Holey Trimmetry

Symmetry arising from symmetry isn’t surprising. But what about symmetry arising from asymmetry? You can find both among the rep-tiles, which are geometrical shapes that can be completely replaced by smaller copies of themselves. A square is a symmetrical rep-tile. It can be replaced by nine smaller copies of itself:

Rep-9 Square

If you trim the copies so that only five are left, you have a symmetrical seed for a symmetrical fractal:

Fractal cross stage #1


Fractal cross #2


Fractal cross #3


Fractal cross #4


Fractal cross #5


Fractal cross #6


Fractal cross (animated)


Fractal cross (static)


If you trim the copies so that six are left, you have another symmetrical seed for a symmetrical fractal:

Fractal Hex-Ring #1


Fractal Hex-Ring #2


Fractal Hex-Ring #3


Fractal Hex-Ring #4


Fractal Hex-Ring #5


Fractal Hex-Ring #6


Fractal Hex-Ring (animated)


Fractal Hex-Ring (static)


Now here’s an asymmetrical rep-tile, a nonomino or shape created from nine squares joined edge-to-edge:

Nonomino


It can be divided into twelve smaller copies of itself, like this:

Rep-12 Nonomino (discovered by Erich Friedman)


If you trim the copies so that only five are left, you have an asymmetrical seed for a familiar symmetrical fractal:

Fractal cross stage #1


Fractal cross #2


Fractal cross #3


Fractal cross #4


Fractal cross #5


Fractal cross #6


Fractal cross (animated)


Fractal cross (static)


If you trim the copies so that six are left, you have an asymmetrical seed for another familiar symmetrical fractal:

Fractal Hex-Ring #1


Fractal Hex-Ring #2


Fractal Hex-Ring #3


Fractal Hex-Ring #4


Fractal Hex-Ring #5


Fractal Hex-Ring (animated)


Fractal Hex-Ring (static)


Elsewhere other-available:

Square Routes Re-Re-Visited

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Game of Zones

The Badminton Game by David Inshaw

David Inshaw, The Badminton Game (1972-3)

I first came across this beautiful and mysterious painting in a book devoted to British art. Then I forgot the name of both artist and painting, and couldn’t get at the book any more. Years later, I’ve found it again on the cover of a paperback in a secondhand shop. I like the way it combines zones: the domestic and the dendric, the lunar and the ludic, the terrestrial and the celestial. And it’s full of fractals: the trees, the clouds and, implicitly, the moon and the two girls playing badminton.

Performativizing Papyrocentricity #58

Papyrocentric Performativity Presents:

Diamond in the DirtDirty Story: A further account of the life and adventures of Arthur Abdel Simpson, Eric Ambler (Bodley Head 1967)

Spin DoctorateGossamer Days: Spiders, Humans and Their Threads, Eleanor Morgan (Strange Attractor Press 2016)

Kid ChaosStill William, Richmal Crompton (1925)

Permission to BlandSomething Fresh, P.G. Wodehouse (1915)

Succulent Selections – for Sizzlingly Serebral Splanchnoscopophilists…

Tempting a Titan – a further exclusive extract from Titans of Transgression (TransVisceral Books, forthcoming)


• Or Read a Review at Random: RaRaR

He Say, He Sigh, He Sow #46

“… for comic effect he also drew on neglected Arabic words, including buldah, or ‘freedom from hair of the space between the eyebrows’, and bahsala, to ‘remove one’s clothes and gamble with them’.” — Christopher de Bellaigue, The Islamic Enlightenment: The Modern Struggle between Faith and Reason (2017), writing of the Lebanese Christian Maronite novelist Ahmad Faris al-Shidyaq (1805-87) (ch. 5, Vortex, pg. 167)

Bent for the Pent

A triangle can be tiled with triangles and a square with squares, but a pentagon can’t be tiled with pentagons. At least, not in the same way, using smaller copies of the same shape. The closest you can get is this:

Pentaflake #1


If you further subdivide the pentagon, you create what is known as a pentaflake:

Pentaflake #2


Pentaflake #3


Pentaflake #4


Pentaflake (animated)


Pentaflake (static)


But if you bend the rules and use irregular smaller pentagons, you can tile a pentagon like this, creating what I called a pentatile:

Pentatile stage 1


Further subdivisions create an interesting final pattern:

Pentatile #2


Pentatile #3


Pentatile #4


Pentatile #5


Pentatile #6


Pentatile (animated)


Pentatile (static)


By varying the size of the central pentagon, you can create other patterns:

Pentatile #1 (animated)


Pentatile #2 (animated)

Pentatile #2







Pentatile with no central pentagon


And here are various pentatiles in an animated gif:


And here are some variations on the pentaflake:







Elsewhere other-posted:

Bent for the Rent (1976) — the title of the incendiary intervention above is of course a reference to the “first and last glitter-rock album” by England’s loudest band, Spinal In Terms Of Tap
Phrallic Frolics — more on pentaflakes

Feel the ’Burne

The Poets at Tea […]

3.—(Swinburne, who let it get cold)

As the sin that was sweet in the sinning
Is foul in the ending thereof,
As the heat of the summer’s beginning
Is past in the winter of love:
O purity, painful and pleading!
O coldness, ineffably gray!
Oh, hear us, our handmaid unheeding,
And take it away!

Barry Pain (1864-1928)


A Melton-Mowbray Pork Pie

Strange pie that is almost a passion,
     O passion immoral for pie!
Unknown are the ways that they fashion,
     Unknown and unseen of the eye.

The pie that is marbled and mottled,
     The pie that digests with a sigh:
For all is not Bass that is bottled,
     And all is not pork that is pie.

Richard Le Gallienne (1866-1947)

Square Routes Re-Revisited

This is a very simple fractal:


It has four orientations:


Any orientation can be turned into any other by a rotation of 90°, 180° or 270°, either clockwise or anticlockwise. If you mix orientations and rotations, you can create much more complex fractals. Here’s a selection of them:

Animated fractal

Static fractal


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Elsewhere other-posted:

Square Routes
Square Routes Revisited

Square on a Three String

222 A.D. was the year in which the Emperor Heliogabalus was assassinated by his own soldiers. Exactly 1666 years later, the Anglo-Dutch classicist Sir Lawrence Alma-Tadema exhibited his painting The Roses of Heliogabalus (1888). I suggested in “Roses Are Golden” that Alma-Tadema must have chosen the year as deliberately as he chose the dimensions of his canvas, which, at 52″ x 84 1/8“, is an excellent approximation to the golden ratio.

But did Alma-Tadema know that lines at 0º and 222º divide a circle in the golden ratio? He could easily have done, just as he could easily have known that 222 precedes the 48th prime, 223. But it is highly unlikely that he knew that 223 yields a magic square whose columns, rows and diagonals all sum to 222. To create the square, simply list the 222 multiples of the reciprocal 1/223 in base 3, or ternary. The digits of the reciprocal repeat after exactly 222 digits and its multiples begin and end like this:

001/223 = 0.00001002102101021212111012022211122022... in base 3
002/223 = 0.00002011211202120201222101122200021121...
003/223 = 0.00010021021010212121110120222111220221...
004/223 = 0.00011100200112011110221210022100120020...
005/223 = 0.00012110002220110100102222122012012120...

[...]

218/223 = 0.22210112220002112122120000100210210102... in base 3
219/223 = 0.22211122022110211112001012200122102202...
220/223 = 0.22212201201212010101112102000111002001...
221/223 = 0.22220211011020102021000121100022201101...
222/223 = 0.22221220120121201010111210200011100200...

Each column, row and diagonal of ternary digits sums to 222. Here is the full n/223 square represented with 0s in grey, 1s in white and 2s in red:

(Click for larger)


It isn’t difficult to see that the white squares are mirror-symmetrical on a horizontal axis. Here is the symmetrical pattern rotated by 90º:

(Click for larger)


But why should the 1s be symmetrical? This isn’t something special to 1/223, because it happens with prime reciprocals like 1/7 too:

1/7 = 0.010212... in base 3
2/7 = 0.021201...
3/7 = 0.102120...
4/7 = 0.120102...
5/7 = 0.201021...
6/7 = 0.212010...

And you can notice something else: 0s mirror 2s and 2s mirror 0s. A related pattern appears in base 10:

1/7 = 0.142857...
2/7 = 0.285714...
3/7 = 0.428571...
4/7 = 0.571428...
5/7 = 0.714285...
6/7 = 0.857142...

The digit 1 in the decimal digits of n/7 corresponds to the digit 8 in the decimal digits of (7-n)/7; 4 corresponds to 5; 2 corresponds to 7; 8 corresponds to 1; 5 corresponds to 4; and 7 corresponds to 2. In short, if you’re given the digits d1 of n/7, you know the digits d2 of (n-7)/7 by the rule d2 = 9-d1.

Why does that happen? Examine these sums:

 1/7 = 0.142857142857142857142857142857142857142857...
+6/7 = 0.857142857142857142857142857142857142857142...
 7/7 = 0.999999999999999999999999999999999999999999... = 1.0

 2/7 = 0.285714285714285714285714285714285714285714...
+5/7 = 0.714285714285714285714285714285714285714285...
 7/7 = 0.999999999999999999999999999999999999999999... = 1.0

 3/7 = 0.428571428571428571428571428571428571428571...
+4/7 = 0.571428571428571428571428571428571428571428...
 7/7 = 0.999999999999999999999999999999999999999999... = 1.0

And here are the same sums in ternary (where the first seven integers are 1, 2, 10, 11, 12, 20, 21):

  1/21 = 0.010212010212010212010212010212010212010212...
+20/21 = 0.212010212010212010212010212010212010212010...
 21/21 = 0.222222222222222222222222222222222222222222... = 1.0

  2/21 = 0.021201021201021201021201021201021201021201...
+12/21 = 0.201021201021201021201021201021201021201021...
 21/21 = 0.222222222222222222222222222222222222222222... = 1.0

 10/21 = 0.102120102120102120102120102120102120102120...
+11/21 = 0.120102120102120102120102120102120102120102...
 21/21 = 0.222222222222222222222222222222222222222222... = 1.0

Accordingly, in base b with the prime p, the digits d1 of n/p correspond to the digits (p-n)/p by the rule d2 = (b-1)-d1. This explains why the 1s mirror themselves in ternary: 1 = 2-1 = (3-1)-1. In base 5, the 2s mirror themselves by the rule 2 = 4-2 = (5-1) – 2. In all odd bases, some digit will mirror itself; in all even bases, no digit will. The mirror-digit will be equal to (b-1)/2, which is always an integer when b is odd, but never an integer when b is even.

Here are some more examples of the symmetrical patterns found in odd bases:

Patterns of 1s in 1/19 in base 3


Patterns of 6s in 1/19 in base 13


Patterns of 7s in 1/19 in base 15


Elsewhere other-posted:

Roses Are Golden — more on The Roses of Heliogabalus (1888)
Three Is The Key — more on the 1/223 square

Wysts and Mellow Flutefulness

(To Randolph Churchill, but not about him)

Broad of Church and broad of mind,
Broad before and broad behind,
A keen ecclesiologist,
A rather dirty Wykehamist.
’Tis not for us to wonder why
He wears that curious knitted tie;
We should not cast reflections on
The very slightest kind of don.
We should not giggle as we like
At his appearance on his bike;
It’s something to become a bore,
And more than that, at twenty-four.
It’s something too to know your wants
And go full pelt for Norman fonts.
Just now the chestnut trees are dark
And full with shadow in the park,
And “Six o’clock!” St. Mary calls
Above the mellow college walls.
The evening stretches arms to twist
And captivate her Wykehamist.
But not for him these autumn days,
He shuts them out with heavy baize;
He gives his Ovaltine a stir
And nibbles at a petit beurre,
And, satisfying fleshy wants,
He settles down to Norman fonts.

John Betjeman (1906-84)

Performativizing Papyrocentricity #57

Papyrocentric Performativity Presents:

Do and DieThe Reason Why, Cecil Woodham-Smith (1953) (posted at O.-o.-t.-Ü)

Liddell im WörterlandLiddell and Scott’s Greek-English Lexicon, Henry George Liddell and Robert Scott (1843)

Lunar or LaterMoon: From 4.5 billion years ago to the present: Owners’ Workshop Manual, David M. Harland (Haynes 2016)

Headlong into NightmareHeadlong Hall (1816) / Nightmare Abbey (1818)

Twisted TalesBiggles’ Big Adventures: Four Classic Stories Starring the British Empire’s Most Fearless Pilot Adventurer, Captain W.E. Johns (Sevenoaks 2007)

Stop the Brott – staying the serial slaying of a sanguinivorous psychoanalyst


• Or Read a Review at Random: RaRaR