Monday, March 2, 2015

High Concepts of Space

Filed under: General — m759 @ 3:05 PM

Related material:

Agents of a Great Despair and A Kirk for Spock —

Tuesday, October 7, 2014

Concepts of Space

Filed under: General — m759 @ 1:28 PM


IMAGE- Rubik's Cube in an ad, and some pure mathematics

Friday, April 8, 2011

Concepts of Space

Filed under: General,Geometry — Tags: — m759 @ 7:35 PM

Part I — Roberta Smith in today's New York Times

"… the argument that painting may ultimately be about
little more than the communication of some quality of
light and space, however abstract or indirect."

– Review of "Rooms With a View" at the Met

Box symbol

Pictorial version
of Hexagram 20,
Contemplation (View)


Space: what you damn well have to see.
– James Joyce, Ulysses

Part II — Window from A Crooked House

"Teal lifted the blind a few inches. He saw nothing, and raised it a little more—still nothing. Slowly he raised it until the window was fully exposed. They gazed out at—nothing.

Nothing, nothing at all. What color is nothing? Don't be silly! What shape is it? Shape is an attribute of something . It had neither depth nor form. It had not even blackness. It was nothing ."

Part III — Not So Crooked: The Cabinet of Dr. Montessori

An April 5 Wall Street Journal  article on Montessori schools, and…


A cabinet from Dr. Montessori's own
explanation of her method

Part IV — Pilate Goes to Kindergarten and The Seven

Thursday, April 9, 2020

Symmetry: Toro, Torino

Filed under: General — Tags: — m759 @ 12:41 PM

For the Toro , see  Pierre Cartier in 2001 on the barber of Seville and
The evolution of concepts of space and symmetry.”

For the Torino , see . . .

“… the ultimate goal of the present essay
which is to illustrate the historic
evolution of the concepts of Space  and Symmetry

Pp. 157-158 of the above book.

See also Fré et al. , “The role of PSL(2,7) in M-theory”
(2018-2019) at  http://arxiv.org/abs/1812.11049v2 ,
esp. Section 4, “Theory of the simple group PSL(2,7)”
on pages 11-27, and remarks on PSL(2,7) in this  journal.

Related material —

Wednesday, April 8, 2020

Follow the Ring

Filed under: General — Tags: — m759 @ 8:45 PM

Click the ring for Pierre Cartier on the barber of Seville
and “The evolution of concepts of space and symmetry.”

Tuesday, November 5, 2019


Filed under: General — Tags: , — m759 @ 4:00 AM

A followup to earlier posts on Trudeau vs. Euclid —

Geometry from July 6, 2014:

IMAGE- Concepts of Space

Monday, March 11, 2019


Filed under: General — m759 @ 10:42 PM

An earlier post discussed other meanings for the commercial brand names
underlined above, but the brand name "Attribution" was omitted from that
earlier discussion.

Hence . . .

Attribution by Foursquare — 'Quantify the impact media has....'

Related material — The unattributed phrase "Concepts of Space"
in the previous post

Attribution — The phrase was from the title of a book by Max Jammer.

A search in this journal for Jammer yields posts now tagged . . .

Page Space

(See an example.)


Ant-Man Meets Doctor Strange

Filed under: General — m759 @ 1:22 PM

IMAGE- Concepts of Space

The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .

"Think outside the tesseract.

Sunday, December 2, 2018

Symmetry at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 6:43 AM

A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018




Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II


Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness.

Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein’s quartic curve, respectively), and Bring’s genus 4 curve arises in Klein’s description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare’s homology 3-sphere, and Kummer’s surface in real dimension 4 also play special roles.

In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay’s binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois’ exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol’d Trinities’.

Motivated originally by the `Eightfold Way’ sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set.

Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered.

Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato’s concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective.

Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones’ genus 70 Riemann surface previously proposed as a completion of an Arnol’d Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston’s highly symmetric 6- and 8-component links, the latter related by Thurston to Klein’s quartic curve.

See also yesterday morning’s post, “Character.”

Update: For a followup, see the next  Log24 post.

Monday, October 24, 2016

Space Review

Filed under: General — m759 @ 8:30 PM

Thursday, July 21, 2016

A Mad Day’s Preprint*

Filed under: General — Tags: — m759 @ 2:02 PM

Pierre Cartier, 'How to take advantage of the blur between the finite and the infinite,' preprint of May 3,2011

See also, from that same day, "24-Part Invention."

* The title is a reference to a 2001 article by Cartier on
   "the evolution of concepts of space and symmetry" —

Tuesday, April 19, 2016

The Folding

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


A recent post about the eightfold cube  suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :

As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding  a 1×8 array from Turyn's
1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again  yields the 2x2x2 eightfold cube .

Those who prefer an entertainment  approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space." 

Monday, October 26, 2015

Expanding the Spielraum

Filed under: General — m759 @ 4:00 PM


Halloween meditation  on  the Tummelplatz  at Innsbruck

"Die Ritter und Knappen des nahegelegenen Ambras
pflegten hier ihre Rosse zu tummeln, woher sich auch
der Name Tummelplatz  schreibt."

"The knights and squires of nearby Ambras used to let their
steeds romp here, whence came the name Tummelplatz ." 

— Quelle: Ludwig von Hörmann, "Der Tummelplatz  bei Amras,"
in: Der Alpenfreund , 1. Band, Gera 1870, S. 72 – 73.

See as well Sigmund Freud, Erinnern, Wiederholen und Durcharbeiten
(1914) —

"Wir eröffnen ihm die Übertragung als den Tummel­platz ,
auf dem ihm gestattet wird, sich in fast völliger Freiheit
zu entfalten, und auferlegt ist, uns alles vorzuführen,
was sich an pathogenen Trieben im Seelenleben des
Analysierten verborgen hat."

"We admit it into the transference as a playground
in which it is allowed to expand in almost complete freedom
and in which it is expected to display to us everything in the
way of pathogenic instincts that is hidden in the patient's mind."

This passage has been discussed by later psychotherapists,
notably Russell Meares.  Dr. Meares, working from a translation
that has "playground" for Freud's Tummelplatz , uses Spielraum  
in place of Freud's own word.

For related material in this  journal, see Expanding the Spielraum.
An illustration from that search —

IMAGE- Concepts of Space

Tuesday, February 3, 2015

Expanding the Spielraum

Filed under: General — Tags: , — m759 @ 11:00 AM

A short poem by several authors:

"The role of
the 16 singular points
on the Kummer surface
is now played by
the 64 singular points
on the Kummer threefold."

— From Remark 2.4 on page 9 of
"The Universal Kummer Threefold,"
by Qingchun Ren, Steven V Sam,
Gus Schrader, and Bernd Sturmfels,
August 6, 2012 — June 12, 2013.

See also "Expanded Field" in this journal.

IMAGE- Concepts of Space

Illustration from "Sunday School," July 20, 2014.

Other Log24 background:  Kummer, Spielraum, Art Space.

Sunday, November 30, 2014

Agents of a Great Despair

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

Or:  Concepts of Space

1976 according to Cullinane:

1976 according to Plotnick:

“Irony and ridicule are entertaining and effective, and . . .
at the same time they are the agents of a great despair
and stasis in U.S. culture.”  — David Foster Wallace,
as quoted by Adam Kirsch today at Salon

Friday, November 14, 2014

Another Opening, Another Show

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

“What happens when you mix the brilliant wit of Noel Coward
with the intricate plotting of Agatha Christie? Set during a
weekend in an English country manor in 1932, Death by Design
is a delightful and mysterious ‘mash-up’ of two of the greatest
English writers of all time. Edward Bennett, a playwright, and
his wife Sorel Bennett, an actress, flee London and head to
Cookham after a disastrous opening night. But various guests
arrive unexpectedly….”

Samuel French (theatrical publisher) on a play that
opened in Houston on September 9, 2011.

Related material:

Friday, October 10, 2014

Cut to Stanley Chase

Filed under: General — m759 @ 9:00 PM

Chase worked for years to make a movie of
‘The Threepenny Opera.’ He finally got it done in 1989 as
Mack the Knife,’ with Menahem Golan directing.”

— David Colker, LA Times  obituary, Oct. 9, 2014

See also, from Tuesday, Oct. 7, 2014, the date of Chase’s death,
the Log24 posts Grids and Space, Concepts of Space, and As Is.

Thursday, October 9, 2014

Nine is a Vine

Filed under: General — m759 @ 11:29 PM

See also Concepts of Space and  “Launched from Cuber.”

Tuesday, October 7, 2014

Cards of Identity

Filed under: General — m759 @ 9:00 PM

Or:  Plan 9 Continues

(Suggested by this afternoon’s post Concepts of Space.)

See also Card

… and Tick Tick Hash.

The image “http://www.log24.com/log/pix05B/051014-Tick.gif” cannot be displayed, because it contains errors.

Grids and Space

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

In honor of the late Sidney Lumet

(See Makom Kadosh , April 9, 2011.)

IMAGE- Christopher Reeve in the 1982 film 'Deathtrap,' illustrating concepts of space

Sunday, July 20, 2014

Sunday School

Filed under: General,Geometry — m759 @ 9:29 AM

Paradigms of Geometry:
Continuous and Discrete

The discovery of the incommensurability of a square's
side with its diagonal contrasted a well-known 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.

IMAGE- Concepts of Space: The Large Desargues Configuration, the Related 4x4 Square, and the 4x4x4 Cube

See Desargues via Galois (August 6, 2013).

Sunday, July 6, 2014

Sticks and Stones

Filed under: General,Geometry — Tags: — m759 @ 6:29 AM

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:

IMAGE- Concepts of Space

See also Desargues via Galois (August 6, 2013).

Tuesday, May 20, 2014


Filed under: General,Geometry — m759 @ 7:47 PM

From a recreational-mathematics 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 —

IMAGE- Concepts of Space: The large Desargues configuration and two figures illustrating Cullinane models of Galois geometry

For further details, see posts on Desargues and Galois.

Saturday, February 2, 2013

Mad Day

Filed under: General — Tags: , , , — m759 @ 11:00 AM

A perceptive review of Missing Out: In Praise of the Unlived Life

IMAGE- The perception of doors

"Page 185: 'Whatever else we are, we are also mad.' "

Related material— last night's Outside the Box and, from Oct. 22 last year

"Some designs work subtly.
Others are successful through sheer force."

Par exemple—

IMAGE- The Cartier diamond ring from 'Inside Man'

See also Cartier in this journal.

The Cartier link leads to, among other things

A Mad Day’s Work: From Grothendieck to Connes and Kontsevich.
The Evolution of Concepts of Space and Symmetry
by Pierre Cartier, Bulletin of the American Mathematical Society ,
Vol. 38 (2001) No. 4, pages 389-408

Tuesday, June 22, 2010

Hermeneutics for Bernstein

Filed under: General — Tags: — m759 @ 8:28 PM

J. M. Bernstein (previous post) has written of moving toward "a Marxist hermeneutic."

I prefer lottery hermeneutics.

Some background from Bernstein—


I would argue that at least sometimes, lottery numbers may be regarded, according to Bernstein's definition, as story statements. For instance—

Today's New York State Lottery— Midday 389, Evening 828.

For the significance of 389, see

A Mad Day’s Work: From Grothendieck to Connes and Kontsevich.
 The Evolution of Concepts of Space and Symmetry
 by Pierre Cartier, Bulletin of the American Mathematical Society,
 Vol. 38 (2001) No. 4, beginning on page 389.

The philosophical import of page 389 is perhaps merely in Cartier's title (see previous post).

For the significance of 828, see 8/28, the feast of St. Augustine, in 2006.

See also Halloween 2007. (Happy birthday, Dan Brown.)

Sunday, October 11, 2009

Sunday October 11, 2009

Filed under: General,Geometry — m759 @ 7:00 PM
Concepts of Space

Today I revised the illustrations
in Finite Geometry of the
Square and Cube

for consistency in labeling
the eightfold cube.

Related material:

Inside the White Cube:
The Ideology of
the Gallery Space

Dagger Definitions

Saturday, April 25, 2009

Saturday April 25, 2009

Filed under: General,Geometry — Tags: — m759 @ 9:22 PM
State of Play

Russell Crowe in 'State of Play'

The Russell Crowe
Hotel Puzzle

by John Tierney

"Russell Crowe arrives at the Hotel Infinity looking tired and ornery. He demands a room. The clerk informs him that there are no vacancies…."

Footprints from California today
(all by a person or persons using Firefox browsers):

7:10 AM
Concepts of Space: Euclid vs. Galois

8:51 AM
Art Wars continued: Behind the Picture

1:33 PM
A Riff for Dave: Me and My Shadow

2:11 PM
A Death of Kings: In Memory of Bobby Fischer

2:48 PM
Art Wars in review– Through the Looking Glass: A Sort of Eternity

3:28 PM and
Annals of Philosophy: The Dormouse of Perception

4:28 PM
Epiphany for Roy, Part I

6:03 PM
At the Still Point: All That Jazz

6:22 PM
Where Entertainment is Not God: The Just Word

7:14 PM
Happy New Yorker Day– Class Galore

7:16 PM
The Politics of Change: Jumpers

"Relax," said the night man.
"We are programmed to receive."
— Hotel California

Tuesday, February 3, 2009

Tuesday February 3, 2009

Filed under: General — m759 @ 7:59 AM

Everything and Nothing

"I know what 'nothing' means…."

— Joan Didion, Play It As It Lays, Farrar, Straus and Giroux, 1990 paperback, page 214

"In 1935, near the end of a long affectionate letter to his son George in America, James Joyce wrote: 'Here I conclude. My eyes are tired. For over half a century they have gazed into nullity, where they have found a lovely nothing.'"

— Lionel Trilling, "James Joyce in His Letters," Commentary, 45, no. 2 (Feb. 1968), abstract

"The quotation is from The Letters of James Joyce, Volume III, ed. Richard Ellman (New York, 1966), p. 359. The original Italian reads 'Adesso termino. Ho gli occhi stanchi. Da più di mezzo secolo scrutano nel nulla dove hanno trovato un bellissimo niente.'"

— Lionel Trilling: Criticism and Politics, by William M. Chace, Stanford U. Press, 1980, page 198, Note 4 to Chapter 9

"Space: what you damn well have to see."

— James Joyce, Ulysses

"What happens to the concepts of space and direction if all the matter in the universe is removed save a small finite number of particles?"

— "On the Origins of Twistor Theory," by Roger Penrose

"… we can look to the prairie, the darkening sky, the birthing of a funnel-cloud to see in its vortex the fundamental structure of everything…"

Against the Day, by Thomas Pynchon (See previous entry.)

"A strange thing then happened."

L. Frank Baum

Tuesday, February 26, 2008

Tuesday February 26, 2008

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

Eight is a Gate (continued)

Tom Stoppard, Jumpers:
"Heaven, how can I believe in Heaven?" she sings at the finale. "Just a lying rhyme for seven!"
"To begin at the beginning: Is God?…" [very long pause]

From "Space," by Salomon Bochner

Makom. Our term “space” derives from the Latin, and is thus relatively late. The nearest to it among earlier terms in the West are the Hebrew makom and the Greek topos (τόπος). The literal meaning of these two terms is the same, namely “place,” and even the scope of connotations is virtually the same (Theol. Wörterbuch…, 1966). Either term denotes: area, region, province; the room occupied by a person or an object, or by a community of persons or arrangements of objects. But by first occurrences in extant sources, makom seems to be the earlier term and concept. Apparently, topos is attested for the first time in the early fifth century B.C., in plays of Aeschylus and fragments of Parmenides, and its meaning there is a rather literal one, even in Parmenides. Now, the Hebrew book Job is more or less contemporary with these Greek sources, but in chapter 16:18 occurs in a rather figurative sense:

O earth, cover not thou my blood, and let my cry have no place (makom).

Late antiquity was already debating whether this makom is meant to be a “hiding place” or a “resting place” (Dhorme, p. 217), and there have even been suggestions that it might have the logical meaning of “occasion,” “opportunity.” Long before it appears in Job, makom occurs in the very first chapter of Genesis, in:

And God said, Let the waters under the heaven be gathered together unto one place (makom) and the dry land appear, and it was so (Genesis 1:9).

This biblical account is more or less contemporary with Hesiod's Theogony, but the makom of the biblical account has a cosmological nuance as no corresponding term in Hesiod. Elsewhere in Genesis (for instance, 22:3; 28:11; 28:19), makom usually refers to a place of cultic significance, where God might be worshipped, eventually if not immediately. Similarly, in the Arabic language, which however has been a written one only since the seventh century A.D., the term makām designates the place of a saint or of a holy tomb (Jammer, p. 27). In post-biblical Hebrew and Aramaic, in the first centuries A.D., makom became a theological synonym for God, as expressed in the Talmudic sayings: “He is the place of His world,” and “His world is His place” (Jammer, p. 26). Pagan Hellenism of the same era did not identify God with place, not noticeably so; except that the One (τὸ ἕν) of Plotinus (third century A.D.) was conceived as something very comprehensive (see for instance J. M. Rist, pp. 21-27) and thus may have been intended to subsume God and place, among other concepts. In the much older One of Parmenides (early fifth century B.C.), from which the Plotinian One ultimately descended, the theological aspect was only faintly discernible. But the spatial aspect was clearly visible, even emphasized (Diels, frag. 8, lines 42-49).


Paul Dhorme, Le livre de Job (Paris, 1926).

H. Diels and W. Kranz, Die Fragmente der Vorsokratiker, 6th ed. (Berlin, 1938).

Max Jammer, Concepts of Space (Cambridge, Mass., 1954).

J. M. Rist, Plotinus: The Road to Reality (Cambridge, 1967).

Theologisches Wörterbuch zum Neuen Testament (1966), 8, 187-208, esp. 199ff.


Related material: In the previous entry — "Father Clark seizes at one place (page eight)
upon the fact that…."

Father Clark's reviewer (previous entry) called a remark by Father Clark "far fetched."
This use of "place" by the reviewer is, one might say, "near fetched."

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