Log24

Friday, May 4, 2018

Art & Design

Filed under: General,Geometry — m759 @ 4:00 PM

http://www.log24.com/log/pix11/110219-SquareRootQuaternion.jpg

A star figure and the Galois quaternion.

The square root of the former is the latter.

See also a passage quoted here a year ago today
(May the Fourth, "Star Wars Day") —

Cube symmetry subgroup of order 8 from 'Geometry and Symmetry,' Paul B. Yale, 1968, p.21

Saturday, January 20, 2018

The Chaos Symbol of Dan Brown

Filed under: General,Geometry — m759 @ 1:30 AM

In the following passage, Dan Brown claims that an eight-ray star
with arrowheads at the rays' ends is "the mathematical symbol for
entropy."  Brown may have first encountered this symbol at a 
questionable "Sacred Science" website.  Wikipedia discusses
some even less  respectable uses of the symbol.

Related news —

Related symbolism —

http://www.log24.com/log/pix11/110219-SquareRootQuaternion.jpg

A star figure and the Galois quaternion.

The square root of the former is the latter.

Saturday, September 23, 2017

The Turn of the Year

Filed under: General,Geometry — m759 @ 11:59 PM

Quioted here  last  year on September 23rd

See also Galois Quaternion.

Friday, August 26, 2016

Structure a Set, Set a Structure

Filed under: General,Geometry — m759 @ 11:32 PM

http://www.log24.com/log/pix11/110219-SquareRootQuaternion.jpg

A star figure and the Galois quaternion.

The square root of the former is the latter.

See also a search in this journal for "Set a Structure."

Thursday, August 13, 2015

Anniversary

Filed under: General,Geometry — m759 @ 10:01 AM

Today is reportedly the anniversary of the death,
in Paris in 1822, of Jean Robert Argand.

Some related material

From MacTutor

"Wessel's fame as a mathematician rests solely
on this paper, which was published in 1799,
giving for the first time a geometrical interpretation
of complex numbers. Today we call this geometric
interpretation the Argand diagram but Wessel's
work came first. It was rediscovered by Argand 
in 1806 and again by Gauss in 1831. ….

Of course it is not unreasonable to call the
geometrical interpretation of complex numbers
the Argand diagram since it was Argand's work
which was influential. It was so named before
the world of mathematics learnt of Wessel's prior
publication. In fact Wessel's paper was not
noticed by the mathematical community until 1895…."

See also Tilting at Whirligigs (Log24 on March 8, 2008)
and The Galois Quaternion.

Tuesday, October 21, 2014

Eerie Twist

Filed under: General,Geometry — m759 @ 9:00 PM

(Continued from Nov. 15, 2011)

Ben Bradlee, legendary Washington Post editor, dies at 93

See also a post of Jan. 20, 2011, and an earlier post on Twelfth Night, 2010.

Commentary:

http://www.log24.com/log/pix11/110219-SquareRootQuaternion.jpg

A star figure and the Galois quaternion.

The square root of the former is the latter.

Thursday, February 6, 2014

The Representation of Minus One

Filed under: General,Geometry — Tags: , — m759 @ 6:24 AM

For the late mathematics educator Zoltan Dienes.

"There comes a time when the learner has identified
the abstract content of a number of different games
and is practically crying out for some sort of picture
by means of which to represent that which has been
gleaned as the common core of the various activities."

— Article by "Melanie" at Zoltan Dienes's website

Dienes reportedly died at 97 on Jan. 11, 2014.

From this journal on that date —

http://www.log24.com/log/pix11/110219-SquareRootQuaternion.jpg

A star figure and the Galois quaternion.

The square root of the former is the latter.

Update of 5:01 PM ET Feb. 6, 2014 —

An illustration by Dienes related to the diamond theorem —

See also the above 15 images in

http://www.log24.com/log/pix11/110220-relativprob.jpg

and versions of the 4×4 coordinatization in  The 4×4 Relativity Problem
(Jan. 17, 2014).

Wednesday, January 15, 2014

Entertainment Theory

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

From "Entertainment," a 1981 story by M. A. Foster—

"For some time, Cormen had enjoyed a peculiar suspicion, which he had learned from his wanderings around the city, and cultivated with a little notebook, in which he had made a detailed series of notes and jottings, as well as crude, but effective, charts and maps of certain districts. 'Cormen's Problem,' as it was known, was familiar to the members of the circle in which he moved; in fact, if he had not been so effective with his productions and so engaging in his personality, they might have considered him a bore.

It seemed, so the suspicion went, that the city was slowly shrinking, as evidenced by abandoned districts along the city edges. Beyond the empty houses were ruins, and beyond that, traces of foundations and street lines. Moreover, it had recently dawned on him that there were no roads out of the city, although there were no restraints. One hardly noticed this—it was the norm. But like many an easy assumption, once broken it became increasingly obvious.

Cormen's acquaintances were tolerant of his aberration, but generally unsympathetic. What he needed was proof, something he could demonstrate in black and white—and color if required. But the city was reluctant, so it appeared, to give up its realities so easily. The Master Entertainment Center, MEC, would not answer direct queries about this, even though it would obediently show him presentations, pictorial or symbolic as he required, of the areas in question. But it was tiresome detail work, in which he had to proceed completely on his own."

Lily Collins in City of Bones  (2013)—

American Folk Art (see August 23, 2011) —

IMAGE- Four Winds quilt block

Art Theory —

IMAGE- The eight Galois quaternions

See as well Ballet Blanc .

Tuesday, January 14, 2014

Release Date

Filed under: General,Geometry — Tags: — m759 @ 10:00 PM

The premiere of the Lily Collins film Abduction 
(see previous post) was reportedly in Sydney, Australia,
on August 23, 2011.

From that date in this journal

IMAGE- The eight Galois quaternions

For the eight-limbed star at the top of the quaternion array above,
see "Damnation Morning" in this journal—

She drew from her handbag a pale grey gleaming 
implement that looked by quick turns to me like 
a knife, a gun, a slim sceptre, and a delicate 
branding iron—especially when its tip sprouted 
an eight-limbed star of silver wire.

“The test?” I faltered, staring at the thing.

“Yes, to determine whether you can live in 
the fourth dimension or only die in it.”

— Fritz Leiber, short story, 1959

Related material from Wikipedia, suggested by the reference quoted
in this morning's post to "a four-dimensionalist (perdurantist) ontology"—

"… perdurantism also applies if one believes there are temporal
but non-spatial abstract entities (like immaterial souls…)."

Saturday, January 11, 2014

Star Wars (continued)

Filed under: General,Geometry — m759 @ 2:29 PM

http://www.log24.com/log/pix11/110219-SquareRootQuaternion.jpg

A star figure and the Galois quaternion.

The square root of the former is the latter.

"… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.
"

– Rubén Darío

Monday, January 6, 2014

Triumph of the Will

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM

"… the human will cannot be simultaneously
triumphant and imaginary."

— Ross Douthat, Defender of the Faith,
     in this afternoon's New York Times  at 3:25* PM ET

Some— even some Catholics— might say the will
cannot be triumphant unless  imaginary.

Related material The Galois Quaternion: A Story.

See also C. S. Lewis on enchantment

* Cf.,  in this  journal,  the most recent 3/25 , 
  and a bareword —

Click image for some context.

Friday, September 2, 2011

Rigged?

Filed under: General,Geometry — m759 @ 1:44 PM

Sarah Tomlin in a Nature  article on the July 12-15 2005 Mykonos meeting on Mathematics and Narrative—

"Today, Mazur says he has woken up to the power of narrative, and in Mykonos gave an example of a 20-year unsolved puzzle in number theory which he described as a cliff-hanger. 'I don’t think I personally understood the problem until I expressed it in narrative terms,' Mazur told the meeting. He argues that similar narrative devices may be especially helpful to young mathematicians…."

Michel Chaouli in "How Interactive Can Fiction Be?" (Critical Inquiry  31, Spring 2005), pages 613-614—

"…a simple thought experiment….*

… If the cliffhanger is done well, it will not simply introduce a wholly unprepared turn into the narrative (a random death, a new character, an entirely unanticipated obstacle) but rather tighten the configuration of known elements to such a degree that the next step appears both inevitable and impossible. We feel the pressure rising to a breaking point, but we simply cannot foresee where the complex narrative structure will give way. This interplay of necessity and contingency produces our anxious— and highly pleasurable— speculation about the future path of the story. But if we could determine that path even slightly, we would narrow the range of possible outcomes and thus the uncertainty in the play of necessity and contingency. The world of the fiction would feel, not open, but rigged."

* The idea of the thought experiment emerged in a conversation with Barry Mazur.

Barry Mazur in the preface to his 2003 book Imagining Numbers

"But the telltale adjective real  suggests two things: that these numbers are somehow real to us and that, in contrast, there are unreal  numbers in the offing. These are the imaginary numbers

The imaginary  numbers are well named, for there is some imaginative work to do to make them as much a part of us as the real numbers we use all the time to measure for bookshelves. 

This book began as a letter to my friend Michel Chaouli. The two of us had been musing about whether or not one could 'feel' the workings of the imagination in its various labors. Michel had also mentioned that he wanted to 'imagine imaginary numbers.' That very (rainy) evening, I tried to work up an explanation of the idea of these numbers, still in the mood of our conversation."

See also The Galois Quaternion and 2/19.

IMAGE- NY Lottery evening numbers Thursday, Sept. 1, 2011 were 144 and 0219

New York Lottery last evening

Tuesday, August 23, 2011

Four Winds

Filed under: General,Geometry — m759 @ 11:07 AM

A Quilt Version

IMAGE- Four Winds quilt block

A Mathematical Version

IMAGE- The eight Galois quaternions

Related remarks —

For the eight-limbed star at the top of the quaternion array above,
see "Damnation Morning" in this journal—

She drew from her handbag a pale grey gleaming implement
that looked by quick turns to me like a knife, a gun, a slim
sceptre, and a delicate branding iron—especially when its
tip sprouted an eight-limbed star of silver wire.

“The test?” I faltered, staring at the thing.

“Yes, to determine whether you can live in the fourth
dimension or only die in it.”

— Fritz Leiber, short story, 1959

See also Feb. 19, 2011.

Sunday, May 1, 2011

Delos

Filed under: General,Geometry — m759 @ 6:00 PM

The late translator Helen Lane in Translation Review , Vol. 5, 1980—

"Among the awards, I submit, should be one for the entire oeuvre  of a lifetime "senior" translator— and  one for the best first  translation…. Similar organization, cooperation, and fund-finding for a first-rate replacement for the sorely missed Delos ."

This leads to one of the founders of Delos , the late Donald Carne-Ross, who died on January 9, 2010.

For one meditation on the date January 9, see Bridal Birthday (last Thursday).

Another meditation, from the date of Carne-Ross's death—

Saturday, January 9, 2010

 

1982 Again

m759 @ 1:00 PM

Rock's top 40 on Jan. 9, 1982

Positional Meaning

m759 @ 11:32 AM

"The positional meaning of a symbol derives from its relationship to other symbols in a totality, a Gestalt, whose elements acquire their significance from the system as a whole."

– Victor Turner, The Forest of Symbols , Ithaca, NY, Cornell University Press, 1967, p. 51, quoted by Beth Barrie in "Victor Turner."

To everything, turn, turn, turn …
– Peter Seeger

The Galois Quaternion:

The Galois Quaternion

Click for context.

See also Delos in this journal.

Saturday, February 19, 2011

Brightness at Noon (continued)

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

From Epiphany Revisited

http://www.log24.com/log/pix11/110219-SquareRootQuaternion.jpg

A star figure and the Galois quaternion.

The square root of the former is the latter.

… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.

Rubén Darío

Tuesday, October 12, 2010

King Solomon’s Mind

Filed under: General,Geometry — m759 @ 2:00 PM

"Always keep a diamond in your mind."

Tom Waits/Kathleen Brennan song performed by Solomon Burke at the Paradiso in Amsterdam

The Galois Quaternion

The Galois Quaternion

"The text is a two-way mirror
that allows me to look into
the life and times of the reader.
Who knows, someday i  may rise
to a text that will compel me
to push through to the other side."

The French Mathematician
   (Galois), by Tom Petsinis

 

http://www.log24.com/log/pix10B/101012-KingSolomonAlbum.jpg

Saturday, August 7, 2010

The Matrix Reloaded

Filed under: General,Geometry — m759 @ 12:00 AM

   For aficionados of mathematics and narrative

Illustration from
"The Galois Quaternion— A Story"

The Galois Quaternion

This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—
Coxeter's 1950 representation in the Euclidean plane of the 9-point affine plane over GF(3)

The quaternion illustration above shows a more natural way to picture this geometry—
not with dots representing points in the Euclidean  plane, but rather with unit squares
representing points in a finite Galois  affine plane. The use of unit squares to
represent points in Galois space allows, in at least some cases, the actions
of finite groups to be represented more naturally than in Euclidean space.

See Galois Geometry, Geometry Simplified, and
Finite Geometry of the Square and Cube.

Saturday, February 13, 2010

Entertainment continued

Filed under: General,Geometry — m759 @ 9:28 AM

Logic is all about the entertaining of possibilities.”

– Colin McGinn, Mindsight: Image, Dream, Meaning,
   Harvard University Press, 2004

Geometry of Language,
continued from St. George's Day, 2009


Professor Arielle Saiber with chess set

Excerpt from Jasper Hopkins's 'Concise Introduction to the Philosophy of Nicholas of Cusa

Related material:

Prima Materia,
The Galois Quaternion,
and The Wake of Imagination.

See also the following from a physicist
(not of the most orthodox sort, but his remarks
  here on Heisenberg seem quite respectable)–

Ian J. Thompson, 7 Dec. 2009

Quantum mechanics describes the probabilities of actual outcomes in terms of a wave function, or at least of a quantum state of amplitudes that varies with time. The public always asks what the wave function is, or what the amplitudes are amplitudes of. Usually, we reply that the amplitudes are ‘probability amplitudes’, or that the wave function is a ‘probability wave function’, but neither answer is ontologically satisfying since probabilities are numbers, not stuff. We have already rehearsed the objections to the natural world being made out of numbers, as these are pure forms. In fact, ‘waves’, ‘amplitudes’ and ‘probabilities’ are all forms, and none of them can be substances. So, what are quantum objects made of: what stuff?

According to Heisenberg [6], the quantum probability waves are “a quantitative formulation of the concept of ‘dynamis’, possibility, or in the later Latin version, ‘potentia’, in Aristotle’s philosophy. The concept of events not determined in a peremptory manner, but that the possibility or ‘tendency’ for an event to take place has a kind of reality—a certain intermediate layer of reality, halfway between the massive reality of matter and the intellectual reality of the idea or the image—this concept plays a decisive role in Aristotle’s philosophy. In modern quantum theory this concept takes on a new form; it is formulated quantitatively as probability and subjected to mathematically expressible laws of nature.” Unfortunately Heisenberg does not develop this interpretation much beyond the sort of generality of the above statements, and the concept of ‘potentiality’ remains awkwardly isolated from much of his other thought on this subject [7]. It is unclear even what he means by ‘potentia’.

Reference

Heisenberg, W. 1961 On Modern Physics, London: Orion Press.

Notes

[6] W. Heisenberg, ‘Planck’s discovery and the philosophical problems of atomic physics’, pp. 3-20 in Heisenberg (1961).

[7] Heisenberg, for example, brings into his thought on quantum physics the Kantian phenomena/noumena distinction, as well as some of Bohr’s ideas on ‘complementarity’ in experimental arrangements.

Saturday, January 9, 2010

Positional Meaning

Filed under: General,Geometry — m759 @ 11:32 AM

"The positional meaning of a symbol derives from its relationship to other symbols in a totality, a Gestalt, whose elements acquire their significance from the system as a whole."

— Victor Turner, The Forest of Symbols, Ithaca, NY, Cornell University Press, 1967, p. 51, quoted by Beth Barrie in "Victor Turner."

To everything, turn, turn, turn…
— Peter Seeger

The Galois Quaternion:

The Galois Quaternion

Click for context.

Wednesday, January 6, 2010

Brightness at Noon, continued

Filed under: General,Geometry — m759 @ 12:00 PM

The Galois Quaternion

From The French Mathematician
by Tom Petsinis (Nov. 30, 1998)–

0

I had foreseen it all in precise detail.
One step led inevitably to the next,
like the proof of a shining theorem,
down to the conclusive shot that still echoes
through time and space.
Facedown in the damp pine needles,
I embraced that fatal sphere
with my whole body. Dreams, memories,
even the mathematics I had cherished
and set down in my last will and testament–
all receded. I am reduced to
a singular point; in an instant
I am transformed to i.

i = an imaginary being

Here, on this complex space,
i am no longer the impetuous youth
who wanted to change the world
first with a formula and then with a flame.
Having learned the meaning of infinite patience,
i now rise to the text whenever anyone reads
about Evariste Galois, preferring to remain
just below the surface,
like a goldfish nibbling the fringe of a floating leaf.
Ink is more mythical than blood
(unless some ancient poet slit his
vein and wrote an epic in red):
The text is a two-way mirror
that allows me to look into
the life and times of the reader.
Who knows, someday i may rise
to a text that will compel me
to push through to the other side.
Do you want proof that i exist? Where am i?
Beneath every word, behind each letter,
on the side of a period that will never see the light.

 

Related material:
The Galois Quaternion

The Galois Quaternion

Click for context.
(See also Nativity and the end
of this morning's post.)

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