S’éteignent, S’encroûtent, S’allument…

by in French, Language, Literature, Philosophy, Quotations and tagged , , , , , | Leave a comment

« Des soleils s’éteignent & s’encroûtent, des planètes périssent & se dispersent dans les plaines des airs ; d’autres soleils s’allument, de nouvelles planètes se forment pour faire leurs révolutions ou pour décrire de nouvelles routes, & l’homme, portion infiniment petite d’un globe, qui n’est lui-même qu’un point imperceptible dans l’immensité, croit que c’est pour lui que l’univers est fait, s’imagine qu’il doit être le confident de la nature, se flatte d’être éternel, se dit le roi de l’univers ! » — Baron d’Holbach, Système de la nature (1770), Partie 1, Chapitre 6

“Suns are extinguished or become corrupted, planets perish and scatter across the wastes of the sky; other suns are kindled, new planets formed to make their revolutions or describe new orbits, and man, an infinitely minute part of a globe which itself is only an imperceptible point in the immense whole, believes that the universe is made for himself, flatters himself that he is eternal, calls himself king of the universe!”

Post-Performative Post-Scriptum…

Mais… Mens Major Est Quam Materia…

Poeta Moquitur

by in Literature, Parodies, Poetry, Swinburne and tagged , , , , , , | Leave a comment

Poeta Loquitur

If a person conceives an opinion
     That my verses are stuff that will wash,
Or my Muse has one plume on her pinion,
     That person’s opinion is bosh.
My philosophy, politics, free-thought!
     Are worth not three skips of a flea,
And the emptiest thoughts that can be thought
        Are mine on the sea.

In a maze of monotonous murmur
     Where reason roves ruined by rhyme,
In a voice neither graver nor firmer
     Than the bells on a fool’s cap chime,
A party pretentiously pensive,
     With a Muse that deserves to be skinned,
Makes language and metre offensive
        With rhymes on the wind.

A perennial procession of phrases
     Pranked primly, though pruriently prime,
Precipitates preachings on praises
     In a ruffianly riot of rhyme
Through the pressure of print on my pages:
     But reckless the reader must be
Who imagines me one of the sages
        That steer through Time’s sea.

Mad mixtures of Frenchified offal
     With insults to Christendom’s creed,
Blind blasphemy, schoolboylike scoff, all
     These blazon me blockhead indeed.
I conceive myself obviously some one
     Whose audience will never be thinned,
But the pupil must needs be a rum one
        Whose teacher is wind.

In my poems, with ravishing rapture
     Storm strikes me and strokes me and stings:
But I’m scarcely the bird you might capture
     Out of doors in the thick of such things.
I prefer to be well out of harm’s way
     When tempest makes tremble the tree,
And the wind with omnipotent arm-sway
        Makes soap of the sea.

Hanging hard on the rent rags of others,
     Who before me did better, I try
To believe them my sisters and brothers,
     Though I know what a low lot am I.
The mere sight of a church sets me yelping
     Like a boy that at football is shinned!
But the cause must indeed be past helping
        Whose gospel is wind.

All the pale past’s red record of history
     Is dusty with damnable deeds;
But the future’s mild motherly mystery
     Peers pure of all crowns and all creeds.
Truth dawns on time’s resonant ruin,
     Frank, fulminant, fragrant, and free:
And apparently this is the doing
        Of wind on the sea.

Fame flutters in front of pretension
     Whose flagstaff is flagrantly fine:
And it cannot be needful to mention
     That such beyond question is mine.
Some singers indulging in curses,
     Though sinful, have splendidly sinned:
But my would-be maleficent verses
        Are nothing but wind.

• Algernon Charles Swinburne viâ Pseudopodium

Elsewhere Other-Accessible…

• Swinburne on Swinburne — “Poeta Loquitur” at Mind of Winter

Squaring the Triangle

by in Circles, Fractals, Geometry, Mathematics, Sierpiński triangle, Triangles and tagged , , , , , | Leave a comment

It’s an interesting little exercise in elementary trigonometry to turn the Sierpiński triangle…

A Sierpiński triangle

…into its circular equivalent:

A Sierpiński trisc

You could call that a trisc, because it’s a triangle turned into a disc. And here’s triangle-and-trisc in one image:

Sierpiński triangle + Sierpiński trisc

But what’s the square equivalent of a Sierpiński triangle? This is:

Square from Sierpiński triangle

You can do that directly, as it were:

Sierpiński triangle → square

Or you can convert the triangle into a disc, then the disc into a square, like this:

Sierpiński triangle → trisc → square

Now try converting the triangle into a pentagon:

Pentagon from Sierpiński triangle

Sierpiński triangle → pentagon

Sierpiński triangle → trisc → pentagon

And a hexagon:

Hexagon from Sierpiński triangle

Sierpiński triangle → hexagon

Sierpiński triangle → trisc → hexagon

But you can also convert the Sierpiński trisc back into a Sierpiński triangle, then into a Sierpiński trisc again:

Sierpiński triangle → trisc → triangle → trisc

Sierpiński triangle → trisc → triangle → trisc (animated at Ezgif)

Sierpiński triangle → trisc → triangle → trisc (b&w)

Sierpiński triangle → trisc → triangle → trisc (b&w) (animated at Ezgif)

After triangles come squares. Here’s a shape called a T-square fractal:

T-square fractal

And here’s the circular equivalent of a T-square fractal:

T-square fractal → T-squisc

T-square fractal + T-squisc

If a disc from a triangle is a trisc, then a disc from a square is a squisc (it would be pentisc, hexisc, heptisc for pentagonal, hexagonal and heptagonal fractals). Here’s the octagonal equivalent of a T-square fractal:

Octagon from T-square fractal

As with the Sierpiński trisc, you can use the T-squisc to create the T-octagon:

T-square fractal → T-squisc → T-octagon (color)

Or you can convert the T-square directly into the T-octagon:

T-square fractal to T-octagon fractal

But using the squisc makes for interesting multiple images:

T-square fractal → T-squisc → T-octagon (b&w)

T-square fractal → T-squisc → T-octagon → T-squisc

T-square fractal → T-squisc → T-octagon → T-squisc (animated at Ezgif)

The conversions from polygon to polygon look best when the number of sides in the higher polygon are a multiple of the number of sides in the lower, like this:

Sierpiński triangle → Sierpiński hexagon → Sierpiński nonagon

Scout the Routes

by in Chaos Game, Fractals, Geometry, Mathematics, Squares and tagged , , , , , , , , | Leave a comment

Triangles? Yes. Squares? No. If you scout the routes with a triangle, you get a beautiful fractal. If you scout the routes with a square, you don’t. Here’s what you get with a triangle:

A Sierpiński triangle

But how do you scout the routes? (That phrase works best in the American dialects where “scout” rhymes with “route”.) Simple: you mark the final positions reached when a point traces all possible ways of jumping, say, eight times 1/2-way towards the vertices of a polygon. Here’s an animation of a point scouting the routes of eight jumps towards the vertices of a triangle (it starts each time at the center):

Creating a Sierpiński triangle by scouting the routes (animated at Ezgif)

If you scout the routes with a square, you don’t get a fractal. Instead, the interior of the square fills evenly (and boringly) with the end-points of the routes:

Scouting the routes with a square (animated at Ezgif)

But you can create fractals with a square if you out routes as you scout routes. That is, if you exclude some routes and don’t mark their end-points. One way to do this is to compare the proposed next jump-vertex (vertex-jumped-towards) with the previous jump-vertex. For example, if the proposed jump-vertex, jv[t], is the same as the previous jump-vertex, jv[t-1], you don’t jump towards jv[t] or you jump towards it in a different way. The test is jv[t] = jv[t-1] + vi. If vi = 0 and you jump towards the clockwise neighbor of jv when the test is true, you get a fractal looking like this:

vi = 0, action = jv → jv + 1

Here’s the fractal if you jump towards the clockwise-neighbor-but-one when the test is true:

vi = 0, action = jv + 2

Now try varying the vi of the jv[t-1] + vi:

vi = 2, action = jv + 2

vi = 2, action = jv + 1

vi = 3, action = jv + 1

Or what about jumping in a different way towards jv when the test is true? If you jump 2/3 of the way rather 1/2, you get his fractal:

vi = 2, action = jump 2/3

And if you jump 4/3 of the way (i.e., you overshoot the vertex jv), you get this fractal:

vi = 0, action = jump 4/3rds to vertex

vi = 0, jump 4/3 (guide-square removed)

vi = 2, jump 4/3rds (guide-square removed)

And in this fractal the point jumps 2/3 of the way to the center of the square when the test is true:

vi = 2, action = jump 2/3rds of way to center of square

But why apply only one test to jv[1] and use only when one alternative jump? If jv[t] = jv[t-1] + 1 or jv[t] = jv[t-1] + 3, jv[t] becomes jv[t]+1 or jv[t]+3, respectively, you get this fractal:

vi = 1, jv + 1; vi = 3, jv + 3

Here are more fractals created by single and double tests:

vi = 1, jv + 1

vi = 0, jump 2/3

vi = 0, jump towards center 2/3rds

vi = 1, jump-center 2/3

vi = 2, jump 1/3; vi = 3, jump 1/1 (i.e, 1)

vi = 0, jv + 2; vi = 2, jump-center 1/2

vi = 0, jv + 2; vi = 2, jump-center 2/3

vi = 0, jv + 2; vi = 2, jump-center 4/3

vi = 0, jv + 1; vi = 2, jump 2/3

vi = 0, jv + 2; vi = 2, jump 2/3

vi = 0, jump 4/3; vi = 2, jv + 2

vi = 0, jump 2/3; vi = 2, jv + 1

vi = 0, jump 4/3; vi = 1, jv + 2

vi = 0, jump 2/3; vi = 2, jump 1/3

vi =0, jump 1/3; vi = 2, jump 2/3

vi = 0, jump 0/1 (i.e, 0); vi = 2, jump 1/3

Rankle Biter

by in French, Language, Latin and tagged , , , , , | Leave a comment

rankle, verb 1. (transitive or intransitive) To cause irritation, bitterness or acrimony. 2. (intransitive) To fester.

Etymology: From Middle English ranklen, ranclen, from Old French rancler, räoncler, draoncler (“to ulcerate, to form a boil”), from Old French draoncle (“a boil”), from Latin dracunculus (“little serpent”), diminutive of Latin dracō (“serpent, dragon”).

• rankle at Wiktionary

Primal Pellicles

by in Arithmetic, Base Behavior, Integer Sequences, Mathematics, Prime numbers and tagged , , , , , , , , | Leave a comment

Numbers have thin skins. And they’re easily replaced. Take 71624133. Here it is permuting its pellicles:

71624133 in base 10 = 100010001001110010111000101 in base 2 = 11222202212211200 in b3 = 10101032113011 in b4 = 121313433013 in b5 = 11035053113 in b6 = 1526536500 in b7 = 421162705 in b8 = 158685750 in b9 = 374802A9 in b11 = 1BBA1199 in b12 = 11AB9B59 in b13 = 9726137 in b14 = 644BE73 in b15 = F3855B7 in b16

But if digits are the skin of 71624133, what are its bones? Well, you could say the skeleton of a number, something that doesn’t change from base to base, is its prime factorization:

71624133 = 32 × 72 × 162413

But the primes themselves are numbers, so they’re wearing pellicles too. And it turns out that, in base 10, the pellicles of the prime factors of 71624133 match the pellicle of 71624133 itself:

71624133 = 32.72.162413

Here’s a list of primal pellicles in base 10:

735 = 3.5.72
3792 = 24.3.79
1341275 = 52.13.4127
13115375 = 53.7.13.1153
22940075 = 52.229.4007
29373375 = 3.53.29.37.73
71624133 = 32.72.162413
311997175 = 52.7.172.31.199
319953792 = 27.3.53.79.199
1019127375 = 32.53.7.127.1019
1147983375 = 3.53.7.11.83.479
1734009275 = 52.173.400927
5581625072 = 24.5581.62507
7350032375 = 53.7.23.73.5003
17370159615 = 34.5.17.59.61.701
33061224492 = 22.33.306122449
103375535837 = 72.37.103.553583
171167303912 = 23.11.172.6730391
319383665913 = 3.133.19.383.6659
533671737975 = 34.52.17.53.367.797
2118067737975 = 32.52.7.79.211.80677
3111368374257 = 3.112.132.683.74257
3216177757191 = 3.73.191.757.21617
3740437158475 = 52.37.4043715847
3977292332775 = 3.52.292.233.277.977
4417149692375 = 53.7.23.4969.44171
7459655393232 = 24.32.72.23.45965539
7699132721175 = 3.52.72.27211.76991
7973529228735 = 3.5.7.972.2287.3529
10771673522535 = 34.5.67.71.107.52253

You can find them at the Online Encyclopedia of Integer Sequences under A121342, “Composite numbers that are a concatenation of their distinct prime divisors in some order.” But what about pairs of primal pellicles, that is, pairs of numbers where the prime factors of each form the pellicle of the other?

35 = 5.7 ↔ 75 = 3.52
1275 = 3.52.17 ↔ 3175 = 52.127
131715 = 32.5.2927 ↔ 329275 = 52.13171
3199767 = 3.359.2971 ↔ 35932971 = 3.19.67.972
14931092 = 22.11.61.5563 ↔ 116155632 = 24.3.109.1492

And here are a few primal pellicles I’ve found in other bases:

Primal Pellicles in Base 2

1111011011110 = 10.1110.110110111 in b2 = 7902 = 2.32.439 in b10
1110001100110111 = 1110.10111.100011001 in b2 = 58167 = 32.23.281 in b10
1111011011011110 = 10.1110.110110110111 in b2 = 63198 = 2.32.3511 in b10
11101001100001101 = 1110.101.101001100001 in b2 = 119565 = 32.5.2657 in b10
1111011011011011110 = 10.1110.110110110110111 in b2 = 505566 = 2.32.28087 in b10
1111011111101111011 = 1110.1011.10111.11011111 in b2 = 507771 = 32.11.23.223 in b10

Primal Pellicles in Base 3

121022 = 210.12.102 in b3 = 440 = 23.5.11 in b10
212212 = 22.21.212 in b3 = 644 = 22.7.23 in b10
20110112 = 210.201.1011 in b3 = 4712 = 23.19.31 in b10
21110110 = 10.212.1101 in b3 = 5439 = 3.72.37 in b10
121111101 = 122.111.1101 in b3 = 12025 = 52.13.37 in b10
222112121 = 22.21.221121 in b3 = 19348 = 22.7.691 in b10
2202122021 = 22.2021.22021 in b3 = 54412 = 22.61.223 in b10
120212201221 = 2.122.21.201.1202 in b3 = 312550 = 2.52.7.19.47 in b10

Primal Pellicles in Base 7

2525 = 2.52.25 in b7 = 950 = 2.52.19 in b10
3210 = 2.34.10 in b7 = 1134 = 2.34.7 in b10
5252 = 2.52.52 in b7 = 1850 = 2.52.37 in b10
332616 = 33.16.326 in b7 = 58617 = 33.13.167 in b10
336045 = 32.5.3604 in b7 = 59715 = 32.5.1327 in b10
2251635 = 22.3.5.16.252 in b7 = 281580 = 22.3.5.13.192 in b10

Primal Pellicles in Base 11

253 = 22.3.52 in b11 = 300 = 22.3.52 in b10
732 = 2.32.72 in b11 = 882 = 2.32.72 in b10
2123 = 23.33.12 in b11 = 2808 = 23.33.13 in b10
3432 = 25.3.43 in b11 = 4512 = 25.3.47 in b10
3710 = 32.72.10 in b11 = 4851 = 32.72.11 in b10
72252 = 23.72.225 in b11 = 105448 = 23.72.269 in b10

Primal Pellicles in Base 15

275 = 24.5.7 in b15 = 560 = 24.5.7 in b10
2D5 = 2.52.D in b15 = 650 = 2.52.13 in b10
2CD5 = 2.52.CD in b15 = 9650 = 2.52.193 in b10
7BE3 = 3.72.BE in b15 = 26313 = 3.72.179 in b10
21285 = 24.52.128 in b15 = 105200 = 24.52.263 in b10