Sunday, November 29, 2020

Osterman Meets Brosterman

Filed under: General — Tags: , — m759 @ 8:25 PM

See the former and the latter.

Wednesday, June 12, 2019

The Osterman Haiku

Filed under: General — m759 @ 11:32 PM

Click on the book cover below for posts tagged "Haiku."

IMAGE- 'Point Omega' by DeLillo

Sunday, May 6, 2018

The Osterman Omega

Filed under: General,Geometry — Tags: , — m759 @ 5:01 PM

From "The Osterman Weekend" (1983) —

Counting symmetries of the R. T. Curtis Omega:

An Illustration from Shakespeare's birthday

Counting symmetries with the orbit-stabilizer theorem

Sunday, March 5, 2017

The Klostermann Weekend

Filed under: General — m759 @ 11:21 PM

A Girl's Best Friend?

Saturday, November 28, 2020

A Poster for Doctor Manhattan

Filed under: General — Tags: — m759 @ 10:32 AM

A nostalgia pill for Watchmen  fans.

For Harvard  Watchmen fans, a link to 2346:

http://m759.net/wordpress/?p=2346 —

Sunday, July 28, 2019


Filed under: General — m759 @ 11:18 AM

From a news article featured on the American Mathematical Society
home page today

A joint Vietnam-USA mathematical meeting in Vietnam on
June 10-13, 2019:

This  journal on June 12, 2019:

Wednesday, June 12, 2019

The Osterman Haiku

Filed under: General — m759 @ 11:32 PM Edit This

Click on the book cover below for posts tagged "Haiku."

'Point Omega' by DeLillo

See also the Twentieth of May, 2008 —

Welcome to the Garden Club, Pilgrim.

Saturday, July 27, 2019

“Am I Still On?”

Filed under: General — m759 @ 1:05 PM

See also the above quote in this  journal.

Saturday, June 1, 2019

What’s in a Name

Filed under: General — m759 @ 12:30 PM

Thursday, July 12, 2018

Kummerhenge Illustrated

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


“… the utterly real thing in writing is the only thing that counts…."

— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935

"Omega is as real  as we need it to be."

— Burt Lancaster in "The Osterman Weekend"

Monday, May 7, 2018


Filed under: General — Tags: , — m759 @ 10:32 AM

(Continued from yesterday's Sunday School Lesson Plan for Peculiar Children)

Novelist George Eliot and programming pioneer Ada Lovelace —

For an image that suggests a resurrected multifaceted 
(specifically, 759-faceted) Osterman Omega (as in Sunday's afternoon
Log24 post
), behold  a photo from today's NY Times  philosophy
column "The Stone" that was reproduced here in today's previous post

For a New York Times  view of George Eliot data, see a Log24 post 
of September 20, 2016, on the diamond theorem as the Middlemarch
"key to all mythologies."

Sunday, March 5, 2017

The Omega Matrix

Filed under: General,Geometry — Tags: — m759 @ 5:00 PM

Richard Evan Schwartz on
the mathematics of the 4×4 square

See also Priority in this journal.

Sunday, December 18, 2016

Manifest O

Filed under: General — m759 @ 10:10 PM

"The Osterman Weekend" (1983) —

IMAGE- Chair from 'Osterman Weekend' ending

“Am I still on?” — Ending line of  The Osterman Weekend  (1983)

Thursday, December 24, 2015

Group Actions…

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

On the Eight

Saturday, August 22, 2015

It’s 10 PM…

Filed under: General — m759 @ 10:00 PM

On an Osterman weekend.

Do you know where your crossbows are?

Sunday, May 24, 2015

Point Omega

Filed under: General — m759 @ 12:00 PM

IMAGE- Chair from 'Osterman Weekend' ending

“Am I still on?” — Ending line of  The Osterman Weekend  (1983)

Sunday, January 11, 2015

The XYZ of Being

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

From a recent Gitterkrieg  post:

"The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being…." — Wallace Stevens

See also the cover of the February 2015
Notices of the American Mathematical Society .

"Omega is as real  as we need it to be."
Burt Lancaster in The Osterman Weekend

Sunday, November 23, 2014

Remarks on Reality

Filed under: General — Tags: — m759 @ 5:09 PM

Wallace Stevens in "An Ordinary Evening in New Haven"
(1950) on "The Ruler of Reality" —

"Again, 'He has thought it out, he thinks it out,
As he has been and is and, with the Queen
Of Fact, lies at his ease beside the sea.'"

One such scene, from 1953 —

Another perspective, from "The Osterman Weekend" (1983) —

Tuesday, August 5, 2014

Omega Post

Filed under: General — m759 @ 2:00 AM

In memory of radio personality Steve Post,
a link to some remarks on the date of his death.

Monday, August 4, 2014

Bunch vs. Bunch

Filed under: General — m759 @ 7:00 PM

“This is a divorce case that was before us on an earlier occasion.”


From the director of The Wild Bunch


From The New York Times —

Sunday, August 3, 2014

Unplatonic Dialogue

Filed under: General — m759 @ 8:12 PM

Dialogue from “The Osterman Weekend”—

01:57:22  “Why did he make us try to believe Omega existed?”
01:57:25    ….
01:57:26  “The existence of Omega has not been disproved.
01:57:28  Don’t you understand that?
01:57:31  Omega is as real  as we need it to be.”

See also Omega elsewhere in this journal.

Update of 9:15 PM ET —

Monday, July 14, 2014

Conversations with an Empty Chair

Filed under: General — m759 @ 1:00 AM

Continued from August 20, 2013

In honor of Sam Peckinpah, the closing shot of his last film:

“Am I still on?” — Ending line of  The Osterman Weekend  (1983)

Tuesday, September 3, 2013

“The Stone” Today Suggests…

Filed under: General,Geometry — m759 @ 12:31 PM

A girl's best friend?

The Philosopher's Gaze , by David Michael Levin,
U. of California Press, 1999, in III.5, "The Field of Vision," pp. 174-175—

The post-metaphysical question—question for a post-metaphysical phenomenology—is therefore: Can the perceptual field, the ground of perception, be released  from our historical compulsion to represent it in a way that accommodates our will to power and its need to totalize and reify the presencing of being? In other words: Can the ground be experienced as  ground? Can its hermeneutical way of presencing, i.e., as a dynamic interplay of concealment and unconcealment, be given appropriate  respect in the receptivity of a perception that lets itself  be appropriated by  the ground and accordingly lets  the phenomenon of the ground be  what and how it is? Can the coming-to-pass of the ontological difference that is constitutive of all the local figure-ground differences taking place in our perceptual field be made visible hermeneutically, and thus without violence to its withdrawal into concealment? But the question concerning the constellation of figure and ground cannot be separated from the question concerning the structure of subject and object. Hence the possibility of a movement beyond metaphysics must also think the historical possibility of breaking out of this structure into the spacing of the ontological difference: différance , the primordial, sensuous, ekstatic écart . As Heidegger states it in his Parmenides lectures, it is a question of "the way historical man belongs within the bestowal of being (Zufügung des Seins ), i.e., the way this order entitles him to acknowledge being and to be the only being among all beings to see  the open" (PE* 150, PG** 223. Italics added). We might also say that it is a question of our response-ability, our capacity as beings gifted with vision, to measure up to the responsibility for perceptual responsiveness laid down for us in the "primordial de-cision" (Entscheid ) of the ontological difference (ibid.). To recognize the operation of the ontological difference taking place in the figure-ground difference of the perceptual Gestalt  is to recognize the ontological difference as the primordial Riß , the primordial Ur-teil  underlying all our perceptual syntheses and judgments—and recognize, moreover, that this rift, this  division, decision, and scission, an ekstatic écart  underlying and gathering all our so-called acts of perception, is also the only "norm" (ἀρχή ) by which our condition, our essential deciding and becoming as the ones who are gifted with sight, can ultimately be judged.

* PE: Parmenides  of Heidegger in English— Bloomington: Indiana University Press, 1992

** PG: Parmenides  of Heidegger in German— Gesamtausgabe , vol. 54— Frankfurt am Main: Vittorio Klostermann, 1992

Examples of "the primordial Riß " as ἀρχή  —

For an explanation in terms of mathematics rather than philosophy,
see the diamond theorem. For more on the Riß  as ἀρχή , see
Function Decomposition Over a Finite Field.

Wednesday, November 14, 2012

Group Actions

Filed under: General,Geometry — Tags: , , — m759 @ 4:30 PM

The December 2012 Notices of the American
Mathematical Society  
has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on “Low-dimensional
Topology, Geometry, and Dynamics”—

(Only the top part of the ad is shown; for further details
see an ICERM page.)

(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)

The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—

The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel’s Third Gift—


Froebel's third gift, the eightfold cube

© 2005 The Institute for Figuring

Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)

The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.

These cubes are not without relevance to the workshops’ topics—
low-dimensional exotic geometric structures, group theory, and dynamics.

See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.

Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—


Friday, December 30, 2011

Quaternions on a Cube

Filed under: General,Geometry — Tags: , , , — m759 @ 5:48 AM

The following picture provides a new visual approach to
the order-8 quaternion  group's automorphisms.

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.

See also…

Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.

* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she co-founded—

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

Tuesday, September 8, 2009

Tuesday September 8, 2009

Filed under: General,Geometry — Tags: — m759 @ 12:25 PM
Magic Box  

Box containing Froebel's Third Gift-- The Eightfold Cube
 Continued from Dec. 7, 2008,
and from yesterday.



Passages from a classic story:

… he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads….


"Your mind has been conditioned to Euclid," Holloway said. "So this– thing– bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."

"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded.
"Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child– especially a baby– might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."

"Hardening of the thought-arteries," Jane interjected.

Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only–"

"Well, look. Let's suppose there are two kinds of geometry– we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid–"

"Poor kid," Jane said.

Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra– they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic–"

"Blocks? What kind?"

Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid."

— "Mimsy Were the Borogoves," Lewis Padgett, 1943

Padgett (pseudonym of a husband-and-wife writing team) says that alphabet blocks are the intuitive "basis of Euclid." Au contraire; they are the basis of Gutenberg.

For the intuitive basis of one type of non-Euclidean* geometry– finite geometry over the two-element Galois field– see the work of…

Friedrich Froebel
 (1782-1852), who
 invented kindergarten.

His "third gift" —

Froebel's Third Gift-- The Eightfold Cube
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring

Go figure.

* i.e., other than Euclidean

Thursday, February 5, 2009

Thursday February 5, 2009

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

Through the
Looking Glass:

A Sort of Eternity

From the new president’s inaugural address:

“… in the words of Scripture, the time has come to set aside childish things.”

The words of Scripture:

9 For we know in part, and we prophesy in part.
10 But when that which is perfect is come, then that which is in part shall be done away.
11 When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.
12 For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known. 

First Corinthians 13

“through a glass”

[di’ esoptrou].
By means of
a mirror [esoptron]

Childish things:

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)


Three planes through
the center of a cube
that split it into
eight subcubes:
Cube subdivided into 8 subcubes by planes through the center
Through a glass, darkly:

A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
that is symmetric under one of
these 8 transformations–
a 180-degree rotation:

Design Cube 2x2x2 for demonstrating Galois geometry

(Click on image
for further details.)

But then face to face:

A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.

Galois age fifteen, drawn by a classmate.

Galois age fifteen,
drawn by a classmate.

These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.

For some generalizations,
see Galois Geometry.

Related material:

The central aim of Western religion– 

"Each of us has something to offer the Creator...
the bridging of
 masculine and feminine,
 life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)

The central aim of Western philosophy–

 Dualities of Pythagoras
 as reconstructed by Aristotle:
  Limited Unlimited
  Odd Even
  Male Female
  Light Dark
  Straight Curved
  ... and so on ....

“Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy.”

— Jamie James in The Music of the Spheres (1993)

“In the garden of Adding
live Even and Odd…
And the song of love’s recision
is the music of the spheres.”

— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)

A quotation today at art critic Carol Kino’s website, slightly expanded:

“Art inherited from the old religion
the power of consecrating things
and endowing them with
a sort of eternity;
museums are our temples,
and the objects displayed in them
are beyond history.”

— Octavio Paz,”Seeing and Using: Art and Craftsmanship,” in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52

From Brian O’Doherty’s 1976 Artforum essays– not on museums, but rather on gallery space:

Inside the White Cube

“We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20th-century art.”


“Space: what you
damn well have to see.”

— James Joyce, Ulysses  

Friday, December 12, 2008

Friday December 12, 2008

Filed under: General,Geometry — m759 @ 3:09 PM
On the Symmetric Group S8

Wikipedia on Rubik's 2×2×2 "Pocket Cube"–


"Any permutation of the 8 corner cubies is possible (8! positions)."

Some pages related to this claim–

Simple Groups at Play

Analyzing Rubik's Cube with GAP

Online JavaScript Pocket Cube.

The claim is of course trivially true for the unconnected subcubes of Froebel's Third Gift:

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring


Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

See also:

MoMA Goes to Kindergarten,

Tea Privileges


"Ad Reinhardt and Tony Smith:
A Dialogue,"
an exhibition opening today
at Pace Wildenstein.

For a different sort
of dialogue, click on the
artists' names above.

For a different
approach to S8,
see Symmetries.

"With humor, my dear Zilkov.
Always with a little humor."

-- The Manchurian Candidate

Sunday, December 7, 2008

Sunday December 7, 2008

Filed under: General,Geometry — m759 @ 11:00 AM
Space and
 the Soul

On a book by Margaret Wertheim:

“She traces the history of space beginning with the cosmology of Dante. Her journey continues through the historical foundations of celestial space, relativistic space, hyperspace, and, finally, cyberspace.” –Joe J. Accardi, Northeastern Illinois Univ. Lib., Chicago, in Library Journal, 1999 (quoted at Amazon.com)

There are also other sorts of space.

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

This photo may serve as an
introduction to a different
sort of space.

See The Eightfold Cube.

For the religious meaning
of this small space, see

Richard Wilhelm on
the eight I Ching trigrams

For a related larger space,
see the entry and links of
 St. Augustine’s Day, 2006.

Saturday, May 10, 2008

Saturday May 10, 2008

MoMA Goes to

"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."

— "Was Modernism Born
     in Toddler Toolboxes?"
     by Trip Gabriel, New York Times,
     April 10, 1997


Figure 1 —
Concept from 1819:

Cubic crystal system
(Footnotes 1 and 2)

Figure 2 —
The Third Gift, 1837:

Froebel's third gift

Froebel's Third Gift

Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.

(Footnote 3)

Figure 3 —
The Third Gift, 1906:

Seven partitions of the eightfold cube in 'Paradise of Childhood,' 1906

Figure 4 —
Solomon's Cube,
1981 and 1983:

Solomon's Cube - A 1981 design by Steven H. Cullinane

Figure 5 —
Design Cube, 2006:

Design Cube 4x4x4 by Steven H. Cullinane

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the two-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

For some mathematical background, see

1. Image said to be after Holden and Morrison, Crystals and Crystal Growing, 1982
2. Curtis Schuh, "The Library: Biobibliography of Mineralogy," article on Mohs
3. Bart Kahr, "Crystal Engineering in Kindergarten" (pdf), Crystal Growth & Design, Vol. 4 No. 1, 2004, 3-9

Wednesday, June 7, 2006

Wednesday June 7, 2006

Filed under: General — Tags: — m759 @ 10:00 AM
Figures of Speech



in memory of
Arnold Newman,
dead on 6/6/6.

TIME magazine, issue dated June 12, 2006, item posted Sunday, June 4, 2006:



"Nervous kids and obscure words are not the stuff of big-time TV, but this year's Scripps National Spelling Bee was an improbable nail-biter. One of the 13 finalists got reinstated after judges made a spelling error, a Canadian came in second–who knew foreign kids could compete?–and KATHARINE CLOSE, 13, prevailed in her fifth year. The eighth-grader from Spring Lake, N.J., won with ursprache. It means protolanguage. Now try to use it in conversation."

John T. Lysaker (pdf)
quoting Heidegger:
"Poetry is the
 originary language

— Heidegger, Erlauterungen
zu Holderlins Dichtung
 Frankfurt am Main:
Klostermann, 1971: 41.

See also a figure from
D-Day morning,

The image “http://www.log24.com/log/pix06A/060604-Roots.jpg” cannot be displayed, because it contains errors.

and a figure from
April 5, 2005:

(Skewed Mirrors
Sept. 14, 2003)

"Evil did not have
the last word."
Richard John Neuhaus,
April 4, 2005

"This is the exact opposite
of what echthroi do in
their X-ing or un-naming."
Wikipedia on
A Wind in the Door

"Lps. The keys to. Given!
 A way a lone a last
 a loved a long the
 — James Joyce,
     Finnegans Wake

"There is never any ending
to Paris."
— Ernest Hemingway    

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