Hour Re-Re-Re-Re-Powered

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In “Dissing the Diamond” I looked at some of the fractals I found by selecting lines from a dissected diamond. Here’s another of those fractals:

Fractal from dissected diamond

It’s a distorted and incomplete version of the hourglass fractal:

Hourglass fractal

Here’s how to create the distorted form of the hourglass fractal:

Distorted hourglass from dissected diamond (stage 1)

Distorted hourglass #2

Distorted hourglass #3

Distorted hourglass #4

Distorted hourglass #5

Distorted hourglass #6

Distorted hourglass #7

Distorted hourglass #8

Distorted hourglass #9

Distorted hourglass #10

Distorted hourglass (animated)

When I de-distorted and doubled the dissected-diamond method, I got this:

Hourglass fractal #1

Hourglass fractal #2

Hourglass fractal #3

Hourglass fractal #4

Hourglass fractal #5

Hourglass fractal #6

Hourglass fractal #7

Hourglass fractal #8

Hourglass fractal #9

Hourglass fractal #10

Hourglass fractal (animated)

Elsewhere other-engageable:

 

• Hour Power
• Hour Re-Powered
• Hour Re-Re-Powered
• Hour Re-Re-Re-Powered

Carved Cascade

by in Art, Free Will, Philosophy, Psychology, Quotations and tagged , , , , , , , , , , , , , , , , , | Leave a comment

Woodcut of a waterfall by Reynolds Stone (1909-79)

It’s the wrong kind of waterfall to go with this passage from Nietzsche, but that can’t be helped dot dot dot colon

Am Wasserfall. — Beim Anblick eines Wasserfalles meinen wir in den zahllosen Biegungen, Schlängelungen, Brechungen der Wellen Freiheit des Willens und Belieben zu sehen; aber Alles ist nothwendig, jede Bewegung mathematisch auszurechnen. So ist es auch bei den menschlichen Handlungen; man müsste jede einzelne Handlung vorher ausrechnen können, wenn man allwissend wäre, ebenso jeden Fortschritt der Erkenntniss, jeden Irrthum, jede Bosheit. Der Handelnde selbst steckt freilich in der Illusion der Willkür; wenn in einem Augenblick das Rad der Welt still stände und ein allwissender, rechnender Verstand da wäre, um diese Pausen zu benützen, so könnte er bis in die fernsten Zeiten die Zukunft jedes Wesens weitererzählen und jede Spur bezeichnen, auf der jenes Rad noch rollen wird. Die Täuschung des Handelnden über sich, die Annahme des freien Willens, gehört mit hinein in diesen auszurechnenden Mechanismus. — Friedrich Nietzsche, Menschliches, Allzumenschliches: Ein Buch für freie Geister (1878)

AT THE WATERFALL.—In looking at a waterfall we imagine that there is freedom of will and fancy in the countless turnings, twistings, and breakings of the waves ; but everything is compulsory, every movement can be mathematically calculated. So it is also with human actions ; one would have to be able to calculate every single action beforehand if one were all-knowing ; equally so all progress of knowledge, every error, all malice. The one who acts certainly labours under the illusion of voluntariness ; if the world’s wheel were to stand still for a moment and an all-knowing, calculating reason were there to make use of this pause, it could foretell the future of every creature to the remotest times, and mark out every track upon which that wheel would continue to roll. The delusion of the acting agent about himself, the supposition of a free will, belongs to this mechanism which still remains to be calculated. — Friedrich Nietzsche, Human, All-Too Human: A Book for Free Spirits (1908)

Dissing the Diamond

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

In “Fractangular Frolics” I looked at how you could create fractals by choosing lines from a dissected equilateral or isosceles right triangle. Now I want to look at fractals created from the lines of a dissected diamond, as here:

Lines in a dissected diamond

Let’s start by creating one of the most famous fractals of all, the Sierpiński triangle:

Sierpiński triangle stage 1

Sierpiński triangle #2

Sierpiński triangle #3

Sierpiński triangle #4

Sierpiński triangle #5

Sierpiński triangle #6

Sierpiński triangle #7

Sierpiński triangle #8

Sierpiński triangle #9

Sierpiński triangle #10

Sierpiński triangle (animated)

However, you can get an infinite number of Sierpiński triangles with three lines from the diamond:

Sierpińfinity #1

Sierpińfinity #2

Sierpińfinity #3

Sierpińfinity #4

Sierpińfinity #5

Sierpińfinity #6

Sierpińfinity #7

Sierpińfinity #8

Sierpińfinity #9

Sierpińfinity #10

Sierpińfinity (animated)

Here are some more fractals created from three lines of the dissected diamond (sometimes the fractals are rotated to looked better):

And in these fractals one or more of the lines are flipped to create the next stage of the fractal:

Previously pre-posted:

• Fractangular Frolics — fractals created in a similar way

• Dissecting the Diamond — fractals from another kind of diamond

Brine Shine

by in Art, The Sea and tagged , , , , , , , , , , , , , , | Leave a comment

Study of waves, wave-crests and foam by the Armenian artist Ivan Aivazovsky (1817-1900)

Aivazovsky was a citizen of Imperial Russia whose name is Հովհաննես Այվազյան in Armenian and Иван Айвазовский in Russian.

Mulch is Less

by in Biology, Botany, Entomology, Quotations, Science and tagged , , , , , , , , , , | Leave a comment

An interesting bio-paradox from the world’s premier papyrocentric purveyor of progressive performativity:

Vast stretches of roadside have been transformed. Where there were thick clumps of grass, there are low-growing wildflowers such as black medic, birds-foot trefoil and red clover. The verges are cut two or three times a year, not 12, saving the council tens of thousands of pounds. Butterflies and other invertebrates have returned in their droves. […]

The process is simple: cut infrequently, ideally, just twice a year in spring and then late summer once plants have bloomed and seeded; remove the clippings to gradually reduce the fertility of the soil and prevent a buildup of mulch; repeat, wait, and enjoy the resurgent wildlife and flowers. […]

“As fertility declines in a soil, biodiversity increases. At first that seems a little counterintuitive because you imagine the more you pour into a soil, the more plants that can grow. That’s not how it works in the natural system. In more fertile systems, a few species dominate and they swamp and smother everything else.”

Grass cuttings are almost always left where they fall along the thousands of miles of road verges that are maintained by law in the UK. Over time, the resulting mulch increases the fertility of the soil, meaning the grass grows with increasing vigour and needs to be cut more frequently. The cut and collect method breaks the cycle. — On the verge: a quiet roadside revolution is boosting wildflowers, The Guardian, 14iii2020

Termed Out Nice Again

by in English Usage, French, In Terms Of, Linguistics and tagged , , , , , , , , , , , , , , , , | Leave a comment

In 2013, I made a key discovery that disturbed and distressed core members of the non-conformist maverick community on a global basis dot dot dot… In America (or so it appeared) a key lexical marker of non-conformist maverickness was rapidly declining in terms of core usage, thusly:

At the same time, the non-conformist maverick community in Britain had maintained their core commitment to this key lexical marker of etc, thusly:

I expressed my puzzlement at the decline of “in terms of” in America. I couldn’t see a linguistic explanation and should (I now realize) have expressed doubts about the reliability of the data. Yes, in 2020 I’m very happy to report to members of the non-conformist maverick community that they need be disturbed and distressed no longer. The term has turned and it seems Google’s nGram wasn’t working properly at that time-period. Key statistics for core usage of “in terms of” are now in core accordance with key expectations, thusly:

“in terms of” (American English)

(open in new tab for larger image)

“in terms of” (British English)

Sadly, however, non-conformist mavericks in French- and Spanish-speaking countries seem to have stopped being non-conformist:

“en termes de” (French)

“en términos de” (Spanish)

Peri-Performative Post-Scriptum

The title of this incendiary intervention radically referencizes a key catchphrase of core comedian George Formby (1904-61), viz, “turned out nice again”. Formby’s home-county of Lancashire (England) was — and remains — a core hotbed of non-conformist maverickness dot dot dot

• Core discussion around “in terms of”…

Seis Segundos de Salvador

by in Art, Language, Quotations, Salvador Dalí and tagged , , , , , , , , , | Leave a comment

“Será tan breve que ya he terminado,” — Salvador Dalí, Con la frase “Ja soc aquí”, Dalí abrió una surrealista conferencia de Prensa, El País, 25×1980

   Salvador Dalí […] once gave the world’s shortest speech – six seconds in duration. He said, “I will be so brief I have already finished,” and he sat down. — Edward O. Wilson

Previously pre-posted

• A Seriously Sizzling Series of Super-Saucy Salvadisms — more good quotes by Salvador Dalí

Circus Trix

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

Here’s a trix, or triangle divided into six smaller triangles:

Trix, or triangle divided into six smaller triangles

Now each sub-triangle becomes a trix in its turn:

Trix stage #2

And again:

Trix #3

Trix #4

Trix #5

Trix divisions (animated)

Now try dividing the trix and discarding sub-triangles, then repeating the process. A fractal appears:

Trix fractal #1

Trix fractal #2

Trix fractal #3

Trix fractal #4

Trix fractal #5

Trix fractal #6

Trix fractal #7

Trix fractal (animated)

But what happens if you delay the discarding, first dividing the trix completely into sub-triangles, then dividing completely again? You get a more attractive and symmetrical fractal, like this:

Trix fractal (delayed discard)

And it’s easy to convert the triangle into a circle, creating a fractal like this:

Delayed-discard trix fractal converted into circle

Delayed-discard trix fractal to circular fractal (animated)

Now a trix fractal that looks like a hawk-god:

Trix hawk-god #1

Trix hawk-god #2

Trix hawk-god #3

Trix hawk-god #4

Trix hawk-god #5

Trix hawk-god #6

Trix hawk-god #7

Trix hawk-god (animated)

Trix hawk-god converted to circle

Trix hawk-god to circle (animated)

If you delay the discard, you get this:

Trix hawk-god circle (delayed discard)

And here are more delayed-discard trix fractals:

Various circular trix-fractals (animated)

Post-Performative Post-Scriptum

In Latin, circus means “ring, circle” — the English word “circle” is actually from the Latin diminutive circulus, meaning “little circle”.