Wednesday, January 17, 2018

“Before Time Began, There Was the Cube”

Filed under: Uncategorized — m759 @ 8:00 AM

See Eightfold Froebel.

The Paradise of Childhood'-- Froebel's Third Gift

Tuesday, April 1, 2014

Kindergarten Geometry

Filed under: Uncategorized — Tags: , — m759 @ 11:22 PM


A screenshot of the new page on the eightfold cube at Froebel Decade:

IMAGE- The eightfold cube at Froebel Decade

Click screenshot to enlarge.

Wednesday, February 5, 2014

Mystery Box II

Filed under: Uncategorized — m759 @ 4:07 PM

Continued from previous post and from Sept. 8, 2009.

Box containing Froebel's Third Gift-- The Eightfold Cube

Examination of the box's contents does not solve
the contents' real mystery. That requires knowledge
of the non-Euclidean geometry of Galois space.

In this case, without that knowledge, prattle (as in
today's online New York Times ) about creativity and
"thinking outside the box" is pointless.

Friday, June 14, 2013

Object of Beauty

Filed under: Uncategorized — Tags: , — m759 @ 2:01 AM

This journal on July 5, 2007 —

The Eightfold Cube and its Inner Structure

“It is not clear why MySpace China will be successful."

— The Chinese magazine Caijing  in 2007, quoted in
Asia Sentinel  on July 12, 2011

This  journal on that same date,  July 12, 2011 —


See also the eightfold cube and kindergarten blocks
at finitegeometry.org/sc.

Friedrich Froebel, Froebel's Chief Writings on Education ,
Part II, "The Kindergarten," Ch. III, "The Third Play":

"The little ones, who always long for novelty and change,
love this simple plaything in its unvarying form and in its
constant number, even as they love their fairy tales with
the ever-recurring dwarfs…."

This journal, Group Actions, Nov. 14, 2012:

"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—


Edwin M. Knowles Fine China Company, 1991

Tuesday, March 26, 2013


Filed under: Uncategorized — m759 @ 12:00 PM


"It should be emphasized that block models are physical models, the elements of which can be physically manipulated. Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics. For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is re-assembled into a linear array, are physical transformations not symbolic transformations. …"

— Storrs McCall, Department of Philosophy, McGill University, "The Consistency of Arithmetic"

"It should be emphasized…."


Storrs McCall at a 2008 philosophy conference .

His blocks talk was at 2:50 PM July 21, 2008.
See also this journal at noon that same day:

Froebel's Third Gift and the Eightfold Cube

Froebel's Third Gift: A cube made up of eight subcubes

The Eightfold Cube: The Beauty of Klein's Simple Group

Wednesday, November 14, 2012

Group Actions

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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

Friday, April 10, 2009

Friday April 10, 2009

Filed under: Uncategorized — m759 @ 8:00 AM

Pilate Goes
to Kindergarten

“There is a pleasantly discursive
 treatment of Pontius Pilate’s
unanswered question
‘What is truth?’.”

— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
 remarks on the “Story Theory
 of truth as opposed to the
Diamond Theory” of truth in
 The Non-Euclidean Revolution

Consider the following question in a paper cited by V. S. Varadarajan:

E. G. Beltrametti, “Can a finite geometry describe physical space-time?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.


“Can a finite geometry describe physical space?”

Simplifying further:

“Yes. VideThe Eightfold Cube.'”

Froebel's 'Third Gift' to kindergarteners: the 2x2x2 cube

Thursday, February 5, 2009

Thursday February 5, 2009

Filed under: Uncategorized — 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 19, 2008

Friday December 19, 2008

Filed under: Uncategorized — Tags: , , — m759 @ 1:06 PM
Inside the
White Cube

Part I: The White Cube

The Eightfold Cube

Part II: Inside
The Paradise of Childhood'-- Froebel's Third Gift

Part III: Outside

Mark Tansey, 'The Key' (1984)

Click to enlarge.

Mark Tansey, The Key (1984)

For remarks on religion
related to the above, see
Log24 on the Garden of Eden
and also Mark C. Taylor,
"What Derrida Really Meant"
(New York Times, Oct. 14, 2004).

For some background on Taylor,
see Wikipedia. Taylor, Chairman
of the Department of Religion
Columbia University, has a
1973 doctorate in religion from
Harvard University. His opinion
of Derrida indicates that his
sympathies lie more with
the serpent than with the angel
in the Tansey picture above.

For some remarks by Taylor on
the art of Tansey relevant to the
structure of the white cube
(Part I above), see Taylor's
The Picture in Question:
Mark Tansey and the
Ends of Representation

(U. of Chicago Press, 1999):

From Chapter 3,
"Sutures* of Structures," p. 58:

"What, then, is a frame, and what is frame work?

This question is deceptive in its simplicity. A frame is, of course, 'a basic skeletal structure designed to give shape or support' (American Heritage Dictionary)…. when the frame is in question, it is difficult to determine what is inside and what is outside. Rather than being on one side or the other, the frame is neither inside nor outside. Where, then, Derrida queries, 'does the frame take place….'"

* P. 61:
"… the frame forms the suture of structure. A suture is 'a seamless [sic**] joint or line of articulation,' which, while joining two surfaces, leaves the trace of their separation."

 ** A dictionary says "a seamlike joint or line of articulation," with no mention of "trace," a term from Derrida's jargon.

Friday, December 12, 2008

Friday December 12, 2008

Filed under: Uncategorized — 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: Uncategorized — 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.

Monday, July 21, 2008

Monday July 21, 2008

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

Knight Moves:

The Relativity Theory
of Kindergarten Blocks

(Continued from
January 16, 2008)

“Hmm, next paper… maybe
‘An Unusually Complicated
Theory of Something.'”

Garrett Lisi at
Physics Forums, July 16


From Friedrich Froebel,
who invented kindergarten:

Froebel's Third Gift: A cube made up of eight subcubes

Click on image for details.

An Unusually
Complicated Theory:

From Christmas 2005:

The Eightfold Cube: The Beauty of Klein's Simple Group

Click on image for details.

For the eightfold cube
as it relates to Klein’s
simple group, see
A Reflection Group
of Order 168

For an even more
complicated theory of
Klein’s simple group, see

Cover of 'The Eightfold Way: The Beauty of Klein's Quartic Curve'

Click on image for details.

Saturday, May 10, 2008

Saturday May 10, 2008

Filed under: Uncategorized — Tags: , , , — m759 @ 8:00 AM
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 a book from 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

Monday, June 25, 2007

Monday June 25, 2007

Filed under: Uncategorized — m759 @ 3:00 PM
Object Lesson
“… the best definition
 I have for Satan
is that it is a real
  spirit of unreality.”

M. Scott Peck,
People of the Lie

“Far in the woods they sang
     their unreal songs,
Secure.  It was difficult
     to sing in face
Of the object.  The singers
     had to avert themselves
Or else avert the object.”

— Wallace Stevens,
   “Credences of Summer”

Today is June 25,
anniversary of the
birth in 1908 of
Willard Van Orman Quine.

Quine died on
Christmas Day, 2000.
Today, Quine’s birthday, is,
as has been noted by
Quine’s son, the point of the
calendar opposite Christmas–
i.e., “Anti-Christmas.”
If the Anti-Christ is,
as M. Scott Peck claims,
a spirit of unreality, it seems
fitting today to invoke
Quine, a student of reality,
  and to borrow the title of
 Quine’s Word and Object


An excerpt from
“Credences of Summer”
by Wallace Stevens:

“Three times the concentred
     self takes hold, three times
The thrice concentred self,
     having possessed

The object, grips it
     in savage scrutiny,
Once to make captive,
     once to subjugate
Or yield to subjugation,
     once to proclaim
The meaning of the capture,
     this hard prize,
Fully made, fully apparent,
     fully found.”

— “Credences of Summer,” VII,
    by Wallace Stevens, from
    Transport to Summer (1947)


From Friedrich Froebel,
who invented kindergarten:

Froebel's Third Gift

From Christmas 2005:

The Eightfold Cube

Click on the images
for further details.

For a larger and
more sophisticaled
relative of this object,
see yesterday’s entry
At Midsummer Noon.

The object is real,
not as a particular
physical object, but
in the way that a
mathematical object
is real — as a
pure Platonic form.

“It’s all in Plato….”
— C. S. Lewis

Saturday, April 7, 2007

Saturday April 7, 2007

Filed under: Uncategorized — m759 @ 12:25 PM
Today's birthdays:
Francis Ford Coppola
and Russell Crowe

Gift of the Third Kind

Art Wars and
Russell Crowe as
Santa's Helper

From Friedrich Froebel,
who invented kindergarten:

Froebel's Third Gift

From Christmas 2005:

The Eightfold Cube

Related material from

Reinventing Froebel's Gifts

… and from Grand Rapids:

Color Cubes

Click on pictures for details.

Related material
for Holy Saturday:

"Hey, Big Spender,"
Santa Versus the Volcano.

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