Log24

* A footnote in memory of a dancer who reportedly died
  yesterday, August 29 —  See posts tagged Paradigm Shift.

"Birthday, death-day — what day is not both?" — John Updike

McCarthy's "materialization of plot and character" does not,
for me, constitute a proof that "there is  being, after all,
beyond the arbitrary flux of existence."

Neither does the above materialization of 281 as the page 
number of her philosophical remark.

See also the materialization of 281 as a page number in
the book Witchcraft  by Charles Williams —

The materialization of 168 as a page number in a 
Stephen King novel is somewhat more convincing,
but less convincing than the materialization of Klein's
simple group of of 168 elements in the eightfold cube.

The title reverses a phrase of Fano —
“costruire (o, dirò meglio immaginare).”

Illustrations of imagining (the Fano plane) and of constructing (the eightfold cube) —
 

The Fano plane and the eightfold cube

Logo from the above webpage —

See also the similar structure of  the eightfold cube,  and …

Related dialogue from the new film "Unlocked" —

1057
01:31:59,926 –> 01:32:01,301
Nice to have you back, Alice.

1058
01:32:04,009 –> 01:32:05,467
Don't be a stranger.

See posts now tagged with the above title.

From Wikipedia's Iceberg Theory —

Related material: 

The Eightfold Cube and The Quantum Identity —

See also the previous post.

Click to enlarge the following (from Cornell U. Press in 1962) —

For a more recent analogical extension at Cornell, see the
Epiphany 2017 post on the eightfold cube and yesterday
evening's post "A Theory of Everything."

Click the above image for remarks on
"deep structure" and binary opposition.

See also the eightfold cube.

Yesterday's post The Eightfold Cube in Oslo suggests a review of
posts that mention The Lost Crucible.

(The crucible in question is from a book by Katherine Neville, 
The Eight . Any connection with Arthur Miller's play  "The Crucible" 
is purely coincidental.)

The Fano Plane —

"A balanced incomplete block design , or BIBD
with parameters b , v , r , k , and λ  is an arrangement
of b  blocks, taken from a set of v  objects (known
for historical reasons as varieties ), such that every
variety appears in exactly r  blocks, every block
contains exactly k  varieties, and every pair of
varieties appears together in exactly λ  blocks.
Such an arrangement is also called a
(b , v , r , k , λ ) design. Thus, (7, 3, 1) [the Fano plane] 
is a (7, 7, 3, 3, 1) design."

— Ezra Brown, "The Many Names of (7, 3, 1),"
     Mathematics Magazine , Vol. 75, No. 2, April 2002

W. Cherowitzo uses the notation (v, b, r, k, λ) instead of
Brown's (b , v , r , k , λ ).  Cherowitzo has described,
without mentioning its close connection with the
Fano-plane design, the following —

"the (8,14,7,4,3)-design on the set
X = {1,2,3,4,5,6,7,8} with blocks:

{1,3,7,8} {1,2,4,8} {2,3,5,8} {3,4,6,8} {4,5,7,8}
{1,5,6,8} {2,6,7,8} {1,2,3,6} {1,2,5,7} {1,3,4,5}
{1,4,6,7} {2,3,4,7} {2,4,5,6} {3,5,6,7}."

We can arrange these 14 blocks in complementary pairs:

{1,2,3,6} {4,5,7,8}
{1,2,4,8} {3,5,6,7}
{1,2,5,7} {3,4,6,8}
{1,3,4,5} {2,6,7,8}
{1,3,7,8} {2,4,5,6}
{1,4,6,7} {2,3,5,8}
{1,5,6,8} {2,3,4,7}.

These pairs correspond to the seven natural slicings
of the following eightfold cube —

Another representation of these seven natural slicings —

These seven slicings represent the seven
planes through the origin in the vector
3-space over the two-element field GF(2).  
In a standard construction, these seven 
planes  provide one way of defining the
seven projective lines  of the Fano plane.

A more colorful illustration —

"It is as if one were to condense
all trends of present day mathematics
onto a single finite structure…."

— Gian-Carlo Rota, foreword to
A Source Book in Matroid Theory ,
Joseph P.S. Kung, Birkhäuser, 1986

"There is  such a thing as a matroid."

— Saying adapted from a novel by Madeleine L'Engle

Related remarks from Mathematics Magazine  in 2009 —

See also the eightfold cube —

 .

Omega is a Greek letter, Ω , used in mathematics to denote
a set on which a group acts. 

For instance, the affine group AGL(3,2) is a group of 1,344
actions on the eight elements of the vector 3-space over the
two-element Galois field GF(2), or, if you prefer, on the Galois
field  Ω = GF(8).

Related fiction:  The Eight , by Katherine Neville.

Related non-fiction:  A remark by Werner Heisenberg
in this journal on Saturday, June 6, 2015, the eightfold cube ,
and the illustrations below —

.

See also Snow White Dance.

For those who prefer mathematics to narrative:

Object of Beauty.

The previous post's
illustration was 
rather complicated.

This is a simpler
algebraic figure.

(Continued)

"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…."

OK:

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

Robert A. Wilson, in an inaugural lecture in April 2008—

Representation theory

A group always arises in nature as the symmetry group of some object, and group
theory in large part consists of studying in detail the symmetry group of some
object, in order to throw light on the structure of the object itself (which in some
sense is the “real” object of study).

But if you look carefully at how groups are used in other areas such as physics
and chemistry, you will see that the real power of the method comes from turning
the whole procedure round: instead of starting from an object and abstracting
its group of symmetries, we start from a group and ask for all possible objects
that it can be the symmetry group of .

This is essentially what we call Representation theory . We think of it as taking a
group, and representing it concretely in terms of a symmetrical object.

Now imagine what you can do if you combine the two processes: we start with a
symmetrical object, and find its group of symmetries. We now look this group up
in a work of reference, such as our big red book (The ATLAS of Finite Groups),
and find out about all (well, perhaps not all) other objects that have the same
group as their group of symmetries.

We now have lots of objects all looking completely different, but all with the same
symmetry group. By translating from the first object to the group, and then to
the second object, we can use everything we know about the first object to tell
us things about the second, and vice versa.

As Poincaré said,

Mathematicians do not study objects, but relations between objects.
Thus they are free to replace some objects by others, so long as the
relations remain unchanged.

Par exemple—

Fano plane transformed to eightfold cube,
and partitions of the latter as points of the former:

* For the "Will" part, see the PyrE link at Talk Amongst Yourselves.

The New York Times  at 9 PM ET June 23, 2011—

ROBERT FANO: I’m trying to think briefly how to put it.

GINO FANO: "On the Fundamental Postulates"—

"E la prova di questo si ha precisamente nel fatto che si è potuto costruire (o, dirò meglio immaginare) un ente per cui sono verificati tutti i postulati precedenti…."

"The proof of this is precisely the fact that you could build (or, to say it better, imagine) an entity by which are verified all previous assumptions…."

Also from the Times  article quoted above…

"… like working on a cathedral. We laid our bricks and knew that others might later replace them with better bricks. We believed in the cause even if we didn’t completely understand the implications.”

— Tom Van Vleck

Some art that is related, if only by a shared metaphor, to Van Vleck's cathedral—

The art is also related to the mathematics of Gino Fano.

For an explanation of this relationship (implicit in the above note from 1984),
see "The Fano plane revisualized—or: the eIghtfold cube."