Category Archives: Mathematics

He Say, He Sigh, He Sow #20

by in Binary numbers, Geometry, π, Mathematics, Pi, Quotations and tagged , , , , , , , , , , , , , , , , , , , , , | 6 Comments

“In 1997, Fabrice Bellard announced that the trillionth digit of π, in binary notation, is 1.” — Ian Stewart, The Great Mathematical Problems (2013).

Blue is the Killer

by in Aesthetics, Art, Book Reviews, Consciousness, Geometry, Optical Illusions, Psychology and tagged , , , , , , , , , , , , , , , , , , | Leave a comment

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

On the M3!

by in Latin, Magistra Mundi, Mathematics and tagged , , , , , , , , , , , , , , , , | Leave a comment

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

Prime Climb Time

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

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.

Prime Time #2

by in Mathematics, Prime numbers and tagged , , , , , , , , , , , , , | Leave a comment

“2n2 + 29 is prime for all values of n for 1 to 28.” — The Penguin Dictionary of Curious and Interesting Numbers, David Wells (1986).

• 31, 37, 47, 61, 79, 101, 127, 157, 191, 229, 271, 317, 367, 421, 479, 541, 607, 677, 751, 829, 911, 997, 1087, 1181, 1279, 1381, 1487, 1597.

Elsewhere other-posted:

• Prime Time #1
• Poulet’s Propellor — Musings on Math and Mathculinity
• La Spirale è Mobile

Performativizing Papyrocentricity #22

by in Über-Transgression..., Book Reviews, Civilization, History, Literature, Mathematics, Philosophy of mathematics, Politics, The Sea and tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Papyrocentric Performativity Presents:

• Plates from the Great – Shots from the Front: The British Soldier 1914-18, Richard Holmes (HarperPress 2008; paperback 2010)

• Math for the Mistress – A Mathematician’s Apology, G.H. Hardy (1940)

• Sinister Sinema – Scalarama: A Celebration of Subterranean Cinema at Its Sleazy, Slimy and Sinister Best, ed. Norman Foreman, B.A. (TransVisceral Books 2015)

• Rick Pickings – Lost, Stolen or Shredded: Stories of Missing Works of Art and Literature, Rick Gekoski (Profile Books 2013/2014)

• Slug is a Drug – Collins Complete Guide to British Coastal Wildlife, Paul Sterry and Andrew Cleave (HarperCollins 2012) (posted @ Overlord of the Über-Feral)

Or Read a Review at Random: RaRaR

He Say, He Sigh, He Sow #19

by in Magistra Mundi, Mathematics, Philosophy of mathematics, Quotations and tagged , , , , , , , , , , , , , , , , , | Leave a comment

“The study of mathematics is the indispensable basis for all intellectual and spiritual progress.” — F.M. Cornford (1874-1943) quoted in The Sacred in Music (see also Pythagoreanism).

This Mortal Doyle

by in A.E. Housman, Biology, Consciousness, Mathematics, Philosophy, Physics, Poetry, Science, Theology and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Challenger chopped and changed. That is to say, in one important respect, Arthur Conan Doyle’s character Professor Challenger lacked continuity. His philosophical views weren’t consistent. At one time he espoused materialism, at another he opposed it. He espoused it in “The Land of Mist” (1927):

“Don’t tell me, Daddy, that you with all your complex brain and wonderful self are a thing with no more life hereafter than a broken clock!”

“Four buckets of water and a bagful of salts,” said Challenger as he smilingly detached his daughter’s grip. “That’s your daddy, my lass, and you may as well reconcile your mind to it.”

But earlier, in “The Poison Belt” (1913), he had opposed it:

“No, Summerlee, I will have none of your materialism, for I, at least, am too great a thing to end in mere physical constituents, a packet of salts and three bucketfuls of water. Here ― here” ― and he beat his great head with his huge, hairy fist ― “there is something which uses matter, but is not of it ― something which might destroy death, but which death can never destroy.”

That story was published just over a century ago, but Challenger’s boast has not been vindicated in the meantime. So far as science can see, matter rules mind, not vice versa. Conan Doyle thought the same as the earlier Challenger, but Conan Doyle’s rich and teeming brain seems to have ended in “mere physical constituents”. To all appearances, when the organization of his brain broke down, so did his consciousness. And that concluded the cycle described by A.E. Housman in “Poem XXXII” of A Shropshire Lad (1896):

From far, from eve and morning
  And yon twelve-winded sky,
The stuff of life to knit me
  Blew hither: here am I.

Now – for a breath I tarry
  Nor yet disperse apart –
Take my hand quick and tell me,
  What have you in your heart.

Speak now, and I will answer;
  How shall I help you, say;
Ere to the wind’s twelve quarters
  I take my endless way. (ASL, XXXII)

Continue reading This Mortal Doyle…

Lat’s That

by in Magic squares, Mathematics, Prime numbers and tagged , , , , , , , , , , , , | Leave a comment

In a magic square of numbers, all rows, columns and diagonals have the same sum, or magic total. Here is an example:

1*5*9
8*3*4
6*7*2

(mt=15)

Here’s another:

06*07*11*10
15*02*14*03
04*13*01*16
09*12*08*05

(mt=34)

And another:

04*25*20*10*06
01*13*11*21*19
23*09*07*08*18
15*16*03*14*17
22*02*24*12*05

(mt=65)

And another:

35*15*10*18*11*22
05*25*33*12*07*29
34*30*04*14*21*08
02*16*27*17*23*26
03*24*09*19*36*20
32*01*28*31*13*06

(mt=111)

In all those magic squares, the magic total is fixed: the sum of all numbers from 1 to 36 is 666, so any individual line in a 6×6 magic square has to equal 666 / 6 or 111. In other kinds of magic figure, this rule doesn’t apply:

2*7*3
4***8
6*5*1

(mt=12)
6*3*4
2***8
5*7*1

(mt=13)
8*5*1
2***6
4*3*7

(mt=14)
8*1*6
4***2
3*5*7

(mt=15)

Continue reading Lat’s That…