Blue is the Killer

Eye Bogglers by Gianni A. Sarcone and Marie-Jo WaeberEye Bogglers: A Mesmerizing Mass of Amazing Illusions, Gianni A. Sarcone and Marie-Jo Waeber (Carlton Books 2011; paperback 2013)

A simple book with some complex illusions. It’s aimed at children but scientists have spent decades understanding how certain arrangements of colour and line fool the eye so powerfully. I particularly like the black-and-white tiger set below a patch of blue on page 60. Stare at the blue “for 15 seconds”, then look quickly at a tiny cross set between the tiger’s eyes and the killer turns colour.

So what’s not there appears to be there, just as, elsewhere, what’s there appears not to be. Straight lines seem curved; large figures seem small; the same colour seems light on the right, dark on the left. There are also some impossible figures, as made famous by M.C. Escher and now studied seriously by geometricians, but the only true art here is a “Face of Fruits” by Arcimboldo. The rest is artful, not art, but it’s interesting to think what Escher might have made of some of the ideas here. Mind is mechanism; mechanism can be fooled. Optical illusions are the most compelling examples, because vision is the most powerful of our senses, but the lesson you learn here is applicable everywhere. This book fools you for fun; others try to fool you for profit. Caveat spectator.

Simple but complex: The café wall illusion

Simple but complex: The café wall illusion

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The Whale’s Way

“Sea Fever” (1902)

I must go down to the seas again, to the lonely sea and the sky,
And all I ask is a tall ship and a star to steer her by,
And the wheel’s kick and the wind’s song and the white sail’s shaking,
And a grey mist on the sea’s face, and a grey dawn breaking.

I must go down to the seas again, for the call of the running tide
Is a wild call and a clear call that may not be denied;
And all I ask is a windy day with the white clouds flying,
And the flung spray and the blown spume, and the sea-gulls crying.

I must go down to the seas again, to the vagrant gypsy life,
To the gull’s way and the whale’s way, where the wind’s like a whetted knife;
And all I ask is a merry yarn from a laughing fellow-rover,
And quiet sleep and a sweet dream when the long trick’s over.


John Masefield (1878–1967)

On the M3!

6 = 2 x 3. And 6 = 1 + 2 + 3. But 6 also equals 3!. That is, 6 = 3 x 2 x 1, or factorial three. If you have three different items, you can arrange them in six different ways. There are three posibilities for the first item, two for the second and one for the third.

You can illustrate this linguistically. All languages are governed by mathematics, but maths manifests itself in different ways. Emphasis is an important part of language, for example, but there are different ways to achieve it. English usually does it with stress or by adding an emphatic word. Other languages can do it by varying the order of words. Latin, for example:

  • Mathematica Magistra Mundi
    — Mathematics is Mistress of the World.
  • Mathematica Mundi Magistra
    — Mathematics of the World is Mistress.
  • Magistra Mathematica Mundi
    — Mistress is Mathematics of the World
  • Magistra Mundi Mathematica
    — Mistress of the World is Mathematics.
  • Mundi Mathematica Magistra
    — Of the World Mathematics is Mistress.
  • Mundi Magistra Mathematica
    — Of the World the Mistress is Mathematics.

Elsewhere other-posted:

Mathematica Magistra Mundi — more on the motto
Moto-Motto — a variant on the motto

Performativizing Papyrocentricity #23

Papyrocentric Performativity Presents:

Face PaintA Face to the World: On Self-Portraits, Laura Cumming (HarperPress 2009; paperback 2010)

The Aesthetics of AnimalsLife: Extraordinary Animals, Extreme Behaviour, Martha Holmes and Michael Gunton (BBC Books 2009)

Less Light, More NightThe End of Night: Searching for Natural Darkness in an Age of Artifical Light, Paul Bogard (Fourth Estate 2013)

The Power of Babel – Clark Ashton Smith, Huysmans, Maupassant


Or Read a Review at Random: RaRaR

Prime Climb Time

The third prime is equal to the sum of the first and second primes: 2 + 3 = 5. After that, for obvious reasons, the prime-sum climbs much more rapidly than the primes themselves:

2, 3, 05, 07, 11, 13, 17, 19, 023, 029...
2, 5, 10, 17, 28, 41, 58, 77, 100, 129...

But what if you use digit-sum(p1..pn), i.e., the sum of the digits of the primes from the first to the nth? For example, the digit-sum(p1..p5) = 2 + 3 + 5 + 7 + 1+1 = 19, whereas the sum(p1..p5) = 2 + 3 + 5 + 7 + 11 = 28. Using the digit-sums of the primes, the comparison now looks like this:

2, 3, 05, 07, 11, 13, 17, 19, 23, 29...
2, 5, 10, 17, 19, 23, 31, 41, 46, 57...

The sum climbs more slowly, but still too fast. So what about a different base? In base-2, the digit-sum(p1..p3) = (1+0) + (1+1) + (1+0+1) = 1 + 2 + 2 = 5. The comparison looks like this:

2, 3, 05, 07, 11, 13, 17, 19, 23, 29...
1, 3, 05, 08, 11, 14, 16, 19, 23, 27...

For primes 3, 5, 11, 19, and 23, p = digit-sum(primes <= p) in base-2. But the cumulative digit-sum soon begins to climb too slowly:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271...

1, 3, 5, 8, 11, 14, 16, 19, 23, 27, 32, 35, 38, 42, 47, 51, 56, 61, 64, 68, 71, 76, 80, 84, 87, 091, 096, 101, 106, 110, 117, 120, 123, 127, 131, 136, 141, 145, 150, 155, 160, 165, 172, 175, 179, 184, 189, 196, 201, 206, 211, 218, 223, 230, 232, 236, 240, 245...

So what about base-3?

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59...
2, 3, 6, 9, 12, 15, 20, 23, 28, 31, 34, 37, 42, 47, 52, 59, 64...

In base-3, for p = 2, 3 and 37, p = digit-sum(primes <= p), while for p = 23, 31, 47 and 59, p = digit-sum(primes < p), like this:

2 = 2.
3 = 2 + (1+0).
37 = 2 + (1+0) + (1+2) + (2+1) + (1+0+2) + (1+1+1) + (1+2+2) + (2+0+1) + (2+1+2) + (1+0+0+2) + (1+0+1+1) + (1+1+0+1) = 2 + 1 + 3 + 3 + 3 + 3 + 5 + 3 + 5 + 3 + 3 + 3.

23 = 2 + (1+0) + (1+2) + (2+1) + (1+0+2) + (1+1+1) + (1+2+2) + (2+0+1) = 2 + 1 + 3 + 3 + 3 + 3 + 5 + 3.
31 = 2 + (1+0) + (1+2) + (2+1) + (1+0+2) + (1+1+1) + (1+2+2) + (2+0+1) + (2+1+2) + (1+0+0+2) = 2 + 1 + 3 + 3 + 3 + 3 + 5 + 3 + 5 + 3.
47 = 2 + (1+0) + (1+2) + (2+1) + (1+0+2) + (1+1+1) + (1+2+2) + (2+0+1) + (2+1+2) + (1+0+0+2) + (1+0+1+1) + (1+1+0+1) + (1+1+1+2) + (1+1+2+1) = 2 + 1 + 3 + 3 + 3 + 3 + 5 + 3 + 5 + 3 + 3 + 3 + 5 + 5.
59 = 2 + (1+0) + (1+2) + (2+1) + (1+0+2) + (1+1+1) + (1+2+2) + (2+0+1) + (2+1+2) + (1+0+0+2) + (1+0+1+1) + (1+1+0+1) + (1+1+1+2) + (1+1+2+1) + (1+2+0+2) + (1+2+2+2) = 2 + 1 + 3 + 3 + 3 + 3 + 5 + 3 + 5 + 3 + 3 + 3 + 5 + 5 + 5 + 7.

This carries on for a long time. For these primes, p = digit-sum(primes < p):

23, 31, 47, 59, 695689, 698471, 883517, 992609, 992737, 993037, 1314239, 1324361, 1324571, 1326511, 1327289, 1766291, 3174029

And for these primes, p = digit-sum(primes <= p):

3, 37, 695663, 695881, 1308731, 1308757, 1313153, 1314301, 1326097, 1766227, 3204779, 14328191

Now try the cumulative digit-sum in base-4:

2, 3, 5, 07, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59...
2, 5, 7, 11, 16, 20, 22, 26, 31, 36, 43, 47, 52, 59, 67, 72, 80... 

The sum of digits climbs too fast. Base-3 is the Goldilocks base, climbing neither too slowly, like base-2, nor too fast, like all bases greater than 3.