Log24

Thursday, January 23, 2020

Exploring Schoolgirl Space…

Filed under: General — Tags: , , , — m759 @ 11:59 pm

Continued .

"Old men ought to be explorers." — T. S. Eliot.

Rose the Hat in her younger days.

See as well Barsotti in this journal.

Saturday, September 21, 2019

Exploring Egypt: The Sequel to Exploring Schoolgirl Space*

Filed under: General — Tags: — m759 @ 9:02 am

     * See Schoolgirl Space in this journal.

Friday, September 13, 2019

Schoolgirl Space…

Filed under: General — Tags: , — m759 @ 4:56 am

According to Wikipedia

See also Schoolgirl Space in this journal.

Thursday, August 15, 2019

Schoolgirl Space — Tetrahedron or Square?

Filed under: General — Tags: , , — m759 @ 9:03 pm

The exercise in the previous post  was suggested by a passage
purporting to "use standard block design theory" that was written
by some anonymous author at Wikipedia on March 1, 2019:

Here "rm OR" apparently means "remove original research."

Before the March 1 revision . . .

The "original research" objected to and removed was the paragraph
beginning "To explain this further."  That paragraph was put into the
article earlier on Feb. 28 by yet another anonymous author (not  by me).

An account of my own (1976 and later) original research on this subject 
is pictured below, in a note from Feb. 20, 1986 —

'The relativity problem in finite geometry,' 1986

Saturday, August 10, 2019

Schoolgirl Space* Revisited:

Filed under: General — Tags: , , — m759 @ 10:51 pm

The Square "Inscape" Model of
the Generalized Quadrangle W(2)

Click image to enlarge.

* The title refers to the role of PG (3,2) in Kirkman's schoolgirl problem.
For some backstory, see my post Anticommuting Dirac Matrices as Skew Lines
and, more generally, posts tagged Dirac and Geometry.

Tuesday, July 16, 2019

Schoolgirl Space for Quantum Mystics

Filed under: General — Tags: , , — m759 @ 2:16 pm

From a post on St. Andrew's Day, 2017

See also "E-Numbers" and "E-Girls."

Schoolgirl Space Continues.

Filed under: General — Tags: , — m759 @ 10:59 am

Monday, July 15, 2019

Arche’s Schoolgirl Space

Filed under: General — Tags: — m759 @ 2:30 pm

  

From Dan Brown's novel Angels & Demons  (2000)—

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

Tuesday, July 9, 2019

Schoolgirl Space: 1984 Revisited

Filed under: General — Tags: , , , , — m759 @ 9:24 pm

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

From "Tomorrowland" (2015) —

From John Baez (2018) —

See also this morning's post Perception of Space 
and yesterday's Exploring Schoolgirl Space.

Monday, July 8, 2019

Exploring Schoolgirl Space

See also "Quantum Tesseract Theorem" and "The Crosswicks Curse."

Thursday, May 19, 2022

Schoolgirl.space

Filed under: General — Tags: , — m759 @ 5:46 am

See also a Log24 post from 2010, "Class of 64."

Saturday, February 17, 2024

Sixteen Finite-Space Topologies

Filed under: General — Tags: — m759 @ 11:18 pm

From Wikipedia, sixteen T0 topologies on a four-set —

 

{∅, {a}, {a, b}, {a, b, c}, {a, b, d}, {a, b, c, d}} (T0)

{∅, {a}, {a, b}, {a, c}, {a, b, c}, {a, b, c, d}} (T0)

{∅, {a}, {b}, {a, b}, {a, c}, {a, b, c}, {a, b, c, d}} (T0)

{∅, {a}, {a, b}, {a, b, c}, {a, b, c, d}} (T0)

 

{∅, {a}, {b}, {a, b}, {a, b, c}, {a, b, c, d}} (T0)

{∅, {a}, {a, b}, {c}, {a, c}, {a, b, c}, {a, b, d}, {a, b, c, d}} (T0)

{∅, {a}, {a, b}, {a, c}, {a, b, c}, {a, b, d}, {a, b, c, d}} (T0)

{∅, {a}, {b}, {a, b}, {a, b, c}, {a, b, d}, {a, b, c, d}} (T0)

 

{∅, {a}, {b}, {a, b}, {a, c}, {a, b, c}, {a, b, d}, {a, b, c, d}} (T0)

{∅, {a}, {b}, {a, b}, {b, c}, {a, b, c}, {a, d}, {a, b, d}, {a, b, c, d}} (T0)

{∅, {a}, {a, b}, {a, c}, {a, b, c}, {a, d}, {a, b, d}, {a, c, d}, {a, b, c, d}} (T0)

{∅, {a}, {b}, {a, b}, {a, c}, {a, b, c}, {a, d}, {a, b, d}, {a, c, d}, {a, b, c, d}} (T0)

 

{∅, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}, {a, b, d}, {a, b, c, d}} (T0)

{∅, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}, {a, d}, {a, b, d}, {a, c, d}, {a, b, c, d}} (T0)

{∅, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}, {a, b, c, d}} (T0)

{∅, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}, {d}, {a, d}, {b, d}, {a, b, d}, {c, d}, {a, c, d}, {b, c, d}, {a, b, c, d}} (T0)

 

Compare and contrast with the finite geometry  of the power set of a four-set.

Friday, February 11, 2022

For Space Groupies

Filed under: General — Tags: , — m759 @ 5:31 pm

A followup to Wednesday's post Deep Space

Related material from this journal on July 9, 2019

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

From "Tomorrowland" (2015) —

From other posts tagged 1984 Cubes

Friday, September 27, 2019

Algebra for Schoolgirls

Filed under: General — Tags: , , — m759 @ 8:37 am

The 15 points of the finite projective 3-space PG(3,2)
arranged in tetrahedral form:

The letter labels, but not the tetrahedral form,
are from The Axioms of Projective Geometry , by
Alfred North Whitehead (Cambridge U. Press, 1906).

The above space PG(3,2), because of its close association with
Kirkman's schoolgirl problem, might be called "schoolgirl space."

Screen Rant  on July 31, 2019:

A Google Search sidebar this morning:

Apocalypse Soon!

Wednesday, June 29, 2022

Mathematical Plays*

Filed under: General — Tags: — m759 @ 12:44 pm

A song whose melody was used in
Westworld, Season 4, Episode 1 —

"Singin' in the old bars
Swingin' with the old stars
Livin' for the fame

Kissin' in the blue dark
Playin' pool and wild darts
Video games"

Lana Del Rey

In memory of a video game executive 
who reportedly died on June 22, 2022

Schoolgirl Space.

* Adapted from a book  title.

Tuesday, May 5, 2020

Appeal

Filed under: General — m759 @ 2:19 pm

See as well Exploring Schoolgirl Space.

Monday, March 16, 2020

Mathematics and Narrative* Continues:

Filed under: General — Tags: — m759 @ 1:27 am

Expanding the Spielraum

Mathematics:  See Tetrahedron vs. Square in this journal
(Notes on two different models of schoolgirl space ).

Narrative:  Replacing the square  from the above posts by
a related cube 

… yields a merchandising inspiration

Dueling Holocrons: 

Jedi Cube vs. Sith Tetrahedron

.

* See also earlier posts on Mathematics and Narrative.

Wednesday, February 26, 2020

“Perfect”

Filed under: General — Tags: — m759 @ 9:59 pm

Usage example —

(Click to enlarge.)

See also the previous post as well as PG(3,2),
Schoolgirl Space, and Tetrahedron vs. Square.

Friday, November 22, 2019

The Virgin Field:

Filed under: General — Tags: — m759 @ 8:45 am

 Representing Schoolgirl Space

From a book reviewed in the April 1923 
Bulletin of the American Mathematical Society —

From a later book —

"Her wall is filled with pictures" — Chuck Berry

Wednesday, September 18, 2019

Powers of X

Filed under: General — Tags: , — m759 @ 11:45 am

Screen Rant  on July 31, 2019 —

The above space PG(3,2), because of its close association with
Kirkman's schoolgirl problem, might be called "schoolgirl space."

See as well a Log24 post from the above Screen Rant  date —

The Epstein Chronicles, or:  Z  is for Zorro .

Monday, August 19, 2019

A Couple of Tots

Filed under: General — Tags: , — m759 @ 10:15 pm

The title is from the post "Child's Play" of May 21, 2012 . . .

"It seems that only one course is open to the philosopher
who values knowledge and truth above all else. He must
refuse to accept from the champions of the forms the
doctrine that all reality is changeless [and exclusively
immaterial], and he must turn a deaf ear to the other party
who represent reality as everywhere changing [and as only
material]. Like a child begging for 'both', he must declare
that reality or the sum of things is both at once  [το όν τε και
το παν συναμφότερα] (Sophist  246a-249d)."

Related material —

"Schoolgirl Space: 1984 Revisited" (July 9, 2019) and
posts tagged Tetrahedron vs. Square.

Friday, August 16, 2019

Nocciolo

Filed under: General — Tags: , , , — m759 @ 10:45 am

(Continued)

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

A revision of the above diagram showing
the Galois-addition-table structure —

Related tables from August 10

See "Schoolgirl Space Revisited."

Sunday, August 11, 2019

The Culture Desk

Filed under: General — m759 @ 12:28 pm

American Songwriters (the Petrusich versions)

The New Yorker  Culture Desk online yesterday —

The New Yorker  Culture Desk onine July 16 —

Related aesthetic meditation from  this  journal on July 16

(From the post "Schoolgirl Space for Quantum Mystics")

Wednesday, July 31, 2019

The Epstein Chronicles, or:  Z is for Zorro

Filed under: General — Tags: , , , — m759 @ 8:54 pm

Zorro Ranch

The geometry of the 15 point-pairs in the previous post suggests a review:

From "Exploring Schoolgirl Space," July 8 —

The date  in the previous post — Oct. 9, 2018 — also suggests a review
of posts from that date now tagged Gen-Z:

Wednesday, July 10, 2019

Artifice* of Eternity …

Filed under: General — Tags: , , , — m759 @ 10:54 am

… and Schoolgirl Space

"This poem contrasts the prosaic and sensual world of the here and now
with the transcendent and timeless world of beauty in art, and the first line,
'That is no country for old men,' refers to an artless world of impermanence
and sensual pleasure."

— "Yeats' 'Sailing to Byzantium' and McCarthy's No Country for Old Men :
Art and Artifice in the New Novel,"
Steven Frye in The Cormac McCarthy Journal ,
Vol. 5, No. 1 (Spring 2005), pp. 14-20.

See also Schoolgirl Space in this  journal.

* See, for instance, Lewis Hyde on the word "artifice" and . . .

Sunday, February 18, 2024

A Scene from “Terminator Zero: Rise of the Chatbots” —

Deepfake Tango in Paris

Filed under: General — Tags: , — m759 @ 12:00 am

Deepfake Image from the Internet:

Not  Emily in Paris.

 

Related Deep Learning and Deepfake  Learning:

The study of finite topological spaces  may be regarded as a
deepfake approach to finite geometry.  See the previous post.

Monday, July 4, 2022

I’ll be seeing you . . .

Filed under: General — m759 @ 4:00 am

This post was suggested by a Hollywood script date — 7/15/19.

Friday, May 27, 2022

Annals of Seductive Branding

Filed under: General — m759 @ 10:52 pm

Also on May 19 —

Wednesday, August 26, 2020

The Book of Ezra

Filed under: General — Tags: — m759 @ 4:48 pm

Other key observations —

Thursday, July 2, 2020

The Maxwell* Enticement

Filed under: General — Tags: , , , — m759 @ 5:20 pm

[Update of Sunday morning, July 12, 2020 —

This July 2 post was suggested in part by the July 1 post Magic Child
and in part by the Sept. 15, 1984, date in the image below. For more
details about that date, possibly the death date of author Richard
Brautigan, see "The Life and Death of Richard Brautigan," by
Lawrence Wright, in Rolling Stone  on April 11, 1985.

From that article:

Marcia called him the next night [Sept. 15, 1984]
in Bolinas. He asked if she liked his mind. "I said,
‘Yes, Richard, I like your mind. You have the ability
to jump in and out of spaces. It’s not linear thinking;
it’s exciting, catalytic, random thinking.’ "

Such thinking, though interesting, is not recommended for the
general public.  Sept. 15, 1984, was perhaps Brautigan's last day alive.]

* See Maxwell in posts tagged Gods and Giants.

Thursday, April 23, 2020

Octads and Geometry

See the web pages octad.group and octad.us.

Related geometry (not the 759 octads, but closely related to them) —

The 4×6 rectangle of R. T. Curtis
illustrates the geometry of octads —

Counting symmetries with the orbit-stabilizer theorem

Curtis splits the 4×6 rectangle into three 4×2 "bricks" —

.

"In fact the construction enables us to describe the octads
in a very revealing manner. It shows that each octad,
other than Λ1, Λ2, Λ3, intersects at least one of these ' bricks' —
the 'heavy brick' – in just four points." . . . .

— R. T. Curtis (1976). "new combinatorial approach to M24,"
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42.

Sunday, March 1, 2020

Notes towards the Ministry of Culture

Filed under: General — Tags: — m759 @ 8:22 pm

Saturday, August 10, 2019

Muse Score

Filed under: General — m759 @ 1:55 pm

Wednesday, August 7, 2019

Arts:  Boom-Booms Meet Pom-Poms

Filed under: General — m759 @ 11:41 am

Exploring  Schoolgirl  Space  (July 8) continues.

"Eh Fatty Boom-Boom
Hit me with the Ching-Ching"

— Song lyric

Wednesday, July 17, 2019

Life in Palermo

Filed under: General — Tags: , — m759 @ 9:55 am

See also other posts tagged Tetrahedron vs. Square, and a related
Log24 search for "Schoolgirl + Space."

Wednesday, July 5, 2017

Imaginarium of a Different Kind

Filed under: General,Geometry — m759 @ 9:00 pm

The title refers to that of the previous post, "The Imaginarium."

In memory of a translator who reportedly died on May  22, 2017,
a passage quoted here on that date —

Related material — A paragraph added on March 15, 2017,
to the Wikipedia article on Galois geometry

George Conwell gave an early demonstration of Galois geometry in 1910 when he characterized a solution of Kirkman's schoolgirl problem as a partition of sets of skew lines in PG(3,2), the three-dimensional projective geometry over the Galois field GF(2).[3] Similar to methods of line geometry in space over a field of characteristic 0, Conwell used Plücker coordinates in PG(5,2) and identified the points representing lines in PG(3,2) as those on the Klein quadric.

— User Rgdboer

Friday, February 17, 2017

Heptads and Heptapods

Filed under: General,Geometry — m759 @ 12:00 am

In the recent science fiction film "Arrival," Amy Adams portrays
a linguist, Louise Banks, who must learn to translate the language of
aliens ("Heptapods") who have just arrived in their spaceships.

The point of this tale seems to have something to do with Banks
learning, along with the aliens' language, their skill of seeing into
the future.

Louise Banks wannabes might enjoy the works of one
Metod Saniga, who thinks that finite geometry might have
something to do with perceptions of time.

See Metod Saniga, “Algebraic Geometry: A Tool for Resolving
the Enigma of Time?”, in R. Buccheri, V. Di Gesù and M. Saniga (eds.), 
Studies on the Structure of Time: From Physics to Psycho(patho)logy,
Kluwer Academic / Plenum Publishers, New York, 2000, pp. 137–166.
Available online at www.ta3.sk/~msaniga/pub/ftp/mathpsych.pdf .

Although I share an interest in finite geometry with Saniga —
see, for instance, his remarks on Conwell heptads in the previous post
and my own remarks in yesterday's post "Schoolgirls and Heptads" —
I do not endorse his temporal speculations.

Wednesday, February 15, 2017

Warp and Woof

Filed under: General,Geometry — m759 @ 3:00 pm

Space —

Space structure —

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Space shuttle —

Related ethnic remarks —

As opposed to Michael  Larsen —

Funny, you don't look  Danish.

Monday, May 30, 2016

Perfect Universe

Filed under: General,Geometry — Tags: — m759 @ 7:00 pm

(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.

Monday, December 21, 2015

The Eppstein Edit

Filed under: General — m759 @ 11:40 pm

A Wikipedia edit today by David Eppstein, a professor
at the University of California, Irvine:

See the Fano-plane page before and after the Eppstein edit. 
Eppstein deleted my Dec. 6 Fano 3-space image as well as 
today's Fano-plane image.  He apparently failed to read the
explanatory notes for both the 3-space model and the
2-space model. The research he refers to was  original
(in 1979) but has been published for some time now in the
online Encyclopedia of Mathematics, as he could have
discovered by following a link in the notes for the 3-space
model.

For a related recent display of ignorance, see Hint of Reality.

Happy darkest night.

Tuesday, July 9, 2013

Vril Chick

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

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Wednesday, March 11, 2009

Wednesday March 11, 2009

Filed under: General,Geometry — Tags: — m759 @ 9:00 am
Sein Feld
in Translation
(continued from
May 15, 1998)

The New York Times March 10–
 "Paris | A Show About Nothing"–

'Voids, a Retrospective,' at the Centre Pompidou in Paris. Photo from NY Times.

The Times describes one of the empty rooms on exhibit as…

"… Yves Klein’s 'La spécialisation de la sensibilité à l’état matière première en sensibilité picturale stabilisée, Le Vide' ('The Specialization of Sensibility in the Raw Material State Into Stabilized Pictorial Sensibility, the Void')"

This is a mistranslation. See "An Aesthetics of Matter" (pdf), by Kiyohiko Kitamura and Tomoyuki Kitamura, pp. 85-101 in International Yearbook of Aesthetics, Volume 6, 2002

"The exhibition «La spécialisation de la sensibilité à l’état matière-première en sensibilité picturale stabilisée», better known as «Le Vide» (The Void) was held at the Gallery Iris Clert in Paris from April 28th till May 5th, 1955." –p. 94

"… «Sensibility in the state of prime matter»… filled the emptiness." –p. 95

Kitamura and Kitamura translate matière première correctly as "prime matter" (the prima materia of the scholastic philosophers) rather than "raw material." (The phrase in French can mean either.)

Related material:
The Diamond Archetype and
The Illuminati Diamond.

The link above to
prima materia
is to an 1876 review
by Cardinal Manning of
a work on philosophy
by T. P. Kirkman, whose
"schoolgirl problem" is
closely related to the
finite space of the
 diamond theorem.

Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 9:26 am

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

— G. H. Hardy, A Mathematician's Apology

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines,  sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

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