 + 6^5) x (3 + 8 + 2 + 8 + 8 + 1 + 6) = 106356 x 36 = 3828816
4801896 = (4^5 + 8^5 + 0^5 + 1^5 + 8^5 + 9^5 + 6^5) x (4 + 8 + 0 + 1 + 8 + 9 + 6) = 133386 x 36 = 4801896
5659875 = (5^5 + 6^5 + 5^5 + 9^5 + 8^5 + 7^5 + 5^5) x (5 + 6 + 5 + 9 + 8 + 7 + 5) = 125775 x 45 = 5659875

Previously pre-posted (please peruse):

• Narcissarithmetic

Pestilent, Pustulent and Pox-Pocked

by in Keyly Committed Core Components of the Counter-Cultural Community, Music, Post-Weird and tagged , , , , , , , , , , , , , , , , , , , , , , , | 2 Comments

I’m sorry, but let’s face facts: you cannot consider yourself a keyly committed core component of the counter-cultural community unless you own at least three copies — a reading copy, a prominent-shelf-of-the-bookcase copy and a wrap-carefully-in-brown-paper-put-away-in-a-cupboard-and-never-touch-or-look-at-again copy — of each of these toxic’n’tenebrose titles:

• Can the Cannibal? Aspects of Angst, Abjection and Anthropophagy in the Music of Suzi Quatro, 1974-1986 (University of Nebraska Press 2004)

• Doubled Slaughter: Barbarism, Brutalism and Bestial Bloodlust in the Music of Simon and Garfunkel, 1965-2010 (Serpent’s Tail 2007)

• Re-Light My Führer: Nausea, Noxiousness and Neo-Nazism in the Music of Take That, 1988-2007 (U.N.P. 2013)

• Base Citizens Raping: Revulsion, Repulsion and Rabidity in the Music of the Bay City Rollers, 1972-2002 (U.N.P. 2014)

• Underground, Jehovahground: Ferality, Fetidity and Fundamentalist Phantasmality in the Music of the Wombles, August 1974-January 1975 (forthcoming)

• Jane in Blood: Castration, Clitoridolatry and Communal Cannibalism in the Novels of Jane Austen (U.N.P. 2018)

All are by Dr Miriam B. Stimbers, of course. And what can I say about them? Simply this: these books hold an uncompromising mirror up before the pestilent, pustulent and pox-pocked features of so-called Western so-called society and say: “Look. That’s you, that is.” Dr Stimbers’ ruthlessly radical research and sizzlingly psychoanalytic scholarship will overturn your preconceptions so hard that, in some cases, they won’t appear to change at all.

Performativizing Papyrocentricity #13

by in Book Reviews, Notices and tagged , , , , , , , , , | Leave a comment

Papyrocentric Performativity Presents:

Brown Study — William in Trouble, Richmal Crompton (1927)

Bleeding Black — Watch You Bleed: The Saga of Guns n’ Roses, Stephen Davis (Michael Joseph 2008)

Toxic Tails — In the Seventies: Adventures in the Counterculture, Barry Miles (Serpent’s Tail 2011)

Leaf Brief — What a Plant Knows: A Field Guide to the Senses of Your Garden — and Beyond, Daniel Chamovitz (Oneworld 2012) (posted @ Overlord of the Über-Feral)

Electrify Your Eyes — The Spark of Life: Electricity in the Human Body, Frances Ashcroft (Penguin 2013) (@ O.o.t.Ü.-F.)

Or Read a Review at Random: RaRaR

Narcissarithmetic

by in Base Behavior, Digit-sums, Mathematics, Powers and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Why is 438,579,088 a beautiful number? Simple: it may seem entirely arbitrary, but it’s actually self-empowered:

438,579,088 = 4^4 + 3^3 + 8^8 + 5^5 + 7^7 + 9^9 + 0^0 + 8^8 + 8^8 = 256 + 27 + 16777216 + 3125 + 823543 + 387420489 + 0 + 16777216 + 16777216 (usually 0^0 = 1, but the rule is slightly varied here)

438,579,088 is so beautiful, in fact, that it’s in love with itself as a narcissistic number, or number that can be generated by manipulation of its own digits. 89 = 8^1 + 9^2 = 8 + 81 and 135 = 1^1 + 3^2 + 5^3 = 1 + 9 + 125 are different kinds of narcissistic number. 3435 is self-empowered again:

3435 = 3^3 + 4^4 + 3^3 + 5^5 = 27 + 256 + 27 + 3125

But that’s your lot: there are no more numbers in base-10 that are equal to the sum of their self-empowered digits (apart from the trivial 0 and 1). To prove this, start by considering that there is a limit to the size of a self-empowered number. 9^9 is 387,420,489, which is nine digits long. The function autopower(999,999,999) = 387,420,489 x 9 = 3,486,784,401, which is ten digits long. But autopower(999,999,999,999) = 387,420,489 x 12 = 4,649,045,868, also ten digits long.

The Metamorphosis of Narcissus by Salvador Dalí

Salvador Dalí, La Metamorfosis de Narciso (1937)

So you don’t need to check numbers above a certain size. There still seem a lot of numbers to check: 438,579,088 is a long way above 3435. However, the search is easy to shorten if you consider that checking 3-3-4-5 is equivalent to checking 3-4-3-5, just as checking 034,578,889 is equivalent to checking 438,579,088. If you self-empower a number and the result has the same digits as the original number, you’ve found what you’re looking for. The order of digits in the original number doesn’t matter, because the result has automatically sorted them for you. The function autopower(3345) produces 3435, therefore 3435 must be self-empowered.

So the rule is simple: Check only the numbers in which any digit is greater than or equal to all digits to its left. In other words, you check 12 and skip 21, check 34 and skip 43, check 567 and skip 576, 657, 675, 756 and 765. That reduces the search-time considerably: discarding numbers is computationally simpler than self-empowering them. It’s also computationally simple to vary the base in which you’re searching. Base-10 produces only two self-empowered numbers, but its neighbours base-9 and base-11 are much more fertile:

30 = 3^3 + 0^0 = 30 + 0 (b=9)
27 = 27 + 0 (b=10)

31 = 3^3 + 1^1 = 30 + 1 (b=9)
28 = 27 + 1 (b=10)

156262 = 1^1 + 5^5 + 6^6 + 2^2 + 6^6 + 2^2 = 1 + 4252 + 71000 + 4 + 71000 + 4 (b=9)
96446 = 1 + 3125 + 46656 + 4 + 46656 + 4 (b=10)

1647063 = 1^1 + 6^6 + 4^4 + 7^7 + 0^0 + 6^6 + 3^3 = 1 + 71000 + 314 + 1484617 + 0 + 71000 + 30 (b=9)
917139 = 1 + 46656 + 256 + 823543 + 0 + 46656 + 27 (b=10)

1656547 = 1^1 + 6^6 + 5^5 + 6^6 + 5^5 + 4^4 + 7^7 = 1 + 71000 + 4252 + 71000 + 4252 + 314 + 1484617 (b=9)
923362 = 1 + 46656 + 3125 + 46656 + 3125 + 256 + 823543 (b=10)

34664084 = 3^3 + 4^4 + 6^6 + 6^6 + 4^4 + 0^0 + 8^8 + 4^4 = 30 + 314 + 71000 + 71000 + 314 + 0 + 34511011 + 314 (b=9)
16871323 = 27 + 256 + 46656 + 46656 + 256 + 0 + 16777216 + 256 (b=10)

66500 = 6^6 + 6^6 + 5^5 + 0^0 + 0^0 = 32065 + 32065 + 2391 + 0 + 0 (b=11)
96437 = 46656 + 46656 + 3125 + 0 + 0 (b=10)

66501 = 6^6 + 6^6 + 5^5 + 0^0 + 1^1 = 32065 + 32065 + 2391 + 0 + 1 (b=11)
96438 = 46656 + 46656 + 3125 + 0 + 1 (b=10)

517503 = 5^5 + 1^1 + 7^7 + 5^5 + 0^0 + 3^3 = 2391 + 1 + 512816 + 2391 + 0 + 25 (b=11)
829821 = 3125 + 1 + 823543 + 3125 + 0 + 27 (b=10)

18453278 = 1^1 + 8^8 + 4^4 + 5^5 + 3^3 + 2^2 + 7^7 + 8^8 = 1 + 9519A75 + 213 + 2391 + 25 + 4 + 512816 + 9519A75 (b=11)
34381388 = 1 + 16777216 + 256 + 3125 + 27 + 4 + 823543 + 16777216 (b=10)

18453487 = 1^1 + 8^8 + 4^4 + 5^5 + 3^3 + 4^4 + 8^8 + 7^7 = 1 + 9519A75 + 213 + 2391 + 25 + 213 + 9519A75 + 512816 (b=11)
34381640 = 1 + 16777216 + 256 + 3125 + 27 + 256 + 16777216 + 823543 (b=10)

It’s easy to extend the concept of self-empowered narcisso-numbers. The prime 71 = 131 in base-7 and the prime 83 = 146 in base-7. If 131[b=7] is empowered to the digits of 146[b=7], you get 146[b=7]; and if 146[b=7] is empowered to the digits of 131[b=7], you get 131[b=7], like this:

71 = 131[b=7] → 1^1 + 3^4 + 1^6 = 1 + 81 + 1 = 83 = 146[b=7]

83 = 146[b=7] → 1^1 + 4^3 + 6^1 = 1 + 64 + 6 = 71 = 131[b=7]

But it’s not easy to find more examples. Are there other-empowering pairs like that in base-10? I don’t know.

Leaf Brief

by in Biology, Book Reviews, Botany, Flowers, Science and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Front cover of What A Plant Knows by Daniel ChamovitzWhat a Plant Knows: A Field Guide to the Senses of Your Garden – and Beyond, Daniel Chamovitz (Oneworld 2012)

This is a brief but burgeoning book, covering a lot of science and a lot of scientific history. Plants stay in one place and don’t seem to suffer pain or discomfort, so they’re good experimental subjects, particularly for introverts. That’s why Charles Darwin devoted even more time to plants than he did to worms and barnacles. Chamovitz describes Darwin’s ingenious experiments and the even more ingenious experiments of the researchers that followed him. Over millions of years the world has set problems of survival for plants; in solving these problems, plants have set puzzles for scientists. How do plants know when to flower and prepare for winter? How do they resist attacks by insects? Or prey on insects? Or invite visits from pollinators? And how do they communicate with each other? The answers aren’t just chemical: they’re electrical too, as research on the world’s most famous carnivorous plant has proved:

Alexander Volkov and his colleagues at Oakwood University in Alabama first demonstrated that it is indeed electricity that causes the Venus flytrap to close. To test the model, they rigged up very fine electrodes and applied an electrical current to the open lobes of the trap. This made the trap close without any direct touch to its trigger hairs … (ch. 6, “What A Plant Remembers”, pp. 147-8)

Acoustics is also at work in the plant kingdom:

In a process known as buzz pollination, bumblebees stimulate a flower to release its pollen by rapidly vibrating their wing muscles without actually flapping their wings, leading to a high-frequency vibration. … In a similar vein, Roman Zweifel and Fabienne Zeugin from the University of Bern in Switzerland have reported ultrasonic vibrations emanating from pine and oak trees during a drought. These vibrations result from changes in the water content of the water-transporting xylem vessels. While these sounds are passive results of physical forces (in the same way that a rock crashing off a cliff makes a noise), perhaps these ultrasonic vibrations are used as a signal by other trees to prepare for dry conditions. (ch. 4, “What A Plant Hears”, pg. 107-8)

All of this is mathematical: a plant is a mechanism that processes not just sun, water and carbon-dioxide, but information from its environment too. But then sun, water and CO2 are all part of that information: sunlight signals plants as well as sustaining them. Its strength and duration are cues for the seasons and time of the day. So is its colour:

By the time John F. Kennedy was elected president, Warren L. Butler and his colleagues had demonstrated that a single photoreceptor in plants was responsible for both the red and far-red effects. They called this receptor “phytochrome”, meaning “plant colour”. In its simplest model, phytochrome is a light-activated switch. Red light activates phytochrome, turning it into a form primed to receive far-red light. Far-red light inactivates phytochrome, turning it into a form primed to receive red light. Ecologically, this makes a lot of sense. In nature, the last light a plant sees at the end of the day is far-red, and this signifies to the plant that it should “turn-off”. In the morning it sees red light and it wakes up. In this way a plant measures how long ago it last saw red light and adjusts its growth accordingly. (ch. 1, “What A Plant Sees”, pg. 21-2)

There’s an obvious analogy with a computer automatically turning itself off and on, which would make phytochrome and its associated chemicals a kind of hardware created by the software of the genes. Plants share some of that software with human beings: in one fascinating section, Chamovitz discusses the links between healthy plants and sick people:

The arabidopsis [A. thaliana, mustard plant] genome contains BRCA, CFTR, and several hundred other genes associated with human disease or impairment because they are essential for basic cellular biology. These important genes had already evolved 1.5 billion years ago in the single-celled organism that was the common evolutionary ancestor to both plants and animals. (ch. 4, “What A Plant Hears”, pg. 105)

What a Plant Knows stimulates human minds as it discusses plant senses. It’s one of the best briefest, or briefest best, books on science I’ve ever read, packing a lot of history and scientific information into six chapters. Plants don’t move much, but they’re a very lively topic and botany is a good way to understand and appreciate biology and scientific research better.

Electrify Your Eyes

by in Biology, Book Reviews, Chemistry, Consciousness, Science and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Front cover of The Spark of Life by Frances AshcroftThe Spark of Life: Electricity in the Human Body, Frances Ashcroft (Penguin 2013)

“Electricity in the Human Body” is the subtitle of this book. Make that the goat, frog, eel, shark, torpedo-ray, snake, platypus, spiny anteater, sooty shearwater and fruit-fly body too. And if Venus flytraps, maize and algae have bodies, throw them in next. Frances Ashcroft gives you a bargeload of buzz for your buck, a shedload of shock for your shekel: The Spark of Life describes the use of electricity by many different forms of life. But it discusses death a lot too, from lightning-strikes and electric chairs to heart-attacks and toxicology. Poisons can be a cheap and highly effective way of interfering with the electro-chemistry of the body:

The importance of sodium and potassium channels in generating the nerve impulse is demonstrated by the fact that a vast array of poisons from spiders, shellfish, sea anemones, frogs, snakes, scorpions and many other exotic creatures interact with these channels and thereby modify the function of nerve and muscle. … The tetrodotoxin contained in the liver and other tissues of this fish [the fugu or puffer-fish, Takifugu spp., Lagocephalus spp., etc] is a potent blocker of the sodium channels found in your nerves and skeletal muscles. It causes numbness and tingling of the lips and mouth within as little as thirty minutes … This sensation of “pins and needles” spreads rapidly to the face and neck, moves onto the fingers and toes, and is then followed by gradual paralysis of the skeletal muscles … Ultimately the respiratory muscles are paralysed, which can be fatal. The heart is not affected, as it has a different kind of sodium channel that is far less sensitive to tetrodotoxin. The toxin is also unable to cross the blood-brain barrier so that, rather horrifyingly, although unable to move and near death, the patient remains conscious. (ch. 3, “Acting on Impulse”, pp. 69-70)

In short, fugu-poisoning is the opposite of electrocution: it’s the absence rather than the excess of electricity that kills its victims. Those “channels” are a reminder that electro-chemistry could also be called electro-mechanics: unlike an electricity-filled computer, an electricity-filled body has moving parts – and in more ways than one. Our muscles move because ions move in and out of our cells. This means that a body has to be wet inside, not dry like a computer, but it’s easy to imagine a human brain controlling a robotic body. But would a brain still be conscious if it became metal-and-plastic too? Perhaps a brain has to be both soggy and sparky to be conscious.

The electrical nature of the brain certainly seems important, though that may be a superstitious conclusion. Electricity is a mysterious phenomenon and so is consciousness, so they seem to go together well. Ashcroft writes a lot about the sense-organs and the data they supply to the brain, but like all scientists she cannot explain how those data are turned into conscious experience as the maths-engine of the brain applies its neuro-functions and neuro-algorithms. However, she does suggest ways in which our consciousness might be expanded in future. Humans have colour vision, based on the three types of cone-cells in our eyes:

Most mammals, such as cats and dogs, have only two types of cone photopigment and so see only a limited range of colour … Other animals live in a world entirely without colour. But humans should not be too complacent, for we are far from having the best colour vision in the animal world and lag far behind the mantis shrimp, which enjoys ten or more different visual pigments. Even tropical fish possess four or five types of cones. (ch. 9, “The Doors of Perception”, pg. 199)

Bio-engineering may one day sharpen and extend all our senses, from sight and hearing to touch, taste and smell. It may also give us new senses, like the ability to form sound-pictures like bats and detect infra-red like pit-vipers. And why not X-rays and radio-waves too? It’s an exciting prospect, but in a sense it won’t be anything new: our new senses, like our old ones, will depend on nerve-impulses and the way they’re mashed and mathed in that handful of “electrified clay” known as the brain.

“Electrified clay” is Shelley’s phrase: like his wife Mary, he was fascinated by the early electric experiments of the Italian scientists Luigi Galvani and Alessandro Volta. Mary turned her fascination into a book called Frankenstein (1818) and her invention is part of the scientific history in this book. The story of bio-electricity is still going strong: there are electric mysteries in all kinds of bodies waiting to be solved. Maybe consciousness is one of them. And if science proves unable to crack consciousness, it’s certainly able to expand it. Reading this book is one way to experience the mind-expanding powers of science, but seeing like a mantis shrimp would be good too.