Log24

Monday, March 14, 2022

Shape Constant

Filed under: General — Tags: , — m759 @ 4:06 am

For Pi Day, see the title in this journal.

Tuesday, July 30, 2024

The Dropped Line

Filed under: General — Tags: — m759 @ 10:42 am

Pythagorean theorem proof by overlapping similar figures

"Drop me a line" — Request attributed to Emma Stone.

Meditation on the dropped line


 

Stone herself might prefer the not-so-mean shapes
of "A Quiet Weekend in Mykonos."

Sunday, February 18, 2024

Bride’s Chair vs. One-Liner

Filed under: General — Tags: , — m759 @ 11:53 am

The "bride's chair" is the figure illustrating Euclid's proof
of the Pythagorean theorem (click image to enlarge) —

A somewhat simpler approach —

Pythagorean theorem proof by overlapping similar figures

"Drop me a line" — Request attributed to Emma Stone

 

Friday, June 30, 2023

Trickster Fuge (German for Joint)

Filed under: General — Tags: , , — m759 @ 12:28 pm
 

Margaret Atwood on Lewis Hyde's 
Trickster Makes This World: Mischief, Myth, and Art

"Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159)

What is "the next world"? It might be the Underworld….

The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation  and art  all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254)  If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist.  Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.

Pythagorean theorem proof by overlapping similar figures

"Drop me a line" — Request attributed to Emma Stone

Saturday, January 21, 2023

One-Liner

Filed under: General — Tags: , — m759 @ 4:11 pm

Pythagorean theorem proof by overlapping similar figures

Saturday, May 7, 2022

“Use Your Noodle” Update

Filed under: General — Tags: , , — m759 @ 3:33 am

Update to yesterday's "Use Your Noodle" post . . . 

Click the above image to enlarge.

Update of 2:40 AM May 7, 2022 —

Flusser's seven "pillars" appear to be the main sections of the Tractatus
— numbered 1 through 7, with many intermediate numbered passages.

For a more geometric meditation on "the shape of things," see other
posts tagged "Shape Constant" in this  journal.

Friday, May 6, 2022

Use Your Noodle (For Byron Gogol*)

Filed under: General — Tags: , , — m759 @ 4:46 pm
 
An essay from . . .

The Shape of Things: A Philosophy of Design
by Vilem Flusser

Wittgenstein’s Architecture

The universe of texts can be seen as a landscape. In it one can make out mountains and valleys, rivers and lakes, castles, farmyards and inner-city slums. On the horizon of the scene visualized in this way, the Bible and Homer appear as gigantic ice-covered mountains. The vast, tranquil lake of Aristotle’s texts, where fishermen idly throw their nets and philologists row their boats, occupies a part of the valley bottom. There, the tumbling waterfall of Nietzsche is captured by the broad river of modern pragmatism. Towering above everything, the Gothic cathedral of St Thomas Aquinas’s Summae dominates the cathedral square of the city, in which the roofs and gables of Baroque speculations jostle one another. In the suburbs of this city, one catches sight of the Romantic, Realist and Modernist housing-blocks and factories of more recent litera¬ ture; somewhat apart from all these stands a small, apparently insignificant house resembling scaffolding more than a finished building: Wittgenstein’s building.

This little house is called the Tractatus. This name isn’t the product of a one-track mind. For when one enters the house, one notices immediately that this is not a place that has lost track of things. Quite the opposite: It is a place of mirror- images. The house stands on six foundation pillars which support one another by means of cross-beams organized in a hierarchy. In the middle, however, there rises a seventh pillar whose function it is to cut through the building and free it from the ground. So the house with all its corners, angles and joints is protected, armoured and impregnable. And yet, and for that very reason, it is threatened with collapse and disappearance without trace – condemned in advance and from the outset.

The building is set out: It consists of propositions. Every proposition presupposes all the preceding ones and is itself the 76 presupposition of all the following propositions. Proposition by proposition, anyone who enters progresses through the prescribed rooms, and his step is supported by consistencies. Suddenly, with one proposition, one single proposition, the ground gives way beneath his feet. He falls head first into the abyss.

Wittgenstein’s house is situated in a suburb of that city whose cathedral square is dominated by the towers of Thomas Aquinas’s cathedral. The small, modest pillars of Wittgenstein’s house support one another according to the same logico- philosophical method as the pillars of the cathedral support one another. But there appears to be a world of difference between the cathedral and the little house: The cathedral is a ship pointing in the direction of heaven, and the little house is a trap-door pointing in the direction of a bottomless abyss. But be careful: May Thomas Aquinas not have been right in saying after his revelation that everything he had written before was like straw? May not the heaven above the cathedral be the same black hole as the abyss beneath the little house? May not Wittgenstein’s little house be the cathedral of today? And those mirrors whose images simultaneously mirror one another, may they not be our equivalent of stained-glass windows?

The landscape portrayed in this essay, it goes without saying, is a metaphor. Is it possible to identify it as Vienna? And is it possible for anyone entering Wittgenstein’s little house in that unlikely place to make out a hint of the unsayable? What we cannot speak about we must pass over in silence. 77

Click the above image to enlarge.

See as well . . .

Update of 2:40 AM May 7, 2022 —

Flusser's seven "pillars" appear to be the main sections of the Tractatus
— numbered 1 through 7, with many intermediate numbered passages.

For a more geometric meditation on "the shape of things," see other
posts tagged "Shape Constant" in this  journal.
 

*Byron Gogol is a tech magnate in the HBO series "Made for Love."
 

Friday, April 1, 2022

Beauty Bare … ?

Filed under: General — Tags: , — m759 @ 12:00 am

Pythagorean theorem proof by similarly divided squares

Update of 11:30 AM ET April 1, 2022 — A simpler version:

The above picture may be used to to introduce the concept of a "shape constant"
in similar figures — like the shape constant pi  in a circle or the square root of 2
in a square. In each of the three similar figures at right above, the ratio of the
triangular area to the area of the attached square is a shape constant  
the same, because of their similarity, for each of the three shapes. Since the
areas of the top two triangles at right sum to that of the enclosed triangle at left,
their attached square areas sum to the area of the bottom square, Q.E.D.

The source of the proof —

Pythagorean-theorem proof using similar triangles and concept of 'shape constant' m

Tuesday, December 28, 2021

Shape as Form (Not the Michael Fried Version)

Filed under: General — Tags: , — m759 @ 10:44 pm

For those who prefer form constants  to shape constants 

Saturday, October 9, 2021

This Side of Paradise

Filed under: General — Tags: , , — m759 @ 3:35 am

Note the book subtitle below: "minutes from an infinite paradise."

For further details, see other posts tagged Revelado.

Wednesday, August 25, 2021

Einstein Revelado

Filed under: General — Tags: , , , — m759 @ 1:36 pm

For those too young to remember the 20th century . . .

Related illustrations —

Pythagorean-theorem proof using similar triangles and concept of 'shape constant' m

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