Log24

Friday, May 6, 2005

Friday May 6, 2005

Filed under: General,Geometry — Tags: , — m759 @ 7:28 PM

Fugues

"To improvise an eight-part fugue
is really beyond human capability."

— Douglas R. Hofstadter,
Gödel, Escher, Bach

The image “http://www.log24.com/theory/images/cube2x2x2.gif” cannot be displayed, because it contains errors.

Order of a projective
 automorphism group:
168

"There are possibilities of
contrapuntal arrangement
of subject-matter."

— T. S. Eliot, quoted in
Origins of Form in Four Quartets.

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Order of a projective
 automorphism group:
20,160

Friday May 6, 2005

Filed under: General — Tags: , — m759 @ 2:56 PM
Involved

Trinity symbol
(See Sequel.)

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Trinity symbol
by Greg Egan
(via John Baez)

Involved:

 

"Difficult to understand because of intricacy: byzantine, complex, complicated, convoluted, daedal, Daedalian, elaborate, intricate, involute, knotty, labyrinthine, tangled."

— Roget's II: The New Thesaurus, Third Edition

See also the previous three entries,
as well as Symmetries.

Friday May 6, 2005

Filed under: General — Tags: — m759 @ 1:01 PM
Apocalypse Wow
continued

 
From the West Wing time slot:
 
Revelations Episode 4
(first airing: 9 PM ET Wednesday,
May 4, 2005)


"It's … extremely weird how the previously-on-Revelations announcer doesn't seem to be able to draw the distinction between what's happening in the real world where Revelations is just a cheesy miniseries that's keeping people from watching Alias and what's happening in the fake world of the miniseries itself, where they keep promising the apocalypse and it keeps not happening. After the wrap-up of all the nothing that's come before, the announcer intones ominously, 'And now, as the end of the world draws near, Revelations continues.' Well, no. Here, where Revelations is continuing, the end of the world is not drawing near. Or is NBC genuinely aiming for the crowd who thinks The Rapture Index is a valuable and educational resource? Does someone involved here have an actual sense of humor?"

The Flick Filosopher

Friday May 6, 2005

Filed under: General — Tags: — m759 @ 10:18 AM
Crystalline

"In Francis Ford Coppola's film, Col. Kurtz tells how after his medics inoculated a small village, the Reds chopped off every child's left arm. 'My God, the genius of that. The genius,' Kurtz said. 'The will to do that. Perfect, genuine, complete, crystalline, pure! And then I realized they were stronger than me because they could stand it.'"

Col. David Hackworth
    on Tuesday, April 9, 2002.
    Col. Hackworth died at 74
    on Wednesday, May 4, 2005.

   Related Log24 entries:

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Click on pictures for details.

Wednesday, May 4, 2005

Wednesday May 4, 2005

Filed under: General,Geometry — Tags: , — m759 @ 1:00 PM
The Fano Plane
Revisualized:

 

 The Eightfold Cube

or, The Eightfold Cube

Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the Fano plane):
 
The image “http://www.log24.com/theory/images/Fano.gif” cannot be displayed, because it contains errors.
 

Every permutation of the plane's points that preserves collinearity is a symmetry of the  plane.  The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group  PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)

The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle.  It does not, however, indicate where the other 162 symmetries come from.  

Shown below is a new model of this same projective plane, using partitions of cubes to represent points:

 

Fano plane with cubes as points
 
The cubes' partitioning planes are added in binary (1+1=0) fashion.  Three partitioned cubes are collinear if and only if their partitioning planes' binary sum equals zero.

 

The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.

 

Fano plane group - generating permutations

For a proof that such permutations generate the 168 symmetries, see Binary Coordinate Systems.

 

(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations.  But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results.  This illustrates the difference between affine and projective spaces over the binary field GF(2).  In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices.  This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices.  Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)

To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.

 

For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.

For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way.  For more on the mathematics of cubic models, see Solomon's Cube.

 

For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.

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