(A review)
Friday, August 14, 2015
Discrete Space
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Monday, June 3, 2019
Art Wars for Spaceheads
Friday, May 3, 2019
The Structure of Story Space
Four Quartets
. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
A Permanent Order of Wondertale Elements
In Vol. I of Structural Anthropology , p. 209, I have shown that this analysis alone can account for the double aspect of time representation in all mythical systems: the narrative is both "in time" (it consists of a succession of events) and "beyond" (its value is permanent). With regard to Propp's theories my analysis offers another advantage: I can reconcile much better than Propp himself his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 9798. p. 108). If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns). 
… Or by congruent quartersections.
Thursday, May 2, 2019
Story Space
Monday, March 25, 2019
Espacement
(Continued from the previous post.)
InBetween "Spacing" and the "Chôra " (Ch. 2 in Henk Oosterling & Ewa Plonowska Ziarek (Eds.), Intermedialities: Philosophy, Arts, Politics , Lexington Books, October 14, 2010) "The term 'spacing' ('espacement ') is absolutely central to Derrida's entire corpus, where it is indissociable from those of différance (characterized, in the text from 1968 bearing this name, as '[at once] spacing [and] temporizing' ^{1}), writing (of which 'spacing' is said to be 'the fundamental property' ^{2}) and deconstruction (with one of Derrida's last major texts, Le Toucher: JeanLuc Nancy , specifying 'spacing ' to be 'the first word of any deconstruction' ^{3})." 1 Jacques Derrida, “La Différance,” in Marges – de la philosophie (Paris: Minuit, 1972), p. 14. Henceforth cited as D . 2 Jacques Derrida, “Freud and the Scene of Writing,” trans. A. Bass, in Writing and Difference (Chicago: University of Chicago Press, 1978), p. 217. Henceforth cited as FSW . 3 Jacques Derrida, Le Toucher, JeanLuc Nancy (Paris: Galilée, 2000), p. 207. . . . . "… a particularly interesting point is made in this respect by the French philosopher, Michel Haar. After remarking that the force Derrida attributes to différance consists simply of the series of its effects, and is, for this reason, 'an indefinite process of substitutions or permutations,' Haar specifies that, for this process to be something other than a simple 'actualisation' lacking any real power of effectivity, it would need “a soubassement porteur ' – let’s say a 'conducting underlay' or 'conducting medium' which would not, however, be an absolute base, nor an 'origin' or 'cause.' If then, as Haar concludes, différance and spacing show themselves to belong to 'a pure Apollonism' 'haunted by the groundless ground,' which they lack and deprive themselves of,^{16} we can better understand both the threat posed by the 'figures' of space and the mother in the Timaeus and, as a result, Derrida’s insistent attempts to disqualify them. So great, it would seem, is the menace to différance that Derrida must, in a 'properly' apotropaic gesture, ward off these 'figures' of an archaic, chthonic, spatial matrix in any and all ways possible…." 16 Michel Haar, “Le jeu de Nietzsche dans Derrida,” Revue philosophique de la France et de l’Etranger 2 (1990): 207227. . . . . … "The conclusion to be drawn from Democritus' conception of rhuthmos , as well as from Plato's conception of the chôra , is not, therefore, as Derrida would have it, that a differential field understood as an originary site of inscription would 'produce' the spatiality of space but, on the contrary, that 'differentiation in general' depends upon a certain 'spatial milieu' – what Haar would name a 'groundless ground' – revealed as such to be an 'inbetween' more 'originary' than the play of differences it informs. As such, this conclusion obviously extends beyond Derrida's conception of 'spacing,' encompassing contemporary philosophy's continual privileging of temporization in its elaboration of a preontological 'opening' – or, shall we say, 'inbetween.' 
For permutations and a possible "groundless ground," see
the eightfold cube and group actions both on a set of eight
building blocks arranged in a cube (a "conducting base") and
on the set of seven natural interstices (espacements ) between
the blocks. Such group actions provide an elementary picture of
the isomorphism between the groups PSL(2,7) (acting on the
eight blocks) and GL(3,2) (acting on the seven interstices).
Espacements
For the Church of Synchronology —
See also, from the reported publication date of the above book
Intermedialities , the Log24 post Synchronicity.
Sunday, March 24, 2019
Espacement: Geometry of the Interstice in Literary Theory
"You said something about the significance of spaces between
elements being repeated. Not only the element itself being repeated,
but the space between. I'm very interested in the space between.
That is where we come together." — Peter Eisenman, 1982
https://www.parrhesiajournal.org/ Parrhesia No. 3 • 2007 • 22–32
(Up) Against the (In) Between: Interstitial Spatiality by Clare Blackburne Blackburne — www.parrhesiajournal.org 24 — "The excessive notion of espacement as the resurgent spatiality of that which is supposedly ‘without space’ (most notably, writing), alerts us to the highly dynamic nature of the interstice – a movement whose discontinuous and ‘aberrant’ nature requires further analysis." Blackburne — www.parrhesiajournal.org 25 — "Espacement also evokes the ambiguous figure of the interstice, and is related to the equally complex derridean notions of chora , différance , the trace and the supplement. Derrida’s reading of the Platonic chora in Chora L Works (a series of discussions with the architect Peter Eisenman) as something which defies the logics of noncontradiction and binarity, implies the internal heterogeneity and instability of all structures, neither ‘sensible’ nor ‘intelligible’ but a third genus which escapes conceptual capture.^{25} Crucially, chora , spacing, dissemination and différance are highly dynamic concepts, involving hybridity, an ongoing ‘corruption’ of categories, and a ‘bastard reasoning.’^{26} Derrida identification of différance in Margins of Philosophy , as an ‘unappropriable excess’ that operates through spacing as ‘the becomingspace of time or the becomingtime of space,’^{27} chimes with his description of chora as an ‘unidentifiable excess’ that is ‘the spacing which is the condition for everything to take place,’ opening up the interval as the plurivocity of writing in defiance of ‘origin’ and ‘essence.’^{28} In this unfolding of différance , spacing ‘insinuates into presence an interval,’^{29} again alerting us to the crucial role of the interstice in deconstruction, and, as Derrida observes in Positions , its impact as ‘a movement, a displacement that indicates an irreducible alterity’: ‘Spacing is the impossibility for an identity to be closed on itself, on the inside of its proper interiority, or on its coincidence with itself. The irreducibility of spacing is the irreducibility of the other.’^{30}"
25. Quoted in Jeffrey Kipnis and Thomas Leeser, eds., 26. Ibid, 25.
27. Derrida, Margins of Philosophy. 28. Derrida, Chora L Works , 19 and 10. 29. Ibid, 203. 30. Derrida, Positions , 94. 
Thursday, January 24, 2019
Name Space
A correction at Wikipedia (Click to enlarge.) —
That this correction is needed indicates that the phrase
"Cullinane space" might be useful. (Click to enlarge.)
Tuesday, January 22, 2019
Monday, December 17, 2018
Tales from Story Space
"Kiernan Brennan Shipka (born November 10, 1999)
is an American actress. She is best known for starring as
Sabrina Spellman on the Netflix supernatural horror series
Chilling Adventures of Sabrina (2018–present)." — Wikipedia
As noted here earlier, Shipka turned 18 on Nov. 10 last year.
From Log24 on that date —
Another 18th birthday in Story Space —
Sunday, December 9, 2018
Quaternions in a Small Space
The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.
Click to enlarge —
Three links from the above finitegeometry.org webpage on the
quaternion group —

Visualizing GL(2,p) — A 1985 note illustrating group actions
on the 3×3 (ninefold) square. 
Another 1985 note showing group actions on the 3×3 square
transferred to the 2x2x2 (eightfold) cube.  Quaternions in an Affine Galois Plane — A webpage from 2010.
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
Sunday, November 4, 2018
Kristen vs. the Space Witch*
* We know the former. There is no shortage of candidates for the latter.
Saturday, November 3, 2018
The Space Theory of Truth
Earlier posts have discussed the "story theory of truth"
versus the "diamond theory of truth," as defined by
Richard Trudeau in his 1987 book The NonEuclidean Revolution.
In a New York Times opinion piece for tomorrow's print edition,*
novelist Dara Horn touched on what might be called
"the space theory of truth."
When they return to synagogue, mourners will be greeted
with more ancient words: “May God comfort you
among the mourners of Zion and Jerusalem.”
In that verse, the word used for God is hamakom —
literally, “the place.” May the place comfort you.
[Link added.]
The Source —
See Dara Horn in this journal, as well as Makom.
* "A version of this article appears in print on ,
on Page A23 of the New York edition with the headline:
American Jews Know This Story."
Tuesday, October 30, 2018
Story Structure, Story Space
Constance Grady at Vox today on a new Netflix series —
We don’t yet have a story structure that allows witches to be powerful for long stretches of time without men holding them back. And what makes the new Sabrina so exciting is that it seems to be trying to build that story structure itself, in real time, to find a way to let Sabrina have her power and her freedom. It might fail. But if it does, it will be a glorious and worthwhile failure — the type that comes with trying to pioneer a new kind of story. 
See also Story Space in this journal.
Monday, October 22, 2018
Story Space
Sunday, July 22, 2018
Wednesday, May 2, 2018
Galois’s Space
(A sequel to Foster's Space and Sawyer's Space)
See posts now tagged Galois's Space.
Sunday, March 4, 2018
The Square Inch Space: A Brief History
Sunday, November 19, 2017
Galois Space
This is a sequel to yesterday's post Cube Space Continued.
Friday, September 15, 2017
Space Art
Silas in "Equals" (2015) —
Ever since we were kids it's been drilled into us that …
Our purpose is to explore the universe, you know.
Outer space is where we'll find …
… the answers to why we're here and …
… and where we come from.
Related material —
See also Galois Space in this journal.
Sunday, April 16, 2017
Art Space Paradigm Shift
This post's title is from the tags of the previous post —
The title's "shift" is in the combined concepts of …
Space and Number
From Finite Jest (May 27, 2012):
The books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
For some details of the shift, see a Log24 search for Boole vs. Galois.
From a post found in that search —
"Benedict Cumberbatch Says
a Journey From Fact to Faith
Is at the Heart of Doctor Strange"
— io9 , July 29, 2016
" 'This man comes from a binary universe
where it’s all about logic,' the actor told us
at San Diego ComicCon . . . .
'And there’s a lot of humor in the collision
between Easter [ sic ] mysticism and
Western scientific, sort of logical binary.' "
[Typo now corrected, except in a comment.]
Thursday, April 13, 2017
Wednesday, March 29, 2017
Art Space Illustrated
Another view of the previous post's art space —
More generally, see Solomon's Cube in Log24.
See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above
cube arrangement from "Affine groups on small binary spaces" (1984).
Art Space, Continued
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson today
From Log24 posts tagged Art Space —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —
Two such considerations —
Monday, September 26, 2016
Tuesday, May 31, 2016
Friday, April 8, 2016
Ogdoads: A Space Odyssey
"Like the Valentinian Ogdoad— a selfcreating theogonic system
of eight Aeons in four begetting pairs— the projected eightfold work
had an esoteric, gnostic quality; much of Frye's formal interest lay in
the 'schematosis' and fearful symmetries of his own presentations."
— From p. 61 of James C. Nohrnberg's "The Master of the Myth
of Literature: An Interpenetrative Ogdoad for Northrop Frye,"
Comparative Literature , Vol. 53 No. 1, pp. 5882, Duke University
Press (quarterly, January 2001)
See also Two by Four in this journal.
Tuesday, January 12, 2016
Harmonic Analysis and Galois Spaces
The above sketch indicates, in a vague, handwaving, fashion,
a connection between Galois spaces and harmonic analysis.
For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.
Monday, January 11, 2016
Space Oddity
It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.
A Google search today for "Galois spaces" + "Boolean spaces"
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.
Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …
"Harmonic Analysis of Switching Functions" ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.
"Galois Switching Functions and Their Applications,"
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 4227 , 1975
D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239249, Mar. 1978
"Switching functions constructed by Galois extension fields,"
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981
An illustration from the Lechner paper above —
"There is such a thing as harmonic analysis of switching functions."
— Saying adapted from a youngadult novel
Wednesday, October 21, 2015
Algebra and Space
"Perhaps an insane conceit …." Perhaps.
Related remarks on algebra and space —
"The Quality Without a Name" (Log24, August 26, 2015).
Friday, August 14, 2015
Space Station 2015
(A sequel to Space Station 1976)
For Kathleen Gibbons* —
* Note Gibbons's work on "Discrete phase space based on finite fields."
Monday, September 22, 2014
Thursday, February 27, 2014
Sacred Space, continued
"An image comes to mind of a white, ideal space
that, more than any single picture, may be the
archetypal image of 20thcentury art."
— Brian O'Doherty, "Inside the White Cube"
Cube spaces exist also in mathematics.
Tuesday, December 3, 2013
Diamond Space
A new website illustrates its URL.
See DiamondSpace.net.
Tuesday, July 16, 2013
Space Itself
"How do you get young people excited
about space? How do you get them interested
not just in watching movies about space,
or in playing video games set in space …
but in space itself?"
— Megan Garber in The Atlantic , Aug. 16, 2012
One approach:
"There is such a thing as a tesseract" and
Diamond Theory in 1937.
See, too, Baez in this journal.
Sunday, March 10, 2013
Galois Space
The 16point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 16_{6} configuration.
Some closely related material:
 Wolfgang Kühnel,
"Minimal Triangulations of Kummer Varieties,"
Abh. Math. Sem. Univ. Hamburg 57, 720 (1986).For the first two pages, click here.
 Jonathan Spreer and Wolfgang Kühnel,
"Combinatorial Properties of the K 3 Surface:
Simplicial Blowups and Slicings,"
preprint, 26 pages. (2009/10) (pdf).
(Published in Experimental Math. 20,
issue 2, 201–216 (2011).)
Monday, March 4, 2013
Occupy Galois Space
Continued from February 27, the day Joseph Frank died…
"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review in 1945, propelled him
to prominence as a theoretician."
— Bruce Weber in this morning's print copy
of The New York Times (p. A15, NY edition)
That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:
See also Galois Space and Occupy Space in this journal.
Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:
"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."
Frank is survived by, among others, his wife, a mathematician.
Thursday, February 21, 2013
Galois Space
The previous post suggests two sayings:
"There is such a thing as a Galois space."
— Adapted from Madeleine L'Engle
"For every kind of vampire, there is a kind of cross."
Illustrations—
Saturday, September 22, 2012
Occupy Space
"The word 'space' has, as you suggest, a large number of different meanings."
— Nanavira Thera in [Early Letters. 136] 10.xii.1958
From that same letter (links added to relevant Wikipedia articles)—
Space (ākāsa) is undoubtedly used in the Suttas
Your second letter seems to suggest that the space 
A simpler metaphysical system along the same lines—
The theory, he had explained, was that the persona
— The Gameplayers of Zan , 
"I am glad you have discovered that the situation is comical:
ever since studying Kummer I have been, with some difficulty,
refraining from making that remark."
— Nanavira Thera, [Early Letters, 131] 17.vii.1958
Thursday, July 12, 2012
Galois Space
An example of lines in a Galois space * —
The 35 lines in the 3dimensional Galois projective space PG(3,2)—
There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3set of linear diagrams
represents the structure of one of the 35 4×4 arrays and also represents a line
of the projective space.
The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.
* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958 [Edinburgh].
(Cambridge U. Press, 1960, 488499.)
(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)
Saturday, February 18, 2012
Symmetry
From the current Wikipedia article "Symmetry (physics)"—
"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.
A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….
"A discrete symmetry is a symmetry that describes noncontinuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."
Note the confusion here between continuous (or discontinuous) transformations and "continuous" (or "discontinuous," i.e. "discrete") groups .
This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.
For an attempt to forestall such confusion, see Noncontinuous Groups.
For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory—
Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3space is that the transformations involved are continuous.
Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself be invariant under rather interesting groups of noncontinuous (and asymmetric) transformations. (These might be called noncontinuous groups, as opposed to socalled discontinuous (or discrete ) symmetry groups. See Weyl's Symmetry .)
For example, the affine group A on the 4space over the 2element field has a natural noncontinuous and asymmetric but symmetrypreserving action on the elements of a 4×4 array. (Details)
(Version first archived on March 27, 2002)
Update of Sunday, February 19, 2012—
The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—
Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous transformations.
This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics (Nirmala Prakash, Imperial College Press)—
"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous (discrete ) symmetry ." — Pp. 235, 236
[* Associated how?]
Monday, August 29, 2005
Monday August 29, 2005
Date: Sun, 28 Aug 2005 12:30:40 0400 From: Alf van der Poorten AM Subject: Vale George Szekeres and Esther Klein Szekeres Members of the Number Theory List will be sad to learn that George and Esther Szekeres both died this morning. George, 94, had been quite ill for the last 23 days, barely conscious, and died first at 06:30. Esther, 95, died a half hour later. Both George Szekeres and Esther Klein will be recalled by number theorists as members of the group of young Hungarian mathematicians of the 1930s including Turan and Erdos. George and Esther's coming to Australia in the late 40s played an important role in the invigoration of Australian Mathematics. George was also an expert in group theory and relativity; he was my PhD supervisor. Emeritus Professor 
AVE
"Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one."
"A very short space of time through very short times of space…. — James Joyce, Ulysses, Proteus chapter A very short space of time through very short times of space…. "It is demonstrated that spacetime should possess a discrete structure on Planck scales." — Peter Szekeres, abstract of Discrete SpaceTime 
Peter Szekeres is the son of George and Esther Szekeres.
"At present, such relationships can at best be heuristically described in terms that invoke some notion of an 'intelligent user standing outside the system.'"
— GianCarlo Rota in Indiscrete Thoughts, p. 152
Thursday, August 25, 2005
Thursday August 25, 2005
Train of Thought
Part I: The 24Cell
From S. H. Cullinane,
Visualizing GL(2,p),
March 26, 1985–
From John Baez, “This Week’s Finds in Mathematical Physics (Week 198),” September 6, 2003: Noam Elkies writes to John Baez:
The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – GianCarlo Rota 
Like footprints erased in the sand….
“Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.”
“A very short space of time through very short times of space….
Am I walking into eternity along Sandymount strand?”
— James Joyce, Ulysses, Proteus chapter
A very short space of time through very short times of space….
“It is demonstrated that spacetime should possess a discrete structure on Planck scales.”
— Peter Szekeres, abstract of Discrete SpaceTime
“A theory…. predicts that space and time are indeed made of discrete pieces.”
— Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004
“… a fundamental discreteness of spacetime seems to be a prediction of the theory….”
— Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity
“Theories of discrete spacetime structure are being studied from a variety of perspectives.”
— Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London
The above speculations by physicists
are offered as curiosities.
I have no idea whether
any of them are correct.
Related material:
Stephen Wolfram offers a brief
History of Discrete Space.
For a discussion of space as discrete
by a nonphysicist, see John Bigelow‘s
Space and Timaeus.
in a Discrete Space
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
(Aug. 4, 2005):
Thursday, June 13, 2019
Tuesday, June 4, 2019
Zen and the Art
Or: Burning Bright
A post in memory of Chicago architect Stanley Tigerman,
who reportedly died at 88 on Monday.
Inside Out
For fans of Space Fleet and of "reclusive but gifted" programmers—
“Hello the Camp”
The title is a quotation from the 2015 film "Mojave."
Monday, June 3, 2019
Jar Story
". . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”
— T. S. Eliot, Four Quartets
From Writing Chinese Characters:
“It is practical to think of a character centered
within an imaginary square grid . . . .
The grid can… be… subdivided, usually to
9 or 16 squares. . . ."
These “Chinese jars” (as opposed to their contents)
are as follows:
.
See as well Eliot's 1922 remarks on "extinction of personality"
and the phrase "egoextinction" in Weyl's Philosophy of Mathematics —
Monday, May 20, 2019
The Bond with Reality
Tuesday, May 7, 2019
Breach
"Honored in the Breach:
Graham Bader on Absence as Memorial"
Artforum International , April 2012
. . . . "In the wake of a century marked by inconceivable atrocity, the use of emptiness as a commemorative trope has arguably become a standard tactic, a default style of public memory. The power of the voids at and around Ground Zero is generated by their origin in real historical circumstance rather than such purely commemorative intent: They are indices as well as icons of the losses they mark.
Nowhere is the negotiation between these two possibilities–on the one hand, the cooptation of absence as tasteful mnemonic trope; on the other, absence's disruptive potential as brute historical scar–more evident than in Berlin, a city whose history, as Andreas Huyssen has argued, can be seen as a 'narrative of voids.' Writing in 1997, Huyssen saw this tale culminating in Berlin's postwall development, defined equally by an obsessive coveringover of the city's lacunae–above all in the elaborate commercial projects then proliferating in the mileslong stretch occupied until 1989 by the Berlin Wall–and a carefully orchestrated deployment of absence as memorial device, particularly in the 'voids' integrated by architect Daniel Libeskind into his addition to the Berlin Museum, now known as the Jewish Museum Berlin." 
See also Breach in this journal.
Monday, May 6, 2019
One Stuff
Saturday, May 4, 2019
Inside the White Cube
See also Espacement and The Thing and I.
Friday, May 3, 2019
Wednesday, April 24, 2019
Critical Visibility
Correction — "Death has 'the whole spirit sparkling…'"
should be "Peace after death has 'the whole spirit sparkling….'"
The page number, 373, is a reference to Wallace Stevens:
Collected Poetry and Prose , Library of America, 1997.
See also the previous post, "Critical Invisibility."
Tuesday, April 23, 2019
Critical Invisibility
From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens 54, 5979 (1992):
"… what is the origin of the unusual name 'symplectic'? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure 'line complex group' the 'symplectic group.'
… the adjective 'symplectic' means 'plaited together' or 'woven.'
This is wonderfully apt…."
On "The Emperor's New Clothes" —
Andersen’s weavers, as one commentator points out, are merely insisting that “the value of their labor be recognized apart from its material embodiment.” The invisible cloth they weave may never manifest itself in material terms, but the description of its beauty (“as light as spiderwebs” and “exquisite”) turns it into one of the many wondrous objects found in Andersen’s fairy tales. It is that cloth that captivates us, making us do the imaginative work of seeing something beautiful even when it has no material reality. Deeply resonant with meaning and of rare aesthetic beauty—even if they never become real—the cloth and other wondrous objets d’art have attained a certain degree of critical invisibility. — Maria Tatar, The Annotated Hans Christian Andersen (W. W. Norton & Company, 2007). Kindle Edition. 
A Certain Dramatic Artfulness
Saturday, March 9, 2019
Weapons of Mass Distraction
"Back to the Future" and . . .
I prefer another presentation from the above
Universal Pictures date — June 28, 2018 —
Friday, March 8, 2019
Photo Opportunity
"I need a photo opportunity . . . ." — Paul Simon
Thursday, February 28, 2019
Previn’s Wake
A search for Previn in this journal yields . . .
"whyse Salmonson set his seel on a hexengown,"
Finnegans Wake , Book II, Episode 2, pp. 296297
Fooling
The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
Note: There is no Galois (i.e., finite) field with six elements, but
the theory of finite fields underlies applications of sixset geometry.
Monday, January 21, 2019
Meditation for the Champ de Mors
"his onesidemissing for an allblind alley
leading to an Irish plot in the Champ de Mors"
— James Joyce, Finnegans Wake
Sunday, January 20, 2019
Scope Resolution
Wikipedia on a programming term —
The scope resolution operator helps to identify
and specify the context to which an identifier refers,
particularly by specifying a namespace. The specific
uses vary across different programming languages
with the notions of scoping. In many languages
the scope resolution operator is written
"::".
In a completely different context, these four dots might represent
a geometric object — the fourpoint plane .
Saturday, January 19, 2019
Friday, January 18, 2019
Location, Location, Location
The Woke Grids …
… as opposed to The Dreaming Jewels .
A July 2014 Amsterdam master's thesis on the Golay code
and Mathieu group —
"The properties of G_{24} and M_{24} are visualized by
four geometric objects: the icosahedron, dodecahedron,
dodecadodecahedron, and the cubicuboctahedron."
Some "geometric objects" — rectangular, square, and cubic arrays —
are even more fundamental than the above polyhedra.
A related image from a post of Dec. 1, 2018 —
Tuesday, December 18, 2018
Shadowhunter Tales
The recent post "Tales from Story Space," about the 18th birthday
of the protagonist in the TV series "Shadowhunters" (2016),
suggests a review of the actual 18th birthday of actress Lily Collins.
Collins is shown below warding off evil with a magical rune as
a shadowhunter in the 2013 film "City of Bones" —
She turned 18 on March 18, 2007. A paper on symmetry and logic
referenced here on that date displays the following "runes" of
philosopher Charles Sanders Peirce —
See also Adamantine Meditation (Log24, Oct. 3, 2018)
and the webpage Geometry of the I Ching.
Sunday, December 16, 2018
Tuesday, November 6, 2018
Bait and Tackle
Monday, November 5, 2018
Sunday, November 4, 2018
“Look Up” — The Breakthrough Prize* Theme This Evening
Looking up images for "The Space Theory of Truth" this evening —
Detail (from the post "Logos" of Oct. 14) —
Saturday, November 3, 2018
For St. Anselm
"… at his home in San Anselmo . . . ."
See also Anselm in this journal, as well as the Devil's Night post Ojos.
Tuesday, October 30, 2018
Sunday, September 9, 2018
Plan 9 Continues.
"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.
Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."
— From p. 192 of "The Phenomenology of Mathematical Proof,"
by GianCarlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183196. Published by: Springer.
Stable URL: https://www.jstor.org/stable/20117627.
Related figures —
Note the 3×3 subsquare containing the triangles ABC, etc.
"That in which space itself is contained" — Wallace Stevens
Friday, July 6, 2018
Something
"… Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."
— T. S. Eliot, "Burnt Norton," 1936
"Read something that means something."
— Advertising slogan for The New Yorker
The previous post quoted some mystic meditations of Octavio Paz
from 1974. I prefer some less mystic remarks of Eddington from
1938 (the Tanner Lectures) published by Cambridge U. Press in 1939 —
"… we have sixteen elements with which to form a groupstructure" —
See as well posts tagged Dirac and Geometry.
Tuesday, May 1, 2018
Wake
Remarks on space from 1998 by scifi author Robert J. Sawyer quoted
here on Sunday (see the tag "Sawyer's Space") suggest a review of
rather similar remarks on space from 1977 by scifi author M. A. Foster
(see the tag "Foster's Space"):
Quoted here on September 26, 2012 —
"All she had to do was kick off and flow."
"I'se so silly to be flowing but I no canna stay."
Another work by Sawyer —
Sunday, April 29, 2018
Amusement
From the online New York Times this afternoon:
Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.
From this journal this morning:
"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to
make a fancy amusementpark ride.
There had to be more—"
— Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168
"The sensation of folding space . . . ."
Or unfolding:
Click the above unfolded space for some background.
Thursday, April 19, 2018
Something to Behold
From a review of a Joyce Carol Oates novel
at firstthings.com on August 23, 2013 —
"Though the Curse is eventually exorcised,
it is through an act of wit and guile,
not an act of repentance or reconciliation.
And so we may wonder if Oates has put this story
to rest, or if it simply lays dormant. A twentyfirst
century eruption of the 'Crosswicks Curse'
would be something to behold." [Link added.]
Related material —
A film version of A Wrinkle in Time —
The Hamilton watch from "Interstellar" (2014) —
See also a post, Vacant Space, from 8/23/13 (the date
of the above review), and posts tagged Space Writer.
Monday, March 12, 2018
“Quantum Tesseract Theorem?”
Remarks related to a recent film and a notsorecent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The SanigaPlanat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
Wednesday, March 7, 2018
Excited
"How do you get young people excited about space?"
— Megan Garber in The Atlantic , Aug. 16, 2012
The above quote is from this journal on 9/11, 2014.
Related material —
Synchronology for the above date — 9/11, 2014 —
A BuzzFeed article with that date, and in reply
"A Personal Statement from Michael Shermer" with that date.
Wednesday, February 28, 2018
A Girl’s Guide to Chaos
The title is that of a play mentioned last night in
a New York Times obituary .
Related recent film lines —
 Thor: How do I escape?

Heimdall: You're on a planet surrounded by doorways.
Go through one.  Thor: Which one?
 Heimdall: The big one!
Related material from this journal on Jan. 20, 2018 —
Mathematics and Narrative
Excerpts from a post of May 25, 2005 —
Above is an example I like of mathematics….
Here is an example I like of narrative:
Kate felt quite dizzy. She didn't know exactly what it was that had just happened, but she felt pretty damn certain that it was the sort of experience that her mother would not have approved of on a first date. "Is this all part of what we have to do to go to Asgard?" she said. "Or are you just fooling around?" "We will go to Asgard...now," he said. At that moment he raised his hand as if to pluck an apple, but instead of plucking he made a tiny, sharp turning movement. The effect was as if he had twisted the entire world through a billionth part of a billionth part of a degree. Everything shifted, was for a moment minutely out of focus, and then snapped back again as a suddenly different world.
— Douglas Adams, The Long Dark TeaTime of the Soul
Image from a different different world —
Hattip to a related Feb. 26 weblog post
at the American Mathematical Society.
Thursday, January 25, 2018
Beware of Analogical Extension
"By an archetype I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."
— Max Black in Models and Metaphors
(Cornell, 1962, p. 241)
"Others … spoke of 'ultimate frames of reference' …."
— Ibid.
A "frame of reference" for the concept four quartets —
A less reputable analogical extension of the same
frame of reference —
Madeleine L'Engle in A Swiftly Tilting Planet :
"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."
See also the phrase Galois tesseract .
Thursday, November 23, 2017
LéviStrauss vs. Propp
Claude LéviStrauss
From his "Structure and Form: To maintain. as I have done. that the permutability of contents is not arbitrary amounts to saying that, if the analysis is carried to a sufficiently deep level, behind diversity we will discover constancy. And, of course. the avowed constancy of form must not hide from us that functions are also permutable. The structure of the folktale as it is illustrated by Propp presents a chronological succession of qualitatively distinct functions. each constituting an independent genre. One can wonder whether—as with dramatis personae and their attributes— Propp does not stop too soon, seeking the form too close to the level of empirical observation. Among the thirtyone functions that he distinguishes, several are reducible to the same function reappearing at different moments of the narrative but after undergoing one or a number of transformations . I have already suggested that this could be true of the false hero (a transformation of the villain), of assigning a difficult task (a transformation of the test), etc. (see p. 181 above), and that in this case the two parties constituting the fundamental tale would themselves be transformations of each other. Nothing prevents pushing this reduction even further and analyzing each separate partie into a small number of recurrent functions, so that several of Propp's functions would constitute groups of transformations of one and the same function. We could treat the "violation" as the reverse of the "prohibition" and the latter as a negative transformation of the "injunction." The "departure" of the hero and his "return" would appear as the negative and positive expressions of the same disjunctive function. The "quest" of the hero (hero pursues someone or something) would become the opposite of "pursuit" (hero is pursued by something or someone), etc.
In Vol. I of Structural Anthropology , p. 209, I have shown that this analysis alone can account for the double aspect of time representation in all mythical systems: the narrative is both "in time" (it consists of a succession of events) and "beyond" (its value is permanent). With regard to Propp's theories my analysis offers another advantage: I can reconcile much better than Propp himself his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 9798. p. 108) If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns). These critical remarks are certainly valid for the method used by Propp and for his conclusions. However. it cannot be stressed enough that Propp envisioned them and in several places formulated with perfect clarity the solutions I have just suggested. Let us take up again from this viewpoint the two essential themes of our discussion: constancy of the content (in spite of its permutability) and permutability of functions (in spite of their constancy).
* Translated from a 1960 work in French. It appeared in English as Chapter VIII 
See also "LéviStrauss" + Formula in this journal.
Some background related to the previous post —
Wednesday, October 25, 2017
To the Egress
The New York Times at 8:22 PM ET —
"Knight Landesman, a longtime publisher of Artforum magazine
and a power broker in the art world, resigned on Wednesday
afternoon, hours after a lawsuit was filed in New York accusing
him of sexually harassing at least nine women in episodes that
stretched back almost a decade."
See as well, in this journal, Way to the Egress.
The Palo Alto Edge
From Stanford — The death on October 9, 2017, of a man who
"always wanted to be at the most cutting of cuttingedge technology."
Related material from Log24 on April 26, 2017 —
A sketch, adapted from Girl Scouts of Palo Alto —
Click the sketch for further details.
Saturday, September 23, 2017
The Turn of the Frame
"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."
— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies No. 55/56 (1977), pp. 94207.
Published by Yale University Press.
See also the previous post and The Galois Tesseract.
Sunday, August 27, 2017
Black Well
The "Black" of the title refers to the previous post.
For the "Well," see Hexagram 48.
Related material —
The Galois Tesseract and, more generally, Binary Coordinate Systems.
Friday, June 16, 2017
Chalkroom Jungle
At MASS MoCA, the installation "Chalkroom" quotes a lyric —
Oh beauty in all its forms funny how hatred can also be a beautiful thing When it's as sharp as a knife as hard as a diamond Perfect 
— From "One Beautiful Evening," by Laurie Anderson.
See also the previous post and "Smallest Perfect" in this journal.
Thursday, June 15, 2017
Building Six
Berkshire tales of May 25, 2017 —
See also, in this journal from May 25 and earlier, posts now tagged
"The Story of Six."
Saturday, June 3, 2017
Expanding the Spielraum (Continued*)
Or: The Square
"What we do may be small, but it has
a certain character of permanence."
— G. H. Hardy
* See Expanding the Spielraum in this journal.
Tuesday, May 23, 2017
Saturday, May 20, 2017
The Ludicrous Extreme
From a review of the 2016 film "Arrival" —
"A seemingly offhand reference to Abbott and Costello
is our gateway. In a movie as generally humorless as Arrival,
the jokes mean something. Ironically, it is Donnelly, not Banks,
who initiates the joke, naming the verbally inexpressive
Heptapod aliens after the loquacious Classical Hollywood
comedians. The squidlike aliens communicate via those beautiful,
cryptic images. Those signs, when thoroughly comprehended,
open the perceiver to a nonlinear conception of time; this is
SapirWhorf taken to the ludicrous extreme."
— Jordan Brower in the Los Angeles Review of Books
Further on in the review —
"Banks doesn’t fully understand the alien language, but she
knows it well enough to get by. This realization emerges
most evidently when Banks enters the alien ship and, floating
alongside Costello, converses with it in their picturelanguage.
She asks where Abbott is, and it responds — as presented
in subtitling — that Abbott 'is death process.'
'Death process' — dying — is not idiomatic English, and what
we see, written for us, is not a perfect translation but a
rendering of Banks’s understanding. This, it seems to me, is a
crucial moment marking the hard limit of a human mind,
working within the confines of human language to understand
an ultimately intractable xenolinguistic system."
For what may seem like an intractable xenolinguistic system to
those whose experience of mathematics is limited to portrayals
by Hollywood, see the previous post —
van Lint and Wilson Meet the Galois Tesseract.
The death process of van Lint occurred on Sept. 28, 2004.
van Lint and Wilson Meet the Galois Tesseract*
Click image to enlarge.
The above 35 projective lines, within a 4×4 array —
The above 15 projective planes, within a 4×4 array (in white) —
* See Galois Tesseract in this journal.
Tuesday, May 2, 2017
Image Albums
Pinterest boards uploaded to the new m759.net/piwigo —
Update of May 2 —
Update of May 3 —
Update of May 8 —
Art Space board created at Pinterest
Thursday, April 27, 2017
Partner, Anchor, Decompose
See also a figure from 2 AM ET April 26 …
" Partner, anchor, decompose. That's not math.
That's the plot to 'Silence of the Lambs.' "
Wednesday, April 26, 2017
A Tale Unfolded
A sketch, adapted tonight from Girl Scouts of Palo Alto —
From the April 14 noon post High Concept —
From the April 14 3 AM post Hudson and Finite Geometry —
From the April 24 evening post The Trials of Device —
Note that Hudson's 1905 "unfolding" of even and odd puts even on top of
the square array, but my own 2013 unfolding above puts even at its left.
Monday, April 24, 2017
The Trials of Device
"A blank underlies the trials of device"
— Wallace Stevens, "An Ordinary Evening in New Haven" (1950)
A possible meaning for the phrase "the trials of device" —
See also Log24 posts mentioning a particular device, the pentagram .
For instance —
Monday, April 17, 2017
Saturday, April 15, 2017
Quanta Dating
For the Church of Synchronology —
See also this journal on July 17, 2014, and March 28, 2017.
Friday, April 14, 2017
Hudson and Finite Geometry
The above fourelement sets of black subsquares of a 4×4 square array
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .
Hudson did not view these 35 tetrads as planes through the origin in a finite
affine 4space (or, equivalently, as lines in the corresponding finite projective
3space).
In order to view them in this way, one can view the tetrads as derived,
via the 15 twoelement subsets of a sixelement set, from the 16 elements
of the binary Galois affine space pictured above at top left.
This space is formed by taking symmetricdifference (Galois binary)
sums of the 15 twoelement subsets, and identifying any resulting four
element (or, summing three disjoint twoelement subsets, sixelement)
subsets with their complements. This process was described in my note
"The 2subsets of a 6set are the points of a PG(3,2)" of May 26, 1986.
The space was later described in the following —
Wednesday, April 12, 2017
Contracting the Spielraum
The contraction of the title is from group actions on
the ninefold square (with the center subsquare fixed)
to group actions on the eightfold cube.
From a post of June 4, 2014 …
At math.stackexchange.com on March 112, 2013:
“Is there a geometric realization of the Quaternion group?” —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).
Expanding the Spielraum
"Cézanne ignores the laws of classical perspective . . . ."
— Voorhies, James. “Paul Cézanne (1839–1906).”
In Heilbrunn Timeline of Art History . New York:
The Metropolitan Museum of Art, 2000–. (October 2004)
Some others do not.
This is what I called "the large Desargues configuration"
in posts of April 2013 and later.
Tuesday, April 4, 2017
White Cube
"We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20thcentury art."
"Space: what you
damn well have to see."
— James Joyce, Ulysses
Monday, April 3, 2017
Wednesday, March 29, 2017
The Crimson Abyss
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson , March 29
1984 —
2010 —
Logo design for Stack Exchange Math by Jin Yang
Recent posts now tagged Crimson Abyss suggest
the above logo be viewed in light of a certain page 29 —
"… as if into a crimson abyss …." —
Update of 9 PM ET March 29, 2017:
Prospero's Children was first published by HarperCollins,
London, in 1999. A statement by the publisher provides
an instance of the famous "muchneeded gap." —
"This is English fantasy at its finest. Prospero’s Children
steps into the gap that exists between The Lion, the Witch
and the Wardrobe and Clive Barker’s Weaveworld , and
is destined to become a modern classic."
Related imagery —
See also "Hexagram 64 in Context" (Log24, March 16, 2017).
Design Abyss
Hexagram 29,
The Abyss (Water)
This post was suggested by an August 6, 2010, post by the designer
(in summer or fall, 2010) of the Stack Exchange math logo (see
the previous Log24 post, Art Space Illustrated) —
In that post, the designer quotes the Wilhelm/Baynes I Ching to explain
his choice of Hexagram 63, Water Over Fire, as a personal icon —
"When water in a kettle hangs over fire, the two elements
stand in relation and thus generate energy (cf. the
production of steam). But the resulting tension demands
caution. If the water boils over, the fire is extinguished
and its energy is lost. If the heat is too great, the water
evaporates into the air. These elements here brought in
to relation and thus generating energy are by nature
hostile to each other. Only the most extreme caution
can prevent damage."
See also this journal on Walpurgisnacht (April 30), 2010 —
Hexagram 29:

Hexagram 30: 
A thought from another Germanspeaking philosopher —
"Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache."
See also The Crimson 's abyss in today's 4:35 AM post Art Space, Continued.
Tuesday, March 28, 2017
Bit by Bit
From Log24, "Cube Bricks 1984" —
Also on March 9, 2017 —
For those who prefer graphic art —
Backstory
Click here to enlarge. Click the image for the source page.
The "this page" reference is to …
Finite Geometry of the Square and Cube.
Also from March 14, 2017 —
Thursday, March 2, 2017
Ageometretos*
Stories
"We tell ourselves stories in order to live." — Joan Didion
The New York Times Magazine online today —
"As a former believer and now a nonbeliever, Carrère,
seeking answers, sets out, in The Kingdom , to tell
the story of the storytellers. He is trying to understand
what it takes to be able to tell a story, any story.
And what he finds, once again, is that you have to find
your role in it."
— Wyatt Mason in The New York Times Magazine ,
online March 2, 2017
Like Tom Hanks?
Click image for related posts.
Wednesday, October 5, 2016
Sources
From a Google image search yesterday —
Sources (left to right, top to bottom) —
Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)
Monday, September 26, 2016
The Embedding
From this morning's 3:33 AM ET post —
Adapted from a post of Dec. 8, 2012, "Defining the Contest" —
From a post of Sept. 22,
"Binary Opposition Illustrated" —
From Sunday's news —
Palmervision
Sunday, September 25, 2016
Introduction to Pragmatism
Stanford Encyclopedia of Philosophy
on the origins of Pragmatism:
"Pragmatism had been born in the discussions at
a ‘metaphysical club’ in Harvard around 1870
(see Menand…*). Peirce and James participated
in these discussions along with some other philosophers
and philosophically inclined lawyers. As we have
already noted, Peirce developed these ideas in his
publications from the 1870s."
From "How to Make Our Ideas Clear," "The very first lesson that we have a right to demand that logic shall teach us is, how to make our ideas clear; and a most important one it is, depreciated only by minds who stand in need of it. To know what we think, to be masters of our own meaning, will make a solid foundation for great and weighty thought. It is most easily learned by those whose ideas are meagre and restricted; and far happier they than such as wallow helplessly in a rich mud of conceptions. A nation, it is true, may, in the course of generations, overcome the disadvantage of an excessive wealth of language and its natural concomitant, a vast, unfathomable deep of ideas. We may see it in history, slowly perfecting its literary forms, sloughing at length its metaphysics, and, by virtue of the untirable patience which is often a compensation, attaining great excellence in every branch of mental acquirement. The page of history is not yet unrolled which is to tell us whether such a people will or will not in the longrun prevail over one whose ideas (like the words of their language) are few, but which possesses a wonderful mastery over those which it has. For an individual, however, there can be no question that a few clear ideas are worth more than many confused ones. A young man would hardly be persuaded to sacrifice the greater part of his thoughts to save the rest; and the muddled head is the least apt to see the necessity of such a sacrifice. Him we can usually only commiserate, as a person with a congenital defect. Time will help him, but intellectual maturity with regard to clearness comes rather late, an unfortunate arrangement of Nature, inasmuch as clearness is of less use to a man settled in life, whose errors have in great measure had their effect, than it would be to one whose path lies before him. It is terrible to see how a single unclear idea, a single formula without meaning, lurking in a young man's head, will sometimes act like an obstruction of inert matter in an artery, hindering the nutrition of the brain, and condemning its victim to pine away in the fullness of his intellectual vigor and in the midst of intellectual plenty. Many a man has cherished for years as his hobby some vague shadow of an idea, too meaningless to be positively false; he has, nevertheless, passionately loved it, has made it his companion by day and by night, and has given to it his strength and his life, leaving all other occupations for its sake, and in short has lived with it and for it, until it has become, as it were, flesh of his flesh and bone of his bone; and then he has waked up some bright morning to find it gone, clean vanished away like the beautiful Melusina of the fable, and the essence of his life gone with it. I have myself known such a man; and who can tell how many histories of circlesquarers, metaphysicians, astrologers, and what not, may not be told in the old German story?" 
Peirce himself may or may not have been entirely successful
in making his ideas clear. See Where Credit Is Due (Log24,
June 11, 2016) and the Wikipedia article Categories (Peirce).
* Menand, L., 2001. The Metaphysical Club : A Story of
Ideas in America , New York: Farrar, Straus and Giroux
Saturday, September 24, 2016
The Seven Seals
From Hermann Weyl's 1952 classic Symmetry —
"Galois' ideas, which for several decades remained
a book with seven seals but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twentyone. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."
Some Galois geometry —
See the previous post for more narrative.
Core Structure
For the director of "Interstellar" and "Inception" —
At the core of the 4x4x4 cube is …
Cover modified.