Sumbertime Views

Friday, 5th Apr, 2013 by Krilling for Company in 666, Digit-sums, Mathematics, Number of the Beast and tagged 153, 666, Arithmetic, base 10, base 11, base 47, base 9, Biblical numbers, Book of Revelation, cube, cubes, Gospel of John, integer bases, math, mathematics, maths, New Testament numbers, number sums, numbers in the Bible, numbers in the New Testament, powers, recreational math, recreational mathematics, recreational maths, square, Squares, sum of integers, sum of successive integers, sumber, sumbers, sums of cubes, sums of powers, sums of squares | Leave a comment

Like 666 (see Revelation 13:18), 153 (see John 21:11) appears in the Bible. And perhaps for the same reason: because it is the sum of successive integers. 153 = 1+2+3+…+17 = Σ(17), just as 666 = Σ(36). So both numbers are sum-numbers or sumbers. But 153 has other interesting properties, including one that can’t have been known in Biblical times, because numbers weren’t represented in the right way. It’s also the sum of the cubes of its digits: 153 = 1^3 + 5^3 + 3^3 = 1 + 125 + 27. So 153 is a cube-sumber or 3-sumber. The other 3-sumbers are 370, 371 and 407. There are 4-sumbers too, like 1,634 = 1^4 + 6^4 + 3^4 + 4^4, and 5-sumbers, like 194,979 = 1^5 + 9^5 + 4^5 + 9^5 + 7^5 + 9^5, and 6-sumbers, like 548,834 = 5^6 + 4^6 + 8^6 + 8^6 + 3^6 + 4^6.

But there are no 2-sumbers, or numbers that are the sum of the squares of their digits. It doesn’t take long to confirm this, because numbers above a certain size can’t be 2-sumbers. 9^2 + 9^2 = 162, but 9^2 + 9^2 + 9^2 = 243. So 2-sumbers can’t exist above 99 and if you search that high you’ll find that they don’t exist at all. At least not in this house, but they do exist in the houses next door. Base 10 yields nothing, so what about base 9?

4^2 + 5^2 = 45_[9] = 41_[10]
5^2 + 5^2 = 55_[9] = 50

And base 11?

5^2 + 6^2 = 56_[11] = 61_[10]
6^2 + 6^2 = 66_[11] = 72

This happens because odd bases always yield a pair of 2-sumbers whose second digit is one more than half the base and whose first digit is the same or one less. See above (and the appendix). Such a pair is found among the 14 sumbers of base 47, which is the best total till base 157 and its 22 sumbers. Here are the 2-sumbers for base 47:

2^2 + 10^2 = 104
3^2 + 12^2 = 153
5^2 + 15^2 = 250
9^2 + 19^2 = 442
12^2 + 21^2 = 585
14^2 + 22^2 = 680
23^2 + 24^2 = 1,105
24^2 + 24^2 = 1,152
33^2 + 22^2 = 1,573
35^2 + 21^2 = 1,666
38^2 + 19^2 = 1,805
42^2 + 15^2 = 1,989
44^2 + 12^2 = 2,080
45^2 + 10^2 = 2,125

As the progressive records for 2-sumber-totals are set, subsequent bases seem to either match or surpass them, except in three cases below base 450:

2 in base 5
4 in base 7
6 in base 13
10 in base 43
14 in base 47
22 in base 157
8 in base 182*
16 in base 268*
30 in base 307
18 in base 443*

Totals for sums-of–squares in bases 4 to 450 (click for larger image)

Appendix: Odd Bases and 2-sumbers

Take an even number and half of that even number: say 12 and 6. 12 x 6 = 11 x 6 + 6. Further, 12 x 6 = 2 x 6 x 6 = 2 x 6^2 = 6^2 + 6^2. Accordingly, 66_[11] = 6 x 11 + 6 = 12 x 6 = 6^2 + 6^2. So 66 in base 11 is a 2-sumber. Similar reasoning applies to every other odd base except base-3 [update: wrong!]. Now, take 12 x 5 = 2 x 6 x 5 = 2 x (5×5 + 5) = 5^2+5 + 5^5+5 = 5^5 + 5^5+2×5. Further, 5^5+2×5 = (5+1)(5+1) – 1 = 6^2 – 1. Accordingly, 56_[11] = 11×5 + 6 = 12×5 + 1 = 5^2 + 6^2. Again, similar reasoning applies to every other odd base except base-3 [update: no — 1^2 + 2^2 = 12_[3] = 5; 2^2 + 2^2 = 22_[3] = 8]. This means that every odd base b, except base-3, will supply a pair of 2-sumbers with digits [d-1][d] and [d][d], where d = (b + 1) / 2.

Live and Let Dice

Thursday, 28th Mar, 2013 by Krilling for Company in Geometry, Mathematics, Probability, Statistics and tagged Big Bang, caltrop, caltrops, center of gravity, centre of gravity, cube, cubic dice, cubic die, death-dice, death-die, death-face, death-penalty, dice odds, dice-roll, dice-rolling, dice-rolls, die-rolling, dodecahedral dice, dodecahedral die, dodecahedron, edge, edges, execution, face, faces, faces and vertices, fair dice, geometric, geometry, hexahedral dice, hexahedral die, hexahedron, icosahedral dice, icosahedral die, icosahedron, life-imprisonment, math, mathematics, maths, octahedral dice, octahedral die, octahedron, odds, Plato, Plato's cosmology, Platonic polyhedra, Platonic polyhedron, polyhedra, polyhedral, polyhedron, probability, probability theory, roll a die, rolling dice, solid shape, solid shapes, spiked weapon, tetrahedral dice, tetrahedral die, tetrahedron, three-dimensional shape, three-dimensional shapes, Tycholatric, Tycholatry, vertex, vertices | Leave a comment

How many ways are there to die? The answer is actually five, if by “die” you mean “roll a die” and by “rolled die” you mean “Platonic polyhedron”. The Platonic polyhedra are the solid shapes in which each polygonal face and each vertex (meeting-point of the edges) are the same. There are surprisingly few. Search as long and as far as you like: you’ll find only five of them in this or any other universe. The standard cubic die is the most familiar: each of its six faces is square and each of its eight vertices is the meeting-point of three edges. The other four Platonic polyhedra are the tetrahedron, with four triangular faces and four vertices; the octahedron, with eight triangular faces and six vertices; the dodecahedron, with twelve pentagonal faces and twenty vertices; and the icosahedron, with twenty triangular faces and twelve vertices. Note the symmetries of face- and vertex-number: the dodecahedron can be created inside the icosahedron, and vice versa. Similarly, the cube, or hexahedron, can be created inside the octahedron, and vice versa. The tetrahedron is self-spawning and pairs itself. Plato wrote about these shapes in his Timaeus (c. 360 B.C.) and based a mathemystical cosmology on them, which is why they are called the Platonic polyhedra.

Tetrahedron

Hexahedron

Octahedron

Dodecahedron

Icosahedron

They make good dice because they have no preferred way to fall: each face has the same relationship with the other faces and the centre of gravity, so no face is likelier to land uppermost. Or downmost, in the case of the tetrahedron, which is why it is the basis of the caltrop. This is a spiked weapon, used for many centuries, that always lands with a sharp point pointing upwards, ready to wound the feet of men and horses or damage tyres and tracks. The other four Platonic polyhedra don’t have a particular role in warfare, as far as I know, but all five might have a role in jurisprudence and might raise an interesting question about probability. Suppose, in some strange Tycholatric, or fortune-worshipping, nation, that one face of each Platonic die represents death. A criminal convicted of a serious offence has to choose one of the five dice. The die is then rolled f times, or as many times as it has faces. If the death-face is rolled, the criminal is executed; if not, he is imprisoned for life.

The question is: Which die should he choose to minimize, or maximize, his chance of getting the death-face? Or doesn’t it matter? After all, for each die, the odds of rolling the death-face are 1/f and the die is rolled f times. Each face of the tetrahedron has a 1/4 chance of being chosen, but the tetrahedron is rolled only four times. For the icosahedron, it’s a much smaller 1/20 chance, but the die is rolled twenty times. Well, it does matter which die is chosen. To see which offers the best odds, you have to raise the odds of not getting the death-face to the power of f, like this:

3/4 x 3/4 x 3/4 x 3/4 = 3/4 ^4 = 27/256 = 0·316…

5/6 ^6 = 15,625 / 46,656 = 0·335…

7/8 ^8 = 5,764,801 / 16,777,216 = 0·344…

11/12 ^12 = 3,138,428,376,721 / 8,916,100,448,256 = 0·352…

19/20 ^20 = 37,589,973,457,545,958,193,355,601 / 104,857,600,000,000,000,000,000,000 = 0·358…

Those represent the odds of avoiding the death-face. Criminals who want to avoid execution should choose the icosahedron. For the odds of rolling the death-face, simply subtract the avoidance-odds from 1, like this:

1 – 3/4 ^4 = 0·684…

1 – 5/6 ^6 = 0·665…

1 – 7/8 ^8 = 0·656…

1 – 11/12 ^12 = 0·648…

1 – 19/20 ^20 = 0·642…

So criminals who prefer execution to life-imprisonment should choose the tetrahedron. If the Tycholatric nation offers freedom to every criminal who rolls the same face of the die f times, then the tetrahedron is also clearly best. The odds of rolling a single specified face f times are 1/f ^f:

1/4 x 1/4 x 1/4 x 1/4 = 1/4^4 = 1 / 256

1/6^6 = 1 / 46,656

1/8^8 = 1 / 16,777,216

1/12^12 = 1 / 8,916,100,448,256

1/20^20 = 1 / 104,857,600,000,000,000,000,000,000

But there are f faces on each polyhedron, so the odds of rolling any face f times are 1/f ^(f-1). On average, of every sixty-four (256/4) criminals who choose to roll the tetrahedron, one will roll the same face four times and be reprieved. If a hundred criminals face the death-penalty each year and all choose to roll the tetrahedron, one criminal will be reprieved roughly every eight months. But if all criminals choose to roll the icosahedron and they have been rolling since the Big Bang, just under fourteen billion years ago, it is very, very, very unlikely that any have yet been reprieved.

Rep-Tile Reflections

Saturday, 23rd Mar, 2013 by Krilling for Company in Astronomy, Geometry, Mathematics, Philosophy, Philosophy of mathematics, Rep-Tiles, Science and tagged alternative realities, alternative reality, √2, Big Bang, Big Crunch, Credo in Piscem Unum et Infinitum et Æternum, double-square, double-triangle, extraterrestrial intelligence, gas-based life, geometric, geometrical, geometry, half-square, hypotenuse, intelligence, intelligent life, math, math-based religion, math-religion, mathematical religion, mathemystic, mathemystic communion, mathemystical, mathemysticism, maths, maths-based religion, mental realm, metaphysical, metaphysics, mind, minds, mystic communism, parallel universe, parallel universes, powers of 9, powers of nine, psycho-math-space, rectangle, rectangles, Religion, rep-tile, Rep-Tiles, rep-triangle, rep-triangles, repeat-tile, repeat-tiles, repeating tile, repeating tiles, repetition-tile, repetition-tiles, right triangle, silicon-based life, speculation, speculatory, square, square root of 2, square root of two, Squares, tessellate, tessellation, tessellations, the fish, the sphinx, trans-biological communion, triangle, Triangles, universe | 5 Comments

A rep-tile, or repeat-tile, is a two-dimensional shape that can be divided completely into copies of itself. A square, for example, can be divided into smaller squares: four or nine or sixteen, and so on. Rectangles are the same. Triangles can be divided into two copies or three or more, depending on their precise shape. Here are some rep-tiles, including various rep-triangles:

Various rep-tiles — click for larger image

Some are simple, some are complex. Some have special names: the sphinx and the fish are easy to spot. I like both of those, particularly the fish. It would make a good symbol for a religion: richly evocative of life, eternally sub-divisible of self: 1, 9, 81, 729, 6561, 59049, 531441… I also like the double-square, the double-triangle and the T-tile in the top row. But perhaps the most potent, to my mind, is the half-square in the bottom left-hand corner. A single stroke sub-divides it, yet its hypotenuse, or longer side, represents the mysterious and mind-expanding √2, a number that exists nowhere in the physical universe. But the half-square itself is mind-expanding. All rep-tiles are. If intelligent life exists elsewhere in the universe, perhaps other minds are contemplating the fish or the sphinx or the half-square and musing thus: “If intelligent life exists elsewhere in the universe, perhaps…”

Mathematics unites human minds across barriers of language, culture and politics. But perhaps it unites minds across barriers of biology too. Imagine a form of life based on silicon or gas, on unguessable combinations of matter and energy in unreachable, unobservable parts of the universe. If it’s intelligent life and has discovered mathematics, it may also have discovered rep-tiles. And it may be contemplating the possibility of other minds doing the same. And why confine these speculations to this universe and this reality? In parallel universes, in alternative realities, minds may be contemplating rep-tiles and speculating in the same way. If our universe ends in a Big Crunch and then explodes again in a Big Bang, intelligent life may rise again and discover rep-tiles again and speculate again on their implications. The wildest speculation of all would be to hypothesize a psycho-math-space, a mental realm beyond time and matter where, in mathemystic communion, suitably attuned and aware minds can sense each other’s presence and even communicate.

Credo in Piscem…

So meditate on the fish or the sphinx or the half-square. Do you feel the tendrils of an alien mind brush your own? Are you in communion with a stone-being from the far past, a fire-being from the far future, a hive-being from a parallel universe? Well, probably not. And even if you do feel those mental tendrils, how would you know they’re really there? No, I doubt that the psycho-math-space exists. But it might and science might prove its existence one day. Another possibility is that there is no other intelligent life, never has been, and never will be. We may be the only ones who will ever muse on rep-tiles and other aspects of mathematics. Somehow, though, rep-tiles themselves seem to say that this isn’t so. Particularly the fish. It mimics life and can spawn itself eternally. As I said, it would make a good symbol for a religion: a mathemysticism of trans-biological communion. Credo in Piscem, Unum et Infinitum et Æternum. “I believe in the Fish, One, Unending, Everlasting.” That might be the motto of the religion. If you want to join it, simply wish upon the fish and muse on other minds, around other stars, who may be doing the same.

Numbered Days

Monday, 18th Mar, 2013 by Krilling for Company in Art, History, Mathematics, Philosophy, Philosophy of mathematics, Poetry and tagged a girl he hardly knew, arrogance, Arthur Rimbaud, Évariste Galois, biographical, biography, Carl Gustav Jacobi, Deus ex Machina, died early, duel, early death, French literature, French poet, French poetry, Galois theory, infamous coquette, literature, math, mathematics, maths, Niels Henrik Abel, nineteenth century, poet, Poetry, rebel, rebellion, Rimbaud's poetry, Srinivasa Ramanujan, stormy adolescence, Verlaine | Leave a comment

Numbered Days: Literature, Mathematics and the Deus Ex Machina

Think French. Think genius. Think rebellious, tormented, iconoclastic. Finally, think dead tragically young in the nineteenth century… And if you’re thinking of anyone at all, I think you’ll be thinking of Rimbaud.

And you’d be right to do so. But only half-right. Because there were in fact two rebellious, tormented, iconoclastic French geniuses who died tragically young in the nineteenth century. One was called Arthur Rimbaud (1854-91) and the other Évariste Galois (1811-32). Rimbaud is still famous, Galois never has been. At least not to the general educated public, though on all objective criteria – but one – you might expect his fame to be greater. In every way – but one – Galois has the more powerful appeal.

Continue reading Numbered Days…

Pest Test

Monday, 18th Mar, 2013 by Krilling for Company in Statistics and tagged false negative, false negatives, false positive, false positives, graph, graphs, infection rate, infection rates, math, mathematical paradox, mathematics, maths, statistical, statistics, test for infection | Leave a comment

Health warning: I am not a mathematician. That said, here is a mathematical question:

Suppose there is a 99% accurate test for a medical condition – say a symptomless infection. You take the test and get a positive result. What are your chances of having the infection?

That obvious answer might seem to be 99%. But the obvious answer is wrong. The accuracy of the test is only half the information you need to answer the question. You also need to know how common the infection is. Say it occurs once in every hundred people. On average, then, if you test a hundred people, one of whom has the infection, you will get two positive results: one that is accurate and one that is inaccurate, i.e., a false positive. Under those circumstances, a positive result means that you have a ½, or 50%, chance of having the infection (see appendix for further discussion). Under some other circumstances, a positive result on an 80% or 90% accurate test would mean that you have a higher chance of having the infection. Here’s a graphic to illustrate this apparent paradox:

The x-axis represents infection rate per 10,000 of the population, the y-axis represents one’s chance of being infected, from 0%, for no chance, to 100%, for complete certainty. The coloured curves represent tests of different accuracy: 1% accurate, for the bottom curve, and 99% accurate, for the uppermost curve. The curves between the two represent tests of 10% to 90% accuracy. Note how the curves mirror each other: the 99% accurate test rises towards certainty very quickly, but takes a long time to finally get there. The 1% accurate test stays near complete uncertainty for a long time, then finally rises rapidly towards certainty. In other words, a positive result on a 99% accurate test is equivalent to a negative result on a 1% accurate test, and vice versa. Ditto for the 90% and 10% accurate tests, and so on. But a positive (or negative) result on a 50% accurate test is useless, because it never tells you anything new: your chance of being infected, given a positive result, is the same as the rate of infection in the population. And when exactly half the population is infected, your chance of being infected, given a positive result, is the same as the accuracy of the test, whether it’s 1%, 50%, or 99%.

Here is a table illustrating the same points:

Accuracy of test → Infection rate ↓	1%	10%	20%	30%	40%	50%	60%	70%	80%	90%	99%
1/100	<1%	0.1%	0.3%	0.4%	0.7%	1%	1.5%	2.3%	3.9%	8.3%	50%
10/100	0.1%	1.2%	2.7%	4.5%	6.9%	10%	14.3%	20.6%	30.8%	50%	91.7%
20/100	0.3%	2.7%	5.9%	9.7%	14.3%	20%	27.3%	36.8%	50%	69.2%	96.1%
30/100	0.4%	4.5%	9.7%	15.5%	22.2%	30%	39.1%	50%	63.2%	79.4%	97.7%
40/100	0.7%	6.9%	14.3%	22.2%	30.8%	40%	50%	60.9%	72.7%	85.7%	98.5%
50/100	1%	10%	20%	30%	40%	50%	60%	70%	80%	90%	99%
60/100	1.5%	14.3%	27.3%	39.1%	50%	60%	69.2%	77.8%	85.7%	93.1%	99.3%
70/100	2.3%	20.6%	36.8%	50%	60.9%	70%	77.8%	84.5%	90.3%	95.5%	99.6%
80/100	3.9%	30.8%	50%	63.2%	72.7%	80%	85.7%	90.3%	94.1%	97.3%	99.7%
90/100	8.3%	50%	69.2%	79.4%	85.7%	90%	93.1%	95.5%	97.3%	98.8%	99.9%
99/100	50%	91.7%	96.1%	97.7%	98.5%	99%	99.3%	99.6%	99.7%	99.9%	>99.9%
100/100	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%

Appendix

We’ve seen that we have to take false positives into account, but what about false negatives? Suppose that the rate of infection is 1 in 100 and the accuracy of the test is 99%. If the population is 10,000, then 100 people will have the disease and 9,900 will not. If the population is tested, on average 100 x 99% = 99 of the infected people will get an accurate positive result and 100 x 1% = 1 will get an inaccurate negative result, i.e., a false negative. Similarly, 9,900 x 1% = 99 of the non-infected people will get a false positive. So there will be 99 + 99 = 198 positive results, of which 99 are accurate. 99/198 = 1/2 = 50%.

Yew and Me

Thursday, 7th Mar, 2013 by Krilling for Company in A.E. Housman, Art, Biology, Book Reviews, Fibonacci Sequence, Fractals, Mathematics, Poetry and tagged A.E. Housman, Alan Mitchell, artist, aspen, aspens, Black Walnut, churches, churchyards, Common Walnut, Common Yew, conifer, conifers, David More, deciduous tree, deciduous trees, dendrophile, dendrophilia, Fibonacci Sequence, fractal structure, furniture, gunstocks, holy site, holy sites, Housman, illustration, illustrations, illustrator, Juglans nigra, Juglans regia, leaf, leaves, math, mathematics, maths, natural fractal, natural fractals, photosynthesis, pine-cone, pine-cones, poisonouns foliage, poisonous leaves, poisonous seeds, Populus tremula, sacred tree, sacred trees, symbol of immortality, symbols of immortality, Taxus baccata, The Pocket Guide to The Trees of Britain and Northern Europe, The Trees of Britain and Northern Europe, tree, tree-poem, tree-poems, tree-poetry, Trees, yew, yews | 3 Comments

The Pocket Guide to The Trees of Britain and Northern Europe, Alan Mitchell, illustrated by David More (1990)

Leafing through this book after I first bought it, I suddenly grabbed at it, because I thought one of the illustrations was real and that a leaf was about to slide off the page and drop to the floor. It was an easy mistake to make, because David More is a good artist. That isn’t surprising: good artists are often attracted to trees. I think it’s a mathemattraction. Trees are one of the clearest and commonest examples of natural fractals, or shapes that mirror themselves on smaller and smaller scales. In trees, trunks divide into branches into branchlets into twigs into twiglets, where the leaves, well distributed in space, wait to eat the sun.

When deciduous, or leaf-dropping, trees go hungry during the winter, this fractal structure is laid bare. And when you look at a bare tree, you’re looking at yourself, because humans are fractals too. Our torsos sprout arms sprout hands sprout fingers. Our veins become veinlets become capillaries. Ditto our lungs and nervous systems. We start big and get small, mirroring ourselves on smaller and smaller scales. Fractals make maximum and most efficient use of space and what’s found in me or thee is also found in a tree, both above and below ground. The roots of a tree are also fractals. But one big difference between trees and people is that trees are much freer to vary their general shape. Trees aren’t mirror-symmetrical like animals and that’s another thing that attracts human eyes and human artists. Each tree is unique, shaped by the chance of its seeding and setting, though each species has its characteristic silhouette. David More occasionally shows that bare winter silhouette, but usually draws the trees in full leaf, disposed to eat the sun. Trees can also be identified by their leaves alone and leaves too are fractals. The veins of a leaf divide and sub-divide, carrying the raw materials and the finished products of photosynthesis to and from the trunk and roots. Trees are giants that work on a microscopic scale, manufacturing themselves from photons and molecules of water and carbon dioxide.

We eat or sculpt what they manufacture, as Alan Mitchell describes in the text of this book:

The name “Walnut” comes from the Anglo-Saxon for “foreign nut” and was in use before the Norman Conquest, probably dating from Roman times. It may refer to the fruit rather than the tree but the Common Walnut, Juglans regia, has been grown in Britain for a very long time. The Romans associated their god Jupiter (Jove) with this tree, hence the Latin name juglans, “Jove’s acorn (glans) or nut”… The wood [of Black Walnut, Juglans nigra] is like that of Common Walnut and both are unsurpassed for use as gunstocks because, once seasoned and worked, neither moves at all and they withstand shock particularly well. They are also valued in furniture for their good colour and their ability to take a high polish. (entry for “Walnuts”, pg. 18)

That’s from the first and longer section, devoted to “Broadleaved Trees and Palms”; in the second section, “Conifers”, devoted to pines and their relatives, maths appears in a new form. Pine-cones embody the Fibonacci sequence, one of the most famous of all number sequences or series. Start with 1 and 1, then add the pair and go on adding pairs: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… That’s the Fibonacci sequence, named after the Italian mathematician Leonardo Fibonacci (c.1170-c.1245). And if you examine the two spirals created by the scales of a pine-cone, clockwise and counter-clockwise, you’ll find that there are, say, five spirals in one direction and eight in another, or eight and thirteen. The scales of a pineapple and petals of many flowers behave in a similar way. These patterns aren’t fractals like branches and leaves, but they’re also about distributing living matter efficiently through space. Mitchell doesn’t discuss any of this mathematics, but it’s there implicitly in the illustrations and underlies his text. Even the toxicity of the yew is ultimately mathematical, because the effect of toxins is determined by their chemical shape and its interaction with the chemicals in our bodies. Micro-geometry can be noxious. Or nourishing:

The Yews are a group of conifers, much more primitive than those which bear cones. Each berry-like fruit has a single large seed, partially enclosed in a succulent red aril which grows up around it. The seed is, like the foliage, very poisonous to people and many animals, but deer and rabbits eat the leaves without harm. Yew has extremely strong and durable wood [and the] Common Yew, Taxus baccata, is nearly immortal, resistant to almost every pest and disease of importance, and immune to stress from exposure, drought and cold. It is by a long way the longest-living tree we have and many in country churchyards are certainly much older than the churches, often thousands of years old. Since the yews pre-date the churches, the sites may have been holy sites and the yews sacred trees, possibly symbols of immortality, under which the Elders met. (entry for “Yews”, pg. 92)

This isn’t a big book, but there’s a lot to look at and read. I’d like a doubtful etymology to be true: some say “book” is related to “beech”, because beech-bark or beech-leaves were used for writing on. Bark is another way of identifying a tree and another aspect of their dendro-mathematics, in its texture, colours and patterns. But trees can please the ear as well as the eye: the dendrophile A.E. Housman (1859-1936) recorded how “…overhead the aspen heaves / Its rainy-sounding silver leaves” (A Shropshire Lad, XXVI). There’s maths there too. An Aspen sounds like rain in part because its many leaves, which tremble even in the lightest breeze, are acting like many rain-drops. That trembling is reflected in the tree’s scientific name: Populus tremula, “trembling poplar”. Housman, a Latin professor as well as an English poet, could have explained how tree-nouns in Latin are masculine in form: Alnus, Pinus, Ulmus; but feminine in gender: A. glutinosa, P. contorta, U. glabra (Common Alder, Lodgepole Pine, Wych-Elm). He also sums up why trees please in these simple and ancient words of English:

Give me a land of boughs in leaf,
A land of trees that stand;
Where trees are fallen, there is grief;
I love no leafless land.

More Poems, VIII.

Angst, Anguish, Abjection

Thursday, 7th Mar, 2013 by Krilling for Company in Album Art, Astronomy, Mathematics, Music, Music reviews and tagged A.W.D.I.M.T.B.F., a3, And What Does It Mean To Be Flesh?, And What Doth It Mean To Be Flesh?, AWDIMTBF, bat-twitter, bat-twitters, bee, bees, Breathing Vacuum, bumblebee, bumblebees, carnal question, Dredge-head, Dredge-heads, Dredgers, ear-worm, ear-worms, fleshtion, gig, gigs, griff, griffs, gruff riff, gruff riffs, industrial noise, infra-sonic, infra-sound, Invention of the Cross, Jerome Daziel, Kargokkult, Kargonauts, lead-singer Jerome Daziel, math, mathematics, maths, Nyctogigas, Ohrwürmer, Ohrwurm, plagiarism suit, pre-biotic, Pyramidion, resonance, resonances, Ruin-Dredger, Scylla / Charybdis, Seventh Sword, silence, star-clusters having tantrums, that A-fucker, Titanomachia, Triple-A, ultra-sonic, ultra-sound, unusual frequencies, unusual frequency, What Does It Mean To Be Flesh?, What Doth It Mean To Be Flesh?, Whilom, wolf-howl, wolves howling, Xoli-Hein | Leave a comment

It’s half tradition, half tic. At every Ruin-Dredger gig, the lead-singer Jerome Daziel asks the same simple question. Sometimes he shouts it and demands a reaction from the audience. Sometimes he whispers it and ignores what the audience does. Depending on the country, he’s asked it in French, Italian, Greek, Russian, Georgian, Mandarin, Thai, Samoan and Quechua. He’s also asked it in complete silence, having written it across his chest and on the palms of his hands in phosph-ink, invisible when the lights are on, glowing ghoulishly when they’re turned off. Occasionally he’s asked it backwards. In English, the question runs like this: “And What Doth It Mean To Be Flesh?”

But you could see the whole of a Ruin-Dredger gig as asking the same searching thing. The band specialize in unusual frequencies that hunt out – and hum out – the resonances of the human body: the lungs, the bones, the blood. And their music sets up strange resonances in the mind. It’s both mindless and masterful, at once tearful and tyrannous. Sometimes it sounds like mathematics trying to come to life, and sometimes like mathematics trying to commit suicide. There’s a lot of science in their music, and a lot of silence too. “Star-clusters having tantrums,” is how one early review ran. “With occasional episodes of narcolepsy.” That mixture of sound and silence is mutually reinforcing: the sounds are sterner, the silence is sharper. They began their career with the albums Xoli-Hein (1992) and Pyramidion (1996), where they forged a series of griffs, or “gruff riffs”, that were often Ohrwürmer, or “ear-worms”, as German calls tunes that stick in your head. Even if you don’t want them to. But I’m not sure “tune” has ever been the right word for the music Ruin-Dredger create. It’s part industrial noise, part wolf-howl, part bat-twitter, but mostly “folded, fused, fissured, fractured, fidgety phonaesthesia.” And if you want to sample it, this album from the turn of the century is a good place to start.

What to call the album is one of the first puzzles it will set you. The band’s website usually calls it “a³” or “a3”; in interviews, the band themselves refer to it as “Triple-A” or “that A-fucker”. The second name comes from a plagiarism suit by the astro-music veterans Kargokkult that put Ruin-Dredger’s career on hold for nearly a year, 2002-3, and allegedly threatened to bankrupt their record-company. In the end the case was thrown out of court and even today some conspiracy-minded Dredge-heads claim it was cooked up for publicity between the ’Dredgers and the Kargonauts. The case might never have got as far as it did without that lunar cover for Triple-A, where the corroded letters of the band’s name and the album’s name hang above a lifeless moon-scape. Only it isn’t our moon. And it isn’t necessarily lifeless. Ruin-Dredger have a bee in their bonnet about the pre-biotic – the conditions necessary for the appearance of life. That’s what the first track on Triple-A, “Invention of the Cross”, is about: the chemicals that gave rise to life. And it literally has bees on it: the band sampled bees and bumblebees in flight and gathering nectar. They then altered the pitch and speed of the buzzing and made it sound both unearthly and unsettling. I’ve known people demand the track be turned off or skipped when it’s played to them.

But skipping track one of Triple-A is a bit like jumping from the frying-pan into the fire, because track two, “Seventh Sword”, is even more unearthly and even more unsettling. Bat-twitters hurtle through the speakers, falling from the ultra-sonic to the infra-sonic, rising in reverse, twisting, turning inside-out, mating, mutating and miscegenating. Then, as though the band have taken mercy on your ears and your mind, everything slows and soothes for track three, “Titanomachia”, which is often preceded in concert by the aforementioned carnal question: “And what doth it mean to be flesh?” This track is one of the last outings for the griffs of their early career: a slow, synth-based triple chord underlain by a sample of waves washing on an unknown shore. Track four, “Breathing Vacuum”, has also been known to provoke a “Turn it off!”, because the mumbling beneath the music is both sinister and sorrowful. You feel as though you should understand the words or, worse, that you will in your dreams. The chimes in the track are sinister too: they sound like a deep-sea, or deep-space, monster tapping on its fangs before putting them to famished use.

Which sets things up nicely, or nastily, for track five, “Scylla / Charybdis”. This is named after a pair of sea-monsters faced by Odysseus on his journey home from Troy and has been described by the ’Dredgers as a “battle-song”. The waves on “Titanomachia” are back, but bigger, badder and in a mood to fight. Daziel’s electronically treated voice wolf-howls a series of unintelligible questions, answered by patches of silence and gong-like drum-rolls. Track six, “Nyctogigas”, starts softly, builds back to the volume and violence of “Scyl/Char”, then breaks apart to allow the bats and bees of “Whilom” to steer your imagination out and up into the freezing star-light on the outer fringes of the solar system, where comets, shorn by the cold and dark, wait to swing sun-ward and regain their blazing locks. I like to listen to “Whilom” in the dark, wearing a blindfold, but then that’s the best way to listen to all of Ruin-Dredger’s music. Listening like that conjures visions and commands the viscera. Not an easy album, nor an unrewarding one, Triple-A isn’t their finest hour, if fan-polls and sales are any guide, but it’s an excellent guide to where they had come from and where they were about to go. If it’s the alpha-and-omega of their career, perhaps that explains the title: the “a” is the alpha (α) and the “3” an omega (ω) tipped on its side. I see it, or hear it, as a bridge between the ’nineties and the ’noughties: they’d give up the griffs and big up the bats, from then on, but they’ve never stopped asking that simple, sinister/sorrowful question of themselves and their listeners: “And What Doth It Mean To Be Flesh?”

a³ / a3 / Triple-A (S.R.K., 2000)

1. Invention of the Cross (5:26)
2. Seventh Sword (3:33)
3. Titanomachia (7:18)
4. Breathing Vacuum (9:03)
5. Scylla / Charybdis (6:11)
6. Nyctogigas (4:20)
7. Whilom (13:37)

Three Is The Key

Saturday, 23rd Feb, 2013 by Krilling for Company in Art, Magic squares, Mathematics, Post-Weird, Prime numbers and tagged 142857, 222, 223, 3, 37, David Wells, magic square, magic squares, math, mathematics, maths, Penguin Dictionary of Curious and Interesting Numbers, prime numbers, prime reciprocal, prime reciprocal magic square, prime reciprocal magic squares, prime reciprocals, primes, reciprocal, reciprocals, recreational math, recreational mathematics, recreational maths, Roses of Heliogabalus, Sir Lawrence Alma-Tadema, two hundred and twenty-two | Leave a comment

If The Roses of Heliogabalus (1888) is any guide, Sir Lawrence Alma-Tadema (1836-1912) thought that 222 is a special number. But his painting doesn’t exhaust its secrets. To get to another curiosity of 222, start with 142857. As David Wells puts it in his Penguin Dictionary of Curious and Interesting Numbers (1986), 142857 is a “number beloved of all recreational mathematicians”. He then describes some of its properties, including this:

142857 x 1 = 142857
142857 x 2 = 285714
142857 x 3 = 428571
142857 x 4 = 571428
142857 x 5 = 714285
142857 x 6 = 857142

The multiples are cyclic permutations: the order of the six numbers doesn’t change, only their starting point. Because each row contains the same numbers, it sums to the same total: 1 + 4 + 2 + 8 + 5 + 7 = 27. And because each row begins with a different number, each column contains the same six numbers and also sums to 27, like this:

1 4 2 8 5 7
+ + + + + +
2 8 5 7 1 4
+ + + + + +
4 2 8 5 7 1
+ + + + + +
5 7 1 4 2 8
+ + + + + +
7 1 4 2 8 5
+ + + + + +
8 5 7 1 4 2

= = = = = =

2 2 2 2 2 2
7 7 7 7 7 7

If the diagonals of the square also summed to the same total, the multiples of 142857 would create a full magic square. But the diagonals don’t have the same total: the left-right diagonal sums to 31 and the right-left to 23 (note that 31 + 23 = 54 = 27 x 2).

But where does 142857 come from? It’s actually the first six digits of the reciprocal of 7, i.e. 1/7 = 0·142857… Those six numbers repeat for ever, because 1/7 is a prime reciprocal with maximum period: when you calculate 1/7, all integers below 7 are represented in the remainders. The square of multiples above is simply another way of representing this:

1/7 = 0·142857…
2/7 = 0·285714…
3/7 = 0·428571…
4/7 = 0·571428…
5/7 = 0·714285…
6/7 = 0·857142…
7/7 = 0·999999…

The prime reciprocals 1/17 and 1/19 also have maximum period, so the squares created by their multiples have the same property: each row and each column sums to the same total, 72 and 81, respectively. But the 1/19 square has an additional property: both diagonals sum to 81, so it is fully magic:

01/19 = 0·0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1…
02/19 = 0·1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2…
03/19 = 0·1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3…
04/19 = 0·2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4…
05/19 = 0·2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5…
06/19 = 0·3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6…
07/19 = 0·3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7…
08/19 = 0·4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8…
09/19 = 0·4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9…
10/19 = 0·5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0…
11/19 = 0·5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1…
12/19 = 0·6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2…
13/19 = 0·6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3…
14/19 = 0·7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4…
15/19 = 0·7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5…
16/19 = 0·8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6…
17/19 = 0·8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7…
18/19 = 0·9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8…

First line = 0 + 5 + 2 + 6 + 3 + 1 + 5 + 7 + 8 + 9 + 4 + 7 + 3 + 6 + 8 + 4 + 2 + 1 = 81

Left-right diagonal = 0 + 0 + 7 + 5 + 5 + 9 + 0 + 3 + 0 + 4 + 2 + 8 + 7 + 5 + 6 + 7 + 5 + 8 = 81

Right-left diagonal = 9 + 9 + 2 + 4 + 4 + 0 + 9 + 6 + 9 + 5 + 7 + 1 + 2 + 4 + 3 + 2 + 4 + 1 = 81

In base 10, this doesn’t happen again until the 1/383 square, whose magic total is 1719 (= 383-1 x 10-1 / 2). But recreational maths isn’t restricted to base 10 and lots more magic squares are created by lots more primes in lots more bases. The prime 223 in base 3 is one of them. Here the first line is

1/223 = 1/22021₃ = 0·

0000100210210102121211101202221112202
2110211112001012200122102202002122220
2110110201020210001211000222011010010
2222122012012120101011121020001110020
0112011110221210022100120020220100002
0112112021202012221011222000211212212…

The digits sum to 222, so 222 is the magic total for all rows and columns of the 1/223 square. It is also the total for both diagonals, so the square is fully magic. I doubt that Alma-Tadema knew this, because he lived before computers made calculations like that fast and easy. But he was probably a Freemason and, if so, would have been pleased to learn that 222 had a link with squares.

Roses Are Golden

Thursday, 14th Feb, 2013 by Krilling for Company in Art, Civilization, History, φ, Mathematics and tagged Aelius Lampridius, Alma-Tadema, ancient Rome, Anglo-Dutch, classical history, Elagabalus, golden ratio, golden section, Heliogabalus, Historia Augusta, φ, Liverpool, math, mathematics, mathematics in art, maths, oil-painting, painting, phi, Roman emperor, Roman history, Sir Laurence Alma-Tadema, Sir Lawrence Alma-Tadema, The Roses of Heliogabalus, Victorian art, Victorian artist, Victorian painting, Vita Heliogabali, Walker Art Gallery | Leave a comment

Sir Lawrence Alma-Tadema’s painting The Roses of Heliogabalus (1888) is based on an apocryphal episode in the sybaritic life of the Roman Emperor Elagabalus (204-222 A.D.), who is said to have suffocated guests with flowers at one of his feasts. The painting is in a private collection, but I saw it for real in an Alma-Tadema exhibition at the Walker Art Gallery in Liverpool sometime during the late 1990s. I wasn’t disappointed: it was a memorable meeting with a painting I’d been interested in for years. Roses is impressively large and impressively skilful. Close-up, the brush-strokes are obvious, obtrusive and hard to interpret as people and objects. It isn’t till you step back, far beyond the distance at which Alma-Tadema was painting, that the almost photographic realism becomes apparent. But you get more of the many details at close range, like the Latin inscription on a bowl below and slightly to the right of that scowling water-mask. Alas, I forgot to take a note of what the inscription was, though perhaps the memory is still locked away somewhere in my subconscious.

The Roses of Heliogabalus (1888)

Whatever it is, I feel sure it is significant, because Roses is rich with meaning. That’s a large part of why I’m interested in it. Yes, I like it a lot as art, but the women would have to be more attractive for it to be higher in the list of my favourite paintings. As it is, I think there are only four reasonably good-looking people in it: the man with the beard on the right; the flautist striding past the marble pillar on the left; the red-headed girl with a crown of white flowers; and Heliogabalus himself, crowned in roses and clutching a handful of grapes beside the overweight man who’s wearing a wreath and sardonically saluting one of the rose-pelted guests in the foreground. When I first wrote about Roses in a pub-zine whose name escapes me, I misidentified the overweight man as Heliogabalus himself, even though I noted that he seemed many years old than Heliogabalus, toppled as a teen tyrant, should have been. It was a bad mistake, but one that, with less knowledge and more excuse, many people must make when they look at Roses, because the overweight man and his sardonic salute are a natural focus for the eye. Once your eye has settled on and noted him, you naturally follow the direction of his gaze down to the man in the foreground, who’s gazing right back.

A comparison between Alma-Tadema's portrayal of Heliogabalus and a bust of Heliogabalus from the Musei Capitolini in Rome

Something Like the Sun

And by following that gaze, you’ve performed a little ratio-ritual, just as Alma-Tadema intended you to do. Yes, Roses is full of meaning and much of that meaning is mathematical. I think the angle of the gaze is one of many references in Roses to the golden ratio, or φ (phi), a number that is supposed to have special aesthetic importance and has certainly been used by many artists and musicians to guide their work. A rectangle with sides in the proportions 8:13, for example, approximates the golden ratio pretty closely, but φ itself is impossible to represent physically, because it’s an irrational number with infinitely many decimal digits, like π or √2, the square root of two. π represents the ratio of a circle’s circumference to its diameter and √2 the ratio of a square’s diagonal to its side, but no earthly circle and no earthly square can ever capture these numbers with infinite precision. Similarly, no earthly rectangle can capture φ, but the rectangle of Roses is a good attempt, because it measures 52″ x 84 ¹/₈". That extra eighth of an inch was my first clue to the painting’s mathematical meaningfulness. And sure enough, 52/84·125 = 416/673 = 0·61812…, which is a good approximation to φ’s never-ending 0·6180339887498948482045868343656…*

That deliberate choice of dimensions for the canvas led me to look for more instances of φ in the painting, though one of the most important and obvious might be called a meta-presence. The Roses of Heliogabalus is dated 1888, or 1666 years after the death of Heliogabalus in 222 AD. A radius at 222º divides a circle in the golden ratio, because 222/360 = 0·616… It’s very hard to believe Alma-Tadema didn’t intend this reference and I also think there’s something significant in 1888 itself, which equals 2 x 2 x 2 x 2 x 2 x 59 = 2⁵ x 59. Recall that 416 is the expanded short side of Roses. This equals 2⁵ x 13, while 673, the expanded long side, is the first prime number after 666. As one of the most technically skilled painters who ever lived, Alma-Tadema was certainly an exceptional implicit mathematician. But he clearly had explicit mathematical knowledge too and this painting is a phi-pie cooked by a master matho-chef. In short, when Roses is read, Roses turns out to be golden.

*φ is more usually represented as 1·6180339887498948482045868343656…, but it has the pecularity that ¹/_φ = φ-1, so the decimal digits don’t change and 0·6180339887498948482045868343656… is also legitimate.

Appendix I

I’ve looked at more of Alma-Tadema’s paintings to see if their dimensions approximate φ, √2, √3 or π, or their reciprocals. These were the results (ε = error, i.e. the difference between the constant and the ratio of the dimensions).

The Roman Wine Tasters (1861), 50" x 69 ²/₃": 150/209 = 0·717… ≈ ¹/_√2 (ε=0·02)
A Roman Scribe (1865), 21 ¹/₂" x 15 ¹/₂": 43/31 = 1·387… ≈ √2 (ε=0·027)
A Picture Gallery (1866), 16 ¹/₈" x 23": 129/184 = 0·701… ≈ ¹/_√2 (ε=0·012)
A Roman Dance (1866), 16 ¹/₈" x 22 ¹/₈": 43/59 = 0·728… ≈ ¹/_√2 (ε=0·042)
In the Peristyle (1866), 23" x 16": 23/16 = 1·437… ≈ √2 (ε=0·023)
Proclaiming Emperor Claudius (1867), 18 ¹/₂" x 26 ¹/₃": 111/158 = 0·702… ≈ ¹/_√2 (ε=0·009)
Phidias and the Frieze of the Parthenon Athens (1868), 29 ²/₃" x 42 ¹/₃": 89/127 = 0·7… ≈ ¹/_√2 (ε=0·012)
The Education of Children of Clovis (1868), 50" x 69 ²/₃": 150/209 = 0·717… ≈ ¹/_√2 (ε=0·02)
An Egyptian Juggler (1870), 31" x 19 ¹/₄": 124/77 = 1·61… ≈ φ (ε=0·007)
A Roman Art Lover (1870), 29" x 40": 29/40 = 0·725… ≈ ¹/_√2 (ε=0·034)
Good Friends (1873), 4 ¹/₂" x 7 ¹/₄": 18/29 = 0·62… ≈ φ (ε=0·006)
Pleading (1876), 8 ¹/₂" x 12 ³/₈": 68/99 = 0·686… ≈ ¹/_√2 (ε=0·041)
An Oleander (1882), 36 ¹/₂" x 25 ¹/₂": 73/51 = 1·431… ≈ √2 (ε=0·017)
Dolce Far Niente (1882), 9 ¹/₄" x 6 ¹/₂": 37/26 = 1·423… ≈ √2 (ε=0·008)
Anthony and Cleopatra (1884), 25 ³/₄" x 36 ¹/₃": 309/436 = 0·708… ≈ ¹/_√2 (ε=0·003)
Rose of All Roses (1885), 15 ¹/₄" x 9 ¹/₄": 61/37 = 1·648… ≈ φ (ε=0·03)
The Roses of Heliogabalus (1888), 52" x 84 ¹/₈": 416/673 = 0·618… ≈ φ (ε<0.001)
The Kiss (1891), 18" x 24 ³/₄": 8/11 = 0·727… ≈ ¹/_√2 (ε=0·039)
Unconscious Rivals (1893), 17 ³/₄" x 24 ³/₄": 71/99 = 0·717… ≈ ¹/_√2 (ε=0·019)
A Coign of Vantage (1895), 25 ¹/₄" x 17 ¹/₂": 101/70 = 1·442… ≈ √2 (ε=0·028)
A Difference of Opinion (1896), 15" x 9": 5/3 = 1·666… ≈ φ (ε=0·048)
Whispering Noon (1896), 22" x 15 ¹/₂": 44/31 = 1·419… ≈ √2 (ε=0·005)
Her Eyes Are With Her Thoughts And Her Thoughts Are Far Away (1897), 9" x 15": 3/5 = 0·6… ≈ φ (ε=0·048)
The Baths of Caracalla (1899), 60" x 37 ¹/₂": 8/5 = 1·6… ≈ φ (ε=0·018)
The Year’s at the Spring, All’s Right with the World (1902), 13 ¹/₂" x 9 ¹/₂": 27/19 = 1·421… ≈ √2 (ε=0·006)
Ask Me No More (1906), 31 ¹/₂" x 45 ¹/₂": 9/13 = 0·692… ≈ ¹/_√2 (ε=0·03)

Appendix II

The Roses of Heliogabalus is based on this section from Aelius Lampridius’ pseudonymous and largely apocryphal Vita Heliogabali, or Life of Heliogabalus, in the Historia Augusta (late fourth century):

XXI. 1 Canes iecineribus anserum pavit. Habuit leones et leopardos exarmatos in deliciis, quos edoctos per mansuetarios subito ad secundam et tertiam mensam iubebat accumbere ignorantibus cunctis, quod exarmati essent, ad pavorem ridiculum excitandum. 2 Misit et uvas Apamenas in praesepia equis suis et psittacis atque fasianis leones pavit et alia animalia. 3 Exhibuit et sumina apruna per dies decem tricena cottidie cum suis vulvis, pisum cum aureis, lentem cum cerauniis, fabam cum electris, orizam cum albis exhibens. 4 Albas praeterea in vicem piperis piscibus et tuberibus conspersit. 5 Oppressit in tricliniis versatilibus parasitos suos violis et floribus, sic ut animam aliqui efflaverint, cum erepere ad summum non possent. 6 Condito piscinas et solia temperavit et rosato atque absentato…

Historia Augusta: Vita Heliogabali

XXI. 1 He fed his dogs on goose-livers. He had pet lions and leopards, which had been rendered harmless and trained by tamers, and these he would suddenly order during the dessert and the after-dessert to get on the couches, thereby causing laughter and panic, for none knew that they were harmless. 2 He sent grapes from Apamea to his stables for the horses, and he fed parrots and pheasants to his lions and other beasts. 3 For ten days in a row, moreover, he served wild sows’ udders with the matrices, at a rate of thirty a day, serving, besides, peas with gold-pieces, lentils with onyx, beans with amber, and rice with pearls; 4 and he also sprinkled pearls on fish and used truffles instead of pepper. 5 In a banqueting-room with a reversible ceiling he once buried his parasites in violets and other flowers, so that some were actually smothered to death, being unable to crawl out to the top. 6 He flavoured his swimming-pools and bath-tubs with essence of spices or of roses or wormwood…

Augustan History: Life of Heliogabalus

Summer-Climb Views

Thursday, 31st Jan, 2013 by Krilling for Company in Mathematics, Palindromic numbers and tagged algorithm, algorithms, amateur math, amateur mathematics, amateur maths, digit sum, digit sums, lychrel number, lychrel numbers, math, mathematical computing, mathematics, maths, palindrome, palindromes, palindromic, recreational math, recreational mathematics, recreational maths | Leave a comment

Simple things can sometimes baffle advanced minds. If you take a number, reverse its digits, add the result to the original number, then repeat all that, will you eventually get a palindrome? (I.e., a number, like 343 or 27172, that reads the same in both directions.) Many numbers do seem to produce palindromes sooner or later. Here are 195 and 197:

195 + 591 = 786 + 687 = 1473 + 3741 = 5214 + 4125 = 9339 (4 steps)

197 + 791 = 988 + 889 = 1877 + 7781 = 9658 + 8569 = 18227 + 72281 = 90508 + 80509 = 171017 + 710171 = 881188 (7 steps)

But what about 196? Well, it starts like this:

196 + 691 = 887 + 788 = 1675 + 5761 = 7436 + 6347 = 13783 + 38731 = 52514 + 41525 = 94039 + 93049 = 187088 + 880781 = 1067869 + 9687601 = 10755470 + 7455701 = 18211171 + 17111281 = 35322452 + 25422353 = 60744805 + 50844706 = 111589511 + 115985111 = 227574622 + 226475722 = 454050344 + 443050454 = 897100798 + 897001798 = 1794102596 + 6952014971 = 8746117567 + 7657116478 = 16403234045 + 54043230461 = 70446464506 + 60546464407 = 130992928913 + 319829299031 = 450822227944 + 449722228054 = 900544455998…

And so far, after literally years of computing by mathematicians, it hasn’t produced a palindrome. It seems very unlikely it ever will, but no-one can prove this and say that 196 is, in base 10, a Lychrel number, or a number that never produces a palindrome. In other words, a simple thing has baffled advanced minds.

I don’t know whether it can baffle advanced minds, but here’s another simple mathematical technique: sum all the digits of a number, then add the result to the original number and repeat. How long before a palindrome appears in this case? Sum it and see:

10 + 1 = 11

12 + 3 = 15 + 6 = 21 + 3 = 24 + 6 = 30 + 3 = 33 (5 steps)

13 + 4 = 17 + 8 = 25 + 7 = 32 + 5 = 37 + 10 = 47 + 11 = 58 + 13 = 71 + 8 = 79 + 16 = 95 + 14 = 109 + 10 = 119 + 11 = 130 + 4 = 134 + 8 = 142 + 7 = 149 + 14 = 163 + 10 = 173 + 11 = 184 + 13 = 197 + 17 = 214 + 7 = 221 + 5 = 226 + 10 = 236 + 11 = 247 + 13 = 260 + 8 = 268 + 16 = 284 + 14 = 298 + 19 = 317 + 11 = 328 + 13 = 341 + 8 = 349 + 16 = 365 + 14 = 379 + 19 = 398 + 20 = 418 + 13 = 431 + 8 = 439 + 16 = 455 + 14 = 469 + 19 = 488 + 20 = 508 + 13 = 521 + 8 = 529 + 16 = 545 (45 steps)

14 + 5 = 19 + 10 = 29 + 11 = 40 + 4 = 44 (4 steps)

15 + 6 = 21 + 3 = 24 + 6 = 30 + 3 = 33 (4 steps)

16 + 7 = 23 + 5 = 28 + 10 = 38 + 11 = 49 + 13 = 62 + 8 = 70 + 7 = 77 (7 steps)

17 + 8 = 25 + 7 = 32 + 5 = 37 + 10 = 47 + 11 = 58 + 13 = 71 + 8 = 79 + 16 = 95 + 14 = 109 + 10 = 119 + 11 = 130 + 4 = 134 + 8 = 142 + 7 = 149 + 14 = 163 + 10 = 173 + 11 = 184 + 13 = 197 + 17 = 214 + 7 = 221 + 5 = 226 + 10 = 236 + 11 = 247 + 13 = 260 + 8 = 268 + 16 = 284 + 14 = 298 + 19 = 317 + 11 = 328 + 13 = 341 + 8 = 349 + 16 = 365 + 14 = 379 + 19 = 398 + 20 = 418 + 13 = 431 + 8 = 439 + 16 = 455 + 14 = 469 + 19 = 488 + 20 = 508 + 13 = 521 + 8 = 529 + 16 = 545 (44 steps)

18 + 9 = 27 + 9 = 36 + 9 = 45 + 9 = 54 + 9 = 63 + 9 = 72 + 9 = 81 + 9 = 90 + 9 = 99 (9 steps)

19 + 10 = 29 + 11 = 40 + 4 = 44 (3 steps)

20 + 2 = 22

I haven’t looked very thoroughly at this technique, so I don’t know whether it throws up a seemingly unpalindromizable number. If it does, I don’t have an advanced mind, so I won’t be able to prove that it is unpalindromizable. But an adaptation of the technique produces something interesting when it is represented on a graph. This time, if s > 9, where s = digit-sum(n), let s = digit-sum(s) until s <= 9 (i.e, s < 10, the base). I call this the condensed digit-sum:

140 + 5 = 145 + 1 = 146 + 2 = 148 + 4 = 152 + 8 = 160 + 7 = 167 + 5 = 172 + 1 = 173 + 2 = 175 + 4 = 179 + 8 = 187 + 7 = 194 + 5 = 199 + 1 = 200 + 2 = 202 (15 steps)

Here, for comparison, is the sequence for 140 using uncondensed digit-sums:

140 + 5 = 145 + 10 = 155 + 11 = 166 + 13 = 179 + 17 = 196 + 16 = 212 (6 steps)

When all the numbers (including palindromes) created using condensed digit-sums are shown on a graph, they create an interesting pattern in base 10 (the x-axis represents n, the y-axis represents n, n1 = n + digit-sum(n), n2 = n1 + digit-sum(n1), etc):

(Please open images in a new window if they fail to animate.)

And here, for comparison, are the patterns created by uncondensed digit-sums in base 2 to 10:

Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral

მეფე (2) • მსოფლიოს (3) • მათემატიკა (5) •

Tag Archives: math