Saturday, July 8, 2017
Related material now available online —
A less businessoriented sort of virtual reality —
For example, "A very important configuration is obtained by
taking the plane section of a complete space fivepoint."
(Veblen and Young, 1910, p. 39)—
Comments Off on Desargues and Galois in Japan
Tuesday, August 6, 2013
The following image gives a brief description
of the geometry discussed in last spring's
Classical Geometry in Light of Galois Geometry.
Update of Aug. 7, 2013: See also an expanded PDF version.
Comments Off on Desargues via Galois
Sunday, November 22, 2020
A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 2329 (7 pages) —
Related material — The Eightfold Cube.
Update at 10:51 PM ET the same day —
Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .
DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.
The arXiv versions —
Comments Off on The GaloisFano Plane
Wednesday, May 2, 2018
(A sequel to Foster's Space and Sawyer's Space)
See posts now tagged Galois's Space.
Comments Off on Galois’s Space
Sunday, November 19, 2017
This is a sequel to yesterday's post Cube Space Continued.
Comments Off on Galois Space
Saturday, May 20, 2017
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.
Comments Off on van Lint and Wilson Meet the Galois Tesseract*
Sunday, August 14, 2016
Continued from earlier posts on Boole vs. Galois.
From a Google image search today for “Galois Boole.”
Click the image to enlarge it.
Comments Off on The BooleGalois Games
Tuesday, May 31, 2016
A very brief introduction:
Comments Off on Galois Space —
Tuesday, January 12, 2016
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.
Comments Off on Harmonic Analysis and Galois Spaces
Tuesday, March 24, 2015
Yesterday's post suggests a review of the following —
Andries Brouwer, preprint, 1982:
"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)
Pages 89:
Substructures of S(5, 8, 24)
An octad is a block of S(5, 8, 24).
Theorem 5.1
Let B_{0} be a fixed octad. The 30 octads disjoint from B_{0}
form a selfcomplementary 3(16,8,3) design, namely
the design of the points and affine hyperplanes in AG(4, 2),
the 4dimensional affine space over F_{2}.
Proof….
… (iv) We have AG(4, 2).
(Proof: invoke your favorite characterization of AG(4, 2)
or PG(3, 2), say DembowskiWagner or Veblen & Young.
An explicit construction of the vector space is also easy….)

Related material: Posts tagged Priority.
Comments Off on Brouwer on the Galois Tesseract
Tuesday, November 25, 2014
For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.
The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.
These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 1617 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
Comments Off on EuclideanGalois Interplay
Thursday, March 20, 2014
Click image for more details.
To enlarge image, click here.
Comments Off on Classical Galois
Friday, April 19, 2013
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6volume
Principles of Geometry .
Pointline incidence in this larger configuration is,
as noted in a post of April 1, 2013, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.
Comments Off on The Large Desargues Configuration
Saturday, April 13, 2013
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
See also the original VeblenYoung figure in context.
Comments Off on Veblen and Young in Light of Galois
Monday, April 1, 2013
Background: Rosenhain and Göpel Tetrads in PG(3,2)
Introduction:
The Large Desargues Configuration
Added by Steven H. Cullinane on Friday, April 19, 2013
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6volume
Principles of Geometry .
Pointline incidence in this larger configuration is,
as noted in the post of April 1 that follows
this introduction, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.

A connection discovered today (April 1, 2013)—
(Click to enlarge the image below.)
Update of April 18, 2013
Note that Baker's Desarguestheorem figure has three triangles,
ABC, A'B'C', A"B"C", instead of the two triangles that occur in
the statement of the theorem. The third triangle appears in the
course of proving, not just stating, the theorem (or, more precisely,
its converse). See, for instance, a note on a standard textbook for
further details.
(End of April 18, 2013 update.)
Update of April 14, 2013
See Baker's Proof (Edited for the Web) for a detailed explanation
of the above picture of Baker's Desarguestheorem frontispiece.
(End of April 14, 2013 update.)
Update of April 12, 2013
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
(End of update of April 12, 2013)
Update of April 13, 2013
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
See also the original VeblenYoung figure in context.
(End of update of April 13, 2013)
Rota's remarks, while perhaps not completely accurate, provide some context
for the above DesarguesRosenhain connection. For some other context,
see the interplay in this journal between classical and finite geometry, i.e.
between Euclid and Galois.
For the recent context of the above finitegeometry version of Baker's Vol. I
frontispiece, see Sunday evening's finitegeometry version of Baker's Vol. IV
frontispiece, featuring the Göpel, rather than the Rosenhain, tetrads.
For a 1986 illustration of Göpel and Rosenhain tetrads (though not under
those names), see Picturing the Smallest Projective 3Space.
In summary… the following classicalgeometry figures
are closely related to the Galois geometry PG(3,2):
Volume I of Baker's Principles
has a cover closely related to
the Rosenhain tetrads in PG(3,2)

Volume IV of Baker's Principles
has a cover closely related to
the Göpel tetrads in PG(3,2)

Foundations
(click to enlarge)

Higher Geometry
(click to enlarge)

Comments Off on Desargues via Rosenhain
Sunday, March 10, 2013
(Continued)
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:
Comments Off on Galois Space
Monday, March 4, 2013
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.
Comments Off on Occupy Galois Space
Thursday, February 21, 2013
(Continued)
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."
— Thomas Pynchon
Illustrations—
(Click to enlarge.)
Comments Off on Galois Space
Sunday, July 29, 2012
(Continued)
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vectorspace structure:
0 
c 
d 
c + d 
a 
a + c 
a + d 
a + c + d 
b 
b + c 
b + d 
b + c + d 
a + b 
a + b + c 
a + b + d 
a + b +
c + d 
(This vectorspace a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)
The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "SelfDual Configurations and Regular Graphs," figures
5 and 6).)
A 1982 discussion of a more abstract form of AG(4, 2):
Source:
The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 ConwaySloane diagram.
Comments Off on The Galois Tesseract
Thursday, July 12, 2012
An example of lines in a Galois space * —
The 35 lines in the 3dimensional Galois projective space PG(3,2)—
(Click to enlarge.)
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.)
Comments Off on Galois Space
Tuesday, July 10, 2012
Comments Off on Euclid vs. Galois
Saturday, September 3, 2011
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
twothirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79TA37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG’s
4×4 square as the affine 4space over the 2element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four “special tetrads” within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 “special tetrads” rather by the parity
of their intersections with the square’s rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The “35 structures” of the abstract were listed, with an application to
Latinsquare orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M_{24}, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was misnamed as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Comments Off on The Galois Tesseract (continued)
Thursday, September 1, 2011
Comments Off on The Galois Tesseract
Friday, September 17, 2010
Yesterday’s excerpt from von Balthasar supplies some Catholic aesthetic background for Galois geometry.
That approach will appeal to few mathematicians, so here is another.
Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace is a book by Leonard Mlodinow published in 2002.
More recently, Mlodinow is the coauthor, with Stephen Hawking, of The Grand Design (published on September 7, 2010).
A review of Mlodinow’s book on geometry—
“This is a shallow book on deep matters, about which the author knows next to nothing.”
— Robert P. Langlands, Notices of the American Mathematical Society, May 2002
The Langlands remark is an apt introduction to Mlodinow’s more recent work.
It also applies to Martin Gardner’s comments on Galois in 2007 and, posthumously, in 2010.
For the latter, see a Google search done this morning—
Here, for future reference, is a copy of the current Google cache of this journal’s “paged=4” page.
Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron’s web journal. Following the link, we find…
For n=4, there is only one factorisation, which we can write concisely as 1234, 1324, 1423. Its automorphism group is the symmetric group S_{4}, and acts as S_{3} on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.
This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.
See also, in this journal, Window and Window, continued (July 5 and 6, 2010).
Gardner scoffs at the importance of Galois’s last letter —
“Galois had written several articles on group theory, and was
merely annotating and correcting those earlier published papers.”
— Last Recreations, page 156
For refutations, see the Bulletin of the American Mathematical Society in March 1899 and February 1909.
Comments Off on The Galois Window
Monday, February 22, 2021
A related image —
Related design theory in mathematics —
http://m759.net/wordpress/?p=9221
Comments Off on Design Theory
Wednesday, December 16, 2020
Comments Off on Kramer’s Cross
Sunday, December 6, 2020
The title phrase is ambiguous and should be avoided.
It is used indiscriminately to denote any system of coordinates
written with 0 ‘s and 1 ‘s, whether these two symbols refer to
the Booleanalgebra truth values false and true , to the absence
or presence of elements in a subset , to the elements of the smallest
Galois field, GF(2) , or to the digits of a binary number .
Related material from the Web —
Some related remarks from “Geometry of the 4×4 Square:
Notes by Steven H. Cullinane” (webpage created March 18, 2004) —
A related anonymous change to Wikipedia today —
The deprecated “binary coordinates” phrase occurs in both
old and new versions of the “Square representation” section
on PG(3,2), but at least the misleading remark about Steiner
quadruple systems has been removed.
Comments Off on “Binary Coordinates”
Sunday, November 15, 2020
Comments Off on Map Methods
Friday, September 11, 2020
Kauffman‘s fixation on the work of SpencerBrown is perhaps in part
due to Kauffman’s familiarity with Boolean algebra and his ignorance of
Galois geometry. See other posts now tagged Boole vs. Galois.
See also “A FourColor Epic” (April 16, 2020).
Comments Off on Kauffman on Algebra
Thursday, April 2, 2020
Friday, February 21, 2020
Tuesday, August 13, 2019
The Matrix of LéviStrauss —
(From his “Structure and Form: Reflections on a Work by Vladimir Propp.”
Translated from a 1960 work in French. It appeared in English as
Chapter VIII of Structural Anthropology, Volume 2 (U. of Chicago Press, 1976).
Chapter VIII was originally published in Cahiers de l’Institut de Science
Économique Appliquée , No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).)
The structure of the matrix of LéviStrauss —
Illustration from Diamond Theory , by Steven H. Cullinane (1976).
The relevant field of mathematics is not Boolean algebra, but rather
Galois geometry.
Comments Off on Putting the Structure in Structuralism
Sunday, June 16, 2019
Comments Off on Master Plan from Outer Space
Tuesday, October 23, 2018
Click the image for some context.
Comments Off on Plan 9 from Inner Space
Sunday, September 9, 2018
"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
Comments Off on Plan 9 Continues.
Sunday, April 29, 2018
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.
Comments Off on Amusement
Monday, March 12, 2018
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."
Comments Off on “Quantum Tesseract Theorem?”
Sunday, March 4, 2018
1955 ("Blackboard Jungle") —
1976 —
2009 —
2016 —
Comments Off on The Square Inch Space: A Brief History
Saturday, February 17, 2018
Michael Atiyah on the late Ron Shaw —
Phrases by Atiyah related to the importance in mathematics
of the twoelement Galois field GF(2) —
 “The digital revolution based on the 2 symbols (0,1)”
 “The algebra of George Boole”
 “Binary codes”
 “Dirac’s spinors, with their up/down dichotomy”
These phrases are from the yearend review of Trinity College,
Cambridge, Trinity Annual Record 2017 .
I prefer other, purely geometric, reasons for the importance of GF(2) —
 The 2×2 square
 The 2x2x2 cube
 The 4×4 square
 The 4x4x4 cube
See Finite Geometry of the Square and Cube.
See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
Comments Off on The Binary Revolution
Thursday, January 25, 2018
"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 .
Comments Off on Beware of Analogical Extension
Friday, September 29, 2017
(Some Remarks for Science Addicts)
Principles —
Personalities —
* See "Tradition Twelve."
Comments Off on Principles Before Personalities*
Saturday, September 23, 2017
"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.
Comments Off on The Turn of the Frame
Friday, September 15, 2017
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.
Comments Off on Space Art
Sunday, August 27, 2017
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.
Comments Off on Black Well
Friday, August 11, 2017
A post suggested by the word tzimtzum (see Wednesday)
or tsimtsum (see this morning) —
Lifeboat from the Tsimtsum in Life of Pi —
Another sort of tsimtsum, contracting infinite space to a finite space —
Comments Off on Symmetry’s Lifeboat
Tuesday, July 11, 2017
The New York TImes reports this evening that
"Jon Underwood, Founder of Death Cafe Movement,"
died suddenly at 44 on June 27.
This journal on that date linked to a post titled "The Mystic Hexastigm."
A related remark on the complete 6point from Sunday, April 28, 2013 —
(See, in Veblen and Young's 1910 Vol. I, exercise 11,
page 53: "A plane section of a 6point in space can
be considered as 3 triangles perspective in pairs
from 3 collinear points with corresponding sides
meeting in 3 collinear points." This is the large
Desargues configuration. See Classical Geometry
in Light of Galois Geometry.)
This post was suggested, in part, by the philosophical ruminations
of Rosalind Krauss in her 2011 book Under Blue Cup . See
Sunday's post Perspective and Its Transections . (Any resemblance
to Freud's title Civilization and Its Discontents is purely coincidental.)
Comments Off on A Date at the Death Cafe
Sunday, July 9, 2017
The title phrase is from Rosalind Krauss (Under Blue Cup , 2011) —
Another way of looking at the title phrase —
"A very important configuration is obtained by
taking the plane section of a complete space fivepoint."
(Veblen and Young, 1910, p. 39) —
For some context, see Desargues + Galois in this journal.
Comments Off on Perspective and Its Transections
Monday, June 26, 2017
Analogies — “A : B :: C : D” may be read “A is to B as C is to D.”
GianCarlo Rota on Heidegger…
“… The universal as is given various names in Heidegger’s writings….
The discovery of the universal as is Heidegger’s contribution to philosophy….
The universal ‘as‘ is the surgence of sense in Man, the shepherd of Being.
The disclosure of the primordial as is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger.”
— “Three Senses of ‘A is B’ in Heideggger,” Ch. 17 in Indiscrete Thoughts
See also Four Dots in this journal.
Some context: McLuhan + Analogy.
Comments Off on Four Dots
Saturday, June 3, 2017
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.
Comments Off on Expanding the Spielraum (Continued*)
Tuesday, May 23, 2017
Comments Off on Pursued by a Biplane
Saturday, May 20, 2017
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.
See this journal on that date.
Comments Off on The Ludicrous Extreme
Tuesday, May 2, 2017
Comments Off on Image Albums
Friday, April 28, 2017
The title is from Don McLean's classic "American Pie."
A Finite Projective Space —
A NonFinite Projective Space —
Comments Off on A Generation Lost in Space
Thursday, April 27, 2017
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.' "
— Greg Gutfeld, September 2014
Comments Off on Partner, Anchor, Decompose
Thursday, April 20, 2017
See also “Romancing the Omega” —
Related mathematics — Guitart in this journal —
See also Weyl + Palermo in this journal —
Comments Off on Stone Logic
Sunday, April 16, 2017
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.]
Comments Off on Art Space Paradigm Shift
Wednesday, October 5, 2016
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)
Comments Off on Sources
Wednesday, August 24, 2016
"That in which space itself is contained" — Wallace Stevens
An image by Steven H. Cullinane from April 1, 2013:
The large Desargues configuration of Euclidean 3space can be
mapped canonically to the 4×4 square of Galois geometry —
On an Auckland University of Technology thesis by Kate Cullinane —
The thesis reportedly won an Art Directors Club award on April 5, 2013.
Comments Off on Core Statements
Tuesday, August 16, 2016
The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —
Boulez and Boole and…
Cross and Boolean lattice.
The improvised cross in the second pair of images
is perhaps being wielded to counteract the
Boole of the first pair of images. See the heading
of the webpage that is the source of the lattice
diagram toward which the cross is directed —
Update of 10 am on August 16, 2016 —
See also Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
Comments Off on Midnight Narrative
Saturday, June 18, 2016
In memory of New Yorker artist Anatol Kovarsky,
who reportedly died at 97 on June 1.
Note the Santa, a figure associated with Macy's at Herald Square.
See also posts tagged Herald Square, as well as the following
figure from this journal on the day preceding Kovarsky's death.
A note related both to Galois space and to
the "Herald Square"tagged posts —
"There is such a thing as a length16 sequence."
— Saying adapted from a youngadult novel.
Comments Off on Midnight in Herald Square
Tuesday, May 24, 2016
The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .
"This famous book is a prototype for the possibility
of explaining and exploring a manyfaceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least,
as an everlasting symbol of mathematical culture."
— Werner Kleinert, Mathematical Reviews ,
as quoted at Amazon.com
Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4space over
the twoelement Galois field GF(2).
Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .
Some related work of my own (click images for related posts)—
Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)
Göpel tetrads as 15 of the 35 projective lines in PG(3,2)
Related terminology describing the Göpel tetrads above
Comments Off on Rosenhain and Göpel Revisited
Sunday, May 8, 2016
Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick’s Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.
A quote from LeWitt indicates the depth of the word “conceptual”
in his approach to “conceptual art.”
From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:
THE SQUARE AND THE CUBE
by Sol LeWitt
“The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two and threedimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed.”
“Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America 55, No. 4 (JulyAugust 1967): 54. (LeWitt’s contribution was originally untitled.)” 
See also the Cullinane models of some small Galois spaces —
Comments Off on The Three Solomons
Friday, May 6, 2016
Thursday, April 14, 2016
On this date in 2005, mathematician Saunders Mac Lane died at 95.
Related material —
Max Planck quotations:
Mac Lane on Boolean algebra:
Mac Lane’s summary chart (note the absence of Galois geometry ):
I disagree with Mac Lane’s assertion that “the finite models of
Boolean algebra are dull.” See Boole vs. Galois in this journal.
Comments Off on One Funeral at a Time
Wednesday, January 13, 2016
(Continued from previous episodes)
Boole and Galois also figure in the mathematics of space —
i.e. , geometry. See Boole + Galois in this journal.
Related material, according to Jung’s notion of synchronicity —
Comments Off on Geometry for Jews
Monday, January 11, 2016
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
Comments Off on Space Oddity
Monday, December 28, 2015
Combining two headlines from this morning’s
New York Times and Washington Post , we have…
Deceptively Simple Geometries
on a Bold Scale
Voilà —
Click image for details.
More generally, see
Boole vs. Galois.
Comments Off on ART WARS Continues
Friday, December 25, 2015
Related material:
The previous post (Bright Symbol) and
a post from Wednesday,
December 23, 2015, that links to posts
on Boolean algebra vs. Galois geometry.
“An analogy between mathematics and religion is apposite.”
— Harvard Magazine review by Avner Ash of
Mathematics without Apologies
(Princeton University Press, January 18, 2015)
Comments Off on Dark Symbol
Wednesday, December 23, 2015
Bleecker Street logo —
Click image for some background.
Related remarks on mathematics:
Boole vs. Galois
Comments Off on Splitting Apart
Sunday, December 13, 2015
“The colorful story of this undertaking begins with a bang.”
— Martin Gardner on the death of Évariste Galois
Comments Off on The Monster as Big as the Ritz
Tuesday, December 1, 2015
See a search for "large Desargues configuration" in this journal.
The 6 Jan. 2015 preprint "Danzer's Configuration Revisited,"
by Boben, Gévay, and Pisanski, places this configuration,
which they call the CayleySalmon configuration , in the
interesting context of Pascal's Hexagrammum Mysticum .
They show how the CayleySalmon configuration is, in a sense,
dual to something they call the SteinerPlücker configuration .
This duality appears implicitly in my note of April 26, 1986,
"Picturing the smallest projective 3space." The sixsets at
the bottom of that note, together with Figures 3 and 4
of Boben et. al. , indicate how this works.
The duality was, as they note, previously described in 1898.
Related material on sixset geometry from the classical literature—
Baker, H. F., "Note II: On the Hexagrammum Mysticum of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219236
Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen (1900), Volume 53, Issue 12, pp 161176
Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125160
Related material on sixset geometry from a more recent source —
Cullinane, Steven H., "Classical Geometry in Light of Galois Geometry," webpage
Comments Off on Pascal’s Finite Geometry
Monday, November 2, 2015
This is a sequel to the previous post and to the Oct. 24 post
Two Views of Finite Space. From the latter —
” ‘All you need to do is give me your soul:
give up geometry and you will have this
marvellous machine.’ (Nowadays you
can think of it as a computer!) “
Comments Off on The Devil’s Offer
"The office of color in the color line
is a very plain and subordinate one.
It simply advertises the objects of
oppression, insult, and persecution.
It is not the maddening liquor, but
the black letters on the sign
telling the world where it may be had."
— Frederick Douglass, "The Color Line,"
The North American Review , Vol. 132,
No. 295, June 1881, page 575
Or gold letters.
From a search for Seagram in this journal —
"The colorful story of this undertaking begins with a bang."
— Martin Gardner on the death of Évariste Galois
Comments Off on Colorful Story
Saturday, October 31, 2015
Stanford Encyclopedia of Philosophy
on the date Friday, April 5, 2013 —
Paraconsistent Logic
“First published Tue Sep 24, 1996;
substantive revision Fri Apr 5, 2013”
This journal on the date Friday, April 5, 2013 —
The object most closely resembling a “philosophers’ stone”
that I know of is the eightfold cube .
For some related philosophical remarks that may appeal
to a general Internet audience, see (for instance) a website
by I Ching enthusiast Andreas Schöter that displays a labeled
eightfold cube in the form of a lattice diagram —
Related material by Schöter —
A 20page PDF, “Boolean Algebra and the Yi Jing.”
(First published in The Oracle: The Journal of Yijing Studies ,
Vol 2, No 7, Summer 1998, pp. 19–34.)
I differ with Schöter’s emphasis on Boolean algebra.
The appropriate mathematics for I Ching studies is,
I maintain, not Boolean algebra but rather Galois geometry.
See last Saturday’s post Two Views of Finite Space.
Unfortunately, that post is, unlike Schöter’s work, not
suitable for a general Internet audience.
Comments Off on Raiders of the Lost Crucible
Saturday, October 24, 2015
The following slides are from lectures on “Advanced Boolean Algebra” —
The small Boolean spaces above correspond exactly to some small
Galois spaces. These two names indicate approaches to the spaces
via Boolean algebra and via Galois geometry .
A reading from Atiyah that seems relevant to this sort of algebra
and this sort of geometry —
” ‘All you need to do is give me your soul: give up geometry
and you will have this marvellous machine.’ (Nowadays you
can think of it as a computer!) “
Related material — The article “Diamond Theory” in the journal
Computer Graphics and Art , Vol. 2 No. 1, February 1977. That
article, despite the word “computer” in the journal’s title, was
much less about Boolean algebra than about Galois geometry .
For later remarks on diamond theory, see finitegeometry.org/sc.
Comments Off on Two Views of Finite Space
Wednesday, October 21, 2015
"Perhaps an insane conceit …." Perhaps.
Related remarks on algebra and space —
"The Quality Without a Name" (Log24, August 26, 2015).
Comments Off on Algebra and Space
Sunday, September 6, 2015
Sarah Larson in the online New Yorker on Sept. 3, 2015,
discussed Google’s new parent company, “Alphabet”—
“… Alphabet takes our most elementally wonderful
generaluse word—the name of the components of
language itself*—and reassigns it, like the words
tweet, twitter, vine, facebook, friend, and so on,
into a branded realm.”
Emma Watson in “The Bling Ring”
This journal, also on September 3 —
* Actually, Sarah, that would be “phonemes.”
Comments Off on Elementally, My Dear Watson
Friday, September 4, 2015
Galois via Boole
(Courtesy of Intel)
Comments Off on Space Program
Thursday, September 3, 2015
For the title, see posts from August 2007 tagged Gyges.
Related theological remarks:
Boolean spaces (old) vs. Galois spaces (new) in
“The Quality Without a Name”
(a post from August 26, 2015) and the…
Related literature: A search for Borogoves in this journal will yield
remarks on the 1943 tale underlying the above film.
Comments Off on Rings of August
Wednesday, August 26, 2015
The title phrase, paraphrased without quotes in
the previous post, is from Christopher Alexander’s book
The Timeless Way of Building (Oxford University Press, 1979).
A quote from the publisher:
“Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself.”
Three paragraphs from the book (pp. xiiixiv):
19. Within this process, every individual act
of building is a process in which space gets
differentiated. It is not a process of addition,
in which preformed parts are combined to
create a whole, but a process of unfolding,
like the evolution of an embryo, in which
the whole precedes the parts, and actualy
gives birth to then, by splitting.
20. The process of unfolding goes step by step,
one pattern at a time. Each step brings just one
pattern to life; and the intensity of the result
depends on the intensity of each one of these
individual steps.
21. From a sequence of these individual patterns,
whole buildings with the character of nature
will form themselves within your thoughts,
as easily as sentences.
Compare to, and contrast with, these illustrations of “Boolean space”:
(See also similar illustrations from Berkeley and Purdue.)
Detail of the above image —
Note the “unfolding,” as Christopher Alexander would have it.
These “Boolean” spaces of 1, 2, 4, 8, and 16 points
are also Galois spaces. See the diamond theorem —
Comments Off on “The Quality Without a Name”
Friday, August 14, 2015
(A review)
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Comments Off on Discrete Space
Tuesday, June 9, 2015
For geeks* —
" Domain, Domain on the Range , "
where Domain = the Galois tesseract and
Range = the fourelement Galois field.
This post was suggested by the previous post,
by a Log24 search for Knight + Move, and by
the phrase "discouraging words" found in that search.
* A term from the 1947 film "Nightmare Alley."
Comments Off on Colorful Song
Sunday, May 31, 2015
To enlarge image, click here.
Comments Off on Foundations
Thursday, March 26, 2015
The incidences of points and planes in the
Möbius 8_{4 } configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.*
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and faceplanes of two mutually inscribed tetrahedra —
Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his
Fig. 24, a schematic representation of the pointplane
incidences, as dotless, and some as hollow dots. The figure,
"Gallucci's version of Möbius's 8_{4}," is shown below.
The hollow dots, representing the 8 points (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.
Here a plane (represented by a dotless intersection) contains
the four points that are represented in the square array as lying
in the same row or same column as the plane.
The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube.
In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.
Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in
Coxeter's labels above may be viewed as naming the positions
of the 1's in (0,1) vectors (x_{4}, x_{3}, x_{2}, x_{1}) over the twoelement
Galois field.^{†} In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "SelfDual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413455
^{†} The subscripts' usual 1234 order is reversed as a reminder
that such a vector may be viewed as labeling a binary number
from 0 through 15, or alternately as labeling a polynomial in
the 16element Galois field GF(2^{4}). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
Comments Off on The Möbius Hypercube
Monday, January 5, 2015
Wednesday, March 13, 2013
From a review in the April 2013 issue of
Notices of the American Mathematical Society—
"The author clearly is passionate about mathematics
as an art, as a creative process. In reading this book,
one can easily get the impression that mathematics
instruction should be more like an unfettered journey
into a jungle where an individual can make his or her
own way through that terrain."
From the book under review—
"Every morning you take your machete into the jungle
and explore and make observations, and every day
you fall more in love with the richness and splendor
of the place."
— Lockhart, Paul (20090401).
A Mathematician's Lament:
How School Cheats Us Out of Our Most Fascinating
and Imaginative Art Form (p. 92).
Bellevue Literary Press. Kindle Edition.
Related material: Blackboard Jungle in this journal.
See also Galois Space and Solomon's Mines.

"I pondered deeply, then, over the
adventures of the jungle. And after
some work with a colored pencil
I succeeded in making my first drawing.
My Drawing Number One.
It looked something like this:
I showed my masterpiece to the
grownups, and asked them whether
the drawing frightened them.
But they answered: 'Why should
anyone be frightened by a hat?'"
— The Little Prince
* For the title, see Plato Thanks the Academy (Jan. 3).
Comments Off on Gitterkrieg*
Monday, December 29, 2014
Recent posts tagged Sagan Dodecahedron
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.
For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book , p. 120:
For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.
Comments Off on Dodecahedron Model of PG(2,5)
Thursday, December 18, 2014
(Five by Five continued)
As the 3×3 grid underlies the order3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.
See posts tagged GaloisPlane Models.
Comments Off on Platonic Analogy
Wednesday, December 3, 2014
Oslo artist Josefine Lyche has a new Instagram post,
this time on pyramids (the monumental kind).
My response —
Wikipedia's definition of a tetrahedron as a
"trianglebased pyramid" …
… and remarks from a Log24 post of August 14, 2013 :
See also some of Burkard Polster's trianglebased pyramids
and a 1983 trianglebased pyramid in a paper that Polster cites —
(Click image below to enlarge.)
Some other illustrations that are particularly relevant
for Lyche, an enthusiast of magic :
From On Art and Magic (May 5, 2011) —
Mathematics
The Fano plane block design

Magic
The Deathly Hallows symbol—
Two blocks short of a design.


(Updated at about 7 PM ET on Dec. 3.)
Comments Off on Pyramid Dance
Sunday, November 30, 2014
The Regular Tetrahedron
The seven symmetry axes of the regular tetrahedron
are of two types: vertextoface and edgetoedge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains
two vertextoface axes and one edgetoedge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three
edgetoedge axes.
(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book , pp. 1617.)
The Cube
There are three reflection planes parallel to faces
of the cube. Take the seven nonempty subsets of
the set of these three planes as the "points" of a
Fano plane. Define the Fano "lines" as those triples
of these seven subsets in which each member of
the triple is the symmetricdifference sum of the
other two members.
(This is the eightfold cube discussed at finitegeometry.org.)
Comments Off on Two Physical Models of the Fano Plane
Wednesday, November 26, 2014
Update of Nov. 30, 2014 —
It turns out that the following construction appears on
pages 1617 of A Geometrical Picture Book , by
Burkard Polster (Springer, 1998).
"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"
—John H. Conway, foreword to the 2004 Princeton
Science Library edition of How to Solve It , by G. Polya
For a similar but more difficult problem involving the
31point projective plane, see yesterday's post
"EuclideanGalois Interplay."
The above new [see update above] Fanoplane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "EuclideanGalois Interplay"
quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.
Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.
Comments Off on A Tetrahedral FanoPlane Model
Update of Nov. 30, 2014 —
For further information on the geometry in
the remarks by Eberhart below, see
pp. 1617 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.
A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:
The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and comer points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of comers, totalling 13 axes (the octahedron simply interchanges the roles of faces and comers); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of comers, totalling 31 axes (the icosahedron again interchanging roles of faces and comers). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former.
[9] I am aware only of a series of inhouse publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie IX.
— Stephen Eberhart, Dept. of Mathematics,
California State University, Northridge,
"Pythagorean and Platonic Bridges between
Geometry and Algebra," in BRIDGES: Mathematical
Connections in Art, Music, and Science , 1998,
archive.bridgesmathart.org/1998/bridges1998121.pdf

Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…
… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled. So 1984 to 2002 I taught math (esp. nonEuclidean geometry) at C.S.U. Northridge. It’s been a rich life. I’m grateful.
Steve

See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.
Comments Off on Class Act
Monday, September 22, 2014
Review of an image from a post of May 6, 2009:
Comments Off on Space
Sunday, September 21, 2014
This post was suggested by Greg Gutfeld’s Sept. 4 remarks on Common Core math.
Problem: What is 9 + 6 ?
Here are two approaches suggested by illustrations of Desargues’s theorem.
Solution 1:
9 + 6 = 10 + 5,
as in Common Core (or, more simply, as in common sense), and
10 + 5 = 5 + 10 = 15 as in Veblen and Young:
Solution 2:
In the figure below,
9 + 6 = no. of V’s + no. of A’s + no. of C’s =
no. of nonempty squares = 16 – 1 = 15.
(Illustration from Feb. 10, 2014.)
The silly educationists’ “partner, anchor, decompose” jargon
discussed by Gutfeld was their attempt to explain “9 + 6 = 10 + 5.”
As he said of the jargon, “That’s not math, that’s the plot from ‘Silence of the Lambs.'”
Or from Richard, Frank, and Marcus in last night’s “Intruders”
(BBC America, 10 PM).
Comments Off on Uncommon Noncore
Sunday, September 14, 2014
Structured gray matter:
Graphic symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Comments Off on Sensibility
Sunday, August 31, 2014
The Folding
Cynthia Zarin in The New Yorker , issue dated April 12, 2004—
“Time, for L’Engle, is accordionpleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”
The geometry of the 4×4 square array is that of the
3dimensional projective Galois space PG(3,2).
This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc. on
15 June 1974). Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.
Some history:
Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.
[Rewritten for clarity on Sept. 3, 2014.]
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Sunday, July 20, 2014
Paradigms of Geometry:
Continuous and Discrete
The discovery of the incommensurability of a square’s
side with its diagonal contrasted a wellknown discrete
length (the side) with a new continuous length (the diagonal).
The figures below illustrate a shift in the other direction.
The essential structure of the continuous configuration at
left is embodied in the discrete unit cells of the square at right.
See Desargues via Galois (August 6, 2013).
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Sunday, July 6, 2014
The title is from this morning’s previous post.
From a theater review in that post—
… “all flying edges and angles, a perpetually moving and hungry soul”
… “a formidably centered presence, the still counterpoint”
A more abstract perspective:
See also Desargues via Galois (August 6, 2013).
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Sunday, June 8, 2014
“The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof.”
— GianCarlo Rota discussing the theorem of Desargues
What space tells us about the theorem :
In the simplest case of a projective space (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel lines and 20 Rosenhain lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (rowbased) lines with 10 corresponding
vertically oriented (columnbased) lines.
Vide Classical Geometry in Light of Galois Geometry.
* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995. The “simplest case” link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:
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Tuesday, May 20, 2014
From a recreationalmathematics weblog yesterday:
“This appears to be the arts section of the post,
so I’ll leave Martin Probert’s page on
The Survival, Origin and Mathematics of String Figures
here. I’ll be back to pick it up at the end. Maybe it’d like
to play with Steven H. Cullinane’s pages on the
Finite Geometry of the Square and Cube.”
I doubt they would play well together.
Perhaps the offensive linking of the purely recreational topic
of string figures to my own work was suggested by the
string figures’ resemblance to figures of projective geometry.
A pairing I prefer: Desargues and Galois —
For further details, see posts on Desargues and Galois.
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Monday, February 10, 2014
(Continued from Mystery Box, Feb. 4, and Mystery Box II, Feb. 5.)
The Box
Inside the Box
Outside the Box
For the connection of the inside notation to the outside geometry,
see Desargues via Galois.
(For a related connection to curves and surfaces in the outside
geometry, see Hudson's classic Kummer's Quartic Surface and
Rosenhain and Göpel Tetrads in PG(3,2).)
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Saturday, August 17, 2013
The following excerpt from a January 20, 2013, preprint shows that
a Galoisgeometry version of the large Desargues 15_{4}20_{3} configuration,
although based on the nineteenthcentury work of Galois* and of Fano,**
may at times have twentyfirstcentury applications.
Some context —
Atkinson's paper does not use the square model of PG(3,2), which later
in 2013 provided a natural view of the large Desargues 15_{4}20_{3} configuration.
See my own Classical Geometry in Light of Galois Geometry. Atkinson's
"subset of 20 lines" corresponds to 20 of the 80 Rosenhain tetrads
mentioned in that later article and pictured within 4×4 squares in Hudson's
1905 classic Kummer's Quartic Surface.
* E. Galois, definition of finite fields in "Sur la Théorie des Nombres,"
Bulletin des Sciences Mathématiques de M. Férussac,
Vol. 13, 1830, pp. 428435.
** G. Fano, definition of PG(3,2) in "Sui Postulati Fondamentali…,"
Giornale di Matematiche, Vol. 30, 1892, pp. 106132.
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Sunday, July 28, 2013
(Simplicity continued)
"Understanding a metaphor is like understanding a geometrical
truth. Features of various geometrical figures or of various contexts
are pulled into revealing alignment with one another by the
demonstration or the metaphor.
What is 'revealed' is not that the alignment is possible; rather,
that the alignment is possible reveals the presence of already
existing shapes or correspondences that lay unnoticed. To 'see' a
proof or 'get' a metaphor is to experience the significance of the
correspondence for what the thing, concept, or figure is ."
— Jan Zwicky, Wisdom & Metaphor , page 36 (left)
Zwicky illustrates this with Plato's diamond figure
from the Meno on the facing page— her page 36 (right).
A more sophisticated geometrical figure—
Galoisgeometry key to
Desargues' theorem:

D

E

F

S'

P

Q

R

S

P'

Q'

R'

O

P_{1}

Q_{1}

R_{1}

For an explanation, see
Classical Geometry in Light of Galois Geometry.
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Sunday, May 19, 2013
Best vs. Bester
The previous post ended with a reference mentioning Rosenhain.
For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).
From the next day, April 2, 2013:
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
– GianCarlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics ,
Vol. 84, Number 1, Nov. 1990, p. 136:
See, too, in the ConwaySloane book, the Galois tesseract …
and, in this journal, Geometry for Jews and The Deceivers , by Bester.
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Sunday, April 28, 2013
… And the history of geometry —
Desargues, Pascal, Brianchon and Galois
in the light of complete npoints in space.
(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)
Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:
"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space fivepoint."
Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black points and dashed lines indicate the
complete space fivepoint and lines connecting it to the plane section
containing the Desargues configuration.
In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six point to construct a configuration of
15 points and 20 lines in the context not of Desargues ' theorem, but
rather of Brianchon 's theorem and of the Pascal hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6point in space can be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large Desargues configuration. See Classical Geometry in Light of
Galois Geometry.)
For this large Desargues configuration see April 19.
For Henderson's complete six –point, see The SixSet (April 23).
That post ends with figures relating the large Desargues configuration
to the Galois geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator and the large Mathieu group M_{24} —
See also Note on the MOG Correspondence from April 25, 2013.
That correspondence was also discussed in a note 28 years ago, on this date in 1985.
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Wednesday, April 10, 2013
"Of course, DeLillo being DeLillo,
it’s the deeper implications of the piece —
what it reveals about the nature of
film, perception and time — that detain him."
— Geoff Dyer, review of Point Omega
Related material:
A phrase of critic Robert Hughes,
"slow art," in this journal.
A search for that phrase yields the following
figure from a post on DeLillo of Oct. 12, 2011:
The above 3×3 grid is embedded in a
somewhat more sophisticated example
of conceptual art from April 1, 2013:
Update of April 12, 2013
The above key uses labels from the frontispiece
to Baker's 1922 Principles of Geometry, Vol. I ,
that shows a threetriangle version of Desargues's theorem.
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
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Wednesday, April 3, 2013
Roberta Smith in 2011 on the American Folk Art Museum (see previous post):
"It could be argued that we need a museum of folk art
the way we need a museum of modern art,
to shine a very strong, undiluted light on
a very important achievement."
Some other aesthetic remarks:
"We have had a gutful of fast art and fast food.
What we need more of is slow art: art that holds time
as a vase holds water: art that grows out of modes
of perception and whose skill and doggedness
make you think and feel; art that isn't merely sensational,
that doesn't get its message across in 10 seconds,
that isn't falsely iconic, that hooks onto something
deeprunning in our natures. In a word, art that is
the very opposite of mass media. For no spiritually
authentic art can beat mass media at their own game."
— Robert Hughes, speech of June 2, 2004,
quoted here June 15, 2007.
Perhaps, as well as museums of modern art and of folk art,
we need a Museum of Slow Art.
One possible exhibit, from this journal Monday:
The diagram on the left is from 1922. The 20 small squares at right
that each have 4 subsquares darkened were discussed, in a different
context, in 1905. They were reillustrated, in a new context
(Galois geometry), in 1986. The "key" square, and the combined
illustration, is from April 1, 2013. For deeper background, see
Classical Geometry in Light of Galois Geometry.
Those who prefer faster art may consult Ten Years After.
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Tuesday, April 2, 2013
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
— GianCarlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics,
Vol. 84, Number 1, Nov. 1990, p. 136:
Related material:
Pascal and the Galois nocciolo ,
Conway and the Galois tesseract,
Gardner and Galois.
See also Rota and Psychoshop.
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The geometry posts of Sunday and Monday have been
placed in finitegeometry.org as
Classical Geometry in Light of Galois Geometry.
Some background:
See Baker, Principles of Geometry , Vol. II, Note I
(pp. 212218)—
On Certain Elementary Configurations, and
on the Complete Figure for Pappus's Theorem
and Vol. II, Note II (pp. 219236)—
On the Hexagrammum Mysticum of Pascal.
Monday's elucidation of Baker's Desarguestheorem figure
treats the figure as a 15_{4}20_{3 }configuration (15 points,
4 lines on each, and 20 lines, 3 points on each).
Such a treatment is by no means new. See Baker's notes
referred to above, and
"The Complete Pascal Figure Graphically Presented,"
a webpage by J. Chris Fisher and Norma Fuller.
What is new in the Monday Desargues post is the graphic
presentation of Baker's frontispiece figure using Galois geometry :
specifically, the diamond theorem square model of PG(3,2).
See also Cremona's kernel, or nocciolo :
Baker on Cremona's approach to Pascal—
"forming, in Cremona's phrase, the nocciolo of the whole."
A related nocciolo :
Click on the nocciolo for some
geometric background.
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Tuesday, February 26, 2013
"I’ve had the privilege recently of being a Harvard University
professor, and there I learned one of the greatest of Harvard
jokes. A group of rabbis are on the road to Golgotha and
Jesus is coming by under the cross. The young rabbi bursts
into tears and says, 'Oh, God, the pity of it!' The old rabbi says,
'What is the pity of it?' The young rabbi says, 'Master, Master,
what a teacher he was.'
'Didn’t publish!'
That cold tenure joke at Harvard contains a deep truth.
Indeed, Jesus and Socrates did not publish."
— George Steiner, 2002 talk at York University
Related material—
See also Steiner on Galois.
Les Miserables at the Academy Awards
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Tuesday, February 5, 2013
The previous post discussed some fundamentals of logic.
The name “Boole” in that post naturally suggests the
concept of Boolean algebra . This is not the algebra
needed for Galois geometry . See below.
Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.
See also yesterday’s Universe of Discourse.
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Monday, August 13, 2012
(An episode of Mathematics and Narrative )
A report on the August 9th opening of Sondheim's Into the Woods—
Amy Adams… explained why she decided to take on the role of the Baker’s Wife.
“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com
Related material—
Amy Adams in Sunshine Cleaning "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro
Compare and contrast…
1. The following item from Walpurgisnacht 2012—
2. The six partitions of a tesseract's 16 vertices
into four parallel faces in Diamond Theory in 1937—
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Tuesday, May 29, 2012
(Continued from May 29, 2002)
May 29, 1832—
Évariste Galois, Lettre de Galois à M. Auguste Chevalier—
Après cela, il se trouvera, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.
(Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)
Martin Gardner on the above letter—
"Galois had written several articles on group theory, and was merely annotating and correcting those earlier published papers."
– The Last Recreations , by Martin Gardner, published by Springer in 2007, page 156.
Commentary from Dec. 2011 on Gardner's word "published" —
(Click to enlarge.)
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Sunday, October 30, 2011
Part I: Timothy Gowers on equivalence relations
Part II: Martin Gardner on normal subgroups
Part III: Evariste Galois on normal subgroups
"In all the history of science there is no completer example
of the triumph of crass stupidity over untamable genius…."
— Eric Temple Bell, Men of Mathematics
See also an interesting definition and Weyl on Galois.
Update of 6:29 PM EDT Oct. 30, 2011—
For further details, see Herstein's phrase
"a tribute to the genius of Galois."
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Thursday, September 8, 2011
"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 8_{3}
are joined by four concurrent lines."
— H. S. M. Coxeter (see below)
Continued from Tuesday, Sept. 6—
The Diamond Star
The above is a version of a figure from Configurations and Squares.
Yesterday's post related the the Pappus configuration to this figure.
Coxeter, in "SelfDual Configurations and Regular Graphs," also relates Pappus to the figure.
Some excerpts from Coxeter—
The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from
an 8cycle in the Galois field GF(9).
The relabeled configuration is used in a discussion of Pappus—
(Update of Sept. 10, 2011—
Coxeter here has a note referring to page 335 of
G. A. Miller, H. F. Blichfeldt, and L. E. Dickson,
Theory and Applications of Finite Groups , New York, 1916.)
Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues configuration—
The Desargues configuration is discussed by GianCarlo Rota on pp. 145146 of Indiscrete Thoughts—
"The value of Desargues' theorem and the reason why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities that relate geometry and algebra in unexpected ways."
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Wednesday, October 20, 2010
"Why the Celebration?"
"Martin Gardner passed away on May 22, 2010."
Imaginary movie poster from stoneship.org
Context— The Gardner Tribute.
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Monday, September 27, 2010
… In the Age of Citation
1. INTRODUCTION TO THE PROBLEM
Social network analysis is focused on the patterning of the social
relationships that link social actors. Typically, network data take the
form of a squareactor by actorbinary adjacency matrix, where
each row and each column in the matrix represents a social actor. A
cell entry is 1 if and only if a pair of actors is linked by some social
relationship of interest (Freeman 1989).
— "Using Galois Lattices to Represent Network Data,"
by Linton C. Freeman and Douglas R. White,
Sociological Methodology, Vol. 23, pp. 127–146 (1993)
From this paper's CiteSeer page—
Citations
Visual Image of the Problem—
From a Google search today:
Related material—
"It is better to light one candle…"
"… the early favorite for best picture at the Oscars" — Roger Moore
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Tuesday, July 6, 2010
or: Combinatorics (Rota) as Philosophy (Heidegger) as Geometry (Me)
“Dasein’s full existential structure is constituted by
the ‘asstructure’ or ‘welljoined structure’ of the riftdesign*…”
— Gary Williams, post of January 22, 2010
Background—
GianCarlo Rota on Heidegger…
“… The universal as is given various names in Heidegger’s writings….
The discovery of the universal as is Heidegger’s contribution to philosophy….
The universal ‘as‘ is the surgence of sense in Man, the shepherd of Being.
The disclosure of the primordial as is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger.”
— “Three Senses of ‘A is B’ in Heideggger,” Ch. 17 in Indiscrete Thoughts
… and projective points as separating rifts—
Click image for details.
* riftdesign— Definition by Deborah Levitt—
“Rift. The stroke or rending by which a world worlds, opening both the ‘old’ world and the selfconcealing earth to the possibility of a new world. As well as being this stroke, the rift is the site— the furrow or crack— created by the stroke. As the ‘rift design‘ it is the particular characteristics or traits of this furrow.”
— “Heidegger and the Theater of Truth,” in Tympanum: A Journal of Comparative Literary Studies, Vol. 1, 1998
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“Simplicity, simplicity, simplicity!
I say, let your affairs be as two or three,
and not a hundred or a thousand;
instead of a million count half a dozen,
and keep your accounts on your thumbnail.”
— Henry David Thoreau, Walden
This quotation is the epigraph to
Section 1.1 of Alexandre V. Borovik’s
Mathematics Under the Microscope:
Notes on Cognitive Aspects of Mathematical Practice
(American Mathematical Society,
Jan. 15, 2010, 317 pages). 
From Peter J. Cameron’s review notes for
his new course in group theory—
From Log24 on June 24—
Geometry Simplified
(an affine space with subsquares as points
and sets of subsquares as hyperplanes)
(a projective space with, as points, sets
of line segments that separate subsquares)
Exercise—
Show that the above geometry is a model
for the algebra discussed by Cameron.
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Monday, July 5, 2010
“Examples are the stainedglass
windows of knowledge.” — Nabokov
Related material:
Thomas Wolfe and the
Kernel of Eternity
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Thursday, June 24, 2010
Geometry Simplified
(a projective space)
The above finite projective space
is the simplest nontrivial example
of a Galois geometry (i.e., a finite
geometry with coordinates in a
finite (that is, Galois) field.)
The vertical (Euclidean) line represents a
(Galois) point, as does the horizontal line
and also the verticalandhorizontal
cross that represents the first two points’
binary sum (i.e., symmetric difference,
if the lines are regarded as sets).
Homogeneous coordinates for the
points of this line —
(1,0), (0,1), (1,1).
Here 0 and 1 stand for the elements
of the twoelement Galois field GF(2).
The 3point line is the projective space
corresponding to the affine space
(a plane, not a line) with four points —
(an affine space)
The (Galois) points of this affine plane are
not the single and combined (Euclidean)
line segments that play the role of
points in the 3point projective line,
but rather the four subsquares
that the line segments separate.
For further details, see Galois Geometry.
There are, of course, also the trivial
twopoint affine space and the corresponding
trivial onepoint projective space —
Here again, the points of the affine space are
represented by squares, and the point of the
projective space is represented by a line segment
separating the affinespace squares.
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