Log24

From the former date above —

From the latter date above —

From March 2018 —

See also yesterday's "For 6/24" and …

The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space  nature.

Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by 
a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem:  to fix objectively
a class of equivalent coordinatizations and to
ascertain the group of transformations S
mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space  coordinates. He describes it
as simply a mapping to a set of reproducible symbols . 

(But Weyl does imply that these symbols should, like vector-space 
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)

The "points" and "lines" of finite  geometry are abstract
entities satisfying only whatever incidence requirements
yield non-contradictory and interesting results. In finite
geometry, neither the points nor the lines are required to
lie within any Euclidean (or, for that matter, non-Euclidean)
space.

Models  of finite geometries may, however, embed the
points and lines within non -finite geometries in order
to aid visualization.

For instance, the 15 points and 35 lines of PG(3,2) may
be represented by subsets of a 4×4 array of dots, or squares,
located in the Euclidean plane. These "lines" are usually finite
subsets of dots or squares and not*  lines of the Euclidean plane.

Example — See "4×4" in this journal.

Some impose on configurations from finite geometry
the rather artificial requirement that both  points and lines
must be representable as those of a Euclidean plane.

Example:  A Cremona-Richmond pentagon —

A square version of these 15 "points" —

A 1905 square version of these 15 "points" 
with digits instead of letters —

See Parametrizing the 4×4 Array
(Log24 post of Sept. 13, 2016).

Update of 8 AM ET Sunday, Sept. 25, 2016 —
For more illustrations, do a Google image search
on "the 2-subsets of a 6-set." (See one such search.)

* But in some models are subsets of the grid lines 
   that separate squares within an array.

(A sequel to the previous post, Perfect Number)

Since antiquity,  six has been known as
"the smallest perfect number." The word "perfect"
here means that a number is the sum of its 
proper divisors — in the case of six: 1, 2, and 3.

The properties of a six-element set (a "6-set") 
divided into three 2-sets and divided into two 3-sets
are those of what Burkard Polster, using the same 
adjective in a different sense, has called 
"the smallest perfect universe" — PG(3,2), the projective
3-dimensional space over the 2-element Galois field.

A Google search for the phrase "smallest perfect universe"
suggests a turnaround in meaning , if not in finance, 
that might please Yahoo CEO Marissa Mayer on her birthday —

The semantic  turnaround here in the meaning  of "perfect"
is accompanied by a model  turnaround in the picture  of PG(3,2) as
Polster's tetrahedral  model is replaced by Cullinane's square  model.

Further background from the previous post —

See also Kirkman's Schoolgirl Problem.

"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—

For the birthday of Marissa Mayer, who turns 41 today —

VOGUE Magazine,
AUGUST 16, 2013 12:01 AM
by JACOB WEISBERG —

"As she works to reverse the fortunes of a failing Silicon Valley
giant, Yahoo’s Marissa Mayer has fueled a national debate
about the office life, motherhood, and what it takes to be the
CEO of the moment.

'I really like even numbers, and
I like heavily divisible numbers.
Twelve is my lucky number—
I just love how divisible it is.
I don’t like odd numbers, and
I really don’t like primes.
When I turned 37,
I put on a strong face, but
I was not looking forward to 37.
But 37 turned out to be a pretty amazing year.
Especially considering that
36 is divisible by twelve!'

A few things may strike you while listening to Marissa Mayer
deliver this riff . . . . "

Yes, they may.

A smaller number for Marissa's meditations:

Six has been known since antiquity as the first "perfect" number.
Why it was so called is of little interest to anyone but historians
of number theory  (a discipline that is not, as Wikipedia notes, 
to be confused with numerology .)

What part geometry , on the other hand, played in Marissa's education,
I do not know.

Here, for what it's worth, is a figure from a review of posts in this journal
on the key role played by the number six in geometry —

My statement yesterday morning that the 15 points
of the finite projective space PG(3,2) are indivisible 
was wrong.  I was misled by quoting the powerful
rhetoric of Lincoln Barnett (LIFE magazine, 1949).

Points of Euclidean  space are of course indivisible: 
"A point is that which has no parts" (in some translations).

And the 15 points of PG(3,2) may be pictured as 15
Euclidean  points in a square array (with one point removed)
or tetrahedral array (with 11 points added).

The geometry of  PG(3,2) becomes more interesting,
however, when the 15 points are each divided  into
several parts. For one approach to such a division,
see Mere Geometry. For another approach, click on the
image below.

See a search for Nocciolo  in this journal.

An image from that search —

Recall also Hamlet's
"O God… bad dreams."

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 Cayley-Salmon configuration , in the 
interesting context of Pascal's Hexagrammum Mysticum .

They show how the Cayley-Salmon configuration is, in a sense,
dual to something they call the Steiner-Plücker configuration .

This duality appears implicitly in my note of April 26, 1986,
"Picturing the smallest projective 3-space." The six-sets 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 six-set 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. 219-236  

Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen  (1900), Volume 53, Issue 1-2, pp 161-176

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. 125-160

Related material on six-set geometry from a more recent source —

Cullinane, Steven H., "Classical Geometry in Light of Galois Geometry," webpage

A useful article on finite geometry,
"21 – 6 = 15: A Connection between Two Distinguished Geometries,"
by Albrecht Beutelspacher, American Mathematical Monthly ,
Vol. 93, No. 1, January 1986, pp. 29-41, is available for purchase
at JSTOR.

This article is related to the geometry of the six-set.
For some background, see remarks from 1986 at finitegeometry.org.

Click to enlarge.

Background: Inscapes and The 2-subsets of a 6-set are the points of a PG(3,2).

Related remarks: Classical Geometry in Light of Galois Geometry.